Part III 3-Manifolds Lecture Notes C Sarah Rasmussen, 2019

Part III 3-manifolds Lecture Notes c Sarah Rasmussen, 2019

Contents Lecture 0 (not lectured): Preliminaries2 Lecture 1: Why not ≥ 5?9 Lecture 2: Why 3-manifolds? + Intro to Knots and Embeddings 13 Lecture 3: Link Diagrams and Alexander Polynomial Skein Relations 17 Lecture 4: Handle Decompositions from Morse critical points 20 Lecture 5: Handles as Cells; Handle-bodies and Heegard splittings 24 Lecture 6: Handle-bodies and Heegaard Diagrams 28 Lecture 7: Fundamental group presentations from handles and Heegaard Diagrams 36 Lecture 8: Alexander Polynomials from Fundamental Groups 39 Lecture 9: Fox Calculus 43 Lecture 10: Dehn presentations and Kauffman states 48 Lecture 11: Mapping tori and Mapping Class Groups 54 Lecture 12: Nielsen-Thurston Classification for Mapping class groups 58 Lecture 13: Dehn filling 61 Lecture 14: Dehn Surgery 64 Lecture 15: 3-manifolds from Dehn Surgery 68 Lecture 16: Seifert fibered spaces 69 Lecture 17: Hyperbolic 3-manifolds and Mostow Rigidity 70 Lecture 18: Dehn’s Lemma and Essential/Incompressible Surfaces 71 Lecture 19: Sphere Decompositions and Knot Connected Sum 74 Lecture 20: JSJ Decomposition, Geometrization, Splice Maps, and Satellites 78 Lecture 21: Turaev torsion and Alexander polynomial of unions 81 Lecture 22: Foliations 84 Lecture 23: The Thurston Norm 88 Lecture 24: Taut foliations on Seifert fibered spaces 89 References 92

1 2

Lecture 0 (not lectured): Preliminaries 0. Notation and conventions. Notation. ∂X – (the manifold given by) the boundary of X, for X a manifold with boundary. th ∂iX – the i connected component of ∂X. ν(X) – a tubular (or collared) neighborhood of X in Y , for an embedding X ⊂ Y . ◦ ν(X) – the interior of ν(X). This notation is somewhat redundant, but emphasises openness. Bn – the compact n-ball with boundary. Dn – the compact n-disk with boundary. Bn and Dn diﬀer in name only. Sn – the n-sphere n n 1 n T – the n torus, T := (S ) . \ – complement. A \ B is the complement of B in A. ∼ – homotopy equivalence. =∼ – homeomorphism. ' – isomorphism. X,→Y – embedding of X in Y . int(X) – interior of X. ◦ X – another notation for interior of X.

Conventions. Unless otherwise indicated... When we say map, we mean a continuous map. When we say embedding, we mean a locally ﬂat embedding. When we say manifold, we mean a connected manifold. When we say H1(X), we mean H1(X; Z). When we say π1(X), we mean π1(X, x0) for some undisclosed basepoint x0 ∈ X.

Lecture 0 attempts to consolidate a few background results that might be useful later in the course. You won’t need to know how to prove any of the theorems from Lecture 0. 3

I. Embeddings. A. Topological and locally flat embeddings. Let X and Y be connected topological manifolds. Definition (topological embedding). A map f : X → Y is a topological embedding, or just embedding, if it is a homeomorphism onto its image. Arbitrary topological embeddings can be quite poorly behaved, with images that fail to admit nice neighborhoods in the ambient manifold, even locally. To remedy this fact, we restrict to what are called locally flat embeddings. We shall also discuss locally flat embeddings in lecture, but to make this section more self-contained, we define them here as well. Definition (locally flat). An embedded submanifold X ⊂ Y , is locally flat at x ∈ X if x has ∼ dim(X) dim(Y ) a neighborhood x ∈ U ⊂ Y in Y with a homeomorphism (U ∩ X,U) = (R , R ). Definition (locally flat embedding). An embedding ι : X,→Y is a locally flat embedding if the embedded submanifold ι(X) ⊂ Y is locally flat at ι(x) for each x ∈ X.

B. Smooth embeddings. Let X and Y be smooth manifolds. Deﬁnition (immersion). A map f : X → Y is an immersion if the induced map on tangent spaces d|xf : TxX → Tf(x)Y is an injection. The inverse function theorem guarantees that Ck immersions from compact sets have local Ck inverses, so this is the condition we want for a Ck or smooth (C∞) notion of embedding. Deﬁnition (smooth embedding). A map f : X → Y is a smooth embedding if it is a smooth immersion which is also a topological embedding. Theorem (variant of Inverse Function Theorem). If the domain X is compact, then a smooth immersion f : X → Y is already locally a smooth embedding. In particular, a smooth immersion from a compact domain is a smooth embedding if and only if it is one-to-one.

C. Proper embeddings. Deﬁnition (proper smooth embedding). Suppose X and Y are oriented compact manifolds, possibly with boundary. The map ι : X → Y is a proper smooth embedding if ι(∂X) ⊂ ∂Y , if ι|int(X) : int(X) → int(Y ) and ι|∂X : ∂X ⊂ ∂Y are smooth embeddings, and ι∗(TxX) 6⊂ Tι(x)(∂Y ) for any x ∈ ∂X.

II. Regular Neighborhoods. Deﬁnition (regular neighborhood). A regular neighborhood means a tubular neighborhood in the smooth category and a collared neighborhood in the topological category.

A. Tubular Neighborhoods. Let X and Y be smooth manifolds. Definition (exponential map). Choose a Riemannian metric on Y . For any y ∈ Y and (y,v) “sufficiently” short vector (y, v) ∈ TyY , let γ : [0, 1] → Y be the (constant speed) geodesic with initial conditions specified by (y, v). That is, γ(y,v)(0) = y, γ˙ (y,v)(0) = v, 4 and by “sufficiently short,” we mean short enough for existence and uniqueness of ODEs to provide existence and uniqueness of such γ(y,v). Then for some neighborhood U ⊂ TY of the zero-section containing only “sufficiently short” vectors, the exponential map sends vectors to the endpoints of the geodesics whose initial conditions were specified by those vectors, exp : (U ⊂ TY ) → Y, exp(y, v) := γ(1)(y,v).

The exponential map has several nice features. Let By,ε ⊂ TyY be the ball of radius ε centered at (y, 0) ∈ TyY , with ε > 0 small enough that the restriction of expy := exp |By,ε is deﬁned and one-to-one onto its image. Then the following statements hold.

(a) For any (y, v) ∈ By,ε ⊂ TyY , the path [0, 1] → Y , t 7→ exp(y, tv) is the geodesic of length |v| and initial conditions (y, v) ∈ TyY . (This gives an alternate deﬁnition for exp.)

(b) The induced map d|(y,v) expy : By,ε → TyY on tangent spaces is in fact the identity map, d|(y,v) expy(y, v) = (y, v). Compare this with the ordinary exponential function:

d f(t) f(t) d d|(y,v) expy(y, v) = (y, v), e = e f(t). dt t=0 dt t=0 The exponential map also has a Taylor-series expansion mimicking that of et.

(c) The restriction expy :(By,ε ⊂ TyY ) → Y is a diﬀeomorphism onto its image, and is smoothly varying in y ∈ Y . (d) When G = Y is a Lie Group, exp is deﬁned on all of TG, and its restiction to the tangent space TidG =: g gives an important map from Lie algebras to Lie groups. For the purposes of this course, we mostly only care about point (c) above.

Normal bundles and tubular neighborhoods. Every smoothly embedded submanifold has a normal bundle, and every compact smoothly embedded submanifold has a tubular neighborhood.

Deﬁnition (normal bundle). If ι : X,→Y is a smooth embedding, the normal bundle NX of X ⊂ Y is given by the quotient bundle NX := TX Y /TX , where TX Y is the pull- ∗ back/restriction TX Y := ι (TY ) = TY |X . Deﬁnition (conormal bundle). In the above case, the conormal bundle is the dual bundle ∗ ∗ ∗ ∨ NX := TX Y/T X ' NX .

Any choice of Riemannian metric defines a splitting TY |X ' NX ⊕ TX, in which case the vectors normal to X at x ∈ X can be regarded as orthogonal to the vectors tangent to X ⊂ Y at x ∈ X. Thus, the normal bundle NX parameterises the most efficient directions in which to escape from X into Y . NX also gives a local prescription for how to thicken X into Y . Definition (tubular neighborhood). A tubular neighborhood of a smoothly embedded submanifold X ⊂ Y is an open submanifold ν(X) ⊂ Y (implying dim ν(X) = dim Y ) such that ◦ ν(X) is diffeomorphic to the open unit disk bundle DNX ⊂ NX of NX , ◦ ∼ (X, ν(X)) =diffeo (ZX , DNX ), for some choice of Riemannian metric on Y and for ZX ⊂ NX the zero section of NX . Definition (unit disk bundle). For a bundle E → X of metric-equipped vector spaces, the unit disk bundle DE → X of E is the bundle with fibers Dx = Bx,1 := {v ∈ Ex : |v| ≤ 1}. ◦ The open unit disk bundle DE is the interior of DE. Theorem. Every compact smoothly embedded submanifold X ⊂ Y has a tubular neighborhood ν(X) ⊂ Y . 5

Proof. Choose a Riemannian metric on Y . The induced splitting TY ' NX ⊕ TX makes NX a subbundle NX ⊂ TY of TY . Write By,r ⊂ TyY for the ball of radius r centered at (y, 0) ∈ TyY . Since the above restrictions expy vary smoothly in y ∈ Y , we can define some smooth function ε : X → (0, 1] ⊂ R such that for each x ∈ X, exp restricts to a well-defined map on Bx,ε(x) ⊂ TxY which is diffeomorphic onto its image. Since X is compact, the function ε has a well-defined, postive minimum, ε0 := minx∈X ε(x) > 0. Rescale the metric by 1/ε0, so that balls which formerly had radius ε0 are now regarded as having radius 1. By definition, the unit disk bundle DNX ⊂ NX is the bundle with fibers S DNX |x = Bx,1 ∩ NX|x and total space DNX = x∈X Bx,1 ∩ NX|x. Thus, the restriction

exp :(DNX ⊂ TY ) → Y DNX ◦ is a diﬀeomorphism onto its image, and seting ν(X) := exp(DNX ) makes ν(X) a tubular neighborhood for X ⊂ Y . Remark. Suppose X ⊂ Y is a compact smoothly embedded submanifold of codimension k. If k = 1 and X and Y are both oriented, and/or if TX and TY are both trivial (which also requires orientable X and Y ), then NX is trivial, and we have ◦ ∼ k (1) ν(X) =diﬀeo X × D . Corollary. If X and Y are oriented, then (1) always holds when dim Y = 3.

B. Collared Neighborhoods. In the category of topological manifolds, the correct analog of tubular neighborhoods is the collared neighborhood, or collar, which comes from a “topological” normal bundle instead of a smooth one. Definition (collar). Let X ⊂ Y be an embedded submanifold of codimension k. A collar ◦ ◦ ∼ k ∼ k−1 ν(X) ⊂ Y of X is a neighborhood of the form ν(X) = X × Bε (or ν(X) = X × [0, ε) × Bε if X ⊂ ∂Y ), with X embedded in ν(X) as the 0-section, X,→X × {0} ⊂ ν(X). Any smooth embedding is also locally flat, and any tubular neighborhood is also a collar. Locally flat embeddings in codimension 1 (Brown [1]) or 2 (Kirby [9]) have collared neighborhoods. In fact, the converse is also true. Theorem (Brown [1], Kirby [9], Collars for Locally flat embeddings). If ι : X,→Y is a codimension-1 or codimension-2 topological embedding of manifolds, then ι is locally flat if and only if ι(X) ⊂ Y admits a collared neighborhood. Theorem (Brown [1], locally flat boundaries). Suppose X and Y are connected oriented topological manifolds with X compact, and that X ⊂ Y is a topologically embedded submanifold that is not necessarily locally flatly embedded. Then ∂X ⊂ Y has a collar (thus, equivalently is locally flatly embedded). ∼ 3 ∼ 3 When X = B and Y = R , the above theorem can be regarded as a converse of Alexander’s Sphere theorem, which crucially requires local flatness or smoothness as a hypothesis. To see how the conclusion of Alexander’s Sphere Theorem fails for non-locally flat embeddings, Google Alexander’s Horned Sphere. 6

III. Homotopy. Let X and Y be topological spaces.

A. Basic deﬁnitions and computations.

Deﬁnition (homotopy). Given maps f1, f2 : X → Y , a homotopy from f0 to f1 is a map F : X × [0, 1] → Y with F |X×{i} = fi. We say f0 is homotopic to f1 if such F exists. Deﬁnition (homotopy equivalence). The map f : X → Y is a homotopy equivalence if there is a map g : Y → X such that g ◦f and f ◦g are homotopic to idX : X → X and idY : Y → Y , respectively. Such g is called a homotopy inverse of f.

Deﬁnition (Fundamental group and Higher homotopy groups). Choose a basepoint x0 ∈ X. For k ≥ 0, πk(X, x0) is deﬁned to be the group of homotopy classes of basepoint-respecting k maps (S , s0) → (X, x0), with multiplication given by concatenation, except when k = 0, in which case π0(X, x0) is only a set, with cardinality equal to the number of path-connected components of X. πk(X, x0) is called the fundamental group of X when k = 1, and a higher homotopy group of X or the kth homotopy group of X when k > 1.

We shall usually suppress the basepoints from notation and write πk(X) for πk(X, x0). Theorem (Fundamental groups of products). If X and Y are path-connected, then

πk(X × Y ) ' πk(X) × πk(Y ) for all k > 0. Theorem (Van Kampen’s Theorem). Suppose the union X ∪ Y identiﬁes the basepoints of X and Y , with X ∩ Y path-connected. Then π1(X ∪ Y ) is the amalgamated free product

π1(X ∪ Y ) ' π1(X) ∗π1(X∩Y ) π1(Y ) := π1(X) ∗ π1(Y )/N, where N C π1(X) ∗ π1(Y ) is the minimal normal subgroup containing all elements of the X Y −1 X form i∗ (γ)i∗ (γ) , γ ∈ π1(X ∩ Y ), for the homomorphisms i∗ : π1(X ∩ Y ) → π1(X) and Y X Y i∗ : π1(X ∩ Y ) → π1(Y ) induced by the inclusions i : X ∩ Y,→X and i : X ∩ Y,→Y . Theorem (Free product case of Van Kampen). If X ∩ Y is path-connected and simply- connected (has trivial π1), then

π1(X ∪ Y ) ' π1(X) ∗ π1(Y ). Theorem (Classiﬁcation of sphere maps). Homotopy classes of sphere maps φ : Sn → Sn n are classiﬁed by the degree deg φ ∈ πn(S ) ' Z of such maps.

B. Homotopy lifting. Proposition (Homotopy lifting property). Given connected manifolds X and Y , a covering space p : Ye → Y , a homotopy ft : X → Y , and a lift fe0 : X → Ye of f0,(i.e., a map fe0 : X → Ye with f0 = p ◦ fe0), there is a unique homotopy fet : X → Ye lifting ft.

Theorem (Homotopy lifting). Given connected manifolds X and Y and a covering space p : Ye → Y , any map f : X → Y has a unique lift fe : X˜ → Ye from the universal cover Xe of X to Ye That is, f ◦ πX = p ◦ fe0 for the universal covering map πX :(X,e x˜0) → (X, x0).

Theorem (Covers respect higher homotopy). If p : Xe → X is a regular cover of X, then p ∼ induces an isomorphism p∗ : πn(X˜) → πn(X) of higher homotopy groups for all n > 1. 7

C. Homotopy versus homology. Let X and Y be path-connected topological spaces. Theorem (Hurewicz Theorem). (a) H1(X; Z) ' π1(X)/[π1(X), π1(X)]. (b) If πi(X) = 0 for all i ∈ {1, . . . , n}, for some n ≥ 1, then Hn+1(X; Z) ' πn+1(X), and Hi(X; Z) = 0 for all i ∈ {0, . . . , n}. Deﬁnition (weak homotopy equivalence). The map f : X → Y is a weak homotopy equivalence if the induced group homomorphisms f∗ : πk(X) → πk(Y ) are isomorphisms for all k > 0. Theorem (Whitehead’s Theorem). Any weak homotopy equivalence of CW complexes is also an ordinary homotopy equivalence. Theorem (Homology variant of Whitehead’s Theorem). Suppose X and Y are simply- connected CW complexes. If the group homomorphisms f∗ : Hk(X; Z) → Hk(Y ; Z) induced by a map f : X → Y are isomorphisms for every k, then f is a homotopy equivalence. I’m not sure whom to cite for this second version. It’s listed as Corollary 4.33 in Hatcher’s textbook [5]. The above theorem also holds if the simply-connected hypothesis is weakened to condition that π1(X) act trivially on all higher homotopy groups of X [5]. The converse holds without restrictions on π1. Theorem (Homotopy invariance of (co)homology). Any homotopy equivalence f : X → Y induces group isomorphisms on all homology groups and a ring isomorphism of cohomology rings, for any choice of homology or cohomology coeﬃcients.

IV. Isotopy, and Isotopy Extension of Embeddings. A. Isotopy.

Deﬁnition (isotopy). An isotopy between two For two maps f0, f1 : X → Y of type ?, an isotopy from f0 to f1 is a homotopy from f0 to f1 through maps of type ?. The term isotopy is most often used in the context of embeddings, possibly with additional conditions on smoothness, properness, etc. A key example for us of maps which are homotopic but not isotopic is the class of knot embeddings, where a knot is a locally ﬂat embedding S1,→S3. The notion knots would be completely trivial if we only considered these embeddings up to homotopy, since a homotopy allows the image S1 to pass through itself, making all knots homotopic. Instead, knots are considered up to isotopy. The condition that isotopy be through embeddings prevents the knot from passing through itself.

B. Ambient Isotopy and Isotopy Extension.

Definition (ambient isotopy). Let X,Y be compact oriented smooth manifolds, and let ι1, ι2 : X,→ Y be smooth embeddings (proper if X has boundary). An ambient isotopy for ι1, ι2 is an isotopy (through diffeomorphisms) ψt : Y → Y with ψ0 = id, such that the composition ψt ◦ ι1 : X,→X is an isotopy (through smooth embeddings) from ι1 to ι2. Theorem (Isotopy Extension Theorem (Palais) [13]). Let X,Y be compact oriented smooth manifolds, and let ι1, ι2 : X,→ Y be smooth embeddings (proper if X has boundary). If ι1, ι2 are isotopic, then they are ambient isotopic. Edwards and Kirby [3] broadened this result to locally flat ambient isotopies covering isotopies of locally flat embeddings. (Smooth implies locally flat, but not the converse.) Rourke proved a converse of Edwards and Kirby’s result, that if a local isotopy of embeddings extends to an ambient isotopy, then these embedded manifolds are collarable. 8

V. Integer Homology and Universal Coeﬃcients.

Convention. Any manifold named M, Mi, etc, is connected, oriented, and compact unless otherwise stated. Deﬁnition (torsion subgroup). Let G be a group. The torsion subgroup of G is n Tors(G) := {g ∈ G | ∃n 6= 0 ∈ Z s.t. g = 1}. Deﬁnition (torsion submodule). Let V be an R-module for R a PID (principal ideal domain: every ideal in R is principal, also implying R has no zero divisors a · b = 0 ∈ R, 0 6= {a, b}). The torsion submodule of V is given by

TorsR(V ) := {a ∈ V | ∃r 6= 0 ∈ R s.t. r · a = 0}. When the ring R is understood, particularly if R = Z, we often just write Tors(V ). Warn- ing: do not confuse this with ExtR(V,R) and TorR(V,R), since TorsR(V ) 6= TorR(V,R)! Torsion submodules can also be defined over non integral domains with the modified condition “∃r ∈ R not a zero divisor such that r · a = 0.” However, torsion submodules over non- PIDs have longer than 2-step resolutions, hence have nontrivial higher Ext and Tor groups. Definition (free quotient module). Let V be an R-module for R a PID. We define the free quotient module of V to be

FreeR(V ) := coker (TorsR(V ) ,→ V ). Theorem (Structure Theorem for ﬁnitely generated modules over a PID). Let V be a ﬁnitely- generated R-module, for R a PID. Then 1. There is a unique integer rank(V ) ∈ Z such that rank(V ) Lrank(V ) FreeR(V ) ' R := i=1 R. 2. The exact sequence

0 −→ TorsR(V ) −→ V −→ FreeR(V ) −→ 0 splits noncanonically, i.e.,

V ' FreeR(V ) ⊕ TorsR(V ). 3. The torsion submodule of V decomposes as

TorsR(V ) ' R/r1R ⊕ · · · ⊕ R/rkR

for some r1, . . . rk ∈ R. Corollary. If V is a ﬁnitely generated R-module for a PID R, then

(a) ExtR(V,R) ' TorsR(V ),

(b) HomR(V,R) ' FreeR(V ), k k (c) ExtR(V,R) ' TorR(V,R) ' 0 ∀ k > 1. Theorem (Free/Torsion Universal Coefficients Theorem for Homology). Suppose R is a PID and X is a compact manifold (implying H∗(X) is finitely generated). Then i H (X; R) ' FreeR(Hi(X; R)) ⊕ TorsR(Hi−1(X; R)). Proof. Apply the Hom/Ext version of the universal coefficients theorem for homology.

Corollary (3-manifold homology determined by H1). If M is a connected closed oriented 3-manifold, then all of H∗(M; Z) is determined by H1(M; Z). In particular, H2(M; Z) ' Free(H1(M; Z)), H0(M; Z) ' H3(M; Z) ' Z. The cohomology ring structure, however, still depends on the linking form of M. 9

Lecture 1: Why not ≥ 5? I. The Poincar´eConjecture. (Early 1900’s)

Question. How does one distinguish S3 from other closed, oriented manifolds?

Strategy. Find an invariant whose characterisation of S3 is unique. First try: Homology? 3 ∼ 3 If H∗(M) ' H∗(S ), does that imply M = S ? ' isomorphism =∼ homeomorphism 3 ∼ 3 Theorem (Poincaré). ∃ closed oriented 3-manifold P with H∗(P ) ' H∗(S ) but P =6 S . Proof. 1. Invent the notion of fundamental group π1(M). 2. Construct “PoincaréSphere” P . (Today: P is “−1 surgery” on the left-handed trefoil.) 3 3. Show that |π1(S )| = 1, whereas |π1(P )| = 120. His method of proof turned out to be more important than the result itself. In fact, Poincaré’snewly-invented fundamental group was so successful at distinguishing 3-manifolds that he grew confident that he had found the right invariant.

Conjecture (Poincar´e,1904). If M if is a connected closed oriented 3-manifold, then M =∼ S3 if and only if π1(M) = 1.

This conjecture went unanswered for nearly a century, until Gregory Perelman presented ideas for a proof in 2002-2003 which took another 2-3 years for the mathematical community to check. In fact, he showed that once you decompose a 3-manifold in a certain way that we shall make precise later, the fundamental group can distinguish all of the resulting pieces except for certain so-called lens spaces.

II. Homotopy (See also Lecture 0). Let X and Y be topological spaces. A. Deﬁnitions.

Deﬁnition (homotopy). Given maps f1, f2 : X → Y , a homotopy from f0 to f1 is a map F : X × [0, 1] → Y with F |X×{i} = fi. We say f0 is homotopic to f1 if such F exists.

k Deﬁnition (Fundamental group and Higher homotopy groups). Let x0 ∈ X and s0 ∈ S be basepoints. k πk(X, x0):'{homotopy classes of maps (S , s0) → (X, x0)} Group under concatenation for k > 0.

Remark: We shall often suppress the basepoints from notation and write πk(X) for πk(X, x0).

Deﬁnition (homotopy equivalence). The map f : X → Y is a homotopy equivalence if there is a map g : Y → X such that g ◦f and f ◦g are homotopic to idX : X → X and idY : Y → Y , respectively. Such g is called a homotopy inverse of f.

Deﬁnition (weak homotopy equivalence). The map f : X → Y is a weak homotopy equivalence if the induced group homomorphisms f∗ : πk(X) → πk(Y ) are isomorphisms for all k > 0. 10

B. Homotopy versus homology. Let X and Y be path-connected topological spaces. Theorem (Hurewicz Theorem). (a) H1(X; Z) ' π1(X)/[π1(X), π1(X)]. (b) If πi(X) = 0 for all i ∈ {1, . . . , n}, for some n ≥ 1, then Hn+1(X; Z) ' πn+1(X), and Hi(X; Z) = 0 for all i ∈ {0, . . . , n}.

Theorem (Whitehead’s Theorem). Any weak homotopy equivalence of CW complexes is also an ordinary homotopy equivalence.

Theorem (Homology variant of Whitehead’s Theorem–Corollary 4.33 in Hatcher’s textbook). Suppose X and Y are simply-connected CW complexes. If the group homomorphisms f∗ : Hk(X; Z) → Hk(Y ; Z) induced by a map f : X → Y are isomorphisms for every k, then f is a homotopy equivalence.

(Suﬃcient that π1(X) act trivially on all higher homotopy groups of X.)

Theorem (Homotopy invariance of (co)homology). Any homotopy equivalence f : X → Y induces group isomorphisms on all homology groups and a ring isomorphism of cohomology rings, for any choice of homology or cohomology coeﬃcients. (No restrictions on π1.)

III. Simpliﬁcations in dimension n≥5 (smooth), n=4 (top) (not examinable). ( smooth n ≥ 5 C = top (with locally ﬂat embeddings) n = 4

Theorem (Whitney Trick: Whitney n ≥ 5, Freedman n = 4). If π1(X) = 1, P,Q ⊂ X are C-embedded and dim P + dim Q = dim X = n, with n ≥ 4, then P,Q can be locally C-isotoped so that geometric intersection #(P,Q) = |algebraic intersection #(P,Q)|.

Deﬁnition (h-cobordism). Let W , with ∂W = X1 q −X2, be a cobordism from X1 to X2. W is an h-cobordism if each embedding Xi ,→ W is a homotopy equivalence.

Theorem (h-cobordism theorem: Smale n ≥ 5, Freedman n = 4). Let W , with ∂W = X1 q −X2, be an h-cobordism from X1 to X2. dim W = n + 1, dim Xi = n. If π1(W ) = π1(X1) = π1(X2) = 1 (i.e., all simply connected) and n ≥ 4, then W is C- isomorphic to X1 × [0, 1], and the restriction of this C-isomorphism to X1 can be taken to be the identity map.

Moral. The Whitney Trick implies manifolds in dimensions ≥ 5 can be given particularly nice cell decompositions (handle decompositions). Consequence: In dimensions ≥ 5, most questions about topological or smooth structures can be answered using purely homotopy-theoretic methods (more algebraic). To give an example of how this plays out, we shall prove the Generlised Poincar´eConjecture in dimensions ≥ 5 using purely homotopy-theoretic methods. 11

IV. Generalised Poincar´eConjecture (not examinable). Conjecture (Generalised Poincar´econjecture). A connected closed oriented n-manifold S is homotopy-equivalent to Sn if and only if it is homeomorphic to Sn.

Theorem (1960s Smale (n ≥ 5), Stallings (n ≥ 7), Zeeman (extended Stallings argument to n = 5, 6); 1982 Freedman (n = 4)). The Generalized Poincar´eConjecture holds for n ≥ 4. Proof of Theorem (n ≥ 5 case). Freedman’s proof for the n = 4 case uses an involved h-cobordism-theorem argument to show that S is h-cobordant to Sn. From now on, we restrict to the simpler case of n ≥ 5. Suppose S ∼ Sn, using the symbol “∼” for homotopy equivalence. ◦ ◦ n n Remove two balls: W := S \ (B1 q B2 ). (“◦” denotes interior.) n−1 n−1 Proposition. W , with ∂W = S1 q −S2 , is an h-cobordism. Assuming this proposition (proved below), we have a homeomorphism ◦ ◦ ◦ ◦ n n ∼ ∼ n−1 ∼ n n n S \ (B 1 q B 2) = W = S × [0, 1] = S \ (B 1 q B 2), and any homeomorphism on a sphere can be extended topologically over the ball it bounds, by taking the cone over the map on the sphere. We thereby extend the above homeomorphism ∼ n to a homeomorphism S = S . n−1 Proof of Proposition. We must show Si ,→W are homotopy equivalences. By Whitehead’s Theorem (homology variant), it’s suﬃcient to show n (i) Bi , W are simply connected, n (ii) Bi ,→ W induces ' on all homology groups. n n n For (i), we already know π1(Bi ) = 1. Moreover, since W ∩ (B1 q B2 ) is path-conected and simply-connected, we can use Van Kampen’s Theorem: n n (2) 1 = π1(S) = π1(B1 ) ∗ π1(W ) ∗ π1(B2 ), implying π1(W ) = 1. We establish (ii) with a Meier-Vietoris sequence argument, where we recall that the (homology) Meier-Vietoris sequence is the long exact sequence of homology groups associated to the short exact sequence of chain complexes,

Meier-Vietoris: 0 → C∗(A ∩ B) → C∗(A q B) → C∗(A ∪ B) → 0, n n A := W, B := B1 q B2 =⇒ ∼ n−1 n−1 ∼ A ∩ B = S1 q S2 ,A q B ∼ W q 2pts,A ∪ B = S.

n−1 n−1 0 −→ Hn(S1 q S2 ) −→ Hn(W q 2pts) −→ Hn(S) ··· n 0 (n − 1 < n) 0 (W not closed) Z (S ∼ S )

n−1 n−1 −→ Hn−1(S1 q S2 ) −→ Hn−1(W q 2pts) −→ Hn−1(S) ··· n Z ⊕ Z ? = Z (by exactness) 0 (S ∼ S ). n−1 n−1 n−1 n−1 The boundary map Hn(S) → Hn−1(S1 q S2 ) sends [S] 7→ ([S1 ], [S2 ]), so the n−1 exactness of the above sequence implies the restrictions Hn−1(Si ) → Hn−1(W ) ' Z are n−1 isomorphisms. The Meier-Vietoris sequence also implies we have Hk(Si ) ' Hk(W ) = 0 for n−1 all k > n − 1 or 0 < k < n − 1, with the isomorphisms H0(Si ) → H0(W ) ' Z for k = 0. It took entirely different methods, and much more work, for Perelman to establish the PoincaréConjecture in dimension 3. Theorem (Perelman, 2003). Poincaré’sConjecture holds (for n = 3). 12

Appendix 1.A. Further details (not examinable). A. Extension of maps from spheres to balls. Here is a more explicit description of how one topologically extends a map from a sphere to the ball it bounds.

Proposition (Alexander’s Sphere Trick). For n ∈ Z≥0, any orientation-preserving homeomorphism f : Sn → Sn can be extended (not necessarily smoothly!) to a homeomorphism ¯ n+1 n+1 ¯ f : B → B with f|Sn = f. n+1 n+1 n+1 Proof. In the pull-back to local coordinates x ∈ R for a standard unit ball B ⊂ R n+1 centered at 0 ∈ R , we can deﬁne ( |x|f(x/|x|) x 6= 0 f¯(x) := . 0 x = 0 n+1 Since |f(x/|x|)| = 1 for all x 6= 0 ∈ R , we have lim |x|f(x/|x|) = 0, |x|→0 proving that f¯ is continuous at x = 0. For x 6= 0, f¯ is a composition of continuous functions, hence continuous. (One could alternatively use generalised polar coordinates instead of |x| and x/|x|. The two notions are equivalent.)

B. The Smooth PoincaréConjecture. Shortly before the Generalized PoincaréConjecture results of the 1960’s, Milnor published a 1956 result [11] revealing that in the smooth category, the behaviour of homotopy spheres is not so straightforward: while the homeomorphism type of the 7-sphere is unique, its diffeomorphism type is not. Milnor and Kervaire went on to demonstrate how to classify the smooth structures on all n-spheres with n > 4, and produced an explicit table for the number of distinct smooth structures on Sn for n ∈ {5,..., 18} [8]. Homotopy one- and two-spheres have long been known to have unique topological and smooth structures as manifolds. Moreover, a 1950’s result of Moise [12] established the equivalence of smooth and topological categories in 3 dimensions, making the original and smooth Poincaréconjectures equivalent in that dimension. Thus, topologists and geometers were once again left with a driving question in a single dimension.

Conjecture (Smooth Poincar´eConjecture). If S is homeomorphic to S4, then it is diﬀeo- morphic to the standard smooth S4. This question is still an area of active research. 13

Lecture 2: Why 3-manifolds? + Intro to Knots and Embeddings

I. 3-manifolds: Active Research Areas (not examinable). A. Interaction with smooth/symplectic/complex structures on 4-manifolds: • gauge-theoretic smooth-structure invariants for 4-manifolds Floer-theoretic 3-manifold invariants + maps induced by cobordism. —– Smooth PoincaréConjecture in 4-d still open. • symplectic form ω on 4-manifold X symplectically-fillable contact structure ξ on 3-manifold Y = ∂X. • Stein (complex-symplectic) structure on 4-manifold X Stein-fillable contact structure ξ on 3-manifold Y = ∂X. • complex analytic data for complex surface singularity (X, ◦) topology of 3-manifold Y = Link(X, ◦), with X a real cone over Y . • complex surface singularity (X, ◦) link Y = Link(X, ◦), with X a real cone over Y .

B. Geometric group theory (Interactions with fundamental group): • There is a bijection, prime, atoroidal, non-lens space fundamental groups of such ↔ , 3-manifolds 3-manifolds so there is research on developing a dictionary properties of / structures on properties of / structures on ↔ . a 3-manifold M π1(M) and π1(covers of M) • Eg., looking for immersed surfaces and surface subgroups.

C. 2-dimensional structures: • contact structures (an everywhere nonintegrable 2-plane field) –classification of tight contact structures unknown for most 3-manifolds. • minimal-genus embedded-surface representatives (Thurston norm and knot genus) –mostly already solved, but still open questions about 4-d analogs of this question. • taut foliations –classification of 3-manifolds with taut foliations unknown.

D. 1-dimensional structures (knots and links): • Every 3-manifold can be realized by Dehn surgery on a link L ⊂ S3. –So 3-manifold theory could almost be regarded as a subset of knot theory. –3-manifold invariants from knot invariants, e.g. Witten-Reshetikhin-Turaev. • Many questions for 3-manifolds have a relative version for knot complements. • Relations between knot invariants and structures from other areas of mathematics: –Chern-Simons for knots in S3 vs. Gromov-Witten invariants on resolved conifold. n 2 –HOMFLY homology of braids vs. Derived category of sheaves on Hilb (C ) –Khovanov homology of braids vs. DC of sheaves on certain other spaces 14

II. Course Themes. A. Decompositions/Constructions of 3-manifolds.

• Surface decompositions/constructions –Decomposition along 2-spheres — prime decomposition. –Decomposition along 2-tori — JSJ decomposition. –Mapping torus construction of ﬁbered manifolds. • Quotient spaces. –Hyperbolic quotients –Quotients of S3 and Seifert ﬁbrations. • Morse-theoretic decompositions/constructions –Handle decompositions. –Heegaard splittings and Heegaard diagrams, especially for knot exteriors. • Dehn surgery on links.

B. Structures and invariants on 3-manifolds.

• Knots, links, –Knot complements and exteriors. –Fundamental group of knot complements. –Alexander polynomial and Turaev torsion. • Essential and incompressible embedded surfaces. –Thurston norm • Foliations 15

III. Embeddings. Convention. In this course, whenever we say “embedding,” we shall mean either a smooth embedding or a locally flat topological embedding, as defined below. 3 `n 1 3 Definition (knots and links). A link L ⊂ S is an embedding L := i=1 Si ,→ S of oriented circles, considered up to isotopy. A knot is a link with exactly one component.

See Lecture 0 for a more detailed collection of deﬁnitions and results associated with embeddings, regular neighborhoods, and isotopies. For now, we write down some essential facts.

A. Smooth = Top in dimensions 1, 2, and 3. Theorem (Moise [12]). There is a canonical bijective correspondence between smooth structures and topological structures for manifolds of dimension ≤ 3. In particular, in dimension ≤ 3 each manifold, considered up to homeomorphism, admits a unique smooth structure. Main idea (Not mentioned in lecture; not examinable) A PL (Piecewise Linear) structure is a triangulation which satisfies the additional condition that the link of each simplex is a PL- sphere. This can fail in any dimension ≥ 4, but is automatically met by all triangulations in dimension ≤ 3. In any dimension, we have smooth structure unique PL structure unique topological manifold structure. In dimensions ≤ 3, we also have Smale Moise unique smooth structure unique PL structure topological manifold structure. There is also a 2-dimensional analog of this result, which we shall revisit later in the course. Theorem (Munkres, Smale and Whitehead). Any homeomorphism between smooth surfaces is isotopic to a diffeomorphism. We shall prove neither Moise’s Theorem, nor the above Munkres-Smale-Whitehead theorem in lecture, but if you are interested, Hatcher has a nice 11-page paper on the arXiv providing new, simplified proofs for each of these two results [6].

Let X and Y be connected topological manifolds.

B. Smooth embeddings. Definition (topological embedding). A map f : X → Y is a topological embedding, or just embedding, if it is a homeomorphism onto its image. Let X and Y be smooth manifolds. Definition (immersion). A map f : X → Y is an immersion if the induced map on tangent spaces d|xf : TxX → Tf(x)Y is an injection. The inverse function theorem guarantees that Ck immersions from compact sets have local Ck inverses, so this is the condition we want for a Ck or smooth (C∞) notion of embedding. Definition (smooth embedding). A map f : X → Y is a smooth embedding if it is a smooth immersion which is also a topological embedding. Theorem (variant of Inverse Function Theorem). If the domain X is compact, then a smooth immersion f : X → Y is already locally a smooth embedding. In particular, a smooth immersion from a compact domain is a smooth embedding if and only if it is one-to-one. 16

C. Locally flat topological embeddings. Definition (locally flat). An embedded submanifold X ⊂ Y , is locally flat at x ∈ X if x has ∼ dim(X) dim(Y ) a neighborhood x ∈ U ⊂ Y in Y with a homeomorphism (U ∩ X,U) = (R , R ). Definition (locally flat embedding). An embedding ι : X,→Y is a locally flat embedding if the embedded submanifold ι(X) ⊂ Y is locally flat at ι(x) for each x ∈ X.

To understand what goes wrong when embeddings are not locally flat, Google “Alexander’s Horned Sphere” for a non-locally-flatly embedded surface. Non-locally-flatly embedded circles are called “wild knots,” an example of which is pictured below.

D. Regular Neighborhoods. Deﬁnition (regular neighborhood). A regular neighborhood means a tubular neighborhood in the smooth category and a collared neighborhood in the topological category. Deﬁnition (tubular neighborhood). For a smooth embedding X,→Y , the restriction

exp :(DNX ⊂ TY ) → Y DNX ◦ is a diffeomorphism onto its image, and seting ν(X) := exp(DNX ) makes ν(X) a tubular neighborhood for X ⊂ Y . (See Lecture 0.II for more detail.) In the category of topological manifolds, the correct analog of tubular neighborhoods is the collared neighborhood, or collar, which comes from a “topological” normal bundle. Definition (collar). Let X ⊂ Y be an embedded submanifold of codimension k. A collar ◦ ◦ ∼ k ∼ k−1 ν(X) ⊂ Y of X is a neighborhood of the form ν(X) = X × Bε (or ν(X) = X × [0, ε) × Bε if X ⊂ ∂Y ), with X embedded in ν(X) as the 0-section, X,→X × {0} ⊂ ν(X). Any smooth embedding is also locally flat, and any tubular neighborhood is also a collar.

Remark. Locally ﬂat embeddings in codimension 1 (Brown [1]) or 2 (Kirby [9]) have collared neighborhoods. In fact, the converse is also true. (See Lecture 0).

Remark. In dimension 3, all regular neighborhoods are products (see Lecture 0). That is, for an embedding X,→Y with X and Y oriented and dim Y = 3, we have ν(X) =∼ D3−dim X × X,→ Y. 17

Lecture 3: Link Diagrams and Alexander Polynomial Skein Relations I. Knot and Link diagrams. 3 `n 1 3 Deﬁnition. A link L ⊂ S is an embedding L := i=1 Si ,→ S of oriented circles, considered up to isotopy (see Lecture 0.IV), or equivalently in this case, up to homeomorphism of S3. Deﬁnition. A link projection (or knot projection, for |L| = 1) of a link L ⊂ S3 is an 2 2 immersion L # ΓL,→R to an embedded planar graph ΓL,→R , induced by a projection

3 ∼ 3 ∼ 2 L S \{x0} = R = R × R

p|L p such that

2 ΓL R

(a) x0 ∈/ L, and 2 (b) p|L → R is an embedding except at isolated double-points. 3 Deﬁnition. A link (or knot) diagram DL = (ΓL, crossings) of a link L ⊂ S is an embedded 2 2 planar graph ΓL ,→ R induced from some link projection L # ΓL ,→ R , together with crossing decoration at each vertex of v of ΓL, indicating which pair of incident edges to v 2 2 has highest preimage in R with respect to the projection p : R × R → R inducing the link projection. Example. Here are some example knot diagrams for the unknot and trefoil, and link diagram for the Hopf link. The crossing decoration is a small gap marking the lower strand.

A given link can be represented by arbitrarily many diﬀerent diagrams, but the operations relating diagrams representing the same link are completely understood. 3 Theorem (Reidemeister). Let D1,D2 be respective diagrams for links L1,L2 ⊂ S . Then L1 and L2 are ambient isotopic if and only if the diagrams D1 and D2 are related by some combination of the following 3 local operations, called Reidemeister moves.

These moves are useful for proving that a given invariant of diagrams is also an invariant of knots. For purposes of this class, however, all I expect you to know about Reidemeister moves is that the intuitive simpliﬁcations we make on knot diagrams can be made rigorous. 18

II. The Alexander Skein relation. Skein relations, on the other hand, refer to relationships among collections of basic local operations which do change the ambient-isotopy type of the corresponding link. In many cases,

including in the Alexander skein relation, these operations are performed at a crossing.

⤸ ⤲ ⤱ ⤹ c(D) ∶ c(D) ∶ c(D) ∶

In the above ﬁgure, we start with the center diagram for the trefoil, and create 3 new ⤸ ⤹ ⤸ ⤹ D ⤲ ⤱ ⤲ ⤱ diagrams, c(D), c(D), and c(D), by performing the operations c, c, and c at the crossing c ∈ D. One of these three operations will always be the identity operation. In the ⤱ above case, c(D) = D, since c is a positive crossing in D.

−1 Deﬁnition. The Alexander Polynomial Δ ∶ {link diagrams D} → ℤ[t , t] is speciﬁed by the following two conditions:

( ) Normalization. Δ(any diagram for the unknot) = 1 ⤸ i ⤹ ⤲ ⤱ 1∕2 −1∕2 (ii) Skein relation. Δ( c(D)) − Δ( c(D)) = (t − t )Δ( c(D)) ∀ c ∈ crossings(D).

Theorem (The Alexander skein relation as an invariant of oriented links). The map Δ as described above is well-deﬁned on diagrams, and Δ(D1) = Δ(D2) for any two diagrams D1,D2 of the same link, or equivalently, for any D1,D2 related by a series of Reidemeister moves. Thus Δ descends to an invariant of oriented links, Δ ∶ {oriented links} → ℤ[t−1, t]. The above theorem arises as a corollary of the following much stronger result.

Theorem (Equivalence of Alexander polynomials). Later in this course, we shall deﬁne the Alexander polynomial Δ3−mfd of a 3-manifold, for any oriented 3-manifold with b1 > 0. We then have 3 Δknot(L) = Δ3−mfd(S ⧵ L) for any oriented link L ⊂ S3. One of the major goals of the next few lectures will be developing the tools necessary to prove the above theorem in the case that that L is a knot, i.e., has only one component. 19

III. Computing ∆(L) via skein relations. The Alexander skein relation relates the Alexander polynomial of a link to that of links of lower complexity with respect to minimal crossing number or number of components. We illustrate how this works with a computation of the Alexander polynomial of the trefoil T .

We ﬁrst perform skein operations at some crossing cT in a minimal-crossing diagram for T .

This relates T to the unknot U1 and the hopf link H. Since ∆(U1) is known, we are reduced to computing ∆(H), requiring a skein relation at some crossing cH in a minimal- crossing diagram for H.

This reduces us to the computation of ∆(U2). The 2-component unlink U2 has no crossings, so instead we combine two components of U2 in an oriented resolution at some cU2 .

Having reduced everything to unknots, we can now solve the resulting skein equations: 1/2 −1/2 (cU2 ) ∆(U1) − ∆(U1) = (t − t )∆(U2) =⇒ ∆(U2) = 0, 1/2 −1/2 1/2 −1/2 (cH ) ∆(U2) − ∆(H) = (t − t )∆(U1) =⇒ ∆(H) = t − t , 1/2 −1/2 −1 1 (cT ) ∆(U1) − ∆(T ) = (t − t )∆(H) =⇒ ∆(T ) = t − 1 + t . 20

Lecture 4: Handle Decompositions from Morse critical points I. Morse functions. One of the major themes in this course is learning diﬀerent ways in which 3-manifolds can be decomposed into simpler pieces. For the moment, we focus on decomposing an n-manifold X into handles—closed tubular neighborhoods of the cells in a cell decomposition for X. Working with k-handles—these n-dimensional tubular neighborhoods of k-cells—instead of with k-dimensional cells themselves, allows us to regard each k-handle as being modeled on the neighborhood of a certain type of singularity called an index k Morse singularity.

Notation. Let f : X → R be a smooth function on a smooth manifold X. Then ∂2 Hessp f := the Hessian at p at f = f , ∂xi∂xj p ij n o crit f := {the critical points of f} = p ∈ X ∂ f = 0 ∀ i , ∂xi p where the right-hand side presents this object in local coordinates x centered at p.

Deﬁnition (Morse function). A smooth function f : X → R on a connected compact oriented manifold X of dimension n is called Morse if (a) every critical point x ∈ crit f of f is isolated, and

(b) Hessp f is nondegenerate at every p ∈ crit f (i.e., has det 6= 0 in local coordinates).

II. Morse singularities. If f is Morse, then for any p ∈ critf and any choice of local coordinates centered at p, f has Taylor series expansion of the form 1 P ∂2 f(x) = f(p) + xixj f + higher degree terms. 2 ij ∂xi∂xj p ij

Since Hessp f is nondegenerate, its eigenvalues are nonzero and can be rescaled to +1 or −1. ∂2 Since partial derivatives commute, Hesspf := f , is symmetric, and can therefore ∂xi∂xj p ij be diagonalised. Thus we can choose coordinates to obtain the new Taylor expansion Pk 2 Pn 2 f(x) = f(p) − i=1 xi + i=k+1 xi + higher degree terms. Using an implicit function theorem argument, Morse showed we can choose coordinates to simplify this expansion even further.

? Lemma (Morse Lemma). One can can choose coordinates x centered at p ∈ crit f such that Pk 2 Pn 2 (∗) Morse Lemma Equation. f(x) = f(p) − i=1 xi + i=k+1 xi .

Deﬁnition (indpf = index at a critical point). If f : X → R is Morse, then for any critical point p ∈ crit f of f, the index indp f of f at p is the number of negative eigenvalues of Hessp f, counted with multiplicity. Equivalently, indpf = k in the Morse Lemma equation (∗).

? Moral. For each k ∈ {0, . . . , n}, there is a unique local model to which the neighborhood of any index k critical point is equivalent. 21

Morse singularities in dimension 3:

Morse singularities in dimension 2: 22

III. Cobordisms through critical points: Handle attachments.

Let X be a smooth n-manifold with Morse function f : X → R. As the “height” or “time” −1 −1 −1 t ∈ R increases, the manifold f ((−∞, t]) grows from f (−∞) = ∅ to f ((−∞, +∞)) = X. We want to discretise this process into the addition of pieces of manifold resembling standard neighborhoods of critical points.

Isotope the Morse function f : X → R so that f(x) 6= f(y) for any distinct x, y ∈ crit f. Partition R into half-open neighborhoods ν h.o.(f(x)) ⊂ R of critical values of f for x ∈ crit f, and let Ix be the closed interval Ix := ν h.o.(f(x)) ⊂ R for x ∈ crit f. We then have ( [ ∅ or [ −1 R = Ix, with Ix ∩ Iy = ∀ x 6= y ∈ crit f =⇒ X = f (Ix). 1 pt x∈crit f x∈crit f − + − + −1 Set tx := inf Ix and tx := sup Ix, so that Ix = [tx , tx ] ⊂ R. Then each f (Ix) is a −1 − −1 + −1 cobordism from −f (tx ) to f (tx ) containing x ∈ crit f. The cobordism f (Ix) is called kx-handle attachment, for kx := indxf.

Deﬁnition (index k handle). An n-dimensional index-k handle (indices are always > 0!) is n ∼ ∼ k n−k Hk = standard neighborhood (index-k Morse critical point) = D × D . Since Dk ×Dn−k =∼ Dj ×Dn−j =∼ Bn for all j, k ∈ {0, . . . , n}, all handles are homeomorphic. What diﬀers about them is our conventions for how they are glued to manifolds.

−1 Theorem/Deﬁnition (handle attachment). The kx-handle-attachment f (Ix) satisﬁes −1 ∼ −1 − n f (Ix) = (f (tx ) × I) ∪ Hk , ∼ 1 k n−k n for I = D a small closed interval, with the union taken along a portion ∂D × D ⊂ ∂Hk n −1 of ∂Hk called the attaching region. Thus f (Ix) is a cobordism from −1 − −1 + ∼ −1 − k n−k k n−k −f (tx ) to f (tx ) = (f (tx ) \ (∂D × D )) ∪ D × ∂D . k n−k n n The new piece of boundary D × ∂D ⊂ ∂Hk is (confusingly) called the belt region of Hk .

In the ﬁgure above, the 2-handle attaching region S1×D0 =∼ S1 and belt region D2×∂D0 =∼ ∅. 23

Dimension n = 3. Cobordism through an Addition of an index k critical point: index k handle:

Dimension n = 2. Cobordism through an Addition of an index k critical point: index k handle:

Passing through a critical point in the positive direction of R corresponds to moving from left to right, both in the critical-point diagrams and in the handle-attachment diagrams. Attaching regions are shown in red; belt regions are purple. 24

Lecture 5: Handles as Cells; Handle-bodies and Heegard splittings I. Handles as neighborhoods of cells.

A. Cell complex interpretation of handles. n Deﬁnitions (handle, core, cocore). An n-dimensional k-handle, or handle of index k, Hk is the product decomposition n k n−k Hk := core × cocore = D × D . of the ball Bn into a k-dimensional core k-cell Dk and (n−k)-dimensional cocore Dn−k. Secretly choosing a metric, we usually regard a handle as having a large core and small cocore, so that a k-handle is the closure of a (narrow) tubular neighborhood of its k-cell core,

n ∼ n ∼ k Hk = ν(core(Hk )) = ν(D ).

• Question: Why do we decompose Bn like this? • Answer: Because we want to think of a handle as handle = (Bn, rules for gluing Bn to the boundary of a manifold). To specify these gluing rules, we specify which part of the boundary gets glued. Decomposing Bn as a product Bn =∼ Dk × Dn−k naturally decomposes the boundary of Bn into 2 pieces: the part that gets glued onto the manifold we are building, and the part that does not.

n Deﬁnitions (attaching/belt region). The boundary of a handle Hk decomposes as n ∼ ∼ ∂(Hk ) = ∂(core × cocore) = ∂(core) × cocore ∪ core × ∂(cocore) =∼ ∂Dk × Dn−k ∪ Dk × ∂Dn−k =∼ attaching region ∪ belt region, with attaching region := ∂(core) × cocore and belt region := core × ∂(cocore). The attaching region isn shown in red in the diagram above, with the belt region shown in purple.

Moral (Cell complex interpretation). In a cell complex, we attach the boundary Sk−1 = ∂Dk of a k-cell to the cell complex skeleton we have built so far. In a handle decomposition we attach the closed neighborhood k ∼ k−1 ∼ k−1 n−k ∼ n ν(∂D ) = ν(S ) = S × D = attaching region(Hk ) of Sk−1 to the manifold we have built so far. The belt region is the left-over part of the handle’s boundary that contributes to the boundary of the manifold formed by handle attachment. 25

B. Handle attachment vs. Cell attachment. Deﬁnition (handle attachment). The attachment of a k-handle to a manifold X to produce a manifold X0 is induced by a k-handle-attachment cobordism Z from −∂X to ∂X0, where n Z =∼ (∂X × I) ∪ n H , attaching reg (Hk ) k for I =∼ D1 a small closed interval. This produces the new boundary 0 ∼ n n ∂X = (∂X \ attaching reg (Hk )) ∪ belt reg (Hk ) =∼ (∂X \ (∂Dk × Dn−k)) ∪ Dk × ∂Dn−k, and we have 0 n X =∼ X ∪ Z =∼ X ∪ n H . ∂X attaching reg (Hk ) k

Deﬁnitions (attaching/belt sphere). attaching sphere := core(attaching region) = ∂(core) × pt =∼ Sk−1, belt sphere := core(belt region) = pt × ∂(cocore) =∼ Sn−k−1. The attaching sphere of a 2-handle is more often called an attaching circle.

n n ∼ k Remark According to our mindset that a k-handle Hk is a narrow thickening Hk = ν(D ) of n k n−k ∼ k its k-cell core, the attaching region of Hk is also a narrow thickening ∂D × D = ν(∂D ) of its Sk−1 attaching sphere core. On the other hand the belt region is a “fat” expansion of the belt sphere. The retraction of the belt region to its belt sphere core is somewhat violent.

n Convention. Even though we attach a handle to the image ϕ(a.r.(Hk )) ⊂ ∂X, under some n k−1 n−k n gluing map ϕ : a.r.(Hk ) → −S × D ⊂ ∂X, of the attaching region of Hk , what we say is that we attach the handle to the image of its attaching sphere. “Attach a 2-handle along this closed curve α ∈ X.”

“Attach a 1-handle to these points x0, x1 ∈ X.” If ∂X is connected, then 1-handle attachment is independent, up to diﬀeomorphism, of the n n image ϕ(a.r.(H1 )) = ({x0} q {x1}) ⊂ ∂X of the attaching sphere of H1 . Thus we simply say, “Attach a 1-handle to X.” 26

II. Morse interpretation of handles, revisted. Definition (gradient vector field). Choose a Riemannian metric g on a smooth n-manifold X, and let f : X → R be a smooth function. The gradient vector field gradf ∈ Vect := Γ(TX) of f is the unique vector field determined by the equation g(gradf, V ) = df(V ),V ∈ Vect(X) := Γ(TX). n ∂ ∂ n n In local coordinates x ∈ R , with basis ∂x1,..., ∂xn for TxR at any point x ∈ R , we have ∂f ∂ grad f = gijP ∈ Γ(T n). ∂xj ∂xi R Definition (flow). The flow of a vector field V ∈ Γ(TX) through a point p ∈ X on a smooth manifold X is a curve γ :(−ε, ε) → X, γ(0) = p, for some sufficiently small ε > 0 such that d dt |t=t0 γ(t) = Vγ(t0) for any t0 ∈ (−ε, ε). Moral (Gradient flow and Handles). For f : X → R Morse and x ∈ crit f, the gradient flow lines of f specify the core, cocore, and attaching/belt regions of ν(x). core := “descending manifold of x” := {points on gradf flow lines flowing into x}. cocore := “ascending manifold of x” := {points on gradf flow lines flowing out of x}. attach region := ∂(ν(x)) ∩ {points on gradf flow lines flowing into ν(x)}. belt region := ∂(ν(x)) ∩ {points on gradf flow lines flowing out of ν(x)}. Points in the interior of the attaching/belt region flow into/out of ν(x), points in the core/cocore flow into/out of x, and indxf is the number of linearly-independent directions of gradf flow into x,

indxf = dim(core), x = core ∩ cocore.

2 2 2 Example. f : R → R, f = x − y .

∂ ∂ grad f = 2x ∂x − 2y ∂y Flow(grad f) Flow lines intersecting • ∈ crit f

−1 2 −1 0 −1 X := f ((−∞, −1]) H1 := ν(0) = f ([−1, 1]) X := f ((−∞, 1]) 2 The attaching/belt regions of H1 are shown in red/purple, and we have

core = , cocore = , 0 = = core ∩ cocore. 27

Remark (Incoming/outgoing vs. below/above). Suppose we regard the image R of a Morse function f : X → R as positioned “vertically” pointing upwards. Then when we consider points in X as traveling along the gradient ﬂow lines of f, there is is a correspondence, incoming = below, outgoing = above.

In particular, if Hx = ν(x) is a handle centered at x ∈ crit f, then we have

core(Hx) lies below x; cocore(Hx) lies above x.

In other words, if we think of Hx as a generalized saddle, the core is the part of the saddle pointing down, and the cocore is the part of the saddle pointing up, hence the respective names descending manifold and ascending manifold for the core and cocore.

3 Warning. Since the graph of f : X → R cannot embed in R for dim X > 2, it is impossible to draw a 3-dimensional handle whose core and cocore lie “below” and “above” the critical 3 ∼ 3 point, respectively. For example, it is impossible to draw a 3-dimensional 0-handle H0 = B whose center is the lowest point of the ball. On the other hand, the notions of incoming and outgoing gradient ﬂow make sense in any dimension. In general, the safest interpretation of the image R of the Morse function f : X → R is that of time, along which we observe the progress of the gradient ﬂow of f. This notion of timing is also one that we can extract from a handle decomposition.

Theorem (Morse functions from handle decompositions). Let X be a smooth n-manifold, and suppose we specify both a smooth handle decomposition H∗(X) for X and the time t(H) ∈ R at which each handle H ∈ H∗(X) is added. Then there is Morse function f : X → R which induces the speciﬁed handle decomposition, so that

H∗(X) = {Hx| Hx = ν(x), x ∈ crit f}, ind(Hx) = indxf, , t(Hx) = f(x) ∈ R ∀ x ∈ crit f.

Sketch of Proof (not examinable). Let x : H∗(X) → X be the map H 7→ core(H)∩cocore(H). n The idea is to choose diffeomorphisms ϕH : H → [−1, 1] compatible with the decomposition of H into core and cocore, define local Morse functions fH := t(H) + find(H) ◦ ϕx(H) for Pk 2 Pn 2 fk := − i=1 xi + i=k+1 xi , and then carefully extend each fH in neighborhood of each ∂H, and patch these extended local Morse functions together with a partition of unity to form a global Morse function f. The compatibility of handle gluing with Morse gradient flow will then imply that at any boundary-neighborhood overlap, gradf will have a nonzero component along the normal direction of at least one handle boundary in that neighborhood. Making this rigorous requires extra care at the corners of handles, but this can be done. 28

Lecture 6: Handle-bodies and Heegaard Diagrams I. Handle-bodies. Deﬁnition (genus). The genus g(Σ) of a closed oriented surface Σ is given by 1 g(Σ) := 2 |H1(Σ)|. If Σ is a compact oriented surface with boundary, then we deﬁne g(Σ) := g(Σ), for Σ the closed oriented surface given by gluing one D2 to each boundary component of Σ.

Deﬁnition (handle-body). The genus g handle-body Ug is the compact oriented 3-manifold obtained from attaching g 1-handles to a 0-handle, making a “fat wedge of circles.” Note that ∂Ug = Σg, the closed oriented surface of genus g. Handle-bodies also have a more intrinsic deﬁnition. Later in the course, we shall learn about a notion of compressibility for surfaces. The genus-g handle-body Ug is the unique ∼ fully-compressible compact oriented 3-manifold Ug with boundary ∂Ug = Σg.

α Proposition/Deﬁnition (Handle-body Ug from 2-handle attachment). Let α1, . . . , αg ⊂ Σ be a collection of g disjoint, embedded, homologically linearly independent curves in Σ. 1 Attaching one 2-handle to Σ × [0, 2 ] along each αi curve, and then attaching one 3-handle to the resulting S2 boundary, forms a genus g handle-body α 1 `g 2 3 U := Σ × [0, ] ∪ D × I ∪ 2 B . g 2 α×I⊂Σ×{0} i=1 αi S

Proof (summary). The linear independence of the homology classes [αi] ∈ H1(Σ) implies none of these curves are separating, and each each handle attachment along αi lowers the genus of the resulting boundary by 1. Thus, attaching g = g(Σ) such 2-handles forms a manifold U with boundary ∂U = Σ q S2, such that U is the exterior of a 3-ball in a genus-g handle-body. α Gluing in a 3-handle then gives the genus-g handle-body Ug . In fact, the above construction is homeomorphic to the construction forming a “standard” α Ug , in which we regard Σg as a wedge of 2-dimensional 1-handles at a 0-handle, and the curves αi as the belt spheres of these 1-handles, so that, for example, if Σ were embedded in 3 some standard way in R , then we could attach 2-handles along these αi curves in a manner 3 respecting this embedding in R . See the Appendix of this lecture for more details both on this latter perspective and on the α 2-handle formation of Ug .

Remark. Note that we would get a diﬀerent answer if one or more of the closed curves αi were separating, i.e., if Σ\α had two components. However, a codimension 1 manifold is separating if and only if it is nullhomologous rel boundary; or in a closed manifold, it is separating if and only if it is nullhomologous. Since the homology classes [α1],..., [αg] ∈ H1(Σ) are linearly independent, no nonempty collection of them gives a separating submanifold.

One important application of handle-bodies is that of Heegaard splittings.

Deﬁnition (Heegaard splitting). Let Ug and Vg be genus g handle-bodies. A Heegaard splitting of genus g for a closed oriented 3-manifold M is a decomposition ∼ M = Ug ∪Σg Vg with respect to some gluing map ϕ : ∂Ug → −∂Vg. A Heegaard splitting is often speciﬁed by a collection of data called a Heegaard diagram. 29

II. Heegaard Diagrams.

Deﬁnition (Heegaard diagram). A Heegaard Diagram is the data H := (Σ, α, β, x0) of • Σ a closed oriented surface,

• α = (α1, . . . , αnα ) an nα-tuple of disjoint embedded closed curves,

• β = (β1, . . . , βnβ ) an nβ-tuple of disjoint embedded closed curves, and • x0 ∈ Σ a basepoint.

Conventions. For purposes of this course, we additionally demand that

• nα = g and nβ ≤ g, and • all curves αi and βj are homologically nontrivial unless otherwise speciﬁed.

There are also Heegaard diagrams with multiple basepoints, but we shall not consider these. Just as with fundamental groups, we shall sometimes suppress the basepoint.

Deﬁnition (3-manifold from Heegaard diagram). A Heegaard diagram H := (Σ, α, β) uniquely speciﬁes a compact oriented 3-manifold MH, as follows. 1. Start with Σ × I,I = [0, 1]. 2a. For each i ∈ {1, . . . , nα}, attach a 2-handle to Σ × I along αi × {0} ⊂ Σ × {0}. 2b. For each j ∈ {1, . . . , nβ}, attach a 2-handle to Σ × I along βj × {1} ⊂ Σ × {1}. 3. Attach a 3-handle to any boundary component =∼ S2 of the resulting manifold.

Proposition. Let H := (Σ, α, β) be a Heegaard diagram obeying our conventions. (i) If nβ = g, then MH is a closed oriented 3-manifold. ∼ 2 (i) If nβ = g−1, then MH is a compact oriented 3-manifold with torus boundary ∂MH = T . 1 Proof. Attaching a 2-handle to Σ × [0, 2 ] along the each αi curve αi × {0} ⊂ Σ × {0} forms α 1 the genus-g handle-body Ug . If nβ = g, then attaching a 2-handle to Σ × [ 2 , 1] βj curve β βj × {1} ⊂ Σ × {1} the genus-g handle-body Ug . MH is then given by the closed union ∼ α β MH = Ug ∪ 1 Ug . Σ×{ 2 } β β The case of nβ = g −1 is similar, except that instead of forming Ug , we form a manifold U β ∼ 2 homeomorphic to the exterior of a solid torus in a genus g handlebody, with ∂U = Σ q T . ∼ α β The union MH = Ug ∪Σ U is therefore the exterior of a solid torus in a closed oriented 2 manifold, making MH a compact oriented 3-manifold with torus boundary ∂MH = T . 30

III. Existence of Heegaard diagrams.

Deﬁnition (self-indexing Morse function). A Morse function f : X → R is called self- indexing if f(x) = indxf for every x ∈ crit f.

Theorem (Existence of self-indexing Morse functions). Any compact oriented smooth n- manifold X admits a self-indexing Morse function with exactly one index 0 critical point if X is connected, and exactly one index n crtical point if X is closed. Proof. See Example Sheet 1. The strategy is to build a handle decomposition compatible with a self-indexing Morse function, and then recover a Morse function from that.

Theorem (Existence of Heegaard Diagrams). Every compact oriented 3-manifold M admits a Heegaard Diagram.

Proof (for nβ = 1). Let f : M → R be a self-indexing Morse function on M with precisely one index-0 and one index-n critical point. Let nk denote the number of index-k critical points of f for each k ∈ {0,..., 3}. −1 Let Σ := f (1.5). We claim that n1 = n2 = g(Σ). That is, since Σ is the boundary −1 of the 3-manifold f (h−∞, 1.5]) formed by attaching n1 1-handles to n0 = 1 0-handle—a genus-n1 handle-body—we have g(Σ) = n1. Similarly, regarding 3 − f as a Morse function −1 on -f ([1.5, +∞i) makes the latter a 3-manifold with boundary Σ, formed by attaching n2 1-handles to n3 = 1 0-handle and therefore a genus-n2 handle-body. Thus g(Σ) = n2. 3 3 Lastly, let H1;i and H2;j denote the ith 1-handle and jth 2-handle speciﬁed by f, for i, j ∈ {1, . . . , g}, and g := g(Σ). We then set 3 αi := Σ ∩ (co-core of H1;i), 3 βj := Σ ∩ (core of H2;j). See diagram on following page. 31

Heegaard diagram from a self-indexing Morse function. Note that various images look distorted in this 4-dimensional graph, since aligning cores and cocores according to the Morse function axis forces parts of our manifold to project into an unseen fourth dimension. 3 3 In particular, near the corresponding critical points, only 2-dimensional slices of H0 and H3 are visible. 32

α Appendix 6.A (not examinable). Ug construction and homological calculations. k In order to describe this 2-handle addition one 2-handle at a time, let Ug denote the the manifold formed by attaching 2-handles D2 × I,...D2 × I to Σ × [0, 1 ] along the respective α1 αk 2 k k curves α1, . . . , αk ⊂ Σ × {0}, and let Σ denote the inner boundary component of Ug , i.e. k 0 1 0 ∼ Σ ∩ Σ × {0} is nonempty. Thus, Ug := Σ × [0, 2 ], and Σ = Σ × {0} = Σ. k k k Note: just as we can regard each Ug as a cobordism from −Σ to Σ , we can also regard Ug k 2 as a cobordism from Σ to Σ. In this case, we regard each Dαi × I as a 1-handle H1,i, with 2 core Ii, whose attaching region Dαi ×({0}q{1} is glued to Σ along the neighborhood of some 0 i i k i i S , say xo q x1 ⊂ Σ . All embedded paths from x0 to x1 are isotopic and contractible, so we k i i can choose an embedded path γi : [0, 1] → Σ from x0 to x1 without loss of generality. This γi then combines with the 1-handle core Ii to form a closed embedded curve ai := Ii ∪ γi.

In the above image, γi and the 1-handle core Ii are drawn as the straight and curved parts of ai, respectively. Note that in this picture, we have already isotoped each ai curve so that k it lies on the outer boundary Σ of Ug . We henceforward regard our ai curves as lying in Σ. The light-blue manifold is hollow, with inner boundary Σk, whereas the 1-handles with belt spheres α1, . . . , αk—or equivalently the 2-handles with attaching spheres α1, . . . , αk—are solid k k k 3-manifolds. Thus, the ∂Ug = Σ q Σ has components of genus g(Σ) = g and g(Σ ) = g − k. We can also compute this directly in homology. Before starting this calculation, we construct ai for i ∈ {k + 1, . . . , g} by temporarily attaching a 2-handle to Σ × {0} along αi, constructing ai as we did before and isotoping it out to the boundary Σ, and then removing the 2-handle we had attached to αi. In particular, these curves ai are well deﬁned up to isotopy regardless of whether we have attached a handle to that particular αi. Since |αi ∩ aj| = δij for all i, j ∈ {1, . . . , g}, the classes [α1],..., [αg], [a1],..., [ag] ∈ H1(Σ) form a basis for H1(Σ). U k k+1 Σ k k Next, observe that the inclusions ιk+1 : Ug ,→Ug and ιk :Σ ,→Ug induce homomorphisms U k k+1 Σ k k ιk+1 ∗ : H1(Ug ) → H1(Ug ), ιk ∗ : H1(Σ ) → H1(Ug ). Each ιU sits inside a Meyer-Vietoris exact sequence for the union of U k with D2 × I, k+1 ∗ g αk+1 where, after splitting oﬀ the exact H0 sequence, we have H (U k+1) → H (α × I) → H (D2 × I) ⊕ H (U k) −→ H (U k+1) → 0 2 g 1 k+1 1 αk+1 1 g 1 g ιU k k+1 ∗ k+1 0 → h[αk+1]i → H1(Ug ) −→ H1(Ug ) → 0, k+1 k 0 from which we compute that H1(Ug ) ' H1(Ug )/ h[αk+1]i. Thus, since Ug = Σ, we have k 2g−k H1(Ug ) ' H1(Σ)/ h[α1],..., [αk]i ' h[αk+1],..., [αg], [a1],..., [ag]i ' Z . 33

k k To compute H1(Σ ) from H1(Ug ), we adopt the perspective of 1-handle addition, with the Σ k k inclusion homomorphism ιk :Σ ,→Ug sitting inside a Meyer-Vietoris exact sequence for the k `k 2 union of Σ × I with i=1 Dαi × Ii, as follows. iΣ k k∗ k d `k i i `k k k 0 → H1(Σ ) → H1(Ug ) → H0( i=1(x0 q x1)) → H0(( i=1 Ii) q Σ ) → H0(Ug ) → 0. The Snake Lemma then tells us that the Meyer-Vietoris boundary map obeys j j d[αi] = 0, daj = [x0] + [x0], ∀ i ∈ {1, . . . , g}, j ∈ {1, . . . , k}. Splitting oﬀ the exact sequence of free modules `k i i D 1 1 k k E `k k k 0 → H0( i=1(x0 q x1))/ [x0] + [x1],..., [x0] + [x1] → H0(( i=1 Ii) q Σ ) → H0(Ug ) → 0 then leaves us with the exact sequence Σ k ik∗ k d D 1 1 k k E 0 → H1(Σ ) → H1(Ug ) → [x0] + [x1],..., [x0] + [x1] → 0, i i which splits via the homomorphism σ :[x0] + [x1] 7→ [ai]. We therefore have k k 2(g−k) H1(Σ ) ' H1(Ug )/ h[a1],..., [ak]i ' h[αk+1],..., [αg], [ak+1],..., [ag]i ' Z . In particular, g(Σk) = g − k. 34

α Appendix 6.B (not examinable). Ug as isotopic to a “standard” handle-body. α α 3 To visualize the handle-body Ug , it helps to know that there is an embedding of Ug in R for which the αi curves are the belt spheres of 1-handles, as a corollary of the following result.

Theorem. Let α1, . . . , αg ⊂ Σα and γ1, . . . , γg ⊂ Σγ be homologically linearly independent disjoint embedded curves in the respective closed oriented genus-g surfaces Σα and Σγ. Then there is an isotopy Φ:Σα × I → Σγ sending αi to γi, for each i ∈ {1, . . . , g}.

The proof of this result is probably less instructive then the preceding homological calculations, but in any case, it helps to reduce the above question to the following result. 1 1 Proposition. Suppose Sα has g embedded closed curves α1, α¯1, . . . , αg, α¯g ⊂ Sα, and let 2 2 2 2 1 Dα1 , Dα¯1,...,Dαg,Dα¯g ⊂ Sα denote the respective embedded disks bounded by these curves: 2 2 2 `g 2 2 ∂Dαi = αi, ∂Dα¯1 =α ¯1. For each i ∈ {1, . . . , g}, let δαi : [0, 1] ,→ Sα\ i=1 int(Dα1 qDα¯i ) be an embedded path from some point xi ∈ αi to some point x¯i ∈ α¯i, such that the paths δα1 , . . . , δαg 2 1 have disjoint images in Sα. In a precisely analogous manner, deﬁne γ1, γ¯1, . . . , γg, γ¯g ⊂ Sγ, 2 2 2 2 1 2 `g 2 2 Dγ1 , Dγ¯1,...,Dγg,Dγ¯g ⊂ Sγ, and δγ1 , . . . , δγg : [0, 1] ,→ Sγ \ i=1 int(Dγ1 q Dγ¯i ). Then there 2 2 is an isotopy Ψ: S × I → S such that, for ψ := Ψ| 2 , we have α γ Sα×{1}

ψ(αi) = γi, ψ(¯αi) =γ ¯i, ψ(δαi ) = δγi ∀ i ∈ {1, . . . , g}.

Proof of Theorem, given Proposition. To make use of the theorem, we want to decompose Σα 2 as the union of a 2g-times punctured Sα—corresponding to Σα cut open along neighborhoods `g `g i=1 ν(αi) of the αi curves—in union with the g left over cylinders i=1 ν(αi). 2 We realize this decomposition by attaching 2-handles Dα × Ii to Σα × I along each of the α,g i αi curves, forming the manifold Ug in the notation of the previous discussion, but where we α,g α,g no longer suppress the dependence on α. Since the inner boundary component Σ of Ug α,g 2 α,g has genus g(Σ ) = g − g = 0, we let Sα denote the inner boundary component of Ug , and α,g then regard Ug as the manifold formed by attaching a 1-handle H1;αi with belt-sphere αi to S2 × {0} ⊂ S2 × [0, 1 ], along what was previously the belt region image D2 × ∂I of the α α 2 αi i 2-handle D2 × ∂I , with boundary ∂D2 × ∂I = α α¯ ⊂ S2 × { 1 }, for each i. αi i αi i q i α 2 α,g 2 2 1 Let ϕα : Ug ,→Sα × [0, 1] be an embedding that restricts to the identity on Sα × [0, 2 ]. (This is possible, since the 1-handles deformation retract to 1-dimensional objects.) Isotope 2 2 ϕα so that under the projection pα : Sα × [0, 1] → Sα, we have pα(H1;αi ) ∩ pα(H1;αj ) = ∅ 2 for any i 6= j. Further isotope ϕα so that on each core(H1;αi ) ⊂ Sα × I, pα restricts to 2 `g 2 2 an embedding onto its image. For each i, let δαi : [0, 1] ,→ Sα \ i=1 int(Dα1 q Dα¯i ) be an 2 2 embedded path with image pα(core(H1;αi )) \ int(Dαi q Dα¯i ). Lastly, construct a precisely γ,g 2 analogous embedding ϕγ : Ug ,→Sγ × [0, 1]. Extend the isotopy Ψ from the Proposition to I 2 2 I α,g an isotopy Ψ :(Sα × I) × I → Sγ × I,Ψ (p, s, t) := (Ψ(p, t), s). Then if we let Ug,0 denote α,g γ,g I Ug with its 1-handles deformation-retracted to their cores, and similarly with Ug,0 , then Ψ α,g γ,g carries Ug,0 to Ug,0 . We extend this by tubular neighborhoods to an isotopy that additionally carries tubular neighborhoods of 1-handle cores to respective tubular neighborhoods of 1- handle cores. Then since any handle is locally isotopic to a tubular neighborhood of its core, we are done. That is, we have shown we can isotope the embeddings ϕα and ϕγ so that I α,g γ,g Ψ |t=1 ◦ ϕα(U ) = ϕγ(U ). −1 I The desired isotopy Φt :Σα → Σγ is then given by restricting ϕγ ◦ Ψt ◦ ϕα to the outer α,g boundary component Σα of U . 35

Proof of Proposition. 0. First, we observe that we may assume the γ-data is as standard as we like, since we could then construct the desired isotopy by factoring through an isotopy to an S2 with standard embedded curves, followed by an isotopy from this S2 with standard embedded curves. 2 2 In particular, we assume (with respect to a standard metric on S ) that the disks Dγ1 , 2 2 2 2 Dγ¯1,...,Dγg,Dγ¯g ⊂ Sγ are each small neighborhoods of points, and that for each i, the 2 2 2 component Dγi ∪δγi ([0, 1])∪Dγ¯i ⊂ ν(yi) where the ν(yi) ⊂ Sγ are small disjoint neighborhoods of chosen points yi ∈ γi. 2 2 We shall construct Ψ as a sequence of isotopies Ψ1,..., Ψ4 : S × I → S , denoting the 2 2 induced diﬀeomorphisms as ψi := Ψi|S2×{1} : S → S .

2 2 2 2 1. Let Ψ1 : S × I → S contract each Dαi and Dα¯i , rel xi, into small neighborhoods ν(xi) and ν(¯xi) of xi andx ¯i, respectively. 2 2 2 2. Let Ψ1 : S × I → S contract each ν(¯xi), rel ψ1(Dα¯i ), along δαi into ν(xi), so that 2 ψ2 ◦ ψ1(Dα¯i ∪ δαi ([0, 1])) ⊂ ν(xi).

2 2 3. Letting idα : ψ2 ◦ψ1(Sα) → Sγ denote the identity map, choose some collection of noninter- 2 −1 2 secting paths ∂i : [0, 1] → ψ2 ◦ψ1(Sα) from xi = ψ2 ◦ψ1(xi) to yi := idα (yi) ∈ ψ2 ◦ψ1(Sα). Let 2 2 2 2 Ψ3 : S ×I → S contract each ν(xi) along δi into ν(yi), recalling that Dγi ∪δγi ([0, 1])∪Dγ¯i ⊂ ν(yi), with the ν(yi) all disjoint. 2 2 3. Let Ψ4 : S × I → S localize to isotopies of ν(yi) that send

ψ3 ◦ ψ2 ◦ ψ1(αi) to γi, ψ3 ◦ ψ2 ◦ ψ1(¯αi) toγ ¯i, ψ3 ◦ ψ2 ◦ ψ1(δαi ) to δγi .

Ψ is then given by performing Ψ1,Ψ2,Ψ3, and Ψ4 in sequence. 36

Lecture 7: Fundamental group presentations from handles and Heegaard Diagrams I. Fundamental group presentations. n Notation (free group). Let hgi denote the group hgi = ({g , n ∈ Z}, ·). The free group generated on the n letters g1, . . . , gn is then denoted

hg1, . . . , gni := hg1i ∗ · · · ∗ hgni . Notation. If G < H is a subgroup, then hhGii denotes the normal subgroup −1 hhGii := h gh, g ∈ G, h ∈ H C H. Such hhGii can alternatively be deﬁned as the smallest normal subgroup in H containing G.

Deﬁnition/Notation. We say a group G has a presentation with generators g1, . . . , gn and relators r1, . . . , rm if

G ' hg1, . . . , gn| r1, . . . , rmi := hg1, . . . , gni / hhr1, . . . , rmii .

II. Fundamental group of 1- and 2-handle attachment.

Deﬁnition (based embedded loop). We say a path γ : [0, 1] → X is an x0-based embedded loop if γ(0) = γ(1) = x0, and the restriction γ|(0,1) : (0, 1) → X is an embedding.

α Lemma (Fundamental group of 1-handle attachment). (Only lectured in special case of Ug .) Let X be an n-manifold, and suppose we attach a 1-handle Hn with core D1 to X at 1 (x0,x1) x q x ⊂ X to form the new manifold X0 := X ∪ (D1 × D2). Let γ : [0, 1] → Hn be 0 1 x0qx0 x0qx1 0 1 a properly embedded path from x to x along the 1-handle core D1 , let γ : [0, 1] → X 0 1 (x0,x1) 1 0 be any properly embedded path from x1 to x0, and let γ : [0, 1] → X be an x0-based embedded 0 loop that traverses ﬁrst γ0 and then γ1. Then π1(X , x0) ' π1(X, x0) ∗ h[γ]i. Proof. We ﬁrst deformation retract Hn to its core D1 . Next, keeping D1 attached 1 (x0,x1) (x0,x1) to x0 and x1, we homotope x1 along γ1 to First homotope x1, together with its attached handle, along γ1 to x0, so that γ is reduced to γ0, which is now an x0-based embedded loop. 0 This gives the homotopy equivalence X ∼ X ∨x0 γ([0, 1]), and so we appeal to the fact that π1(A ∨x0 B, x0) ' π1(A, x0) ∗ π1(B, x0).

Lemma (Fundamental group of 2-handle attachment). Let X be an n-manifold, and suppose n 2 we attach a 2-handle H2 with core Dγ to X along a closed embedded curve γ ,→ ∂X, to form 0 2 X := X ∪γ×I Dγ × I. Let x0 ∈ γ be a basepoint, and let γ∗ denote an x0-based embedded loop 0 with image γ and homotopy class [γ∗] ∈ π1(X). Then π1(X ) ' π1(X)/ hh[γ∗]ii. Proof. This follows immediately from Van Kampen’s Theorem. Details. (Not lectured, not examinable). Applying Van Kampen’s Theorem gives 0 2 2 π1(X , x0) ' π1(X, x0) ∗π1(γ×I,x0) π1(Dγ × I, x0) ' π1(X, x0) ∗ π1(Dγ × I, x0)/N −1 2 where N := i∗(a)j∗(b) , a ∈ π1(X, x0), b ∈ π1(Dγ × I, x0) (imposing the relations i∗(a) = 2 j∗(b)), for i∗ : π1(γ ×I, x0) → π1(X, x0), j∗ : π1(γ ×I, x0) → π1(Dγ ×I, x0) the homomorphisms 2 2 induced by the inclusions i : γ×I,→X and j : γ ×I,→Dγ ×I. Since π1(Dγ ×I, x0) = 1, we then 2 have N = hhi∗(π1(γ × I, x0)ii ' hh[γ∗]ii, with π1(X, x0) ∗ π1(Dγ × I, x0) ' π1(X, x0). 37

α III. Fundamental group presentation of π1(Ug , x0). α Wg 0 Choose orientations for each αi. Ug deformation retracts onto a wedge j=1 aj at x0 of 0 α images aj of x0-based embedded loops aj : [0, 1] → Ug such that each ai is dual to each 2 αi. That is, we have the signed intersection Di · aj = +δij for any i, j ∈ {1, . . . , g} and any 2 α 2 Wg 0 properly-embedded disk Di ,→Ug with ∂Di = αi. The deformation retraction j=1 aj then induces the isomorphism α Wg 0 π1(Ug , x0) ' π1( j=1aj, x0) ' ha1, . . . , agi . α Proposition (Spelling elements of π1(Ug , x0)). Choose orientations for each of α1, . . . , αg, let 2 2 α α 2 D1,...,Dg ,→ Ug \{x0} be properly embedded disks with disjoint images in Ug and ∂Di = αi, α and suppose γ is an x0-based embedded loop in Ug , isotoped so that all intersections of γ with 2 α these Di are transverse. Let p1, . . . , pn ∈ Ug denote the n successive points of intersection of 2 γ with qiDi as one starts at x0 and traverses γ once in the positive direction. Then γ = ae1 ··· aen ∈ π (U α), i1 in 1 g with multiplication by concatenation on the right, where γ intersects the disks D2 ,...,D2 in i1 in 2 that order, and where ek := D · γ ∈ {±1} is the sign of this intersection. ik pk α Wg 0 2 Proof. Again, Ug deformation retracts to the wedge of circles j=1 aj, with each disk Di 0 0 retracting to a single point xi ∈ ai \ x0. If γ denotes this deformation retracted γ, then 0 γ passes through xi in the positive, respectively negative, direction at time t ∈ [0, 1] if and 2 only if γ passes through Di in the positive, respectively negative, direction at time t ∈ [0, 1]. 0 `g Homotope γ rel i=1 xi so that it meets x0 once between any two successive intersections with `g x . This decomposes γ0 as the product γ0 = ae1 ··· aen . i=1 i i1 in α 2 α Remark. If γ ⊂ ∂Ug , the same result holds, but with Di ·|Ug γ = αi ·|Σ γ. 38

IV. Fundamental group presentation of π1(MH, x0).

Let H = (Σ, α, β, x0) be a Heegaard diagram specifying a 3-manifold MH. We can compute the fundamental group π1(MH, x0) of MH directly from its Heegaard diagram. 1. Orientation and basepoint choices. For the correspondence between our Heegaard diagram and our fundamental group presentation to be well deﬁned, and our computations self-consistent, we must make some choices.

• Choose a basepoint xj ∈ βj in each βj. • Choose an orientation for each αi and βj. • Σ is already oriented as part of the data of H = (Σ, α, β, x0), with the standard convention that as drawn on paper, Σ is oriented according to the “righthand rule.”

2. Presentation of π1(MH, x0). Our Fundamental group of 2-handle attachment Lemma tells us that if all the βj curves met α α x0 ∈ Ug , then the kernel of the epimorphism π1(Ug , x0) → π1(MH, x0) would be generated by the classes of x0-based loop representatives of the βj curves, along with their conjugates. ` In our case, x0 ∈/ j βj, but this is only a minor problem. For each j ∈ {1, . . . , nβ}, j j α let a1, . . . , ag : [0, 1] → Ug denote a collection of xj-embedded loops respectively dual to the j j α curves α1, . . . , αg, just as we did for the x0-based embedded loops a1, . . . , ag : [0, 1] → Ug . Let α βj∗ denote an xj-based embedded loop with image βj. As usual, we can spell [βj∗] ∈ π1(Ug , xj) j j α as an element of ha1, . . . , agi ' π1(Ug , xj) by successively detecting, for k ∈ {1, . . . , n}, the th j ej th k successive factor (a ) of [βj∗] by recording the k signed intersection ek := αi · βj ik k pk `g th of βj∗ with i=1 αi at its k point of intersection pk, for n total points of intersection. On the other hand, after choosing a path γj from xj to x0, we can push a ﬁnger of βj along 0 0 γj to form a new embedded curve βj intersecting x0. Let βj∗ denote an x0-based embedded 0 D j j E loop with image βj , and let ψj : a1, . . . , ag → ha1, . . . , agi denote the group homomorphism j sending each ai 7→ ai. Then as words in ha1, . . . , agi, 0 −1 ψj(βj∗ ) = wγj [βj∗]wγj ∈ ha1, . . . , agi , where wγj denotes the abstract word in ha1, . . . , agi corresponding to successive points of `g signed intersection of γ with i=1 αi, as we would compute for the homotopy class of γ if it were a loop instead of a path. Thus, since our 2-handle attachment Lemma gives α 0 0 π1(Ug , x0) = ha1, . . . , agi / [β1∗],..., [βg∗] , we have proved the following result.

Theorem (Fundamental group presentation from a Heegaard diagram). Let H = (Σ, α, β, x0), β1∗, . . . , βg∗, and ψ1, . . . , ψg be as above. Then the fundamental group of the 3-manifold MH associated to H is given by

π1(MH, x0) ' ha1, . . . , an | ψ1([β1∗]), . . . , ψg([βg∗])i.

0 *** Figure with βj∗, βj∗, and γj. 39

Lecture 8: Alexander Polynomials from Fundamental Groups I. Motivation. 3 Theorem (Knot determined by its exterior—Gordon). Two knots K1,K2 ⊂ S are isotopic 3 if and only if their exteriors XKi := S \ ν(K) are homeomorphic. Theorem (Decomposed knot determined by its exterior’s fundamental group—Thurston). If two knot exteriors XK1 ,XK2 have no embedded essential 2-spheres, annuli, or tori, (conditions ∼ we shall discuss more later in the course), then XK1 = XK2 if and only if π1(XK1 ) ' π1(XK2 ). 3 Thus, a knot K ⊂ S is completely determined by its exterior XK , which, in turn (assuming it is suﬃciently decomposed into primitive pieces), is completely determined by its fundamental group π1(XK ). 40

II. Construction and definition of the Alexander polynomial. A. Geometric group theory: covering spaces. Every connected, path connected, and semilocally simply-connected space, hence every ˜ connected manifold X, admits a universal simply-connected cover πuc : Xuc → X, constructed as the space of homotopy classes of paths initiating at a fixed basepoint x0 ∈ X, ˜ Xuc = {γ : [0, 1] → X, γ(0) = x0}, πuc(γ) → γ(1) ∈ X, with basepointx ˜0 := [γ ≡ x0] given by the constant path at x0. This construction makes ˜ π1(X, x0) act naturally on Xuc by deck transformations: ˜ π1(X, x0) → Homeo(Xuc), g 7→ [γ 7→ gγ]. Convention. For now, we adopt the convention of concatenating paths in the fundamental group by acting on the right. This means that deck transformations act on the left. ˜ Quotienting Xuc by the deck transformation action of π1(X, x0) gives X. Alternatively, ˜ one can quotient Xuc by any normal subgroup N C π1(X, x0), as specified by the cokernel ϕ : π1(X, x0) → π1(X)/N of its inclusion, so that ker ϕ = N. This gives the cover ˜ ˜ ˜ ˜ Xϕ → X, Xϕ := Xuc/ker ϕ, π1(Xϕ) ' ker ϕ. ˜ The deck transformation action of π1(X, x0) on Xuc descends to an action of π1(X, x0) on the ˜ ˜ quotient Xϕ := Xuc/ker ϕ, but with kernel ker ϕ, yielding the deck transformation action

π1(X, x0)/ker ϕ → Homeo(X˜ϕ).

B. Abelianization. Hurewicz Theorem provides the abelianization homomorphism ϕ : π1(X) → H1(X; Z), with ker ϕ = [π1(X), π1(X)]. Thus the group H1(X) acts by deck transformations on the ˜ ˜ cover Xϕ := Xuc/ker ϕ → X speciﬁed by ϕ : π1(X) → H1(X; Z). From now on, we restrict to the case H1(X) ' Z, as for knot exteriors. Thus H1(X) has a canonical (up to sign) generator m ∈ H1(X) with H1(X) = hmi ' Z, generating an action of ˜ ˜ H1(X) on Xϕ by translations, which further induces an action of H1(X; Z) on H1(Xϕ; Z), ˜ ˜ H1(X; Z) → Homeo(Xϕ), H1(X; Z) → Aut(H1(Xϕ; Z)), ˜ ˜ ˜ ˜ m 7→ (τ : Xϕ → Xϕ) m 7→ (τ∗ : H1(Xϕ; Z) → H1(Xϕ; Z))

C. Group ring modules. Write H1(X) ,→ Z[H1(X)], m 7→ [m] =: t, for the inclusion of H1(X) = hmi ' Z into its group −1 ˜ ring Z[H1(X)] ' Z[t , t]. We want to promote the group action H1(X; Z) → Aut(H1(Xϕ; Z)), ˜ m 7→ τ∗, to a group-ring action of Z[H1(X)] on H1(Xϕ; Z), regarded as a Z[H1(X)]-module. Proposition (Group-ring modules from representations). Let R be a commutative ring with identity, VR an R-module, G a group, R[G] the associated group ring, and ρ : G → AutR(VR) an R-linear action of G on VR. Then

VR[G] := R[G] ⊗R VR / hg ⊗ v − 1 ⊗ ρ(g)(v)i is an R[G]-module such that VR[G] ' VR as R-modules. n Corollary. If VR is a ﬁnitely-generated free R-module, VR ' R , then AutR(VR) ' GL(n, R), and our R-linear action of G on VR becomes ρ : G → GL(n, R), g 7→ Bg ∈ GL(n, R), so that n VR[G] ' R[G] / h(g Idn − Bg), g ∈ Gi . (This corollary is for later reference; we ignore it for now.) Following the prescription of ˜ −1 the above proposition, we regard H1(Xϕ; Z) as the Z[t , t]-module ˜ −1 ˜ H1(Xϕ; Z)Z[t−1,t] := Z[t , t] ⊗ H1(Xϕ; Z) / (t = τ∗). 41

D. Module tools and invariants.

Theorem (Structure Theorem for finitely generated modules over a PID). (See Lecture 0.V). Let V be a finitely-generated R-module, for R a PID. Then k V ' FreeR(V ) ⊕ TorsR(V ) ' R ⊕ R/r1R ⊕ · · · ⊕ R/rkR for some r1, . . . rk ∈ R. Definition (PID: principal ideal domain). A ring R is a PID (principal ideal domain) if every ideal in R is principal (can be generated by a single element of R). Remark. Every PID R is also an integral domain: R has no zero divisors a · b = 0 ∈ R auch that 0 6= {a, b}.

Definition (order). Let T ' R/r1R ⊕ · · · ⊕ R/rkR be an R-torsion module over a PID R. Then T has R-order Qk OrdR(T ) := i=1ri and is well-defined up to multiplication by units in R. The order of a non-torsion module is defined to be zero, regarding R as R/0. Remark. The above definition also makes sense for a module V over an integral domain non- −1 PID R (such as Z[t , t]) in the case that V is the quotient of a free module by a principal −1 −1 ideal. Alternatively, we can appeal to the PID Q[t , t] ' Z[t , t] ⊗ Q.

Deﬁnition (Alexander polynomial, H1(X) ' Z case). Let X be a 3-manifold with H1(X) ' Z and inﬁnite cyclic cover X˜ := X˜ϕ → X, π1(X˜) ' ker ϕ, induced by the surjection ϕ : π1(X) → ˜ −1 −1 −1 H1(X) ' Z, Regard H1(X; Q) as a Q[t , t] ' Z[t , t] ⊗ Q-module, for Z[t , t] ' Z[H1(X)] the group ring of H1(X) and t := [m] ∈ Z[H1(X)] the image of the (canonical up to sign) generator m ∈ H1(X). Then X has Alexander polynomial ˜ ∆(X) := OrdQ[t−1,t](H1(X; Q)).

Definition (Alexander polynomial, general case). Suppose X is a 3-manifold with b1(X) ≥ 1 and finitely presented fundamental group. The Alexander polynomial ∆(X) is defined the same as above, if one replaces • the epimorphism π1(X) → H1(X) with a specified epimorphism ϕ : π1(X) → Z, • the cover X˜ → X with the cover X˜ϕ → X specified by ϕ, and • H1(X) with im ϕ ' Z.

Theorem (Alexander). If X is a knot exterior or knot complement, then ∆(X) also satisﬁes ˜ ∆(X) = OrdZ[t−1,t](H1(Xϕ; Z)).

Remark. Any epimorphism ϕ : π1(X) → Z necessarily factors through the abelianization ab homomorphism π1(X) → H1(X) and is therefore also speciﬁed by a choice of epimorphism H1(X) → Z. In particular, if b1(X) = 1, then im ϕ ' H1(X), and ϕ is canonical up to sign. 42

E. Matrix tools. Definition (presentation matrix). If V is a finitely presented module over a commutative ring n m with identity R, then V ' coker ψA, for ψA : R → R , v 7→ A·v, the homomorphism induced by matrix multiplication by some m × n matrix A over R. Such A is called a presentation matrix for V , and is not unique. Corollary (Corollary of Structure Theorem for finitely generated modules over a PID). If V is a finitely generated module over a PID R, and V ' coker ψA as above, then

OrdR (TorsR (coker ψA)) = gcd{determinants of maximal-rank minors of A}.

Deﬁnition (Fox derivative). Let X be a topological space. Suppose π1(X) admits (i) an “augmentation”: an epimorphism ϕ : π1(X) → Z, and (ii) a ﬁnite presentation π1(X) ' ha1, . . . , am| w1, . . . , wni. −1 Let [·] : im ϕ ,→ Z[im ϕ] ' Z[t , t], m 7→ [m] =: t, denote the inclusion of im ϕ ' Z =: hmi into its group-ring. Then the Fox derivative ∂/∂ai =: “∂i” is the map ∂ −1 : {words in a1, . . . , am} → Z[imϕ] ' Z[t , t] ∂ai determined by the rules

∂i(aj) = δij, ∂i(xy) = ∂i(x) + [ϕ(x)]∂i(y). Exercise. The above Fox calculus rules further imply the rules −1 −1 e e ∂i(x ) = −[ϕ(x)] ∂i(x), ∂i(ai · (word without ai)) = ∂(ai ).

Remark 1. Note that if b1(X) > 1, then the Fox derivative depends on our choice of ϕ (if H1(X) 6= Z), but we suppress this dependence in our notation. Remark 2. The Fox derivative computes the abelianization via ϕ of the action of π1(X) on the Q[t−1, t]-module H1(Xeϕ; Q), in the following sense.

Deﬁnition. Given an augmentation ϕ : π1(X) → Z the Alexander matrix associated to a ﬁnite presentation ha1, . . . , am|w1, . . . , wni is the m × n matrix ∂ A := wj , ∂ai ij for ∂/∂ai the Fox derivative.

Theorem (Fox). Let X be a 3-manifold with b1(X) > 1 and ﬁnitely presented fundamental group. Let X˜ϕ → X, with π1(X˜ϕ) ' ker ϕ, be the cover associated to an augmenta- cell ˜ −1 tion ϕ : π1(X) → Z, and let C∗ (Xϕ) denote the cell complex, regarded as a Q[t , t] := cell Q[im ϕ], module, lifting the cell complex C∗ (X) associated to a presentation π1(X, x0) ' ∂ ha1, . . . , am| w1, . . . , wni, with Alexander matrix A := wj . ∂ai ij cell ˜ Then C∗ (Xϕ) has boundary maps cell ˜ cell ˜ cell ˜ cell ˜ d2 = ψA : C2 (Xϕ) → C1 (Xϕ), d1 : C1 (Xϕ) → C0 (Xϕ), v 7→ A · v, a¯i 7→ ([ϕ(ai)] − 1)˜x0, for a¯i an appropriate lift of ai. In particular, ∆(X) = gcd{determinants (m − 1) × (m − 1) minors of A}.

Corollary. Let X, A, m, and n be as above. If rank A = m − 1, then ˜ H1(Xϕ; Q) ' TorsQ[t−1,t](coker ψA). 43

Lecture 9: Fox Calculus I. Fox Calculus: definition and results. Corollary (Corollary of Structure Theorem for finitely generated modules over a PID). If V is a finitely generated module over a PID R, then n m (i) V ' coker ψA, for ψA : R → R , v 7→ A · v, the homomorphism induced by matrix multiplication by some m × n matrix A. (ii) For any such ψA, we have

OrdR (TorsR (coker ψA)) = gcd{determinants of maximal-rank minors of A}.

Remark 1. Note that if b1(X) > 1, then the Fox derivative depends on our choice of ϕ (if H1(X) 6= Z), but we suppress this dependence in our notation. Remark 2. The Fox derivative computes the abelianization via ϕ of the action of π1(X) on −1 the Q[t , t]-module H1(Xeϕ; Q), in the following sense.

Corollary. Let X, A, m, and n be as above. If rank A = m − 1, then ˜ H1(Xϕ; Q) ' TorsQ[t−1,t](coker ψA). 44

II. Trefoil example: direct computation. A. Cell decomposition of X. 3 3 Let T ⊂ S be the trefoil knot, and let X := XT = S \ ν(T ) be its exterior. For now, we simply assert that X deformation retracts onto a 2-skeleton X2 given by attaching a disk 2 −1 −1 −1 with boundary ∂D = abab a b , to a wedge at x0 of loops a([0, 1]) and b([0, 1]) at x0. def ret 2 X −→ X2 = D ∪∂D2 X1, 2 −1 −1 −1 −1 −1 −1 X1 = a([0, 1]) ∨x0 b([0, 1]), ∂D = abab a b , π1(X, x0) ' a, b| abab a b .

B. Cell decomposition of X˜ := X˜ϕ. Let ϕ : π1(X, x0) → H1(X) ' Z be the abelianization homomorphism. Then

H1(X) ' hϕ(a), ϕ(b)i /(ϕ(a) + ϕ(b) + ϕ(a) − ϕ(b) − ϕ(a) − ϕ(a)) ' hϕ(a), ϕ(b)i /(ϕ(a) − ϕ(b)), and H1(X) is generated by m := ϕ(a) = ϕ(b).

Let X˜ := X˜ϕ → X, be the cover associated to ϕ, with π1(X,˜ x˜0) ' ker ϕ and deck transformation group H1(X) → Homeo(X˜), m 7→ (τ : X˜ → X˜). 2 2 Fix a liftx ˜0 ∈ X˜ of x0, determining the liftsa ˜, ˜b, and D˜ of a, b, and D , so that ∼ ` i ˜ 2 ˜ i ˜ 2 i ˜ 2 X = iτ D ∪ X1, ∂(τ D ) = τ (∂D ), for X˜1 the 1-skeleton of X˜, as depicted below.

Here, only the solid red and blue curves are among the 1-cells of X1. The dotted curves are merely paths in the 1-skeleton X1, not part of X1 itself. Thus, if we lift the boundary path 2 −1 −1 −1 ˜2 ∂D = abab a b ∈ π1(X, x0) to ∂D ∈ π1(X,˜ x˜0), we obtain ∂D˜ 2 = (˜a)(τ 1˜b)(τ 2a˜)(τ 2˜b−1)(τ 1a˜−1)(˜b−1). The individual factorsa, ˜ τ 1˜b, τ 2a,˜ τ 2˜b−1, τ 1a˜−1, ˜b−1 of this product are elements of the fundamental groupoid Π(X˜1) of homotopy classes of paths in X˜1, but they can be multiplied because consecutive factors agree between the endpoint of one factor and the starting point of the next factor. 45

For any subspace Y ⊂ X˜1, the fundamental groupoid restricts to the relative fundamental groupoid Π(X˜1,Y ) of homotopy classes of paths in X˜1 with startpoints and endpoints in Y . In particular, for the relative fundamental groupoid ˜ ˜ ` ˜ i j Π(X1, X0) := i,j π1(X1, τ x˜0, τ x˜0) ˜ ˜ i ˜ of paths in X1 between points in the 0-skeleton X0 = {τ x˜0}i∈Z ⊂ X1, we have H1(X) → Aut Π(X˜1, X˜0) ,

k i j i+k j+k k km 7→ τ : π1(X˜1, τ x˜0, τ x˜0) → π1(X˜1, τ x˜0, τ x˜0) , w 7→ τ w.

This is just a fancy way of saying that—like everything else in X˜1—paths between 0-cells in X˜1 get translated by the deck transformation action of H1(X).

C. Cellular chain complex computations. To ﬁx some notation, note that we have the abelianization map (see Appendix 9.A) ˜ ˜ cell ˜ k ±1 k k˜±1 k¯ ϕ˜cell : Π(X1, X0) −→ C1 (X), ϕ˜cell(τ a˜ ) = ±t a,¯ ϕ˜cell(τ b ) = ±t b, ¯ cell ˜ wherea, ¯ b ∈ C1 (X) are the respective cellular 1-chains associated to the images, counted with multiplicity, of the pathsa ˜ and ˜b. We also have the inclusion homomorphism −1 [·]: H1(X) ,→ Z[H1(X)] =: Z[t , t], ϕ(˜a) = ϕ(˜b) = m 7→ [m] =: t.

cell ˜ −1 C∗ (X) is ﬁnitely generated as a Z[H1(X)] ' Z[t , t]-module: cell ˜ d2 cell ˜ d1 cell ˜ 0 −→ C2 (X) −→ C1 (X) −→ C0 (X) −→ 0, 2 d2 d1 0 −→ D¯ −→ a,¯ ¯b −→ hx¯0i −→ 0,

d1a¯ = tx¯0 − x¯0 = (t − 1)¯x0,

d1¯b = tx¯0 − x¯0 = (t − 1)¯x0,

ker d1 = ha¯ −¯bi,

2 1˜ 2 2˜−1 1 −1 ˜−1 d2D¯ =ϕ ˜cell(˜a · τ b · τ a˜ · τ b · τ a˜ · b ) =a ¯ + t1¯b + t2a¯ − t2¯b − t1a¯ − ¯b = (1 − t + t2)(¯a − ¯b). Thus, we compute that

H1(X˜) ' ker d1/im d2 ' a¯ − ¯b /(1 − t + t2)(¯a − ¯b) −1 2 ' Z[t , t]/(1 − t + t ), −1 1 which has Z[t , t ] order ∆(X) = t−1 − 1 + t2, 2 −1 −1 −1 after multiplying 1 − t + t by the unit +t ∈ Z[t , t] to achieve ∆(t) = ∆(t ) and ∆(t)|t=1 = 1. 46

D. Result in matrix notation. Note that in the language of matrices and basis elements, with respect to the isomorphisms cell ˜ ∼ −1 1 cell ˜ ∼ −1 2 cell ˜ ∼ −1 1 C2 (X) −→ Z[t , t] ,C1 (X) −→ Z[t , t] ,C0 (X) −→ Z[t , t] , 1 0 D¯ 2 7−→ (1), a¯ 7→ , ¯b 7→ , x¯ 7−→ (1), 0 1 0 the above boundary maps, as computed, would be written 1 + t2 − t (3) d = , d = [t − 1 t − 1]. 2 t − t2 − 1 1

III. Trefoil example by Fox calculus. ∂ 2 ∂ 2 To determine the Alexander matrix, we compute ∂a (∂D ) and ∂b (∂D ) as follows. ∂ ∂ −1 −1 −1 ∂a (1) = ∂a (abab a b ) ∂ ∂ −1 ∂ −1 = ∂a (a) + [ϕ(ab)] ∂a (a) + [ϕ(abab )] ∂a (a ) = (1) + t · t · (1) + t · t · t · t−1 · (−t−1) = 1 + t2 − t,

∂ 2 ∂ −1 −1 −1 ∂b (∂D ) = ∂b (abab a b ) ∂ ∂ −1 −1 −1 ∂ −1 = [ϕ(a)] ∂b (b) + [ϕ(aba)] ∂b (b ) + [ϕ(abab a )] ∂b (b ) = t · (1) + t · t · t · (−t−1) + t · t · t · t−1 · t−1 · (−t−1) = t − t2 − 1.

Thus our d2 boundary map is given by the Alexander matrix " # ∂ (∂D2) 1 + t2 − t d = A = ∂a = , 2 ∂ 2 t − t2 − 1 ∂b (∂D ) and the d1 boundary map is given by

d1 = [[ϕ(a)] − 1 [ϕ(b)] − 1] = [t − 1 t − 1], which we compare with (3), above. Remark. Strictly speaking, we stated Fox’s Theorem in terms of the cellular chain complex −1 −1 over the ring Q[t , t] ⊃ Z[t , t], since the former is a PID and the latter is not, requiring more complicated deﬁnitions for torsion and order in the latter case. It is still the case, however, that if the cokernel of the Alexander matrix, as applied to a −1 −1 −1 Z[t , t]-module, gives the quotient of a free Z[t , t]-module by a principal ideal in Z[t , t], −1 then Fox’s Theorem also holds over Z[t , t]. Alexander showed that this latter situation always occurs if X is the exterior of a knot. 47

cell Appendix 9.A (not examinable). C1 (X) from Π(X1). Let d1 0 −→ C1(X1) −→ C0(X1) −→ 0, dγ = end(γ) − start(γ) be the cellular chain complex associated to the 1-skeleton X1 of a cell complex X. Then since H1(X1) ' ker d1, Hurewicz’ Theorem provides the homomorphism ab (4) π1(X1, x0) −→ ker d1 ⊂ C1(X1) ' C1(X), which sends an x0-based loop in X1 to the cellular 1-chain associated with its image, counted with multiplicity. Any commutator path has vanishing image when counted with multiplicity.

Observation. If X has only one 0-cell, X0 = x0, then end(γ) = start(γ) = x0 for all γ ∈ C1(X1), so d1 ≡ 0, implying ker d1 = C1(X1), so that (4) becomes the epimorphism ab π1(X1, x0) −→ C1(X).

If |X0| > 1, however, we must look to the fundamental groupoid if we want to get a surjective homomorphism to C1(X). The fundamental groupoid a Π(X) = π1(X, x1, x2)

x1,x2∈X is the set of homotopy classes of paths in X, together with multiplication by concatenation of paths, which is only deﬁned for pairs of paths that can be concatenated: γ1 · γ2 ∈ Π(X) when γ1, γ2 ∈ Π(X) satisfy γ1(1) = γ2(0). In addition, for any subspace Y ⊂ X, the fundamental groupoid restricts to the subgroupoid given by the relative fundamental groupoid a Π(X,Y ) := π1(X, y1, y2) < Π(X)

y1,y2∈Y of homotopy classes of paths in X with startpoints and endpoints in Y . In particular, let a Π(X1,X0) := π1(X1, y1, y2)

y1,y2∈X0 denote the groupoid of homotopy classes of paths in the 1-skeleton X1 between 0-cells in X0. Π(X ,X ) Let 2 1 0 denote set of subsets of Π(X1,X0). Then there is a natural surjective map

Π(X1,X0) ϕcell : 2 → C1(X) which respects multiplication, has abelian image, and restricts on π1(X1, x0) to a homomorphism ϕcell|π1(X1,x0) : π1(X1, x0) → ker d1 coinciding with the homomorphism in (4).

Π(X1,X0) Remark. ϕcell is determined by its restriction to 1-element subsets of 2 , i.e., by its restriction to Π(X1,X0): ϕcell : Π(X1,X0) → C1(X) In the preceding lecture, we used this map for X replaced with X˜, which has 0-skeleton ˜ k X0 = {τ x˜0}k∈Z. Even so, none of this is particularly important. It is just a formal way of describing a map that is intuitively clear: it sends a path between 0-cells in X1 to the cellular 1-chain associated to the image of that path, counted with multiplicity. The point of writing this map down formally is to clarify what it does to multiplication. 48

Lecture 10: Dehn presentations and Kauffman states I. Based Link Diagrams (covered in Example Sheet 1 instead of lecture). Here For the following discussion, we assume that D is the diagram of a non-split link 3 2 L ⊂ S , so that its underlying embedded graph Γ ,→ R is connected. A. Regions, crossings, and edges. 3 2 Recall that any link diagram D for a link L ⊂ S has an underlying embedded graph Γ,→R , 2 2 2 which we can regard as an embedded graph Γ,→S by compactifying R with a point y∞ ∈ S 2 ∼ 2 2 at inﬁnity, so that R = S \{y∞}. The graph Γ then partitions S into (contractible) regions, edges, and vertices. By slight abuse of terminology, we shall also refer to these as regions, edges, and crossings, respectively, of the diagram D.

Deﬁnition (outer region). We call the unique region containing y∞ the outer region r∞. Like all the regions associated to D, the outer region r∞ has the topology of a disk. Deﬁnition (adjacent). We say a region r is adjacent to an edge e or a crossing c if the closure of r has nonempty intersection with e or with the vertex corresponding to c, respectively. Thus, keeping in mind that the outer region is still a region, we observe that • Each crossing has exactly 4 adjacent regions. • Each edge has exactly 2 adjacent regions.

Deﬁnition (outer edge). An outer edge is any edge adjacent to the outer region r∞.

B. Based link diagrams.

Definition (based link diagram). A based link diagram (D,?) is a diagram D with a choice of basepoint, ?, lying on the interior of some outer edge of D. Remark. In principle, one could consider based diagrams with basepoints on non-outer edges, but since we are free to move the point at infinity y∞ to a different region, there is no loss in generality in demanding that our basepoint, ?, lie on an outer edge.

Deﬁnition (based edge). In a based diagram, the based edge e? is the edge containing the basepoint ?. Remark. One can use an Euler characteristic argument to show that (5) #crossings(D) = #regions(D) − 2. Thus, for a based diagram (D,?), we have

#crossings(D) = #{regions(D) not adjacent to the based edge e?}. 49

II. Dehn presentation Heegaard Diagrams. A. Deﬁnition.

Deﬁnition. In the Dehn-presentation Heegaard diagram HDL = (Σ, α, β, x0) for the exterior 3 2 of a link L ⊂ S , associated to a based link diagram (DL,?) with underlying graph ΓL,→R , • Σ := −∂(ν(ΓL)), • αi := ∂(ith non-outer region of ΓL) 2 • βj := ∂(Dj at jth crossing, separating top link strand from bottom link strand), • x0 ∈ −∂(ν(edge containing ?)) ∩ −∂(ν(ΓL)).

Remark. We think of forming M by attaching αi to (−Σ) × {1} and βj to (−Σ) × {0}, HDL which is equivalent to attaching αi to Σ × {0} and βj to Σ × {1}.

Proposition. M =∼ X := S3 \ ν(L). HDL L

Proof by construction of XL. 1. Re-embed 2 ∼ 2 2 3 ∼ 3 ΓK ,→R = R × {0} ,→ R × R ,→ R ∪ {∞} = S . 2 Remember that this R factor in R × R points towards us out of the diagram, with the positive direction of R distinguishing the heights of strands at crossings. 3 Take a tubular neighborhood ν(ΓK ) ⊂ S , set

Σ := ∂(ν(ΓK )), noting that g(Σ) = b1(ΓK ) = 1 + |{crossings(DK )}|,

and draw this Σ on the page, with x0 ∈ Σ positioned as in ΓK if ΓK is based. The above righthand equality is due the fact that in a closed embedded graph with only degree-4 vertices, the number of cycles is one more than the number of vertices. This is straightforward to prove inductively or via Euler characteristic arguments.

th 2. Draw a curve αi ⊂ Σ along each i cycle of ΓK . Attach a 2-handle to Σ × I along each 2 3 ◦ αi ⊂ Σ × {1}. Attach a 3-handle to the resulting S boundary, to form S \ ν(ΓK ). 3. Draw the knot K on Σ (say in green), so that the top strands of crossings go along the top of Σ and the bottom strands of crossings go along the underside of Σ. At each jth crossing, draw a curve βj whose top half parallels the upper strand on each side and whose bottom half parallels the bottom strand on either side. Attach a 2-handle along each βj ⊂ Σ×{0}.

These disks prevent the upper strand of K from passing through the lower strand of K at each crossing. Exercise: forms boundary a disjoint union of tori. Thus, we have formed ∼ 3 ◦ `g−1 2 XK = (S \ ν(ΓK )) ∪β i=1 D × I. Remark. Since m − 1 = n, we take n × n minors of the Alexander polynomial A. That is, if

A¯i := (n × n minor of A with row Ai deleted), then ¯ ∆(XL) = gcd{det(Ai)}i∈{1,...,m}.

Alternatively, if the meridian m positively links αi? , so that ai? = m, then ¯ ¯ ¯ ¯ ∆(XL) = det(Ai? ) = det(A), A := Ai? . 50

B. Trefoil example.

The knot diagram for the trefoil knot T ⊂ S3 gives rise to the following

3 ◦ Dehn presentation Heegaard diagram for the knot’s exterior XT := S \ ν(T ).

Orient the αi and βj as shown, and choose basepoints on each βj. I generally orient my curves so that their visible part winds counterclockwise, and I generally place basepoints at the top right corner of each crossing, but there are other conventions. We then compute the respective relation word wj for each βj by reading oﬀ ai exponents from successive signed intersections αi · βj, starting at our chosen basepoint for βj. In the above case, this gives −1 −1 −1 −1 w1 = 1a1a2 a4, w2 = a2 a1a3 a4, w3 = a3 a11a4, where I have placed a 1 for the (non)intersection with the “missing” α curve around the outer region. This is just an optional trick for keeping track of the alternating sign. We then have

π1(XT , x0) = ha1, a2, a3, a4|w1, w2, w3i . −1 We compute our meridian µ to be µ = a4 ∈ π1(XT , x0). Abelianizing, we obtain 0 0 0 0 0 0 0 0 0 0 0 0 0 0 H1(XT ; Z) ' a1, a2, a3, a4 / a1 − a2 + a4, −a2 + a1 − a3 + a4, −a3 + a1 + a4 , 0 0 with m := ϕ(µ) = −a4 = −ϕ(a4), where we temporarily wrote ai := ϕ(ai) for brevity. Solving the above equations for each ϕ(ai) in terms of m, we then obtain

ϕ(a1) = m, ϕ(a2) = 0, ϕ(a3) = 0, ϕ(a4) = −m. 51

III. Alexander polynomial from Dehn presentation. Since we have already discussed how to compute the Alexander polynomial from Fox calculus in the general case, we focus on the speciﬁc example of our trefoil diagram. Using Fox calculus, we compute the Alexander matrix associated to our fundamental group presentation: −1 −1 −1 −1 w1 = a1a2 a4 w2 = a2 a1a3 a4 w3 = a3 a1a4

 −1 −1  ∂1 1 [ϕ(a2 )] [ϕ(a3 )]    −1 −1  ∂ ∂2  −[ϕ(a1a2) )] −[ϕ(a2 )] 0  A := wj =   . ∂ai ij  −1 −1 −1  ∂3  0 −[ϕ(a2 a1a3 )] −[ϕ(a3 )]   −1 −1 −1  ∂4 [ϕ(a1a2 )] [ϕ(a2a1a3 )] [ϕ(a3 a1)]

We next take the minor A¯ := A¯4 of A that excludes the row corresponding to ∂/∂a4, since α4 −1 bounds the region adjoining our based edge (or equivalently, since a4 ∈ {µ, µ }). Expressing A¯ in terms of t := [m] = [ϕ(µ)], we have

w1 w2 w3   ∂1 1 1 1   ¯   A := ∂2  −t −1 0  .   ∂3 0 −t −1

Lastly, to compute the Alexander polynomial ∆(XT ), we take the determinant det A¯, which has 3 nonzero summands:

∆(XT ) = det A¯ = A11A22A33 + A21A32A13 − A21A12A33 = (1)(−1(−1) + (−t)(−t)(1) − (−t)(1)(−1) = t(t−1 − 1 + t). We can depict the 3 permutations corresponding to these 3 nonzero summands pictorially on the diagram for the trefoil by placing a dot in the ith region by the jth crossing for each factor Aij in a nonzero summand of the determinant, as shown.

A11A22A33 A21A32A13 − A21A12A33

Such diagrams are called Kauﬀman state diagrams, and they can be used to compute the Alexander polynomial directly, as following section. 52

IV. Kauffman States (covered in Example Sheet 1 instead of lecture). Definition (Kauffman State). A Kauffman state s of a based diagram (D,?) is a matching,

s : crossings(D) → {regions(D) not adjacent to the based edge e?}, such that the region s(c) is adjacent to c for each c ∈ crossings(D). (If D is a split link, hence with nonconnected underlying graph Γ, we say that (D,?) has zero Kauﬀman states.) Convention. To depict a Kauﬀman state s, we decorate (D,?) with # crossings(D) dots. For each crossing c ∈ crossings(D), we place a dot in s(c) near c.

The 3 Kauﬀman States for the based diagram .

Non-Kauﬀman States.

(a) 2 dots share the same region. (b) A dot occupies a region adjacent to the based edge e?. (c) A dot occupies the outer region r∞, which by our conventions is always adjacent to e?. Definition (Kauffman state crossing monomial). Given a Kauffman state s of an oriented based diagram (D,?), the Kauffman state crossing monomial of a Kauffman-state-labeled ori- − 1 1 ented crossing c is a monomial ks,c(t) ∈ Z[t 2 , t 2 ], which depends on whether c is a positive or negative crossing, and on the location of s(c) relative to this oriented crossing, as shown.

In other words, we assign monomials according to the diagram

Definition (Kauffman state monomial). The Kauffman state monomial of a Kauffman state s of an oriented based diagram (D,?) is the monomial Y − 1 1 ks(t) := ks,c(t) ∈ Z[t 2 , t 2 ]. c ∈ crossings(D) 53

−1/2 1/2 Definition (Kauffman state sum). The Kauffman state sum kD(t) ∈ Z[t , t ] of a based diagram (D,?) is the polynomial X −1/2 1/2 kD(t) := ks(t) ∈ Z[t , t ]. s∈{Kauffman states(D,?)} Definition (Alexander grading). The t-degree of a Kauffman state monomial or crossing monomial is called its Alexander grading.

V. Equivalence of Alexander polynomial definitions. We have already demonstrated the equivalence of the Kauffman state sum and Skein- relation Alexander polynomials as an exercise in Example Sheet 1. Theorem. Given a based oriented diagram (D,?) for a link L ⊂ S3, the Skein-relation Alexander polynomial ∆D(t) and Kauffman state sum kD(t) are equivalent, up to multiplica- k tion by ±t for some k ∈ Z: k ∆D(t) = ±t kD(t). Proof. See Example Sheet 1.

We next return to our observation from the end of Section III. Let HD = (Σ, α, β, x0) be the Dehn presentation Heegaard diagram associated to a based diagram (D,?) of a link 3 L ⊂ S , with x0-based meridian linking e?, so that XL = MHD . Let A be the Alexander matrix associated to the fundamental group presentation of π1(MHD , x0) associated to HD, and A¯ the minor of A excluding the ith row, for αi bounding the inner region adjoining e?. Proposition. With respect to the sum ¯ X ¯ det A = sign(σ)Ai,σ(i), σ∈Sn there is a bijection {non-zero summands of det(A¯)} ↔ {Kauﬀman states of D}.

To show that Kauffman state monomials also have the correct Alexander grading (t-degree), Kauffman compared the gradings of states related by a move he called a “transposition.” Definition (Kauffman state transposition (not examinable)). Two Kauffman states

r, s : crossings(D) → {regions(D) not adjacent to the based edge e?}, differ by a transposition if (i) r and s differ at precisely two crossings v 6= w ∈ crossings(D); that is, r(v) = s(w), r(w) = s(v), and r(z) = s(z) ∀z∈ / {v, w}; and (ii) there is a path P along L from v to w, not containing ?, such that {∂r(v) ∩ ∂s(v) ∩ edges(v), ∂r(w) ∩ ∂s(w) ∩ edges(w)} = {first edge of P, last edge of P } keeping in mind that the first and last edge of P might coincide. Theorem (Kauffman’s Clock Theorem). 1. Any 2 Kauffman states differ by a sequence of transpositions among Kauffman states. 2. If the Kauffman states r, s differ by a transposition, then k (t)k (t) A A r,v r,w = − iv jw , ks,v(t)ks,w(t) AjvAiw where αi = ∂r(v) = ∂s(w), αj = ∂r(w) = ∂s(v), and βv and βw are the β curves associated to the respective crossings v and w. k k Corollary. For an appropriate ±t , we have ∆(XL) = ±t kD(t). 54

Lecture 11: Mapping tori and Mapping Class Groups I. Mapping tori. A. Motivation: applications of surface homeomorphisms. There are two primary contexts in which we need to know about the available class of homeomorphisms, or equivalently, of diﬀeomorphisms, from Σ → Σ from a surface to itself:

(i) A gluing map ϕ : ∂iM1 → −∂jM2, for a union M1 ∪ M2 of 3-manifolds. So far, we have primarily encountered such gluing maps with Heegaard splittings, where we glue two handle ∼ α β bodies together along a Heegaard surface, M = Ug ∪Σg Ug . 1 (ii) Any 3-manifold fibered over S can be expressed as the mapping torus Mϕ of some homeomorphism or diffeomorphism ϕ :Σ → Σ, where Σ is the fiber of the fibration.

B. Fibered manifolds and mapping tori. Deﬁnitions (mapping torus, monodromy). Suppose φ : Y → Y is a homeomorphism (respectively diﬀeomorphism) of a manifold Y . Then the mapping torus Mφ of φ is the topological (respectively smooth) manifold given by the quotient

Mφ := (Y × I)/(y, 1) ∼ (φ(y), 0).

The map φ : Y → Y is called the monodromy of Mφ.

Remark. We shall particularly be interested in 3-manifold mapping tori Mφ of surface homeomorphisms φ :Σ → Σ.

Definitions (fibered 3-manifold, fiber). In the case of 3-manifolds, topologists call a 3- manifold M fibered if it can be realized as a (locally trivial) fiber bundle over S1. That is, there is a map π : M → S1 such that π−1(I) =∼ Σ × I for any interval I ⊂ S1, for some fixed surface Σ called the fiber of the fibration π.

Definition (fibered link). A link L ⊂ M in a 3-manifold M is called a fibered link if (i) its exterior M \ ν(L) is fibered (or equivalently its complement M \ K is fibered), and (ii) L =∼ ∂Σ, for Σ the fiber of the fibration (M \ ν(L)) → S1. Such Σ is called a Seifert surface for L ⊂ M.

Remark. If H1(M) = 0, then (ii) is automatically satisﬁed for a link with ﬁbered exterior. In particular, this holds for knots and links in S3.

Proposition. A 3-manifold M is fibered if and only if it can be realized as the mapping torus 1 Mφ of some monodromy homeomorphism φ :Σ → Σ. A fibration π : M → S specifies Σ and φ :Σ → Σ uniquely up to homeomorphism and conjugation by homeomorphism, respectively.

Theorem (Alexander polynomial as characteristic polynomial). Let φ :Σ → Σ be a homeomorphism of an oriented surface Σ, with mapping torus Mφ := (Σ × I)/((x, 1) ∼ (φ(x), 0)), 1 1 and let π : Mφ → S be the associated ﬁbration over S . Then the Alexander polynomial 1 ∆(Mφ) of Mφ (associated to the augmentation π∗ : π1(Mφ) → π1(S ), if b1(Mφ) > 1) is the characteristic polynomial of the induced homomorphism φ∗ : H1(Σ) → H1(Σ). That is,

∆(Mφ) = det(φ∗ − t · id). Proof. See Example Sheet 2. 55

II. Mapping Class Group. ? Definition (mapping class group). The mapping class group MCG(Σ) (often denoted Mod(Σ)) of a surface Σ is the group MCG(Σ) := {orientation-preserving diffeomorphisms Σ → Σ}/isotopy. The following two results give us extra flexibility in how we treat this group. Theorem (Munkres, Smale and Whitehead). Any homeomorphism between smooth surfaces is isotopic to a diffeomorphism.

Proof (not examinable). If interested, see Hatcher’s more recent proof [6].

Theorem (Dehn-Nielsen-Baer). Two diﬀeomorphisms f1, f2 :Σ → Σ are isotopic if and only if they are homotopic.

Proof (not examinable). Standard proofs invoke something called the “curve complex.”

? Corollary. As a corollary of the two above theorems, we also have MCG(Σ) '{orientation-preserving homeomorphisms Σ → Σ}/homotopy. It turns out that this homotopy-theoretic formulation has the consequence that we can compute Mapping Class Group of a surface in terms of its fundamental group. Before stating and (sketchily) proving this, we temporarily introduce two new versions of mapping class group that will play a role in the statement and proof of this result. Deﬁnition (unoriented mapping class group). Let Σ be a closed oriented surface. The unoriented mapping class group MCG±(Σ) is the group

MCG±(Σ) := {(not-necessarily-oriented) diﬀeomorphisms of Σ}/isotopy

Note that MCG(Σ) C MCG±(Σ) as an index-2 subgroup: MCG(Σ) ' ker (deg : MCG±(Σ) → {±1}). We can also consider based, or basepoint-preserving, diﬀeomorphisms/homeomorphisms.

Deﬁnition ((unoriented) based mapping class group). Let (Σ, x0), be a closed oriented surface with basepoint x0 ∈ Σ. The unoriented based mapping class group MCG±(Σ, x0) is the group

MCG±(Σ, x0) := {based diﬀeomorphisms (Σ, x0) → (Σ, x0)}/isotopy

The (ordinary) based mapping class group MCG(Σ, x0) C MCG±(Σ, x0)) has the same deﬁ- nition except that we restrict to oriented diﬀeomorphisms.

Remark. The groups MCG±(Σ), MCG±(Σ, x0)), and MCG(Σ, x0) can also be formulated in terms of homeomorphisms up to homotopy. 56

III. Presentation of MCG as outer automorphisms of the fundamental group. A. Main results. Deﬁnitions (inner/outer automorphisms). Let G be a group. The inner automorphism group Inn(G) of G is the normal subgroup −1 Inn(G) := {φh ∈ Aut(G)| φh : g 7→ h gh, h ∈ G} C Aut(G) of automorphisms of G induced by conjugation. The outer automorphism group Out(G) of G is then the quotient Out(G) := Aut(G)/Inn(G).

Theorem. If (Σ, x0) is a based closed oriented surface of genus g > 0, then the map ∼ (6) ρ : MCG±(Σ, x0) −→ Aut(π1(Σ, x0)),

[ϕ : (Σ, x0) → (Σ, x0)] 7→ (ϕ∗ : π1(Σ, x0) → π1(Σ, x0)) is an isomorphism of groups which descends to an isomorphism, ∼ (7) MCG±(Σ) −→ Out(π1(Σ)). Sketch of Proof (more detail here than provided in lecture; not examinable). 1. We claim such ρ is (a) well-defined and (b) a homomorphism. (a) Any based map ϕ : (Σ, x0) → (Σ, x0) induces a homotopy-invariant homomorphism ϕ∗ : π1(Σ, x0) → π1(Σ, x0), [γ] 7→ [ϕ ◦ γ]. Since ϕ is a homeomorphism, it has an inverse map −1 ϕ , so the homomorphism ϕ∗ is invertible, hence is an automorphism. (b) Such ρ is a homomorphism since ϕ∗ ◦ ψ∗ = (ϕ ◦ ψ)∗. 2. We claim that (6) implies (7). That is, forgetting the basepoint on the lefthand side corresponds to identifying homeomorphisms previously distinguished by sending basepoints to different places, which corresponds on the righthand side to quotienting Aut(π1(Σ, x0)) by the action of conjugation by π1(Σ, x0). 3. To sketch a proof of (6), we construct an inverse of the homomorphism ρ, using the facts 2 (a) For g(Σ) > 0, (Σ, x0) has a contractible universal cover, π :(R , x˜0) → (Σ, x0); (b) Every homotopy equivalence (Σ, x0) → (Σ, x0) is homotopic to a homeomorphism. You have hopefully already seen fact (a). Fact (b) is a deep result which we shall not prove. 2 (a) For any manifold, we have an embedding ι : π1(Σ, x0),→R , g 7→ gx˜0, into the universal cover. When the universal cover is contractible (so that we have a so-called K(π, 1) space), one can use this ι to promote any homomorphism f : π1(Σ, x0) → π1(Σ, x0) to a based, π1- ˜ 2 2 equivariant map f :(R , x˜0) → (R , x˜0) determined up to homotopy, such that the induced map on quotient spaces f¯ : (Σ, x0) → (Σ, x0) satisfies f¯∗ = f. If we have invertible homomorphisms f, g : π1(Σ) → π1(Σ), with f ◦ g = id, then the induced maps f,¯ g¯ : (Σ, x0) → (Σ, x0), are homotopy inverses: f¯◦ g,¯ g¯ ◦ f¯ ∼ id. Thus, our K(π, 1) methods have produced a map

ρ¯ : Aut(π1(Σ, x0)) → {homotopy equivalences (Σ, x0) → (Σ, x0)}/homotopy, f 7→ f.¯ (b) Fact (b) then produces a canonical isomorphism, ∼ “homeo” : {homotopy equivalences (Σ, x0) → (Σ, x0)}/homotopy → MCG±(Σ, x0), −1 such that “homeo” ◦ ρ¯ = ρ .

Deﬁnition. For g(Σ) > 0, let σ : Aut(π1(Σ, x0)) → {±1}, be the homomorphism, descending to σOut : Out(π1(Σ, x0)) → {±1}, determined by whether f ∈ Aut(π1(Σ, x0)) preserves or reverses the sign of the intersection form on H1(Σ; Z) This gives rise to index-2 subgroups

Aut+(π1(Σ, x0)) := ker σ, Out+(π1(Σ, x0)) := ker σOut. Corollary. The two isomorphisms in the above theorem restrict to isomorphisms ∼ ∼ ρ+ : MCG(Σ, x0) −→ Aut+(π1(Σ, x0)),MCG(Σ) −→ Out+(π1(Σ)). 57

2 2 B. MCG for genus 0 or 1 (S or T ). 2 2 Theorem (MCG for g ≤ 1). The respective mapping class groups of S and T are 2 2 (a) MCG(S ) ' 1, (b) MCG(T ) ' SL(2, Z). Proof. (a) First, recall that homotopy classes of sphere maps φ : Sn → Sn are classiﬁed by the n degree deg φ ∈ πn(S ) ' Z of such maps. An orientation-preserving homeomorphism S2 → S2 has degree +1. Thus, any two orientation- preserving homeomorphisms are homotopic. 2 2 (b) Since T has genus g(T ) = 1 > 0, we can compute its mapping class group from its fundamental group π1(T2) ' Z ⊕ Z. 2 2 MCG±(T ) ' Out(π1(T )) ' Out(Z ⊕ Z) ' GL(2, Z), 2 2 MCG±(T ) ' Out+(π1(T )) ' Out+(Z ⊕ Z) ' SL(2, Z). 58

Lecture 12: Nielsen-Thurston Classification for Mapping class groups 2 2 I. Classifying elements of MCG(T ) = Mod(T ). Last time, we discussed an isomorphism ∼ MCG±(Σ, x0) −→ Aut(π1(Σ, x0)), ϕ 7→ ϕ∗ ∼ that descends to an isomorphism ρ : MCG±(Σ) −→ Out(π1(Σ)). This further restricts to an isomorphism on the subgroups ∼ ρ : MCG(Σ) −→ Out+(π1(Σ)), where the group Out+(π(Σ)), of “orientation-preserving outer automorphisms” is the group of outer automorphisms of π1(Σ) which preserves the sign of the intersection pairing on H1(Σ). Alternatively, it is the image of ρ under its restriction to MCG(Σ). ∼ 2 For general Σ, our construction of an inverse to ρ was abstract. For Σ = T , this inverse is −1 2 easy to describe explicitly. Let ρ : SL(2, Z) → MCG(T ) be the homomorphism sending −1 2 2 2 2 −1 2 2 2 ∼ 2 φ 7→ ρ (φ): R /Z → R /Z ), ρ (φ)(x) := φ(x) + Z ∈ R /Z = T . −1 2 −1 2 2 Then ρ (φ) represents an element of MCG(T ), and φ = ρ (φ)∗ : π1(T ) → π1(T ).

2 Proposition (MCG(T ) Classification). Each φ ∈ SL(2, Z) satisfies one of the following. (i) |tr φ| ∈ {0, 1}— φ is periodic: φk = id for some k > 0. 2 (ii) |tr φ| = 2 — φ is reducible: φ(vR)=vR for some line vR ⊂ R , −1 2 (ρ (φ))(γ) = γ for some closed curve γ ⊂ T . λ 0 (iii) |tr φ| > 2 — φ is Anosov: φ ∼ , |λ| > 1 ∈ . 0 λ−1 R Proof. Since φ has det φ = 1, it has characteristic polynomial 2 2 χφ(λ) = det(λId − φ) = λ − (tr φ)λ + det φ = λ − (tr φ)λ + 1, so the qualitative behaviour of the eigenvalue roots of χφ(λ) depends on the discriminant 2 2 1 1 p 2 disc(χφ) = b − 4ac = (tr φ) − 4, with λ = 2 trφ ± 2 (trφ) − 4. Cases (i), (ii), and (iii) correspond to disc(χφ) < 0, = 0, and > 0, respectively: (i) (trφ)2 − 4 < 0, (ii) (trφ)2 − 4 = 0, (iii) (trφ)2 − 4 > 0, λ 0 λ ∗ λ 0 φ ∼ , φ ∼ , φ ∼ , 0 λ−1 0 λ 0 λ−1 iθ 2πi 2πi 2πi −1 λ = e , θ ∈ { 4 , 3 , 6 }, λ = λ = ±1, |λ| > 1 ∈ R. ±1 ±iθ For (i), we deduce that λ = e because χφ has distinct complex conjugate roots with product 1. To solve for θ given trφ ∈ {0, −1, 1}, we recall that Cayley-Hamilton’s Theorem 2 2 3 3 tells us 0 = χφ(φ) = φ − (trφ)φ + Id, implying φ = −Id, φ = Id and φ = −Id in the respective cases trφ = 0, trφ = −1, and trφ = 1, respectively corresponding to λ±1 = e±iθ 0 −1 −1 −1 1 1 with θ ∈ { 2πi , 2πi , 2πi }, and representative matrices φ ∼ , , . 4 3 6 −1 0 1 0 −1 0 1 n 1 1 n 1 0 1 0 n For (ii), all such φ are conjugate to = or = , for n ∈ , 0 1 0 1 n 1 1 1 Z −1 ∗ preserving (1, 0)> and (0, 1)> respectively. Note that while does not preserve a 0 −1 2 > vector in R , it preserves the line (1, 0) R. n n − 1 For (iii) we have representative matrices φ ∼ ± . 1 1 2 Remark. Any ϕ ∈ MCG(T ) has a Seifert fibered or graph manifold mapping torus Mϕ. 59

II. Dehn Twists. Deﬁnition (local Dehn twist). Let A := S1 × [0, 1] be an annulus, parameterized by (θ, t) ∈ ∼ R/2πZ × [0, 1] = A, and regarded as the regular neighborhood of an embedded curve α := 1 1 S × 2 ⊂ A. The Dehn twist on the annulus is the map, T : A → A, (θ, t) 7→ (θ + 2πt, t), which ﬁxes α setwise but rotates it by π.

Deﬁnition (Dehn twist on a surface). Let α ⊂ Σ be an embedded closed curve in an oriented surface Σ, and let ψ : ν(α),→Σ be an embedding of a regular neighborhood of α in Σ. The Dehn twist about α in Σ is the map ( ψT ψ−1(x) x ∈ ψ(ν(α)) τα :Σ → Σ, τα(x) := . x x∈ / ψ(ν(α))

Proposition. A Dehn twist τα along a curve α ⊂ Σ induces the homomorphism

τα∗ : H1(Σ) → H1(Σ), γ 7→ γ + (γ · α)α.

2 Corollary (Dehn twist on a torus). If α, β ⊂ T are homologically linearly independent embedded closed curves with α · β = 1, then we can take α 7→ (1, 0)> and β 7→ (0, 1)> as a 2 2 2 2 ∼ 2 basis for Z ' H1(T ) ' π1(T ). The isomorphism MCG(T ) → Out+(π1(T )) then sends 1 −1 1 0 τ 7→ , τ 7→ , and these generate SL(2, ) ' MCG( 2). α 0 1 β 1 1 Z T

Theorem (Dehn, Lickorish, et al). For Σ a closed oriented surface of genus g, MCG(Σ) is generated by a ﬁnite collection of Dehn twists. For g > 1 (not examinable), a generating set of Dehn twists must contain at least 2g + 1 curves, and the below collection of curves suﬃces.

History of Proof for g > 1 (not examinable). Dehn showed that MCG(Σ) is ﬁnitely generated by Dehn twists. Lickorish rediscovered this result, and bounded the cardinality of this generating set by 3g − 1. Humphries showed that a generating set of Dehn twists must contain at least 2g + 1 elements, and that such a 2g + 1-element generating set always exists. Johnson showed that the above 2g + 1 curves generate MCG(Σ). If one does not insist on using Dehn twists for generators, Wajnryb showed one can generate MCG(Σ) with just 2 homeomorphisms of Σ. 60

III. Higher genus MCG: Nielsen-Thurston Classiﬁcation.

Theorem (Nielsen-Thurston Classification of MCG elements). Let Σ be a connected, oriented, closed surface of genus g > 1, and let Mφ denote the mapping torus of a diffeomorphism φ :Σ → Σ. Then φ satisfies one of the following up to isotopy. (i) φ is periodic: φk = id for some k > 0. =⇒ Mφ is Seifert fibered. (ii) φ is reducible: φ setwise-preserves some closed curve γ ⊂ Σ. =⇒ Mφ has an embedded essential torus Tγ ⊂ Mφ. s (iii) φ is pseudo-Anosov:(not examinable) there are transverse measured foliations (F , µs) u and (F , µu) on Σ, and λ > 1 ∈ R so that −1 −1 φ(Fs) = Fs, φ(Fu) = Fu; φ(µs) = λ (µs), φ(µu) = λ (µu),

=⇒ Mφ is hyperbolic. Remarks (reducible case). For a closed curve γ ⊂ Σ fixed setwise by φ, the mapping torus of φ restricted to γ is in fact an embedded essential torus, Tγ := γ × I/(x, 1) ∼ (φ(x), 0) ⊂ Mφ. (We shall learn more about essential and/or incompressible embedded surfaces later in the course. A positive-genus surface is an essential embedded surface if its embedding induces an injetion on π1.) If γ ⊂ Σ is separating (i.e., if |π0(Σ \ ν(γ))| = |π0(Σ)| + 1—which occurs if and only if γ is ∼ nullhomologous rel boundary, if any— then Tγ is separating, and we have Mφ = Mφ1 ∪Tγ Mφ2 , ∼ for φi := φ|Σi and Σ1 q Σ2 := Σ \ ν(γ). If γ ⊂ Σ is not separating, then neither is Tγ, but 0 we can still take the restriction φ|Σ0 ,Σ := Σ \ ν(γ). In particular, if the disjoint curves γ1, . . . , γk ⊂ Σ are the complete set of curves fixed setwise by φ, then the restriction of φ `k to each connected component of Σ \ i=1 ν(γi) is either periodic or pseudo-Anosov, and the resulting mapping tori are each either Seifert fibered or hyperbolic. Remark (pseudo-Anosov case). The above description for psuedo-Anosov elements is not examinable, but you will be expected to remember about pseudo-Anosov elements that (a) for g(Σ) > 1, any φ ∈ MCG(Σ) that is neither periodic nor reducible is pseudo-Anosov; (b) the mapping torus Mφ of a pseudo-Anosov element φ is hyperbolic. Remark (Seifert fibered/hyperbolic). Of course, we have not yet defined what it means for a 3-manifold to be Seifert fibered or hyperbolic. We shall learn this later in the course. Warning. Later in the course, we shall learn a technical meaning of the term reducible: a compact oriented 3-manifold M is called reducible if it has an embedded S2 that does 3 not bound an embedded B , or equivalently (as we shall learn), if π2(M) 6= 0. A reducible φ ∈ MCG(Σ) does not have a reducible mapping torus Mφ in this sense! 61

Lecture 13: Dehn filling I. Dehn filling as handle attachment. A. Dehn filling. Definition. Let M be a compact oriented 3-manifold with a toroidal boundary component ∼ 2 1 2 ∂iM = T . A Dehn filling of M is a union M ∪∂iM (S × D ) of M along ∂iM with a solid torus.

1 2 ∼ β Observation. Dehn ﬁlling can be realized by handle attachment. Since S × D = Ug=1, for 2 1 2 1 2 β isotopic to [pt] × ∂D ⊂ ∂(S × D ), we can realize M ∪∂iM (S × D ) by • 2-handle attachment along a closed embedded curve β ⊂ ∂iM, plus • 3-handle attachment along the resulting S2 boundary component.

Remark. The resultiing manifold is determined up to diﬀeomorphism by the image β ⊂ ∂iM 3 ∼ 1 of the attaching sphere AS(H2 ) = S in ∂iM, considered as an unoriented embedded curve up to isotopy.

B. Slopes. ∼ 2 Proposition. Suppose ∂iM = T . Then there is a bijective correspondence, closed unoriented primitive classes primitive classes embedded curves β ⊂ ∂iM in π1(∂iM, x0) in H1(∂iM; Z) ↔ ↔ isotopy ±1 ±1 ↔ P(H1(∂iM; Z)) := (H1(∂iM; Z) \{0})/Z Proof. Exercise.

∼ 2 Deﬁnition. Let ∂iM = T be a toroidal boundary component. A slope of ∂iM (or of M, if the speciﬁed boundary component is clear from context) is an element of P(H1(∂iM; Z)), or alternatively one of the above objects bijectively corresponding to it.

Deﬁnition. The Dehn ﬁlling of M of slope β, denoted M(β), is the 3-manifold produced by attaching a 2-handle along the closed embedded curve in ∂iM determined by the slope β, followed by a 3-handle.

Corollary. The Dehn ﬁlling M(β) has respective fundamental group and ﬁrst homology

π1(M(β)) ' π1(M)/hhβii, H1(M(β); Z) ' H1(M)/ hβi . Proof. The ﬁrst statement is just our 2-handle attachment Lemma, proved by Van Kampen’s Theorem. The second statement follows from a Meier-Vietoris sequence argument. 62

II. Marking ∂M. ∼ α Remark. In a Heegaard diagram, we not only specify curves β1, . . . , βg ⊂ Σ = ∂Ug along α ∼ α which to attach 2-handles to Ug . We also specify curves α1, . . . , αg ⊂ Σ = ∂Ug to tell us α how Σ, and hence β1, . . . , βg ⊂ Σ, relate to Ug . 2 ∼ Similarly, we need some sort of canonical basis for T = ∂iM, relative to M, to specify how β ⊂ ∂iM relates to M. Q: How do we generalize the notion of α-curves to non-handle-bodies? Deﬁnition (rank of group). The rank of a group G is the minimal number of generators in a presentation for G. Remark. There is also a relative notion of rank for normal subgroups, but its terminology is less standard. For us, the normal rank of a normal subgroup H C G is the minimal number of generators for a subgroup H0 < G with hhH0ii = H. α Observation. A handle-body Ug is characterized by the fact that

π1 α α 1 α normal rank(ker (i∗ : π1(∂Ug , x0) → π(Ug , x0))) = 2 rank (π1(∂Ug , x0)) = g. The αi curves play the role of (normally) generating this normal subgroup:

π1 α α ker (i∗ : π1(∂Ug , x0) → π(Ug , x0)) = hhα1∗, . . . αg∗ii , α for α1∗, . . . αg∗ x0-based loops associated to α1, . . . αg ⊂ ∂Ug Since each αi is nullhomotopic α α in Ug , it is contractible in Ug , which, by Dehn’s Lemma (from later in the course) implies it bounds an embedded disk. An analogous phenomenon occurs in rational homology for more general 3-manifolds. Proposition. Let M be a compact oriented 3-manifold with nonempty, not necessarily con- Q nected, boundary ∂M, and let i∗ : H1(∂M; Q) → H1(M; Q) be the homomorphism induced by the inclusion i : ∂M,→M. Then Q 1 1 rank ker i∗ = 2 rank H1(∂M; Q) = 2 b1(∂M). Proof. First, note that

3−k b3−k(M) Hk(M, ∂M; Q) 'pd H (M; Q) 'uc H3−k(M; Q) = Q , where the first equality is from Poincaréduality and the second is from the universal coefficients theorem for Hom. Consider the relative long exact sequence over Q associated to the inclusion i : ∂M,→M. b (M) b (∂M) b (M) b (M) 0 0 Q 0 Q 2 Q 2 Q 1 0 →H3(∂M) →H3(M) →H3(M, ∂M) →H2(∂M) →H2(M) →H2(M, ∂M) → · · ·

b (∂M) b (M) b (M) b (∂M) b (M) Q 1 Q 1 Q 2 Q 0 Q 0 0 Q d2 i∗ ··· −→H1(∂M) −→H1(M) →H1(M, ∂M) →H0(∂M) →H0(M) →H0(M, ∂M) → 0. Exactness then implies

rank im d2 = b1(M) − b2(M) + b2(∂M) − b0(M), Q rank ker i∗ = b1(∂M) − b1(M) + b2(M) − b0(∂M) + b0(M) = b1(∂M) − rank im d2, Q which, since ker i∗ = im d2, implies Q 2 rank ker i∗ = b1(∂M). 63

Corollary. There are embedded (multi-) curves α1, . . . , α 1 ⊂ M (multi- if ∂M has 2 b1(∂M) multiple components), such that (i)[α1],..., [α 1 ] ∈ H1(∂M; Z) are linearly independent, 2 b1(∂M) Q (ii) each αi is rationally nullhomologous in M; that is, i∗ [αi] = 0 ∈ H1(M; Q).

α α Remark. Thus, whereas the αi curves in ∂Ug are nullhomotopic in Ug and bound embedded α disks in Ug , The αi curves in ∂M are rationally nullhomologous in M and therefore rationally bound H2 classes which turn out to have embedded surface representatives. (A curve α ⊂ ∂X “rationally bounds” Σ ⊂ X if there exists k ∈ Z+ such that ∂Σ = {k disjoint copies of α}, implying k[α] = 0 ∈ H1(X; Z).)

α *** Insert ﬁgure comparing αi curves in ∂Ug , bounding disks, to αi curves in ∂M, rationally bounding higher genus surfaces.

III. Rational longitudes and Dehn ﬁlling bases. ∼ 2 Deﬁnition. Suppose ∂M = T (so ∂M has just one component now). Then Q 1 rank (ker (i∗ : H1(∂M; Q) → H1(M; Q))) = 2 b1(∂M) = 1. Q The rational longitude of M is the slope generating ker i∗ . For example if we regard this slope slope as an embedded curve up to isotopy, it is the embedded curve ` ⊂ ∂M such that Q −1 h[`]i = ker i∗ = i∗ (Tors H1(M; Z)).

∼ 2 Definition (Dehn filling basis). Suppose ∂M = T .A Dehn-filling basis for M is an ordered 2 pair (m, l) ∈ (H1(∂M; Z)) such that • l ∈ H1(∂M; Z) is a signed rational longitude for M, and • m · l = 1.

Remark. In the case of a homology basis, we regard the homology element l ∈ H1(∂M; Z) as a signed element, because we must keep track of its sign relative to m. Warning. Do not be misled by the dot “·” notation for the intersection pairing of curves on a surface: it is an antisymmetric product, computed via determinant. That is, if we write 1 0 m = (m , m )>, l = (l , l )> with respect to some basis , ∈ 2 ' H (∂M, ), then 1 2 1 2 0 1 Z 1 Z m l m · l = det[m l] = det 1 1 . m2 l2 64

Lecture 14: Dehn Surgery I. Motivation and definitions for Dehn surgery. A. Non-uniqueness of Dehn-filling bases. Last lecture, we tried to develop a Dehn-filling homology basis that would specify Dehn fillings as uniquely as possible. This was a partial success. ∼ 2 Observation. Suppose M is a compact oriented 3-manifold with boundary ∂M = T , and 2 let (m1, l1), (m2, l2) ∈ (H1(∂M; Z)) be two Dehn filling bases for M. Then • (m , l ) = ±(τ k (m ), τ k (l )) = ±(m + kl , l ) for some k ∈ ; 1 1 l2∗ 2 l2∗ 2 2 2 2 Z • H1(M(am1 + bl1); Z) ' H1(M(am2 + b2); Z), although not canonically, but ∼ • M(am1 + bl1) =6 M(am2 + bl2) for generic M. ∼ The prototypical exception would be M = Ug=1, since U1(am1 + bl1) = U1(am2 + bl2).

B. Dehn surgery. Thus, to specify a Dehn filling uniquely, we need another distinguished slope in ∂M. This is accomplished by choosing a reference Dehn filling slope µ, and then regarding all other Dehn fillings of M as “surgery” on M(µ). If you hand someone the manifold Y =∼ M(µ) to begin with, however, then you still need a way to specify how Y decomposes as Y = M ∪ (S1 × D2). Definition (knot core). The knot core of a Dehn filling M(α) is given by the embedded core K := core(M(α) \ M) ⊂ M of the Dehn-filling solid torus interior, so that M(α) = M ∪ ν(K). Remark. In particular, if Y = M(α), and K ⊂ Y is the knot core of the Dehn filling M(α), then M is recovered by taking the exterior M = Y \ ν(K) of K ⊂ Y .

Definition (Dehn surgery on a knot). Let K ⊂ Y be a knot in a compact oriented 3-manifold Y . The Dehn surgery of slope β along K is the manifold XK (β), for XK := Y \ ν(K), where β is understood to be a slope on ∂XK . In other words, to perform Dehn surgery on a knot, one removes the solid torus neighborhood of a knot, and then reglues the solid torus to the manifold along the remaining torus boundary, but with a new gluing map, as specified (up to differences that don’t change the homeomorphism type of the resulting manifold) by the surgery slope.

II. Dehn surgery bases: meridians and longitudes. A. Knots case. Deﬁnition (meridian of a solid torus). The merdian µ ⊂ ∂(S1 × D2) of a solid torus S1 × D2 is the slope µ = [pt] × ∂D2 ⊂ ∂(S1 × D2) (up to isotopy).

Remark. The meridian of a solid torus is simply a special name for the rational longitude of the solid torus, earned for being nullhomotopic and bounding a disk, as opposed to merely being nullhomologous or rationally nullhomologous.

Deﬁnition (meridian of a knot). Let K ⊂ Y be a knot in a compact oriented 3-manifold Y, with exterior XK := Y \ν(K). The meridian of K is the slope µ ⊂ ∂XK such that XK (µ) = Y. The meridian of K is also the image of the meridian of the Dehn-ﬁlling solid torus in XK (µ). 65

Definition (Dehn-surgery basis, longitude). Let K ⊂ Y be a knot in a compact oriented 3- manifold Y, with exterior XK .A Dehn-surgery basis for K ⊂ Y is an ordered pair (µ, λ) ∈ 2 (H1(∂XK ; Z)) such that • µ ∈ H1(∂XK ; Z) is a signed meridian for K ⊂ Y , • µ · λ = 1. Such λ is called a longitude for K ⊂ Y . Remark. One specifies a Dehn surgery basis by stating its longitude λ. (Sometimes the choice of longitude is called a choice of framing for K ⊂ Y , regarding the framing as a ribbon running between K ⊂ ν(K) and the longitude λ ⊂ ∂ν(K), since framings are classified by longitude.) Definition (canonical Dehn-surgery basis, Seifert longitude). Let K ⊂Y be a nullhomologous knot in a closed oriented 3-manifold Y, with exterior XK . The canonical Dehn-surgery basis 2 for K ⊂Y is the unique (up to over-all sign) ordered pair (µ, λ) ∈ (H1(∂XK ; Z)) such that • (µ, λ) is a Dehn-surgery basis for K ⊂ Y , • (µ, λ) is a Dehn-filling basis for XK . In other words, λ is both a longitude for K ⊂ Y and a rational longitude for XK . Such λ is called the Seifert longitude of K. Remark. The above nullhomologous hypothesis is necessary. A knot K ⊂ Y admits a canonical Dehn-surgery basis if and only if it is nullhomologous, i.e., if and only if [K] = 0 ∈ H1(Y ; Z). As a consequence, the Seifert longitude λ is nullhomologous in XK . (Exercise.) Definition (Seifert surface). Let K ⊂ Y be a nullhomologous knot in a closed oriented 3- 2 manifold Y , with exterior XK and Seifert longitude λ ∈ (H1(∂XK ; Z)) . Then any properly- embedded surface (Σ, ∂Σ),→(XK , ∂XK ) with ∂Σ = λ is called a Seifert surface of K. (Such surfaces always exist for nullhomologous K.)

B. Links case. Whereas an arbitrary manifold with multiple toroidal boundary components does not have a unique rational longitude, in the case of a link exterior in a closed manifold we can cheat by taking the exterior of one link component at a time. Note that for the exterior XL := Y \ν(L) of a link L ⊂ Y , we have canonical isomorphisms Ln H1(∂XL) ' i=1H1(∂iXL; Z),H1(∂iXL; Z) ' H1(∂(Y \ ν(Li)); Z). Definition. An ordered 2n-tuple Ln 2 2 (µ, λ) = ((µ1, λ1),..., (µn, λn)) ∈ i=1(H1(∂(Y \ ν(Li)); Z)) ' (H1(∂XL; Z)) 2 is called a Dehn-filling basis or Dehn-surgery basis if each (µi, λi) ∈ H1(∂(Y \ ν(Li)); Z)) is such a basis. The same almost holds for canonical Dehn surgery bases, except that we impose an extra condition on the relative sign of component-wise bases. Definition (canonical Dehn-surgery basis for a link). Let L ⊂ Y be a nullhomologous n-component link in a closed oriented manifold Y , with exterior XL.A canonical Dehn-surgery basis 2 for L ⊂ Y is the unique (up to over-all sign) ordered 2n-tuple (µ, λ) ∈ (H1(∂XL; Z)) such that 2 • each (µi, λi) ∈ H1(∂(Y \ ν(Li)); Z)) is a canonical Dehn-surgery basis; • each µi · Σ > 0, for some properly embedded surface (Σ, ∂Σ),→(XL, ∂XL). 66

III. Integer Dehn surgery. Definition (Dehn-surgery coefficient). Let L ⊂ Y be an n-component link in a compact 2 oriented manifold Y , with exterior XL. Then any Dehn-surgery basis (µ, λ) ∈ (H1(∂XL; Z)) n determines a map from nonzero homology to slopes, regarded as elements of (Q ∪ {∞}) , n Qn π(µ,λ) : H1(∂XL) \{0} −→ (Q ∪ {∞}) ' i=1P(H1(∂iXL; Z)), α = (a µ + b λ , . . . , a µ + b λ ) 7−→ ( a1,..., an). 1 1 1 1 n n n n b1 bn The Dehn surgery X (α) then has surgery coefficient π (α) = ( a1,..., an), and we write L (µ,λ) b1 bn X ( a1,..., an) := X (α). L b1 bn L

Remark. Dehn surgery on a link component K ⊂ Y with coeﬃcient ∞ is the identity operation, replacing Y = XK (µ) with XK (µ). This is sometimes called trivial Dehn surgery.

2 Proposition. If (µ1, λ1), (µ2, λ2) ∈ (H1(∂XL; Z)) are two Dehn surgery bases, then n π(µ2,λ2)(α) − π(µ1,λ1)(α) ∈ Z for any α ∈ H1(∂XL; Z) \{0}. Proof (left as an exercise in lecture). Restricting to some ith component, and suppressing the i from notation, the conditions µ1 · λ1 = µ2 · λ2 = 1 imply −k −k (µ1, λ1) = ±(τµ2∗(µ2), τµ2∗(λ2)) = ±(µ2, λ2 + kµ2) for some k ∈ Z, −k =⇒ α =: aµ1 + bλ1 = ±τµ2∗(aµ2 + bλ2) = ±((a + bk)µ2 + bλ2), so that we have a + bk a π(µ2,λ2)(α) − π(µ1,λ1)(α) = − = k ∈ Z. b b

Deﬁnition (integer Dehn surgery). Let L ⊂ Y be a link with exterior XL and signed meridian µ ∈ H1(∂XL; Z), and let α ∈ H1(∂XL; Z) be a signed slope. The following are equivalent: n • XL(α) has surgery coeﬃcient π(µ,λ)(α) ∈ Z with respect to some longitude λ, n −ki • any longitude λ has Z 3 π(µ,λ)(α) =: k, with each αi = ±τµi∗ (λi) = ±(kiµi + λi), • there exists a longitude λ such that α ∼ λ as slopes, • each µi · αi ∈ {±1}. Such XL(α) is called an integer Dehn surgery.

Proposition (Dehn-surgery cobordism). Integer Dehn surgery on a knot K ⊂ Y is induced by the Dehn-surgery cobordism of 4-dimensional 2-handle attachment. 4 Proof (left as an exercise in lecture). The boundary of H2 decomposes as 4 ∼ 2 2 ∼ 1 2 2 1 ∂H2 := ∂(D × D ) = (S × D ) ∪ (D × S ), 4 ∼ 1 2 4 ∼ 2 1 with attaching region AR(H2 ) = S ×D and belt region BR(H2 ) = D ×S . For the integer Dehn surgery XK (α), choose the sign for the meridian µ so that µ · α = 1, and then attach 4 1 2 2 1 H2 to Y via a gluing map ϕ : −(S × D ) → Y sending −[pt] × ∂D 7→ µ and −S × [pt] 7→ α. 4 This makes Y ∪ϕH2 is a cobordism from Y to XK (α), since it removes a torus with meridian µ and core K (isotopic to α) from Y , and it adds in the solid torus with meridian α, attached 4 4 ∼ 2 to the same torus boundary ∂AR(H2 ) = ∂BR(H2 ) = T . 67

IV. 3-manifolds from integer Dehn surgery on S3. Theorem (Dehn, Lickorish). Any oriented 3-manifold in Y can be realized by integer Dehn surgery on a link in S3. The main idea of the proof is to present each of Y and S3 as the union of two genus-g handle-bodies with the mapping cylinder of a genus-g surface homeomorphism, and then to relate these two mapping cylinders by integer Dehn surgery, one Dehn twist at at time. In particular, we need to introduce the notion of mapping cylinders.

c Deﬁnition (mapping cylinder). The mapping cylinder Mφ of a map φ : X → Y is the union c Mφ := (X × [0, 1]) q Y/ (x, 1) ∼ φ(x).

Proposition. If φ : X → X is a homeomorphism, then for small ε > 0, we have c ∼ ((torus) Mφ =rel ∂ Mφ \ ((0, ε) × X).

Remark. We always consider mapping cylinders up to homeomorphism relative to boundary. c ∼ Otherwise, for any orientation-preserving homeomorphism φ : X → X, we have Mφ = X × I. 68

Lecture 15: 3-manifolds from Dehn Surgery 69

Lecture 16: Seifert fibered spaces 70

Lecture 17: Hyperbolic 3-manifolds and Mostow Rigidity 71

Lecture 18: Dehn’s Lemma and Essential/Incompressible Surfaces I. Loop Theorem and Dehn’s Lemma. The following statements of the Loop Theorem and Dehn’s Lemma are loosely modeled on those from Danny Calegari’s Notes on Manifolds lecture notes [2], which additionally provide many of the (nonexaminable) proofs missing from this course material. ι Theorem (Loop Theorem, Papakyriakopoulos, [14]). Let M be a 3-manifold, and let Σ ,→ ∂M be an embedded compact surface. Suppose that N C π1(Σ) is a normal subgroup of π1(Σ) such that N 6⊃ ker [ι∗ : π1(Σ) → π1(M)]. 2 2 Then there is a (locally flat) embedding f :(D , ∂D ) → (M, Σ) such that the loop [f|∂D2 : 2 ∂D → Σ] represents a conjugacy class in π1(Σ) which is not contained in N. If the role of the normal subgroup N seems confusing, then it might be instructive to replace N with the identity element id ∈ π1(Σ, x0), for some basepoint x0 ∈ Σ. This corresponds to the modified hypothesis that the induced homomorphism ι∗ : π1(Σ, x0) → π1(M, x0) fails to be injective. Now, for any element γ ∈ ker ι∗, the fact that ι∗(γ) is null-homotopic means that 1 there is a homotopy ϕ :(S , s0) × [0, 1] → (M, x0), from a loop (Γ, x0) ⊂ (Σ, x0) ⊂ (M, x0) 1 representing ι∗(γ), to the constant loop x0 ⊂ M. Since ϕ collapses the annulus S × [0, 1] to a disk, it provides a continuous map h :(D2, ∂D2) → (M, Σ) restricting on ∂D2 to a loop representing γ. However, this continuous map h can be arbitrarily horrible. The content of the Loop Theorem, analogous to the Papakyriakopoulos’s Sphere Theorem we shall encounter later in the course, is that we can choose nontrivial γ ∈ ker ι∗ such that the above h is homotopic to an embedding, i.e., so that M has an embedded disk f :(D2, ∂D2) → 2 (M, Σ) whose boundary restriction f|∂D2 :(∂D , x0) → (Σ, x0) represents γ. If we return to the original statement of the Loop Theorem, the replacement of id ∈ π1(Σ) with N Cπ1(Σ), and of nontrivial elements in π1(Σ) with conjugacy classes not in N, provides a stronger statement which takes into account the movement of basepoints. This strengthened statement is useful for working with not-explicitly-basepointed fundamental groups up to conjugacy, instead of with fundamental groups with specified basepoints.

The Loop Theorem has the following important corollary. Theorem (Dehn’s Lemma). Let M be a manifold, and suppose that f :(D2, ∂D2) → (M, ∂M) 2 is a continuous map such that the restriction f|∂D2 : ∂D → ∂M is a (locally ﬂat) embedding, thereby guaranteeing the existence of some one-sided-collar neighborhood A of ∂D2 on which 2 f restricts to an embedding, f|A :(A, ∂D ) → (M, ∂M). Then there is an embedding f 0 :(D2, ∂D2) → (M, ∂M) such that f 0 and f agree on A. 2 Proof. By performing a suitable small isotopy, relative to ∂D , of the embedding f|A : A → M, one can push the annulus f(A) into ∂M as an embedded submanifold ι :(A0, ∂D2) ,→ (∂M, f(∂D2)). Let γ : S1 → ∂D2 ⊂ A0 be a loop embedding S1 in ∂D2. Since the original 2 2 continuous map f :(D , ∂D ) → (M, ∂M) makes the loop f|∂D2 ◦ γ = ι ◦ γ contractible in 0 M, we have ι∗([γ]) = id ∈ π1(M). But since [γ] generates π1(A ) ' Z, this implies that 0 0 ker [ι∗ : π1(A ) → π1(M)] = π1(A ). 0 Thus, when N is the trivial subgroup in π1(A ), we have N 6⊃ ker ι∗, and the Loop Theorem 0 2 2 0 0 0 guarantees the existence of an embedding f :(D , ∂D ) → (M,A ) such that [f |∂D2 ] ∈ π1(A ) 0 represents a conjugacy class in π1(A ) not contained in N, which in our case means that 0 0 0 0 [f |∂D2 ] 6= id ∈ π1(A ). Since f is an embedding, we know that [f |∂D2 ] is primitive in 0 π1(A ) ' Z, hence is one of the two generators ±[γ]. Thus, possibly after orientation reversal, 0 1 2 0 we have [f |∂D2 ] = [γ] = [S ,→∂D ,→A ]. Since every embedded representative of [γ] is isotopic both to the core of A’ and to γ itself, we can isotope f 0 to a locally ﬂat embedding 0 0 0 with f |∂D2 = f|∂D2 . Finally, inside some neighborhood of A containing A, we can isotope f 0 2 2 0 to an embedding f :(D , ∂D ) → (M, ∂M) with f |A = f|A. 72

II. Incompressible / Essential Definitions. Definition (proper smooth embedding). Suppose X and Y are oriented compact manifolds, possibly with boundary. The map ι : X → Y is a proper smooth embedding if ι(∂X) ⊂ ∂Y , if ι|int(X) : int(X) → int(Y ) and ι|∂X : ∂X ⊂ ∂Y are smooth embeddings, and ι∗(TxX) 6⊂ Tι(x)(∂Y ) for any x ∈ ∂X. Remark. More generally, since one can always turn an embedded codimension-1 submanifold Z ⊂ Y into a boundary component by considering Y \ ν(Z), the notion of proper embedding extends to embeddings (X, ∂X),→(Y,Z). Definition (compressing disk). Suppose Σ ⊂ M is a closed oriented embedded surface in M with an embedded curve γ ⊂ Σ that does not bound an embedded disk in Σ (implying the homomorphism π1(γ) → π1(Σ) induced by inclusion is nonzero). A compressing disk is a properly embedded disk (D2, ∂D2),→(M, Σ) with boundary γ. Corollary (Compressing disks from Dehn’s Lemma). Suppose Σ ⊂ M is a closed oriented embedded surface in M with an embedded curve γ ⊂ Σ which does not bound an embedded disk in Σ but is contractible in M. That is, of the inclusion-induced homomorphisms π1(γ) → π1(Σ) and π1(γ) → π1(M), the former is nonzero and the latter is zero, sending [γ] → 1. Then Σ has a compressing disk D2 ⊂ M with boundary γ. One can also define compressing disks for compact Σ with boundary. In this case, Σ,→M is properly embedded, and ∂D bounds neither an embedded disk nor an embedded annulus in Σ. In the latter case, we usually say ∂D does not cobound an annulus, since it would only do half of the work in bounding an annulus, with some other boundary component in Σ providing the other boundary component of the annulus.

Deﬁnition (incompressible surface). A closed oriented properly embedded surface Σ,→M is called incompressible if (i)Σ has genus g > 0 and has no compressing disks, or (ii)Σ =∼ S2 and Σ does not bound an embedded ball B3 ⊂ M.

Deﬁnition (essential surface). A closed oriented properly embedded surface ι :Σ,→M is called essential if (i)Σ has genus g > 0 and the induced homomorphism ι∗ : π1(Σ) → π1(M) is injective, or ∼ 2 (ii)Σ = S and is Σ is homotopically nontrivial: [Σ] 6= 0 ∈ π2(M).

Deﬁnition (boundary compressible). M is boundary compressible if some boundary component ∂iM of ∂M has a compressing disk in the connect summand containing ∂iM in its prime decomposition. 73

III. Incompressible / Essential Equivalence. Theorem (Equivalence of incompressible and essential surfaces). A closed oriented embedded surface Σ,→M is essential if and only if it is incompressible. Proof (not examinable). Defering the proof of the genus g = 0 case to a later lecture, we henceforth assume Σ has genus g > 0. The proof that an incompressible surface is inessential is the easy direction. Suppose ι :Σ,→M is compressible. Then Σ has some compressing disk (D2, ∂D2),→(M, Σ) with 2 2 essential boundary in Σ. That is, 0 6= [∂D ] ∈ π1(Σ). On the other hand, since ∂D bounds 2 a disk in M, the induced homomorphism ι∗ : π1(Σ) → π1(M) sends ι∗ :[∂D ] 7→ 0 ∈ π1(M). 2 Thus 0 6= [∂D ] ∈ ker ι∗, implying ι∗ is not injective, and so Σ ⊂ M is not essential. The converse statement, often called Kneser’s Lemma, is a deeper result and invokes Dehn’s Lemma. Suppose ι∗ : π1(Σ) → π1(M) is not injective.

(1) Then there is some loop γ ⊂ Σ with 0 6= [γ] ∈ π1(Σ) but ι∗(γ) = 0 ∈ π1(M). Such γ therefore bounds an immersed disk D ⊂ M. (2) We next ensure that a collar neighborhood of ∂D in D is properly embedded in (M \ Σ, ∂(M \ Σ). That is, since Σ is oriented, ν(Σ) =∼ Σ × (−ε, +ε), so we can push γ off of the zero-section Σ,→Σ × (−ε, +ε) of ν(Σ) into, say the +ε direction. The isotopy of this push-off from γ to, say, γ 0 forms a properly embedded annulus (A, γ),→(M \Σ, ∂(M \Σ) with boundary γ q γ 0. We also simultaneously push off a neighborhood of ∂D in D into the +ε direction, so 0 0 that ∂D = γ . The new disk D := A ∪γ 0 D now has an embedded collar neighborhood A of γ = ∂D0 which is properly embedded in (M \ Σ, ∂(M \ Σ). Rename D0 as D. (3) If D ∩ Σ = γ, then D \ Σ is connected, and we can apply Dehn’s Lemma to assert the existence of a properly embedded disk ι0 : (int(D0), ∂D0),→(M \ Σ, ∂(M \ Σ)) with ι0 and ι agreeing on A (and on γ). This D0 is then a compressing disk, making Σ compressible. (4) If D∩Σ has multiple components, then we can rethink our original choice of γ. We start by assuming that we originally deformed ι so that D intersects Σ transversely in a collection of simple closed curves, of which γ was one. If any innermost curve γi is homotopically trivial in Σ, then we can replace the component of D \ Σ bounded by γi with an embedded disk Di ⊂ Σ with boundary γi, and push this disk off in the direction of the remaining part of D in M. This operation removes γi from our collection of curves. Performing this operation repeatedly removes all innermost curves which are homotopically trivial in Σ, so that we are left with some innermost curve which is homotopically nontrivial in Σ. Call this curve γ, and let the component of the disk bounded by γ be our new D. (5) A neighborhood of ∂D in D lies entirely in one side of ν(Σ), so perform (2) pushing γ off onto the same side as D. We then have D ∩ Σ = γ, so we perform (3) and are done.

There are also notions of essential and incompressible for properly-embedded surfaces with boundary. 74

Lecture 19: Sphere Decompositions and Knot Connected Sum I. Extending spheres to balls. n n+1 Theorem (Jordan-Schoenflies Theorem). Any locally-flatly embedded n-sphere S ⊂ R n+1 n+1 bounds a locally flatly embeddded B ⊂ R . Equivalently, any locally-flatly embedded n- n n+1 n+1 n+1 n+1 sphere S ⊂ S splits S into a union of two locally flatly embedded balls B1 ,B2 ⊂ Sn+1. For n ≤ 2, the analogous statements also hold for smooth embeddings. Remark. The n = 2 case of the above theorem is called the Alexander’s Sphere Theorem. The problem was originally posed as the Jordan curve/Schoenflies problem for n = 1. Alexander solved it for n = 2, and Morton Brown solved it for arbitrary n in 1960. Alexander is also known for a much simpler sphere result, often called his sphere “trick.”

Theorem (Alexander’s Sphere Trick: isotopy with the identity map). If, for some n ≥ 0, the map f : Bn+1 → Bn+1 is an orientation-preserving homeomorphism restricting to the n n n+1 ∼ n identity f|∂Bn+1 = id : S → S on the boundary ∂B = S , then f is isotopic (through homeomorphisms restricting to the identity on ∂Bn+1) to the identity map id : Bn+1 → Bn+1. n+1 n+1 Proof. Pulling back to local coordinates x ∈ R sending B to the standard unit ball in n+1 n+1 n+1 R centered at 0, we construct an isotopy H : B × I → B from id to f. Set ( tf( x ) 0 ≤ |x| < t H(x, t) := t . x t ≤ |x| ≤ 1 n+1 Starting out as the identity map on B , the homotopy Ht slowly inflates an origin-centered bubble in the image Bn+1. This bubble is filled with a rescaled image of f, which reaches full size once the boundary of this bubble reaches the unit-sphere boundary of Bn+1. We first note H satisfies the required boundary conditions. That is, Ht|∂Bn+1 = Ht|{|x|=1} ≡ idSn for all t ∈ I, and we have H0 = idBn+1 , H1 = f. It remains to verify that H is continuous and that each Ht is a homeomorphism. For continuity, H is a composition of continuous functions in a neighborhood of each (x, t) except along the submanifold {(x, t)| t = |x|} ⊂ Bn+1 × I. In this latter case, however, we have still have x lim H(x, t) = lim t · id( t ) = x = H(x, |x|) = lim H(x, t) t→|x|+ t→|x|+ t→|x|− n+1 −1 for all x ∈ B . Lastly, each Ht is a homeomorphism, since each Ht has inverse Ht given by replacing f with f −1 in the above definition of H, and the same continuity argument holds −1 for H as the one we applied for the continuity of H. Remark. In the case of n = 2, the above result is equivalent to the statement that the mapping class group of the disk, MCG(D2), is trivial. Corollary (Uniqueness of extension of sphere maps). If f, g : Bn+1 → Bn+1 are homeomorphisms with f|∂Bn+1 = g|∂Bn+1 , then f and g are isotopic. −1 n+1 n+1 Proof. Consider the composition f ◦ g : B → B . Since f|∂Bn+1 = g|∂Bn+1 , we know −1 2 2 that (f ◦ g )|∂Bn+1 = idS2 : S → S . This means we can apply Alexander’s Trick to obtain −1 n+1 n+1 n+1 n+1 that f ◦ g : B → B is isotopic to the identity map idBn+1 : B → B . Thus −1 (f ◦ g ) ◦ g = f is isotopic to idBn+1 ◦ g = g. ◦ Moral. In the topological category, we can treat Xˆ := X \Bn and X on an equal footing, since there is only one way to refill Xˆ with Bn. Contrast this to the situation of a knot exterior 3 ◦ XK := S \ ν(K), where there are Q ∪ {∞} choices for how to glue a solid torus back to XK . 75

II. Connected sum.

Definition (connected sum). Let X1 and X2 be compact oriented n-manifolds, choose orient- n n ation-preserving embeddings ιi : B ,→ Xi, and let each Xî := Xi \ ιi(B ) be the respective complement. The connected sum X1 #X2 is the union X1 #X2 := Xb1 ∪ϕ Xb2, where the orientation-reversing gluing homeomorphism ϕ : ∂Xb1 → −∂Xb2 used to identify the boundaries −1 n−1 n−1 n of X1 and X2 is the reflection map; that is, ι2 |∂X2 ◦ϕ◦ι1|∂B : S → S is the antipodal map. Smooth connected sum is the same, but with diffeomorphisms and smooth embeddings. Remarks. One can also define connected sum for other codimension-1 submanifolds, but the submanifold is taken to be Sn−1 by default. In dimension n ≤ 3, every orientation-reversing homeomorphism (or diffeomorphism) ϕ : ∂Xb1 → −∂Xb2 is isotopic to the reflection map.

Theorem (Kervaire-Milnor [8]). The connected sum X1#X2 is independent, up to diﬀeomor- n phism, of the choice of embeddings ιi : B ,→ Xi. The connected sum operation is associative and commutative, with unit Sn. To prove this, we ﬁrst need to say a few things about isotopy and ambient isotopy.

Theorem (Isotopy Extension Theorem (Cerf, Palais) [13]). Let X,Y be compact oriented smooth manifolds, and let ι1, ι2 : X,→ Y be smooth embeddings (proper if X has boundary). If ι1, ι2 are isotopic, then they are ambient isotopic. Edwards and Kirby [3] showed this result also holds for locally flat embeddings. For the Kervaire-Milnor Theorem, we also need nicely embedded paths. Lemma. If γ : [0, 1] → X is a path, i.e., only a continuous map, in a smooth, respectively topological, manifold X of dimension n ≥ 3, then γ is homotopic rel boundary to a smoothly, respectively locally flatly, embedded path. Proof of lemma (not examinable). First, homotope γ rel endpoints to remove any self intersections. This is possible since γ([0, 1]) has codimension n − 1 ≥ 2. Choose a Riemannian metric for X, and cover this new γ([0, 1]) with open balls Uα small enough that each preim- −1 −1 age γ (Uα) has only one component. Since γ ([0, 1]) is compact, we take a finite subcover of these balls, and then take a finite refinement {Ui} such that the Ui cover γ([0, 1]), but n have no triple intersections. Writing Bi := U i, we now homotope γ rel boundary, and rel ` n i ∂Bi ∩ γ([0, 1]), to a piecewise linear path. This makes γ a locally flat embedding, which can be made smooth by smoothing the corners of the piecewise linear path. Proof of Kervaire-Milnor Theorem (not examinable). Commutativity and associativity follow directly from the first statement, since a judicious choice of ball embeddings makes performing the gluing operations in a chosen order equivalent to performing them all at once. Since ◦ Sn \ Bn =∼ Bn, the map X → X#Sn is the identity operation. 0 1 n For the first statement, suppose ιi , ιi : B ,→Xi are different orientation-preserving smooth embeddings. Since Xi is path connected, the above Lemma gives us a smoothly embedded path 0 1 n γ : [0, 1] → Xi from ιi (p) to ιi (p), for some p ∈ B . After choosing a tubular neighborhood 0 n 1 n 0 1 ν of ιi (B ) ∪ γ([0, 1]) ∪ ιi (B ), we can smoothly isotope ιi to ιi along γ, inside ν. The Cerf- Palais theorem extends this to an ambient isotopy, inducing a diffeomorphism fi : Xi → Xi 0 1 such that fi ◦ ιi = ιi . 76

Deﬁnition (prime). A connected compact oriented 3-manifold M is called prime if any de- ∼ composition M = M1 #M2 of M as a connected sum of compact oriented 3-manifolds M1 and 3 M2 implies that S ∈ {M1,M2}.

Deﬁnition (irreducible). A connected compact oriented 3-manifold M is irreducible if any embedded S2 in M bounds an embedded B3 in M. A non-irreducible manifold is called reducible.

Deﬁnition (splitting). Given a closed embedded surface Σ ⊂ M in a compact 3-manifold M, ◦ the splitting M |Σ of M along Σ is the (compact) complement M |Σ := M \ ν(Σ).

Deﬁnition (separating). A connected closed embedded surface Σ ⊂ M is called separating if the splitting M |Σ has one more connected component than M.

Deﬁnition (rank of a group). The rank of a ﬁnitely-generated group G is the minimum number of generators appearing in any presentation of G.

Theorem (Papakyriakopoulos’s Sphere Theorem [14]). If M is a compact oriented 3-manifold 2 with π2(M) 6= 0, then there is a homotopically-nontrivial smooth embedding ι : S ,→M. (That 2 is, within the set π2(M) \{0} of nontrivial homotopy classes of maps ι : S ,→M, at least one such homotopy class has an actual embedding as a representative.) Theorem (Kneser-Milnor’s Theorem [10]). Any connected compact oriented 3-manifold M 3 admits a prime decomposition: either M = S =: M1, or M decomposes as a connected sum ∼ ∼ 3 M = M1# ··· #Mk of prime oriented 3-manifolds Mi =6 S . This decomposition is unique up to orientation-preserving homeomorphism and reordering of summands.

Theorem (Van-Kampen’s Theorem for #). If M1 and M2 are connected compact oriented 3-manifolds, then the fundamental group of their connected sum decomposes as the free product

π1(M1#M2) ' π1(M1) ∗ π1(M2).

Theorem (Grushko’s Theorem). If G1 and G2 are ﬁnitely generated groups, then

rank(G1 ∗ G2) = rank(G1) + rank(G2).

Theorem (Kneser-Stallings’ Theorem, or “Kneser’s Conjecture”). If π1(M) of a connected closed oriented 3-manifold M decomposes as a free product π1(M) ' G1 ∗ G2, then there exist ∼ (connected closed oriented) 3-manifolds M1, M2 such that π1(Mi) ' Gi and M = M1#M2. Heil extended the above result from closed manifolds to compact manifolds with boundary [7]. Theorem (Perelman’s proof of the Poincar´eConjecture). If M is a closed connected 3- ∼ 3 manifold, then π1(M) ' 1 if and only if M = S . 77

IV. Knot connected sum. 78

Lecture 20: JSJ Decomposition, Geometrization, Splice Maps, and Satellites I.Notions of Toroidalness. Just like with incompressible/essential embedded surfaces, the notion of being toroidal has both a topological and a fundamental-group-theoretic interpretation, but due to the occasional role of π1-injective immersions, these two notions do not always coincide.

Deﬁnition (peripheral subgroup). Let X be a compact manifold with nonempty boundary. A peripheral subgroup is conjugate to the image of ι∗ : π1(∂iX) → π1(X) induced by the inclusion ι : ∂iX,→X, for some boundary component ∂iX of ∂X.

Deﬁnition (“group-theoretically” toroidal). M is “group-theoretically” toroidal if π1(M) has no non-peripheral Z ⊕ Z subgroup. Deﬁnition (boundary parallel). An embedded surface Σ,→M is boundary parallel if it can be isotoped into a boundary component of M.

Deﬁnition (“topologically” toroidal). M is “topologically” toroidal if it has an embedded incompressible torus that is not boundary-parallel.

Deﬁnition (boundary compressible). M is boundary compressible if some boundary component ∂iM of ∂M has a compressing disk in the connect summand of M containing ∂iM.

Deﬁnition (toroidal boundary). M has toroidal boundary if its boundary is a disjoint union ∼ `n ∼ `n 2 of tori ∂M := i=1 ∂iM = i=1 Ti . Toroidal boundary does not imply toroidal manifold!

Theorem. Suppose M is compact, oriented, irreducible, and with toroidal or empty boundary. Then M is group-theoretically atoroidal if and only if it is a quotient of S3 by a finite subgroup of SO(4) (hence is a genus-0 Seifert fibered space with 3 or fewer exceptional fibers and an orbifold base with positive orbifold fundmental group), or M is hyperbolic. Group-theoretically atoroidal manifolds are topologically atoroidal. If M is group-theoretically toroidal, then either it is topologically toroidal or it is a small Seifert fibered space with infinite fundamental group. In other words, if M is not a small Seifert fibered space with infinite fundamental group, then M is group-theoretically toroidal if and only if topologically toroidal. Recall. A small Seifert fibered space is a genus-0 Seifert fibered space M = M(g = 0; β1 , β2 , β3 ) with precisely 3 exceptional fibers. Such M has infinite fundamental group α1 α2 α3 if and only if M is not a quotient of S3 by a finite subgroup of SO(4), the latter of which occurs if and only if its orbifold base Σ = S2( β1 , β2 , β3 ) has positive orbifold fundmental α1 α2 α3 group, i.e., if and only if χorb(Σ) := χ(S2) − 3 + P3 1 > 0. i=1 αi 2 Remark. The issue here is that whereas any π2-injective map S → M is homotopic to an 2 embedding, not every π1-injective map T → M is homotopic to an embedding. Such ϕ is, 0 2 0 however, homotopic to an immersion ϕ : T # M such that ϕ lifts to an embedding in some finite-index cover Mf → M of M. Many topologically toroidal manifolds have immersed tori not homotopic to embedded tori, but among these, small Seifert fibered spaces with infinite fundamental group are the only such manifolds that do not additionally have essential embedded tori. 79

II. JSJ decompositions and Geometrization.

A. JSJ Decompositions. Theorem (Jaco-Shalen, Johannson). Let M be a compact oriented irreducible 3-manifold, 2 2 with empty or toroidal boundary. Then there is a possibly-empty collection T1,..., Tn of 2 n embedded incompressible non-boundary-parallel tori such that the splitting M|{Ti }i=1 consists of pieces which are either Seifert ﬁbered or group-theoretically atoroidal (or both).

B. Geometrization: Haken Case. Deﬁnition (Haken). A connected compact oriented 3-manifold M is called Haken if it is irreducible and it contains a properly embedded connected incompressible surface that is not boundary parallel.

This notion is particularly relevant to manifolds with toroidal boundary (really for any boundary with no S2 component, but we shall focus on the toroidal case). Theorem (Seifert et al). If M has toroidal boundary, it has an essential embedded Seifert surface.

Proof. For any toroidal boundary component ∂iM, there is an essential curve λ ⊂ ∂iM which is nullhomologous in M: ιi ∗([λ]) = 0 ∈ H1(M). Thus the 1-chain λ ∈ C1(M) is the boundary of some 2-chain, say σ 6= 0 ∈ C2(M), with ∂σ = λ and [σ] 6= 0 ∈ H1(M, ∂iM). In a 3- manifold, any nontrivial homology class has an embedded representative, and this also holds for homology rel boundary. Thus the homology class [σ] has a properly embedded representative, (Σ, ∂Σ) ⊂ (M, ∂iM). A minimal-genus such embedded representative is incompressible, presumably because if it had a compressing disk, one could compress along this to form a lower genus embedded surface of the same homology class. (If ∂iM is a higher genus surface, the story is similar, except that there are now g homologocially linearly independent essential curves λj ⊂ ∂iM which are nullhomologous in M. Corollary. Every compact oriented irreducible boundary-incompressible M with toroidal boundary is Haken.

Theorem (Thurston’s hyperbolization theorem). Let M be an compact, oriented, irreducible, manifold. If M is Haken, then M \∂Mb is hyperbolic, where ∂Mb ⊂ ∂M is the union of toroidal boundary components of M. Warning! Tori are ﬂat and do not admit nonsingular hyperbolic structures. Thus, for a manifold with toroidal boundary, the hyperbolic metric can only extend over the interior of the manifold. If one prefers to keep the boundary intact, then the manifold is called cusped hyperbolic, because the hyperbolic metric becomes singular (develops a cusp) on each toroidal boundary component.

Deﬁnition (hyperbolic group). A group Γ is hyperbolic if it acts properly discontinuously and cocompactly by isometries on a proper, geodesic hyperbolic metric space. Hyperbolicity of a group can also be formulated as condition on the Cayley graph of the group. There is a sense in which its triangles must be thin, just as triangles are thin in an ordinary hyperbolic metric space. We will say (a tiny bit) more about hyperbolic geometry and hyperbolic fundamental groups later in the course. 80

2 C. Geometrization: Case {Ti } = ∅ is empty and M is closed. Theorem (Perelman). If M is a closed, oriented, irreducible, group-theoretically atoroidal 3-manifold, then (i) M has a metric of constant sectional curvature; 3 (ii) if M has constant positive curvature, then π1(M) is finite, and M is a quotient of S by a (finite) rigid subroup of SO(4), implying M is Seifert fibered; 3 (iii) if M has constant negative curvature, then M is hyperbolic, i.e, a quotient of H by the action of a hyperbolic group. Idea of proof. Run Ricci flow until you get constant curvature. Use special techniques to remove any singularities in the flow that develop. III. Splice Maps: Reformulating JSJ decompositions as splice diagrams.

Deﬁnition (splice). Let K1 ⊂ M1 and K2 ⊂ M2 be knots in closed 3-manifolds, and for each ◦ i ∈ {1, 2}, let Mci := Mi \ ν(Ki) denote the exterior of Ki in Mi and let µi, λi ∈ H1(∂Mci) be a conventional surgery basis for this exterior. The splice of M1 and M2 along K1 and K2 is the union ˆ ˆ Mc1 ∪ϕ Mc2, ϕ : ∂M1 → −∂M2 such that ϕ∗(µ1) = λ2 and ϕ∗(λ1) = µ2, for ϕ∗ : H1(∂Mc1) → H1(∂Mc2). Such ϕ is called a splice map, or is sometimes itself called a splice.

Problem: A splice map produces a closed manifold. Where is the new knot? Have to specify some third knot, say K3 ⊂ M1 or K3 ⊂ M2, so no longer have a binary operation. Need to make one of the 3 knots canonical. Leaves 2 options: (1) Declare K3 to be a curve on the newly formed incompressible torus in X1 ∪ϕ X2 This is like a JSJ decomposition with a speciﬁed slope on the torus, which we sometimes do without giving the construction a special name. 3 (2) Make either K1 ⊂ M1 or K2 ⊂ M2 be the unknot U ⊂ S . This second choice is called a satellite construction.

IV. Satellites. ◦ Deﬁnition (satellite). Suppose that Kc ⊂ M is a knot with exterior Xc := M \ ν(Kc), and 3 ◦ that Kp ⊂ XU is a knot in the unknot exterior XU := S \ ν(U). The satellite of Kc by Kp is the knot Kp Kc := Kp ⊂ Xc ∪ϕ XU , ϕ : ∂Xc → −∂XU a splice map. induced from the composition of the embeddings Kp,→XU and XU ,→Xc ∪ϕ XU . In this construction, Kc ⊂ M is called the companion knot and Kp ⊂ XU is called the pattern knot. In other words, a satellite embeds the pattern knot into the tubular neighborhood of the companion knot. One of the most common satellite examples is the cable. Deﬁnition (cable of a knot). The (p, q) cable of a knot K ⊂ S3 is the knot K(p,q) ⊂ S3 (p,q) 3 induced from the embedding K := Tp,q,→ν(K) ⊂ S of the (p, q) solid-torus torus knot 1 2 3 Tp,q ⊂ S × D into the solid-torus neighborhood of K in S . 81

Lecture 21: Turaev torsion and Alexander polynomial of unions I. Turaev Torsion. The Turaev torsion is closely related to the Alexander polynomial, but behaves more naturally under gluing. This torsion is also sometimes called the Reidemeister-Turaev torsion, since it is a special case of Reidemeister torsion τReid(C∗, b∗), which is an invariant, up to chain homotopy, of an acyclic complex C∗, together with a speciﬁed basis, b∗. Deﬁnition (acyclic complex). A complex is called acyclic if it is everywhere exact, or equivalently, if all of its homology groups vanish.

The strategy (as reinterpreted by Milnor) of the Reidemeister torsion τReid(C∗, b∗) is to 1 ri compare the volume of the speciﬁed basis bi = (bi , . . . , bi ), for ri = rank Ci, to the volume of 1 ri a comparison basis ai = (ai , . . . , ai ) chosen in a certain natural way. The construction n Q Y i [det(bi)/ det(ai)] τ (C , b ) = [det(b )/ det(a )](−1) = i even Reid ∗ ∗ i i Q [det(b )/ det(a )] i=1 j odd j j of the Reidemeister torsion as an alternating product of volume ratios then serves to cancel out any volume-ratio changes due permissible change of comparison basis. The expressions (bi) and (ai) should be interpreted as ri × ri matrices written with respect to some arbitrary ﬁxed basis of Ci. Since determinant is a group homomorphism, the ratio det(bi)/ det(ai) is independent of this choice of third basis. In particular, if we take ai for this third basis, then n k X jk j [det(bi)/ det(ai)] = det Bi/ det id = det Bi, for Bi satisfying bi = Bi ai . j=1

The prescription we shall describe for choosing a comparison basis ai is motivated by the following observation.

Proposition. Let (C∗, d) be a bounded (i.e., Ci = 0 for i < 0 or i > n), acyclic complex of R-modules, di+1 di di−1 ··· Ci+1 −→ Ci −→ Ci−1 −→ · · · for R satisfying ********, and let each Zi := ker di ⊂ Ci denote the subgroup of cycles. di Then the induced homomorphism Ci/Zi −→ Zi−1 is an isomorphism for each i. Proof. We ﬁrst note that the induced homomorphism is well-deﬁned and surjective, since

di di di−1 im[Ci/Zi −→ Zi−1] = im[Ci −→ Zi−1] = ker[Ci−1 −→ Ci−2] =: Zi−1, with the second equality following from the acyclicity of C∗. f The homomorphism is also injective, since for any homomorphism A −→ B, the induced f homomorphism (A/kerf) −→ B is injective. 82

II. Turaev torsion computations. ∼ 2 Theorem 0.1. Suppose Y is a compact oriented 3-manifold with torus boundary ∂Y = T and ﬁrst homology H1(Y ; Z) ' Z ': hmi, for some generator m ∈ H1(Y ; Z). Then (1 − [m])τ(Y ) = ∆(Y ) ∈ Z[H1(Y ; Z)], where τ denotes Turaev torsion, ∆ denotes Alexander polynomial, and [m], often denoted −1 t ∈ Z[t , t] ' Z[H1(Y ; Z)], is the image of m under the inclusion H1(Y ) ,→ Z[H1(Y )].

3 Corollary. If K ⊂ S is a knot with exterior XK , then −1 (1 − t)τ(XK ) = ∆(K) ∈ Z[t , t] ' Z[H1(XK )], 3 ι∗ for t the image of the meridian of K ⊂ S under the inclusion H1(∂Y ) ,→ H1(Y ) ,→ Z[H1(Y )].

Theorem 0.2. Suppose X and Y , with χ(X) = χ(Y ) = 0, are compact oriented 3-manifolds with boundary, with ∼ ∼ ∼ X ∩ Y = I × Σ = I × (∂iX) = I × (∂jY ) for certain connected components ∂iX of ∂X and ∂jY of ∂Y . Then X Y τ(X)τ(Y ) ι∗ τ(X)ι∗ τ(Y ) τ(X ∪ Y ) = , or more precisely, τ(X ∪ Y ) = X∩Y , τ(X ∩ Y ) ι∗ τ(X ∩ Y ) X Y X∩Y for ι∗ , ι∗ , and ι∗ the homomorphisms induced on homology by the respective inclusion maps ιX : X,→X ∪ Y , ιY : Y,→X ∪ Y , and ιX∩Y : X ∩ Y,→X ∪ Y .

◦ ∼ 2 `n 2 Theorem 0.3. If Σ = S \ i=1 Di for some n ∈ Z≥0, then (∗) τ(S1 × Σ) = (1 − [S1])−χ(Σ), 1 1 1 where [S ] ∈ Z[H1(S × Σ)] denotes the image of the homology class of S × {point} under 1 1 the inclusion homomorphism H1(S × Σ) ,→ Z[H1(S × Σ)]. Note: Theorem 0.3 was stated without proof in lecture, but for more general Σ. Time permitting, I will add a (non-examinable) proof of the above result to our lecture notes. 83

III. Alexander Polynomial of Unions. ******************

IV. References. The standard short reference on this subject is Turaev’s paper https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/1997/0004/0005/ MRL-1997-0004-0005-a006.pdf Turaev also has a more accessible book, Introduction to Combinatorial Torsions, part of which can be found in Google Books. One can also refer to https://www3.nd.edu/~lnicolae/Torsion.pdf for deﬁnitions and results. 84

Lecture 22: Foliations I. Foliations: Motivation and Definition. A. Introduction and Motivation. Foliations can be studied in arbitrary codimension k on manifolds of arbitrary dimension n. In real (versus complex) geometry and topology, foliations are most often studied in codimension 1 or n−1. For this course, our interest lies mainly in codimension-1 foliations on 3-manifolds. Every 3-manifold admits codimension-1 foliations, but when we impose an extra rigidity condition, called tautness, this is no longer the case. Taut foliations are fundamental topological structures in the study of 3-manifolds. Like knots, they are only minimally captured by (ordinary) homology. Unlike knots, since taut foliations are of dimension 2 instead of 1, their relationship with the fundamental group is still poorly understood. This is true even though the fundamental group completely charac- terises any JSJ-reduced, non-lens-space, oriented 3-manifold. Thus, at least in principle, all information about taut foliations is somehow captured by the fundamental group. Taut foliations have implications for contact structures (which generalise boundary conditions for symplectic structures), real dynamics, quasigeodesic and pseudo-Anosov flows, the Thurston norm, taut sutured decompositions of manifolds (another type of decomposition into topologically-basic pieces), representations of the fundamental group, and gauge theoretic invariants. In some cases, the relationship of taut foliations with one of these other fields is only partly, or perhaps only minimally, understood, and is an area of active research.

B. Foliations as a form of manifold decomposition. One of the major themes of this course has been methods of decomposing 3-manifolds. For example, we have have encountered: 1. Decomposition into topologically basic pieces, but with more complicated gluing maps: – Handle decompositions/cell decompositions, – Heegaard splittings. 2. Decomposition along simple essential surfaces (spheres or tori) with simple gluing maps, into hyperbolic or Seifert fibered pieces: – Connected sums (decomposition along essential S2s), 2 – JSJ decomposition (decomposition along essential T s). A foliation is yet another type of manifold decomposition, but this time into a family of manifolds called leaves, which are all of the same codimension. For example, a fiber bundle π : E → B over a base B of dimension k is a codimension k foliation. In this case, all of the leaves of this foliation are homeomorphic to the same manifold, namely, the fiber F =∼ π−1(b), b ∈ B. Foliations are more general than fiber bundles, however. In a fiber bundle, the neighborhood of a fiber is a trivial fibration, whereas in a foliation, the neighborhood of a point is a trivial fibration. We now make this idea more precise. 85

C. Definitions. Definition (product foliation). A codimension-k product foliation F on a manifold X is a trivial fiber bundle structure π : X → B over a base B of dimension k. The leaves of F are the fibers π−1(b) =∼ F , b ∈ B, and we sometimes call F the “foliation by F .” We then have a F = F. b∈B A choice of global section σ : B,→X, π ◦ σ = id, for π : X → B realises a homeomorphism X =∼ F × B. Remark 1. Just as we often, by mild abuse of notation, write X =∼ F ×B to denote a trivial F -fibration over B, with the understanding that this homeomorphism depends on a choice of global section, we also sometimes write F = F × B to denote the product foliation of X by F Remark 2. Since any manifold X satisfies X =∼ X × {pt}, any manifold X admits the codimension-0 product foliation given by a foliation of X by points. However, since most manifolds do not admit a product decomposition X =∼ B × F for dim B > 0, most manifolds do not admit a codimension-k product decomposition for k > 0.

Deﬁnition. A codimension-k foliation F on a manifold X of dimension n is a globally compatible decomposition of X into codimension-k submanifolds, speciﬁed by charts (on neighborhoods of points) equipped with codimension-k product foliations respected by gluing maps. ∼ S That is, for some covering {Uα ⊂ X}α∈A of X = α∈A Uα, we have charts a n−k ∼ k n−k ϕα : Uα −→ R = R × R , x∈Rk with transition functions ϕ := ϕ | ◦ ϕ |−1 : k × n−k −→ k × n−k αβ α Uα∩Uβ β Uα∩Uβ R R R R n−k n−k which work like R -bundle transition functions, but for R as a manifold, not as a vector space. k That is, the transition function ϕαβ decomposes as ϕαβ(x) = (ϕαβ(π(x)), (φαβ(x))(y)), k k k k n−k n−k for ϕαβ : R → R and φαβ : R → Homeo(R ), where Homeo(R ) is the group n−k of homeomorphisms of R . We call F a co-oriented foliation if we always have φαβ : k n−k n−k R → Homeo+(R ), where Homeo+(R ) is the group of orientation-preserving homeo- n−k morphisms of R .

II. Foliation Examples. Read the wikipedia article’s collection of examples: https://en.wikipedia.org/wiki/Foliation. 86

III. Taut foliations. Deﬁnition (transversal). Let F be a codimension-1 foliation on a compact oriented manifold X with closed or toroidal boundary. For any p ∈ X, let Lp denote the leaf of F intersecting p. A transversal to F is an embedded arc with open endpoints, a properly-embedded compact arc, or an embedded closed curve τ ⊂ X which intersects F transversely. That is, for every p ∈ intτ ⊂ X, we have Tpτ 6⊂ TpLp. A transversal to F which is also a closed curve is called a closed transversal.

A. Deﬁnition. Deﬁnition (taut foliation). Let X be a closed oriented manifold of dimension n. A codimension-1 foliation F on X is a taut foliation if there is a closed transversal to F through each point p ∈ X. There is also a notion of taut foliation for compact oriented manifolds with toroidal boundary, for which we demand a closed transversal at every point p ∈ int(X), and demand that the foliation meet the boundary transversely.

B. Examples of taut foliations on 3-dimensional manifolds. Proposition (fibered manifold). Let M be a compact oriented fibered manifold π : M → S1 with fiber Σ, and let F be the codimension-1 foliation on M whose leaves are given by the fibers π−1(s) =∼ Σ, s ∈ S1. Then F is a taut foliation.

Proof. For any p ∈ M, there is a homeomorphism ψ : M → Mφ of M with the mapping ∼ torus Mφ = (Σ × [0, 1])/((x, 1) ∼ (φ(x), 0)) of some basepoint-preserving homeomorphism φ : (Σ, x) → (Σ, x) such that ψ(p) = (x, 0). The embedding s : {x} × S1 ,→ M then gives a global section of the ﬁbration π : M → S1, and this global section s is a closed transversal to F through (x, t) for any t ∈ I. In particular, it is a closed transversal to F through p. ∼ When M = Mφ has boundary, its boundary components arise from orbits of the boundary m components {∂iΣ}i=1 of Σ under φ. That is, the monodromy φ ﬁxes each boundary component setwise, up to overall permutation, say, σ ∈ Sm of the boundary components of Σ. The boundary components of M then take the form ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

When Σ has boundary, we demand that φ ﬁx these boundary components ∂iΣ up to permutation. 87

but in either case they give rise to toroidal boundary components of M, and the leaves of F meet these toroidal boundary components transversely. Note: Later, when we define what it means for a foliation to be smooth, we will be able to state the result that for a smooth codimension-1 foliation F on M, the foliation F is taut if and only if there is some closed transversal intersecting every leaf of F at least once. IV. Reebless foliations. 1 2 Definition (Reeb foliation). The Reeb foliation FReeb of the solid torus S ×D is a foliation of the solid torus described in the above cited wikipedia article. (The below figure is from there as well.) The boundary of the solid torus is also a leaf of the foliation.

(Figure by Ilya Voyager, via wikicommons.)

Deﬁnition (Reeb component). A Reeb component of a codimension-1 foliation F on M is 1 2 a restriction F|S1×D2 = FReeb of F to some embedded solid torus S × D ⊂ M. Deﬁnition (Reebless). A codimension-1 foliation F on M is Reebless if it has no Reeb components. Reeb components are an obstruction to tautness, because a transversal spirals into the Reeb component but then gets stuck and cannot get out. Reeblessness does not guarantee tautness unless M is atoroidal. Theorem. If M is closed and atoroidal, then a Reebless foliation on M is taut.

V. Review of Lecture 20. • Theorem and proof that a fibered manifold is a foliation by fibers, and this foliation is taut. • Theorem that a (not necessarily taut) foliation is C2 if and only if its tangent 2-plane field can be realized as the kernel of a one-form C2 α satisfying α ∧ α = 0. We only justified the converse direction of this theorem, that if a 2-plane field ξ ∈ Γ(TM V TM) is the kernel of α for α ∧ α = 0, then ξ is integrable. 88

Lecture 23: The Thurston Norm Curtis McMullen’s The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology http://www.math.harvard.edu/~ctm/papers/home/text/papers/alex/alex.pdf might be a useful reference.

II. Review of Lecture 22. • Deﬁnition of Thurston Norm. • Result not mentioned in former Lecture 22. Theorem (Gabai). Suppose M is irreducible and has empty or toroidal boundary. Then for each primitive class α ∈ H2(M), there is a taut foliation on M containing a genus-minimizing representative of α as a closed leaf. 89

Lecture 24: Taut foliations on Seifert fibered spaces

I. Orbifold Fundamental Group. Suppose Σ is an orbifold with singular points s1, . . . , sn orb of respective orders d1, . . . , dn. The orbifold fundamental group π1 (Σ) is a certain extension of π1(Σ)e by homotopy classes of “orbifold loops.” Since we have restricted to the case of 2-dimensional orbifolds with elliptic (quotients by cyclic rotation action) singularities, it is probably easiest to deﬁne the orbifold fundamental group in terms of a more constructive procedure.

Deﬁnition. Let Σb := Σ\{s1, . . . , sn} denote the complement of singular points in Σ, ﬁx a basepoint s0 ∈ Σb, and choose meridians µ1, . . . , µn in Σb for the respective punctures by s1, . . . , sn. d1 dn d1 dn Let hhµ1 . . . , µn ii denote the normal subgroup in π1(Σb, s0) generated by µ1 . . . , µn . Then orb the orbifold fundamental group π1 (Σ) of Σ can be constructed as follows: orb ∼ d1 dn π1 (Σ, s0) = π1(Σb, s0)/hhµ1 . . . , µn ii. Up to conjugacy, this group is independent of the choices of meridians and basepoint.

In fact, we can make this construction even more explicit. First, recall that one can obtain a closed genus-g surface Sg by taking a disk with 4g-sided polygon boundary and making certain pairwise identiﬁcations of edges. For example, we could identify edges according to the consecutive oriented edge labels −1 −1 −1 −1 α1, β1, α1 , β1 , . . . , αg, βg, αg , βg .

This construction of Sg gives rise to the fundamental group presentation Qg π1(Sg) ' α1, β1, . . . , αg, βg i=1[αi, βi] = 1 . Thus, if the orbifold Σ is closed and of genus g, we have Qn Qg π1(Σ)b ' α1, β1, . . . , αg, βg, µ1, . . . , µn ( k=1µk)( i=1[αi, βi]) = 1 , and for the orbifold fundamental group of Σ, we have E orb Qn Qg d1 dn π1 (Σ) ' α1, β1, . . . , αg, βg, µ1, . . . , µn ( k=1µk)( i=1[αi, βi]) = 1, µ1 = ··· = µn = 1 .

In particular, if Σ0 = Σ has genus 0, then E orb Qn d1 dn π1 (Σ0) ' µ1, . . . , µn k=1µk = 1, µ1 = ··· = µn = 1 . 90

II. Fundamental Groups of Seifert Fibered Spaces. A. Homotopy long exact sequence for fibrations. (Not examinable). Definition (fibration). A map E → B is a fibration such that any map f : X → B, for any toplogolical space X, satisfies the homotopy lifting property (see Lecture 0). A fibration can be regarded as interpolating between the notions of fiber bundle and covering −1 ∼ space. Whereas a fiber bundle E → B with fiber F has homeomorphic fibers π (b1) = −1 ∼ −1 π (b2) = F for any b1, b2 ∈ B, a fibration has homotopy equivalent preimages π (b1) and −1 π (b2) for any b1, b2 ∈ B, and this homotopy equivalence class of preimages is also sometimes called the fiber. The homotopy lifting property enables us to consider the action of π1(B) on E. In fact, a stronger statement is true. Theorem (homotopy exact sequence). Suppose E → B is a fibration with fiber F , and that B is connected and path connected. then the homotopy groups of B, E, and F satisfy a long exact sequence of groups,

· · · → π2(F ) → π2(E) → π2(B) → π1(F ) → π1(E) → π1(B) → 1. The above hypotheses can be considerably weakened. For example, only a weaker notion of fibration called a “Serre fibration” is necessary. While we shall not make use of the homotopy exact sequence in the exact form shown above, it is good to have the above result in mind when considering the following analogous result for orbifold fundamental groups in the context of Seifert fibered spaces.

B. Seifert fibered Fundamental Groups as central extensions. Theorem (Eisenbud-Hirsch-Neumann). If M is a Seifert fibered space with orbifold base Σ, 1 then the class ϕ ∈ π1(M) of the S fiber is central in π1(M), and there is a short exact sequence orb 1 → π1(ϕ) → π1(M) → π1 (Σ) → 1 (with π1(ϕ) ' Z). Corollary. The fundamental groups of Seifert fibered spaces with orbifold base Σ are classified orb by the central extensions by Z of π1 (Σ). Thus, if Σ has more than 2 singular points, then orb the actual Seifert fibered spaces over Σ are classified by central extensions by Z of π1 (Σ). 91

III. Transverse foliations. A. Transverse foliations on bundles. Notation. For F a connected, oriented manifold, we deﬁne

Homeo+F := Group of orientation-preserving homeomorphisms of F Deﬁnition (transversely-foliated bundle). Suppose F has contractible universal cover. Given a representation ρ : π1(M) → Homeo+F , the transversely foliated F -bundle with holonomy ρ, (Eρ, Fρ), is the bundle πρ : Eρ → M with ﬁber F and total space −1 (8) Eρ := (M˜ × F )/π1(M), (x, t) · g := (x · g, ρ(g )(t)), g ∈ π1(M) and projection

πρ : Eρ → M, [(x, t) · g]g∈π1(M) 7→ [x · g]g∈π1(M), ` ˜ ˜ equipped with the foliation Fρ on Eρ induced by the product foliation t∈F M × {t} of M × F .

Remark. If F is replaced with a vector space V of dimension n, and Homeo+F is replaced with GL(V ), then the above construction gives the vector bundle Eρ → M with holonomy ρ, together with the codimension-n foliation of Eρ whose leaves are the orbits under parallel transport of the vectors, with respect to the ﬂat connection of holonomy ρ.

Proposition. Let sh(y) ∈ Homeo+R denote the homeomorphism sh(y): R → R, x 7→ x + y. 1 Then the universal central extension Homeo^ +S is isomorphic to the centralizer of sh(1) in Homeo+R, and satisfies the exact sequence ι 1 1 1 → Z → Homeo^ +S → Homeo+S → 1, ι : 1 7→ sh(1). Proof. Exercise. B. Transverse foliations of Seifert fibered spaces. Theorem (Eisenbud-Hirsch-Neumann [4]). Let M be a closed oriented Seifert fibered space. A foliation on M is taut if and only if it is isotopic to a fiber-transverse foliation. Moreover, there is a bijection,   Non-trivial representations Fiber-transverse ∼   /isotopy ←→ 1 /conjugacy. foliations on M π1(M) → Homeo^ +(S )  sending ϕ 7→ sh(1)  92

References [1] Morton Brown. Locally flat embeddings of topological manifolds. In Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), pages 83–91. Prentice-Hall, Englewood Cliffs, N.J., 1962. [2] Danny Calegari. Notes on 3-Manifolds. Lecture notes from Winter 2014 Course at U Chicago, 2014. https://math.uchicago.edu/~dannyc/courses/3manifolds_2014/3_manifolds_notes.pdf. [3] Robert D. Edwards and Robion C. Kirby. Deformations of spaces of imbeddings. Ann. Math. (2), 93:63–88, 1971. https://www.jstor.org/stable/1970753. [4] David Eisenbud, Ulrich Hirsch, and Walter Neumann. Transverse foliations of Seifert bundles and self- homeomorphism of the circle. Comment. Math. Helv., 56(4):638–660, 1981. [5] Allen Hatcher. Algebraic Topology. Cambridge University Press, Cambridge, UK, 2002. https://www. math.cornell.edu/~hatcher/AT/ATpage.html. [6] Allen Hatcher. The Kirby Torus Trick for Surfaces. arXiv:1312.3518, 2013. [7] Wolfgang Heil. On Kneser’s conjecture for bounded 3-manifolds. Proc. Cambridge Philos. Soc., 71:243–246, 1972. [8] Michel A. Kervaire and John W. Milnor. Groups of homotopy spheres. I. Ann. of Math. (2), 77:504–537, 1963. https://www.jstor.org/stable/1970128. [9] Robion C. Kirby. Codimension-two locally flat imbeddings have normal bundles. In Topology of Mani- folds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), pages 416–423. Markham, Chicago, Ill., 1970. https://math.berkeley.edu/∼kirby/papers/. [10] J. Milnor. A unique decomposition theorem for 3-manifolds. Amer. J. Math., 84:1–7, 1962. https://www. jstor.org/stable/2372800. [11] John W. Milnor. On Manifolds Homeomorphic to the 7-Sphere. Ann. of Math. (2), 64(2):399–405, 1956. https://www.jstor.org/stable/1969983. [12] Edwin E. Moise. Affine Structures in 3-Manifolds: V. The Triangulation Theorem and Hauptvermutung. Annals of Mathematics, 56(1):96–114, 1952. http://www.jstor.org/stable/1969769. [13] Richard S. Palais. Extending diffeomorphisms. Proc. Amer. Math. Soc., 11:274–277, 1960. http://www. ams.org/journals/proc/1960-011-02/S0002-9939-1960-0117741-0/. [14] C. D. Papakyriakopoulos. On Dehn’s lemma and the asphericity of knots. Ann. of Math. (2), 66:1–26, 1957. https://www.jstor.org/stable/1970113.