751-2 Lecture Notes

JWR Jan 14, 2005

1 Preliminaries

1.1. Unless otherwise speciﬁed, the word space means topological space and the word map means continuous function between two topological spaces. The notation f :(X,A) → (Y,B) means f : X → Y , A ⊂ X, B ⊂ Y , and f(A) ⊂ B. It is often convenient to ﬁx a base point x0 ∈ X. The pair (X, x0) is then called a pointed space and the notation f :(X, x0) → (Y, y0) means f : X → Y , x0 ∈ X, y0 ∈ Y , and f(x0) = y0.

1.2. The disjoint union of an indexed collection {Yα}α∈Λ is by deﬁnition the set G Yα := {(y, α): y ∈ Yα}. α∈Λ F S There is a projection α∈Λ Yα → α∈Λ Yα :(y, α) 7→ y which is a bijection if and only if the sets Yα are pairwise disjoint. 1.3. By a disk we understand a space homeomorphic to the standard closed unit disk n n D := {x ∈ R : kxk ≤ 1}. We may on occasion use other models for the disk, e.g. the unit cube

n n I := {(x1, x2, . . . , xn) ∈ R : 0 ≤ xi ≤ 1} or the standard simplex

n n+1 ∆ := {(x0, x1, . . . , xn) ∈ I : x0 + x1 + ··· + xn = 1}.

The spaces Dn, In, and ∆n are homeomorphic; homeomorphisms can be constructed using radial projection. We use the terms n-disk, n-cube, or n-simplex to indicate the dimension. When D is an n-disk we denote by ∂D the image of

n−1 n n S := ∂D := {x ∈ R : kxk = 1} under a homeomorphism Dn → D. By the Brouwer Invariance of Domain Theorem the subset ∂D ⊂ D is independent of the choice of the homeomorphism used to deﬁne it but for the deﬁnitions in this section the reader may take the

1 term disk to signify one of Dn, In, or ∆n and deﬁne ∂D in each of these cases separately. The set ∂D is called the boundary of the disk but the reader is cautioned that in point set topology this term depends on the ambient space (in this case Rn). Thus n n [ n−1 ∂∆ := ιk(∆ ) k=0 n−1 n where the inclusion ιk : ∆ → ∆ is deﬁned by

ιk(x1, . . . , xk) = (x1, . . . , xk−1, 0, xk, . . . , xn). (In point set topology the boundary of a set is the intersection of its closure with the closure of its complement so the set theoretic boundary of ∆n as a subset of Rn+1 is ∆n itself.) We may call a disk D a closed disk to distinguish it from the open disk D \ ∂D.

1.4. A path in X is a map γ : I → X and a loop in X is a path with γ(0) = γ(1). The words are also used more generally, for example a map γ :[a, b] → X might also be called a path and a map γ : S1 → X might also be called a loop. A space X is called path connected iff for all x, y ∈ X there is a path γ : I → X with γ(0) = x and γ(1) = y. A space is said to have a property locally iff every point has arbitrarily small neighborhoods which have the property. For example, a space X is locally path connected iff for every x ∈ X and every neighborhood V of x in X there is a path connected neighborhood U of x with U ⊂ V . Exercise 1.5. For any space X define an equivalence relation by x ≡ y iff here is a path γ : I → X with γ(0) = x and γ(1) = y. The equivalence classes are called the path components of X. Show that the following are equivalent.

(i) The space X is locally path connected, i.e. for every x0 ∈ X and every neighborhood V of x0 there is an open set U with x0 ∈ U ⊂ V and such that any two points of U can be joined by a path in U. (ii) For every x ∈ X and every neighborhood V of x in X there is a a neighborhood U of x with U ⊂ V and such that any two points of U can be joined by a path in V . (iii) The path components of every open set are open. Solution. That (i) =⇒ (ii) is immediate: a path in U is a fortiori a path in V . Assume (ii). Choose an open set V and a path component C of V . (We will show C is open.) Choose x0 ∈ C. By (ii) there is an open set U containing x0 such that any two points of U lie in the same path component of V . In particular any point of U lies in the same path component of C as x0, i.e. U ⊂ C. Thus C is open. This proves (iii). Finally assume (iii). Choose x0 ∈ X and a neighborhood V of x0. Let U be the path component of V containing x0. By (iii), U is open. This proves (i). Remark 1.6. The whole space X is an open set so it follows from (iii) that if X is locally path connected, then the path components are open.

2 Remark 1.7. It is a theorem of point set topology that a compact, connected, and locally connected metric space is path connected. (See [4] page 116.) How- ever, a compact metric space can be quite nasty. Consider for example the exercises in page 18 Chapter 0 of [3]. Pathological examples are important for understanding proofs, but for the most we will concentrate on nice spaces (ﬁnite cell complexes – See Deﬁnition 4.2 below).

1.8. A homotopy is a family of maps {ft : X → Y }t∈I such that the evaluation map X × I → Y :(x, t) 7→ ft(x) is continuous. We do not distinguish between the homotopy and the corresponding evaluation map. We say two maps f, g : X → Y are homotopic and write f ' g iff there is a homotopy with f0 = f and f1 = g. When we say that f, g :(X,A) → (Y,B) are homotopic it is understood that each stage ft of the homotopy is also a map of pairs, i.e. ft(A) ⊂ B. If A ⊂ X we say f and g are homotopic relative to A and write f ' g (rel A) iff there is a homotopy {ft}t between f and g with ft(a) = f0(a) for all a ∈ A and t ∈ I. Remark 1.9. When X is compact and the space C(X,Y ) of all continuous maps from X to Y is endowed with the compact open topology, two maps f, g ∈ C(X,Y ) are homotopic if and only if they belong to the same path component of C(X,Y ). Thus (in this situation at least) a homotopy is truly a path of maps. Definition 1.10. We say spaces X and Y are homotopy equivalent or have the same homotopy type and write X ' Y iff there are maps f : X → Y and g : Y → X such that g ◦ f ' idX and f ◦ g ' idY . Definition 1.11. A space X is called contractible iff it is homotopy equivalent to a point, i.e. iff there is a homotopy ft : X → X with f0 = idX and f1 a constant. Definition 1.12. Let X be a space. A subset A ⊂ X. is called a retract of X iff there is a retraction of X to A, i.e. a left inverse to the inclusion A → X, i.e. a map r : X → A such that r(a) = a for a ∈ A. The subset A ⊂ X. is called a deformation retract of X iff there is a deformation retraction of X to

A, i.e. a homotopy {rt : X → X}t∈I such that r0(x) = x for x ∈ X, r1(X) = A, and rt(a) = a for a ∈ A and t ∈ I.

Remark 1.13. If {x0} is a deformation retract of X for some x0 ∈ X, then cer- tainly X is contractible. A still stronger condition is that {x0} is a deformation retract of X for every x0 ∈ X. In general the reverse implications do not hold (see Exercises 6 and 7 on page 18 of [3]) but they do hold for cell complexes (see Corollary 0.20 page 16 of [3]). See Section 4 for the deﬁnition of cell complex. Exercise 1.14. Construct explicit homotopy equivalences between the wedge of circles 1 1 X = (−1, 0) + S ∪ (1, 0) + S ,

3 the spectacles 1 1 Y = (−2, 0) + S ∪ (2, 0) + S ∪ [−1, 1] × 0 , and the space 1 Z = S ∪ 0 × [−1, 1] obtained from a circle by adjoining a diameter. Solution. Let p = (−1, 0), q = (1, 0), and C be the convex hull of X. It suffices to show that each of the spaces X, Y , Z is homeomorphic to a deformation retract of C \{p, q}. 1. The space X is a deformation retract of C \{p, q}. To see this radially project the insides of the two circles from their centers, radially project the portion of E outside the circles and above the x-axis from (0, 2), and radially project the portion of E outside the circles and below the x-axis from (0, −2). 0 2 2 2 2. The space Y := {(x, y) ∈ R :(x ± 1) + y = 1/4} ∪ [−1/2, 1/2] × 0 is a deformation retract of C \{p, q}. To see this radially project the insides of the two circles from their centers, radially project the points of C outside the two circles and outside the square [−1, 1]2 from the corresponding center, and project the remaining points vertically. 3. The space Z0 := ∂C ∪ 0 × [−1, 1] is a deformation retract of C \{p, q}. To see this radially project the left half from p and the right half from q. Each fiber of each of these retractions is a closed or half open interval so it is easy to see that each of these retractions is a deformation retraction. 1.15. Suppose that E is a topological space, X is a set and g : E → X is a surjective map. The quotient topology on X is defined by the condition that U ⊂ X is open iff g−1(U) ⊂ E is open. That this defines a topology is an immediate consequence of the identities ! ! −1 [ [ −1 −1 \ \ −1 g Uα = g (Uα), g Uα = g (Uα) α∈Λ α∈Λ α∈Λ α∈Λ for any indexed collection {Uα}α∈Λ of subsets of X. Since g is surjective, X can be interpreted as the set of equivalence classes of an equivalence relation:

X = E/ ∼ where x ∼ y ⇐⇒ g(x) = g(y).

In most applications X is an identiﬁcation space. Typically

E = Z t Y,X = Z tφ Y where Y t Z denotes the disjoint union and the notation means that there is a continuous map φ : A → Z with A ⊂ Y and X = E/∼ where the equivalence relation is the minimal equivalence relation satisfying y = φ(x) =⇒ y ∼ x for x ∈ A and y ∈ Z. Typically Y = Dn and A = ∂Dn := Sn−1. In this case we say that X is obtained by attaching an n cell to Z along φ.

4 Remark 1.16. If A ⊂ X the notation X/A indicates the space X/∼ of equivalence classes where x ∼ y ⇐⇒ either x = y or else both x ∈ A and y ∈ A. If A 6= ∅, the space X/A has a distinguished point A/A. However, X/∅ = X or more precisely X/∅ = {x} : x ∈ X .

2 Categories

2.1. A category C consists of • a collection called the objects of the category; • for each pair X, Y of objects a set C(X,Y ) called the morphisms from X to Y ;

• for each object X a morphism idX ∈ C(X,X) called the identity morphism; • for each triple X, Y , Z, of objects a map

C(X,Y ) × C(Y,Z) → C(X,Z):(f, g) 7→ g ◦ f

called composition; satisfying (associative law) (h ◦ g) ◦ f = h ◦ (g ◦ f), and

(identity laws) idY ◦ f = f ◦ idX = f for f ∈ C(X,Y ). In most examples the objects are sets with some additional structure and the morphisms are maps which preserve that structure; we write f : X → Y instead of f ∈ C(X,Y ). 2.2. A morphism g : Y → X is a left inverse (resp. right inverse) to the morphism f : X → Y iﬀ g ◦ f = idX (resp. f ◦ g = idY ). An isomorphism is a morphism which has a left inverse and a right inverse. If g1 is a left inverse to f and g2 is a right inverse to f, then g1 = g2 and g1 is the only left inverse to f and the only right inverse. Hence an isomorphism f : X → Y has a unique inverse denoted f −1 : Y → X characterized by (either of) the equations

−1 −1 f ◦ f = idX , f ◦ f = idY .

2.3. Here are a few of the categories we will study. 1. The category of groups and group homomorphisms. 2. The category of rings and rings homomorphisms. 3. The category of topological spaces and continuous maps. (The isomorphisms are called homeomorphisms.)

5 4. The category of topological spaces and homotopy class of maps. (The isomorphisms are called homotopy equivalences.) 5. The category of topological spaces with base point and continuous base point preserving maps f :(X, x0) → (Y, y0). 2.4. A covariant functor from a category C to a category D is an operation which assigns to each object X of C and object Φ(X) of D and to each morphism f ∈ C(X,Y ) of C a morphism Φ(f) ∈ D(Φ(X), Φ(Y )) such that

Φ(idX ) = idΦ(X), Φ(g ◦ f) = Φ(g) ◦ Φ(f).

The notation f∗ = Φ(f) is often used. A contravariant functor from a category C to a category D is an operation which assigns to each object X of C and object Φ(X) of D and to each morphism f ∈ C(X,Y ) of C a morphism Φ(f) ∈ D(Φ(Y ), Φ(X)) such that

Φ(idX ) = idΦ(X), Φ(g ◦ f) = Φ(f) ◦ Φ(g). The notation f ∗ = Φ(f) is often used. Example 2.5. The operation which assigns to each pointed topological space (X, x0) the fundamental group π1(X, x0) is a covariant functor from the category of pointed topological spaces to the category of groups. For f :(X, x0) → (Y, y0) the induced map f∗ : π1(X, x0) → π1(Y, y0) sends the homotopy class of a loop γ to the homotopy class of f ◦ γ. Example 2.6. The operation which assigns to each topological space X the ring C(X, R) of continuous real valued functions on X is a contravariant functor from the category of topological spaces to the category of rings. For a continuous map f : X → Y the induced morphism f ∗ : C(Y, R) → C(X, R) is deﬁned by f ∗u = u ◦ f for u ∈ C(Y, R).

3 Surfaces

Exercise 3.1. Let P be a polygon with an even number of sides. Suppose that the sides are identified in pairs in any way whatsoever. Prove that the quotient space is a compact surface. A surface is a space which is locally homeomorphic to R2. That the sides are identified in pairs means the following. There is an enumeration α1, β1, . . . , αn, βn of the edges of P (not necessarily in cyclic order but without repetitions) and for each k = 1, 2 . . . , n a homeomorphism φk : αk → βk so that desired identification space S is obtained from P by identifying x ∈ αk ⊂ ∂P with φk(x) ∈ βk ⊂ ∂P . Solution. Choose p ∈ S. We must construct a neighborhood U of p in S and a homeomorphism h : U → R2. There are three cases. Case 1. Assume p lies in the interior of P . Then there is an open disk in P centered at p. Any open disk is homeomorphic to R2.

6 Case 2. Assume that p = {a, b} where a ∈ α := αk, b ∈ β := βk, and φ(a) = b where φ := φk for some k = 1, 2 . . . , n. Abbreviate R := (−1, 1) × (−1, 1), R1 := (−1, 1) × [0, 1), R2 := (−1, 1) × (−1, 1], I := R1 ∩ R2. Let f and g be a homeomorphisms from R1 and R2 onto neighborhoods U1 and U2 of a and b in P respectively. Thus a ∈ f(I) ⊂ α and b ∈ g(I) ⊂ β. Shrinking and modifying as necessary we may assume φ ◦ f(I) = g(I). Deﬁne ψ : I → I −1 by ψ(x, 0) = g (φ(f(x, 0)) and Ψ : R− → R− by Ψ(x, y) = ψ(x, 0) + (0, y). Replacing g by g ◦ Ψ we may assume that φ ◦ f = g. Now deﬁne h : R → S by h(x, y) = f(x, y) if y ≥ 0 and h(x, y) = g(x, y) if y ≤ 0.

Case 3. Assume that p = {v1, v2, . . . , vk} is a set of vertices of P . Renaming −1 and replacing some of the φj by φj we may assume that vj is the common 2 vertex of βj−1 and αj where β0 := βk. Let R be the open unit disk in C = R and write R = R1 ∪ · · · ∪ Rk where Rj is the sector 2(j − 1)π 2jπ R := r exp(iθ) : 0 ≤ r < 1, ≤ θ ≤ . j k k

It is easy to construct homeomorphisms fj : Rj → Uj where Uj is a neighborhood of vj in P , for example we make take fj(r exp(iθ) = vj +cjr exp(i(ajθ+bj)) where cj > 0 is small and aj and bj are judiciously chosen. Reﬂecting if necessary we can even achieve fj(Ij−1) ⊂ αj and fj(Ij) ⊂ βj where 2jπ I := r exp(iθ) : 0 ≤ r < 1, θ = . j k To get a homeomorphism from R to a neighborhood of p in S we must ﬁnd such homeomorphisms fj satisfying the additional condition that

φj ◦ fj+1|Ij = fj|Ij. (∗) For this we try

fj(r exp(iθ)) = vj + ρj(θ, r) exp(i(ajθ + bj)).

As in case 2 we modify fj+1|Ij to achieve (∗). This determines ρj(2(j −1)π/k, ·) and ρj(2π/k, ·). These last two maps are homeomorphisms (i.e. strictly increas- ing functions) from from the unit interval to two other intervals and we define ρj(θ, ·) for intermediate values of θ by linear interpolation. Exercise 3.2. Let S0 be the surface obtained by modifying the surface S of Exercise 3.1 by replacing each homeomorphism φk : αk → βk by a different −1 homeomorphism ψk : αk → βk. Assume each homeomorphism ψk ◦ φk : αk → 0 αk fixes the endpoints of the side αk. Show that S and S are homeomorphic. Remark 3.3. The following representations are called standard: (I) P is a square with boundary

∂P = α1 ∪ β1 ∪ α2 ∪ β2, the sides are are enumerated and oriented in the clockwise direction, and the identiﬁcations reverse orientation.

7 (II) P is a 4g-gon with boundary

∂P = α1 ∪ α2 ∪ β1 ∪ β2 ∪ · · · ∪ α2g−1 ∪ α2g ∪ β2g−1 ∪ β2g, the sides are are enumerated and oriented in the clockwise direction, and the identiﬁcations reverse orientation. (III) P is a 2k-gon with boundary

∂P = α1 ∪ β1 ∪ · · · ∪ αk ∪ βk. the sides are are enumerated and oriented in the clockwise direction, and the identifications preserve orientation. The Classification Theorem for Surfaces (See [2] Page 236, [6] page 9, or [5] page 204) implies that any compact surface is homeomorphic to one of these. One can transform any space S = P/∼ as in Exercise 3.1 to one of the standard representations with a sequence of elementary moves. There are two kinds of elementary moves: one which removes adjacent edges α and β that are identified reversing orientation and another which cuts a polygon in two along a diagonal producing a new pair (α, β) and then reassembling along an old pair (α, β). This process produces a proof of the Classification Theorem from the assertion that any compact surface is homeomorphic to some P/∼ (not necessarily a standard one). The process has nothing to do with proving that P/∼ is a compact surface. Exercise 3.4. We say surface S is said to be the connected sum of surfaces S1 and S2 and write S = S1 # S2 iff there are disks Di ⊂ Si (i = 1, 2) and a homeomorphism φ : ∂D2 → ∂D1 with S = (S1 \ E1) tφ (S2 \ E2) where Ei := Di \ ∂Di is the interior of Di and ∂Di is the boundary of Di. Show that 1. The connected sum of any surface S with the sphere S2 is homeomorphic to S. 2. A surface as in part (II) of Remark 3.3 is homeomorphic to a connected sum gT 2 := T 2 # T 2 # ··· # T 2 | {z } g of g tori T 2 := S1 × S1. 3. A surface as in part (III) of Remark 3.3 is homeomorphic to a connected sum kP 2 := P 2 # P 2 # ··· # P 2 | {z } k of k projective planes.

8 Exercise 3.5. One says that the surface S # T 2 results from S by adding a handle and that the surface S # P 2 results from S by adding a cross cap. It is a consequence of the previous exercise that every compact surface is homeomorphic to the surface obtained from a sphere by adding handles and cross caps. Given g, k > 0, ﬁnd n so that (gT 2)#(kP 2) is homeomorphic to nP 2.

4 Cell Complexes

4.1. Throughout this section {Φα : Dα → X}α∈Λ denotes an indexed collection of maps into a topological space X; the domain Dα of each map is a (space homeomorphic to the standard) closed disk of dimension nα. Such a collection determines and is determined by a map G Φ: D → X, D := Dα, Φ|Dα := Φα α∈Λ from the disjoint union of the disks Dα to X. Denote by G E := Eα ⊂ D,Eα := Dα \ ∂Dα α∈Λ the (disjoint) union of the interiors of the disks Dα and introduce the notations G Dk := Dα, Ek := E ∩ Dk.

nα≤k

Definition 4.2. Such a collection {Φα}α∈Λ as in 4.1 is called a cell structure on X iff it satisfies the following: (1) The restriction Φ|E is a bijection between E and X.

(2) Φ(Dk \Ek) ⊂ Φ(Dk−1). (3) The space X has the quotient topology, i.e. a subset U ⊂ X is open if −1 and only if its preimage Φ (U) intersects each component Dα of D in an open set. A space X equipped with a cell structure is called a cell complex. 4.3. The following notations and terminology are used for cell complexes. For each nonnegative integer k the set

(k) X := Φ(Dk) is called the k-skeleton of X. The sets X(k) give a ﬁltration of X, i.e.

∞ [ X = X(k),X(0) ⊂ X(1) ⊂ X(2) ⊂ · · · . k=0

9 Introduce the further notations

e¯α := Φα(Dα), eα := Φα(Eα), φα := Φα|∂Dα.

Thus [ X = eα, eα ∩ eβ = ∅ for α 6= β. α∈Λ

The sets eα are called cells, the setse ¯α are called closed cells, the maps Φα are called characteristic maps, and the maps φα are called attaching maps. Sometimes a cell is called an open cell for emphasis, but a cell need not be an n open subset of X. It is customary to write eα to indicate that the dimension of the cell eα is n = nα so

(k) [ n [ n k k (k−1 X = eα = e¯α, ande ¯α \ eα ⊂ X . n≤k n≤k

n n We may also write Dα to indicate that Dα = D i.e. that n = nα. A cell complex is called finite (resp. countable) iff there are only finitely many (resp. countably many) cells, i.e. iff the index set Λ is finite (resp. countable). A cell complex is called n-dimensional iff X(n) = X and X(n−1) 6= X and is called finite dimensional iff it is n-dimensional for some n. It is customary to denote a cell complex by the same letter X as its underlying topological space rather than by the more correct notation {Φα : Dα → X}α∈Λ. and to introduce the other notations of this paragraph as needed. Example 4.4. Exercise 3.1 shows how to construct a finite cell complex whose underlying topological space X is a surface. This cell complex has only one 2-cell. For the surface of type (II) in Remark 3.3 there is one 0-cell, one 2-cell, and 2g 1-cells, while for the surface of of type (III) there is one 0-cell, one 2-cell, and k 1-cells.

Exercise 4.5. Construct an explicit homeomorphism between the sphere Sn and the cell complex with one 0-cell, one n-cell, and no other cells. Proposition 4.6. A map Φ: D → X as in 4.1 with Λ finite and which satisfies (1) and (2) also satisfies (3), i.e. it is a (finite) cell complex. Proposition 4.7. A compact set in a cell complex intersects only finitely many n cells of X. In particular, each closed cell e¯α intersects only finitely many other m cells e¯β . Proof. See [3] Proposition A.1 page 520. Proposition 4.8. A subset of a cell complex X is closed if and only if it intersects each n-skeleton X(n) in a closed set. Proof. See [3] Proposition A.2 page 521.

10 Remark 4.9. Cell complexes were invented by G. H. C. Whitehead [11] who called them CW complexes. He defined a CW complex as a space X equipped S with a decomposition X = α eα which arises from some cell structure {Φα}α, whereas for us a cell complex is a space X equipped with a cell structure {Φα}α. (Different cell structures can give rise to the same decomposition.) The C stands for closure finite which signifies the conclusion of Proposition 4.7, i.e. that n m each closed celle ¯α intersects only finitely many other cellse ¯β with m ≤ n. The W stands for weak topology, the property in the conclusion of Proposition 4.8, that a subset of X is closed if and only if it intersects each n-skeleton X(n) in a closed set. Whitehead viewed a CW complex as being constructed inductively: the n-skeleton X(n) is constructed from the (n−1)-skeleton X(n−1) by attaching n n (n−1) the n-cellse ¯α along their boundary. This is why the maps φα : ∂Dα → X are called attaching maps. The equivalence of Whitehead’s definition with the inductive definition is Proposition A.2 page 521 of [3]. Definition 4.2 is easily seen to be equivalent to the inductive definition. Following [3] we use the terms cell complex and CW complex synonymously, but other authors use the former term to signify a structure satisfying parts (1) and (2) of Definition 4.2 but not necessarily (3). In these notes we are mostly concerned with finite cell complexes where part (3) is superfluous by Proposition 4.6. Example 4.10. The Hawaiian earring

∞ [ 2 2 2 X = {(x, y) ∈ R : x − 2x/n + y = 0}. n=1 inherits a topology as a subset of R2 which is weaker (fewer open sets) than its topology as a countable wedge sum of circles. The former topology satisfies parts (1) and (2) of Definition 4.2 but not (3). The latter topology is its topology as a cell complex. Calling the quotient topology the weak topology is confusing because of this example. Definition 4.11. A map f : X → Y between cell complexes is called cellular iff it preserves skeletons, i.e. iff f(X(n)) ⊂ Y (n) for all n. The cell complexes and the cellular maps between them form a category: The identity map of a cell complex is cellular, and the composition of cellular maps is cellular. A cellular isomorphism is an isomorphism in this category, i.e. a cellular map f : X → Y such that there is a cellular map g : Y → X satisfying g ◦ f = idX and f ◦ g = idY . Proposition 4.12. A map f : X → Y between cell complexes is cellular if and only if the image of each cell of X is a subset of a finite union of cells of Y of the same or lower dimension. Proof. ‘If’ is immediate and ‘only if’ follows from the image of a closed set is compact and therefore a subset of a finite subcomplex of Y . Remark 4.13. A cellular map f need not satisfy the stronger condition that the image of each cell of X is a subset of a cell of Y of the same or lower dimension.

11 If X = Y = I, X(0) = {0, 1}, Y 0 = {0, 1/2, 1}, and f is the identity map, then f(X(0)) ⊂ Y (0) but f(X) is not contained in a cell of Y . The cell complexes and the cellular maps between them form a category somewhat analogous to the category simplicial complexes and the simplicial maps between them. (See 13.6.) However, for a simplicial map the image of a simplex is a subset of a simplex of the same or lower dimension. Proposition 4.14. A map f : X → Y between cell complexes is a cellular isomorphism if and only if f is a homeomorphism and maps each cell of X onto a cell of Y (necessarily of the same dimension). Proof. A cellular map between cell complexes is continuous so a cellular isomorphism f is (in particular) a homeomorphism. For each k map f induces a homeomorphism X(k)/X(k−1) → Y (k)/Y (k−1) and the map g := f −1 induces the inverse homeomorphism. But X(k)/X(k−1) is a wedge of spheres and be- comes disconnected if the wedge point is removed. Since the wedge point of X(k)/X(k−1) is mapped to the wedge point of Y (k)/Y (k−1) the components of the complement are preserved, i.e. f maps each open k-cell of X homeomorphicly onto a k-cell of Y and thus determines a bijection between the k-cells of X and the k-cells of Y . (Homeomorphic cells must have the same dimension by Brouwer’s Invariance of Domain Theorem but that theorem is not used here.)

S n Deﬁnition 4.15. A subcomplex of a cell complex X = α∈Λ eα is a closed subset A of form [ n A = eα

α∈Λ0 where Λ ⊂ Λ. The condition that A is closed implies that A = S e¯n. Thus 0 α∈Λ0 α A inherits a cell structure from X and the k-skeleton of A is A(k) = A ∩ X(k). In particular, the inclusion map A → X is cellular. The k-skeleton X(k) of X is a subcomplex. A cellular pair is a pair (X,A) consisting of a cell complex X and a subcomplex A ⊂ X. Remark 4.16. The deﬁnitions do not require that a closed cell be a subcomplex, but this is usually the case. Proposition 4.17. A compact subset of a cell complex lies in a ﬁnite subcomplex. Proof. As in Proposition 4.7. 4.18. Certain operations on cell complexes produce new cell complexes as explained on page 8 of chapter 0 of [3]. The product X × Y of two countable cell complexes is a cell complex. The quotient X/A obtained from a cellular pair (X,A) by collapsing A to a point is a cell complex. The cone CX on a cell complex X is a cell complex. The suspension SX of a cell complex is a cell complex. The join X ∗ Y of two cell complexes is a cell complex. The wedge sum X ∨ Y of two cell complexes is a cell complex (provided that the wedge

12 point is a zero cell of each). As noted in [3] the product topology is the quotient topology (i.e. Item (3) of Deﬁnition 4.2 holds) for countable complexes but this can fail if one of the factors is uncountable. Remark 4.19. There is an example of a product X × Y of two cell complexes (with X countable but Y not) which is not a cell complex in the product topology.

5 Some Homotopies

Lemma 5.1 (Mailed Fist Homotopy). Let (X,A) be a cullular pair. Then (X × 0) ∪ (A × I) is a deformation retract of X × I.

Proof. We must show that there is a homotopy {Rt = (rt, τt): X×I → X×I}t∈I satisfying r0(x) = x, rt(a) = a, r1(A) = A, and τ0(x) = 0, τt(a, s) = s, for x ∈ X, a ∈ A, t, s ∈ I. See Proposition 0.16 on page 26 of [3]. Corollary 5.2. Assume that (X,A) is a cellular pair and that A is contractible. Then the projection X → X/A is a homotopy equivalence.

Proof. Define K :(X × 0) ∪ (A × I) → (X × 0) ∪ (A × I) by K(x, 0) = x and K(a, t) = kt(a) for x ∈ X, a ∈ A, t ∈ I. Let {Rt : X × I → (X × 0) ∪ (A × I)}t∈I be as in the proof of Lemma 5.1. Then R1 : X × I →:(X × 0) ∪ (A × I) is a retraction, i.e. R1(x, 0) = (x, 0) and R1(a, t) = (a, t). Define ft : X → X by ft(x) = K(R1(x, t)) so f0(x) = K(x, 0) = x and f1(x) = K(a, 1) = k1(a) = a0. Let π : X → X/A be the projection and define σ : X/A → X by σ(x) = f1(x) for x ∈ X \ A and σ({A}) = a0. Then σ ◦ π = f1 ' idX and π ◦ σ ' idX/A by the homotopy ht(x) = π(ft(x)) for x ∈ X \ A and ht({A}) = {A}. Corollary 5.3. Assume that (X,A) is a cellular pair and that f, g : A → Y are homotopic. Then Y ∪f X ' Y ∪g X.

Proof. By hypothesis there is a map H : A × I → Y with H(·, 0) = f and H(·, 1) = g. There are inclusions

ι0 : Y tf X → Y tH (X × I), ι1 : Y tg X → Y tH (X × I) induced from the inclusions X × 0 ⊂ X × I and X × 1 ⊂ X × I; it is enough to show that ι0 and ι1 are homotopy equivalences. We prove the former; the same argument (read 1 − t for t) proves the latter.

Step 1. We deﬁne F : Y tH (X × I) → Y tf X. Let {Rt}t∈I be deformation retraction from X ×I onto (X ×0)∪(A×I) as in Lemma 5.1 and let Rt = (rt, τt). Deﬁne F : Y tH (X × I) → Y tf X by F (y) = y for y ∈ Y and ( H(R (x, s)) ∈ Y if R (x, s) ∈ A × , F (x, s) = 1 1 I r1(x) ∈ X if R1(x, s) ∈ X × 0.

13 The map F is well deﬁned as H(R1(x, s)) = f(r1(x)) ∼ r1(x) for R1(x, s) ∈ A × 0.

Step 2. We show F ◦ ι0 ' idY tf X . Define {Φt : Y tf X → Y tf X}t∈I by Φt(y) = y,Φt(x) = rt(x). The map Φt is well defined because rt(x) = x ∼ f(x) for x ∈ A. One easily checks that Φ0 = idY tf X and Φ1 = F ◦ ι0. Step 3. We show ι ◦ F ' id . Define {Ψ : Y t X → Y t X} 0 Y tH (X×I) t H H t∈I by Ψt(y) = y for y ∈ Y and Ψt(x, s) = Rt(x, s) ∈ X × I for (x, s) ∈ X × I. The map Ψt is well defined because Rt(x, s) = (x, s) ∼ H(x, s) for (x, s) ∈ A × I.

One easily checks that Ψ0 = idY tH X and Ψ1 = ι0 ◦ F .

Remark 5.4. Let Y = A, f = idA : A → Y and g : A → Y be a constant map: g(a) = a0. Clearly the space Y tf X is homeomorphic to the space X and the space Y tg X is homeomorphic to the wedge sum A ∨ (X/A) where the point a0 ∈ A is identiﬁed. Hence if A is contractible, then f and g are homotopic so X and A ∨ (X/A) are homotopy equivalent. By Corollary 5.6 below, {a0} is a deformation retract of A. Hence the wedge sum A ∨ (X/A) is homotopy equivalent to X/A. Thus Corollary 5.2 is a consequence of Corollary 5.3. Theorem 5.5. Let (X,A) be a cellular pair. Then the inclusion map ι : A → X is a homotopy equivalence if and only if A is a deformation retract of X. Proof. (This is Corollary 0.20 page 16 of [3]. The following proof is from [12] page 25.) ’If’ is trivial. For ’only if’ let f : X → A be a homotopy inverse to ι.

Let {φt : A → A}t∈I be a homotopy with φ1 = idA and φ0 = f ◦ ι. Extend to a homotopy {Φt : X → A}t∈I with Φ0 = f. Then Φ1 : X → A is a retraction. Hence, replacing f by Φ1, we may assume w.l.o.g. that the homotopy inverse f to ι is a retraction, i.e. that f|A = idA. Let {ψt : X → X}t∈I be a homotopy from ψ0 = idX to ψ1 = ι ◦ f. Let (Y,B) be the cellular pair deﬁned by

Y = I × X,B = ({0, 1} × X) ∪ (I × A).

Deﬁne h :({0} × Y ) ∪ (I × B) → X by

h(0, t, x) = ψt(x)

h(s, 0, x) = ψ0(x) = x

h(s, t, a) = ψ(1−s)t(a)

h(s, 1, x) = ψ1−s(f(x)) for s, t ∈ I, x ∈ X, and a ∈ A. By Lemma 5.1, extend h to H : I × Y → X and deﬁne {gt : X → X}t∈I by gt(x) = H(1, t, x). Then

gt|A = idA, g0 = idX , g1 = f, as gt(a) = H(1, t, a) = h(1, t, a) = ψ0(a) = a, g0(x) = H(1, 0, x) = h(1, 0, x) = ψ0(x) = x, and g1(x) = H(1, 1, x) = ψ0(f(x)) = f(x), for a ∈ A and x ∈ X. Corollary 5.6. For a cell complex X the following are equivalent:

14 (i) X is contractible.

(ii) For some a0 ∈ X, {a0} is a deformation of X.

(iii) For every a0 ∈ X, {a0} is a deformation of X. Remark 5.7. Of course, the implications (iii) =⇒ (ii) =⇒ (i) in Corollary 5.6 hold for any space. Exercises 6 and 7 on page 18 of [3] show that the implications (ii) =⇒ (iii) and (i) =⇒ (ii) do not hold for general spaces. Proposition 5.8. Let (X,A) be a cellular pair. Then there is a neighborhood N of A in X such that A is a deformation retract of N. Proof. This is Proposition A.5 page 523 of [3].

Corollary 5.9. A cell complex is locally contractible, i.e. for every x0 ∈ X and every neighborhood V of x0 there is a neighborhood U ⊂ V of x0 and a deformation retraction of U onto {x0}. Problem 5.10. It is plausible that a retraction whose fibers are (homeomorphic to) intervals is a deformation retraction, but I doubt that this is true in complete generality. However, a theorem of Smale [9] combined with Whitehead’s Theorem (Theorem 4.5 page 346 of [3]) implies that map between finite cell complexes with contractible fibers is a homotopy equivalence. It then follows from Theorem 5.5 that if (X,A) is a cellular pair and r : X → A is a retraction with contractible fibers, then A is a deformation retract of X. Prove (or disprove) the following slightly stronger statement: If (X,A) is a cellular pair then a retraction r : X → A with contractible fibers is a deformation retraction.

Exercise 5.11. Let X = D2 × I be a solid cylinder at α : I → X be a polygonal arc, possibly knotted, with α(0) = (0, 0, 1) ∈ D2 ×1 and α(1) = (0, 0, 0) ∈ D2 ×0. Show that the image A := α(I) is a deformation retract of X. Exercise 5.12. Given positive integers v, e, and f satisfying v − e + f = 2 construct a cell complex homeomorphic to S2 having v 0-cells, e 1-cells, and f 2-cells. Solution. If v − e + f = 2 then

(v, e, f) = (1, 0, 1) + m(1, −1, 0) + n(0, −1, 1) where m = v − 1 and n = f − 1. The sphere is a cell complex with v = f = 1 and e = 0 in the only way possible: the attaching map is constant. To increase m by one without changing n add a new vertex and edge and modify the corresponding characteristic map by composing with complex square root. To increase n without changing m insert a loop at a vertex. The result follows by induction.

15 6 The Fundamental Group

6.1. When γ : I → X is a path, we denote by [γ] the homotopy class of γ (rel ∂I); here ∂I = {0, 1}. Two paths α, β : I → X with α(1) = β(0) determine a path α · β : I → X via the formula

( 1 α(2t), 0 ≤ t ≤ 2 ; (α · β)(t) := 1 β(2t − 1), 2 ≤ t ≤ 1.

The fundamental group of the pointed space (X, x0) is

π1(X, x0) = {[γ]: γ(0) = γ(1) = x0}.

The group operation is well deﬁned by [α][β] := [α · β]. The identity element is the constant path and the inverse operation is deﬁned by

[γ]−1 := [¯γ], γ¯(t) := γ(1 − t).

Note that any path α : I → X gives an isomorphism

π1(X, x0) → π1(X, x1):[γ] 7→ [¯α · γ · α] where x0 = α(0), x1 = α(1), and (as before)γ ¯(t) := γ(1 − t). In this sense, if X is path connected, the fundamental group is independent of the choice of the base point. Remark 6.2. A groupoid is an algebraic structure consisting of two sets B (the objects) and G (the morphisms) and ﬁve maps s, t : G → B (source and target), e : B → G (identity), m : G s×t G → G (composition), and i : G → G (inverse) satisfying the following axioms:

(Identity Law) g · ex = g, ey · g = g where s(g) = x, t(g) = y.

(Associative Law) g1 · (g1 · g3) = (g1 · g1) · g3 for g1, g2, g3 ∈ G s×t G.

−1 −1 (Inverse Law) g · g = gey, g · g = ex where s(g) = x, t(g) = y. Here we are using the abbreviations

G s×t G := {(g1, g2) ∈ G × G : t(g2) = s(g1)},

−1 ex := e(x), g1 · g2 := m(g1, g2), g := i(g). If the map i and the Inverse Law are dropped we recover the deﬁnition of a category. In a category a morphism which has a (necessarily unique) two-sided inverse is called an isomorphism (or an automorphism if source and target are the same). Thus a groupoid is a category in which every morphism is an isomorphism. For a category (B,G) and x, y ∈ B denote the set of morphisms from x to y by Gx,y := {g ∈ G : s(g) = x, t(g) = y}.

16 The set Gx := {g ∈ Gx,x : g is an isomorphism} is called the automorphism group of the object x. With these deﬁnitions π1(X, x0) is the automorphism group x0 in the fundamental groupoid deﬁned by

B = X,G = { [γ] | γ : I → X}, s([γ]) = γ(0), t([γ]) = γ(1), e(x) = [x] (the homotopy class of the constant path at the point x), m([β], [α]) = [α · β], and i(γ) =γ ¯. Deﬁnition 6.3. A path connected space is called simply connected iﬀ its fundamental group π1(X, x0) is trivial.

1 n Theorem 6.4. π1(S ) = Z and S is simply connected for n > 1.

7 Covering Spaces

7.1. A map p : E → B is called locally trivial iff every x0 ∈ B there is a −1 neighborhood U of x0 ∈ B and a homeomorphism φ : U × F → p (U) such that p(φ(x, v)) = x for (x, v) ∈ U × F . The map p is called the projection, E is called the total space, B is called the base, and the topological space F is called the fiber. It is easy to see that if B is connected, then the fiber is unique up to homeomorphism. A locally trivial map is also called a fiber bundle, but sometimes the latter term has a more restricted meaning. A covering space is a locally trivial map with discrete fiber. Usually we denote covering spaces with notation like p : Y → X or p : X˜ → X. Note that the projection p of a covering space is a local homeomorphism. Example 7.2. The archetypal example of a covering space is the exponential map p : R → S1 defined by p(θ) = eiθ. The restriction of p to an open interval is a local homeomorphism but not a covering.

Example 7.3. The map p : S1 → S1, p(z) = zn is an n-sheeted covering space. Example 7.4. The various covers of a wedge S1 ∨S1 of circles shown on page 58 of [3] are a good source of examples. 7.5. When p : X˜ → X and f : Y → X we call a map f˜ : Y → X˜ a lift of f iﬀ ˜ p ◦ f = f. If p :(X,˜ x˜0) → (X, x0) and f :(Y, y0) → (X, x0) a pointed lift of ˜ ˜ f is a lift f such that f(y0) =x ˜0.

Proposition 7.6 (Path Lifting). Let p :(X,˜ x˜0) → (X, x0) is a covering −1 ˜ space, γ : I → X, and y0 ∈ p (γ(0)). Then there is a unique lift γ˜ : I → X of γ with γ˜(0) = y0. Corollary 7.7 (Homotopy Lifting). Let p : X˜ → X is a covering space, {ft : Y → X}t∈ be a homotopy, and g : Y → X˜ of be a lift of f0. Then there I ˜ ˜ ˜ ˜ is a unique homotopy {ft : I → X}t∈I such that f0 = g and ft is a lift of ft.

17 ˜ Proof. The uniqueness of ft(y) follows from 7.6 applied to the path I 7→ X : ˜ t 7→ ft(y) with starting point g(y) ∈ X˜ and this also deﬁnes ft(y) for y ∈ Y and ˜ t ∈ I. We show that (t, y) 7→ ft(y) is continuous at (t0, y0). Bu compactness write write the interval [0, t0] as a union of closed intervals [0, t0] = I1 ∪ · · · ∪ Ik so that ft(y0) ∈ Ui for t ∈ Ii where Ui ⊂ X is such that p is trivial over Ui. By induction on i there are open sets U˜i ⊂ X˜ such that p maps U˜i homeomorphically ˜ onto Ui and ft(y0) ∈ U˜i for t ∈ Ii. By continuity there is a neighborhood V of y0 ∈ Y such that ft(y) ∈ Ui for t ∈ Ii and y ∈ V . By induction on i, the fact that p : U˜i → Ui is a homeomorphism, and uniqueness of path lifting, ˜ ft(y) ∈ U˜i for t ∈ Ii and y ∈ V . Now, since p : U˜i → Ui is a homeomorphism, ˜ the continuity of [0, t0] × V → X˜ :(t, y) 7→ ft(y) at points t ∈ [0, t0] and y ∈ V follows from the continuity of [0, t0] × V → X :(t, y) 7→ ft(y)

Corollary 7.8 (Injectivity). Let p :(X,˜ x˜0) → (X, x0) is a covering space. Then the induced map p∗ : π1(X,˜ x˜0) → π1(X, x0) is injective.

Theorem 7.9 (Lifting Criterion). Assume p :(X,˜ x˜0) → (X, x0) be a covering space and let f :(Y, y0) → (X, x0) be a map. Assume Y is path connected and locally path connected. Then (i) Two pointed lifts of the same map f are equal.

(ii) There is a pointed lift of f if and only if f∗π1(Y, y0) ⊂ p∗π1(X,˜ x˜0).

Remark 7.10. Since p∗ is injective, we have f∗π1(Y, y0) ⊂ p∗π1(X,˜ x˜0) if and only if there is a homomorphism h : π1(Y, y0) → π1(X,˜ x˜0) such that p∗ ◦h = f∗. ˜ When these equivalent conditions hold we have h = f∗ as follows: If p∗ ◦ h = f∗, ˜ ˜ then p∗ ◦ h = f∗ = p∗ ◦ f∗ so h = f∗ by the injectivity of p∗. Thus, in the two diagrams

(X,˜ x˜0) π1(X,˜ x˜0) 9 8 f˜ ss h ppp ss p p p∗ ss ppp sss ppp s f p f∗ (Y, y0) / (X, x0) π1(Y, y0) / π1(X, x0) we can find f˜ so that the diagram on the left commutes if and only if we can find h so that the diagram on the right commutes. ˜ Proof of Theorem 7.9. When f is a pointed lift of f and α is a path from y0 to ˜ ˜ y, then f ◦ α is a path from f(y0) to f(y) and f ◦ α is a path from f(y0) to f˜(y), so uniqueness of the lift follows from uniqueness of path lifting. ’Only if’ ˜ ˜ in part (ii) follows from (p ◦ f)∗ = p∗ ◦ f∗. We prove ’if’. ˜ Assume that f∗π1(Y, y0) ⊂ p∗π1(X,˜ x˜0). As just noted, we must define f(y) ˜ by f(y) =γ ˜(1) whereγ ˜ is the lift of γ := f ◦ α withγ ˜(0) =x ˜0 and α is a path ˜ from y0 to y in Y . We must show that f is well defined (i.e. independent of the choice of α) and continuous. ¯ Suppose that β is another path from y0 to y. Then [αβ] ∈ π1(Y, y0) so ¯ [(f ◦ α)(f ◦ β)] ∈ f∗π1(Y, y0) ⊂ p∗π1(X,˜ x˜0) so there is an element of [˜γ] ∈

18 ¯ ¯ π1(X,˜ x˜0) which projects to [(f ◦α)(f ◦β)], i.e. p◦γ˜ is homotopic to (f ◦α)(f ◦β) with endpoints ﬁxed so the liftγ ˜ of which starts atx ˜0 has the same endpoint ¯ as the lift of (f ◦ α)(f ◦ β) of which starts atx ˜0. But this endpoint isx ˜0 asγ ˜ is ¯ closed. Hence the lift of (f ◦ α)(f ◦ β) of which starts atx ˜0 is closed so the lifts of f ◦ α and f ◦ β which start atx ˜0 have the same endpoint as required. ˜ Now we show that f is continuous. Choose y1 ∈ Y and a neighborhood W˜ ˜ ˜ of f(y1) in X˜. We must construct a neighborhood V of y1 such that f(V ) ⊂ W˜ . As p is a covering we may shrink W˜ so that p maps W˜ homeomorphicly onto a neighborhood W of f(y1). By the continuity of f there is a neighborhood V of y1 such that f(V ) ⊂ U. By local path connectedness there is a path connected neighborhood U of y1 with U ⊂ V . For y ∈ V there is a path β from y1 to y in ˜ V . Then β lies in U so f ◦ β lies in W so the lift of f ◦ β starting at f(y1) lies in W˜ . As Y is path connected there is a path α from y0 to y1. Then αβ is a path from y0 to y and the lift of (f ◦ α)(f ◦ β) starting at f(y0) ends the same ˜ ˜ ˜ point as the lift of f ◦ β starting at f(y1). This lift is f(y) so f(y) lies in W˜ as required.

7.11. Let Cov(X, x0) be the category whose objects are covering spaces p : (Y, y0) → (X, x0). A morphism in this category from the covering space p : (Y, y0) → (X, x0) to the covering space q :(Z, z0) → (X, x0) is a map φ : (Y, y0) → (Z, z0) such that q ◦ φ = p. For any group G let Sub(G) denote the category whose objects are subgroups H of G and where the morphisms are inclusions K ⊂ H. There is a functor

Cov(X, x0) → Sub(π1(X, x0)). which assigns the subgroup p∗π1(Y, y0) ⊂ π1(X, x0) to each covering space p : (Y, y0) → (X, x0) and assigns the inclusion

p∗π1(Y, y0) ⊂ p∗φ∗π1(Z, z0) = q∗π1(Z, z0) to the morphism φ :(Y, y0) → (Z, z0).

Corollary 7.12. Let p :(Y, y˜0) → (X, x0) and q :(Z, z0) → (X, x0) be pointed covering spaces over the same pointed space (X, x0). Then (i) There is at most one morphism from p to q.

(ii) There is a morphism from p to q if and only if p∗π1(Y, y0) ⊂ q∗π1(Z, z0). Proof. This is a special case of Theorem 7.9. Exercise 7.13. Let φ : Y → Z be a morphism from the covering space p : Y → X to the covering space q : Z → X, as in 7.11. Assume that X (and hence also Y and Z) is locally connected. Show that φ : Y → Z is a covering space. Give an example showing that the hypothesis that X is locally connected cannot be dropped.

19 Solution. Choose z0 ∈ Z and let x0 = q(z0). We must ﬁnd a neighborhood −1 V of z0 and a trivialization of the restriction φ to φ (V ). Trivialize p and q over a neighborhood of x0 and then shrink this neighborhood so it is connected. Restricting to this neighborhood we may assume w.l.o.g. that

Y = X × A, Z = X × B, p(x, a) = q(a, b) = x where A and B are discrete and X is connected. The condition q ◦φ = p implies that φ(x, a) = (x, f(x, a)) where f : X × A → B is continuous. Let z0 = (x0, b0) and C = {a ∈ A : f(x0, a) = b0}. Let V = X × {b0}. Then for (x, a) ∈ X × A we have

φ(x, a) ∈ V ⇐⇒ f(x, a) = b0 ⇐⇒ f(x0, a) = b0 ⇐⇒ a ∈ C where the middle ⇐⇒ holds because f is locally constant and X is connected. Thus φ−1(V ) = X × C so the inclusion X × C → X × A is a local trivialization. 1 1 1 Consider X = {0, 1, 2 , 3 , 4 ,...}, A = N, B = {1, 2, }, and f(x, n) = 1 or 2 according as nx < 1 or xn ≥ 1. Then the fiber φ−1(x, 1) is finite for x 6= 0 and infinite for x = 0 so φ is not trivial over any neighborhood of (0, 1). 7.14. A covering space p : X˜ → X of a path connected space X is called universal iff X˜ is connected and simply connected. A space X is called semi locally simply connected iff every point x0 ∈ X has arbitrarily small neighborhoods U such that the homomorphism π1(U, x0) → π1(X, x0) is trivial. Theorem 7.15. A path connected space and locally path connected space X has a universal cover if and only if it is semi locally simply connected.

Proof. For “only if” assume that p : X˜ → X is universal. Choose x0 ∈ X and a neighborhood V of x. Choose a neighborhood U of x0 in V so that P is trivial over U. Let U˜ ⊂ X˜ map homeomorphicly to U by p andx ˜0 ∈ U˜ be defined by p(˜x0) = x0. The inclusion π1(U, x0) → π1(X, x0) factors as π1(U, x0) → π1(U,˜ x˜0) → π1(X,˜ x˜0) → π1(X, x0) so it is trivial as π1(X,˜ x˜0) is trivial. For “if” choose x0 ∈ X and define p : X˜ → X by ˜ X := {[γ]: γ : I → X, γ(0) = x0}, p([γ]) = γ(1). For each [γ] ∈ X˜ and each path connected neighborhood U of γ(1) such that π1(U, γ(1)) → π1(X, γ(1)) is trivial define

U[γ] := {[γη]: η : I → U, η(0) = γ(1)}.

(The set U[γ] is clearly independent of the choice of the representative γ of [γ].) ˜ 0 These sets form a basis for a topology on X. If U[γ] ∩ U[γ0] 6= ∅ then [γ] = [γ ] so U[γ] = U[γ0]. This deﬁnes a homeomorphism

−1 U × {[γ]: γ(1) = x0} → p (U)

20 so p : X˜ → X is a covering. A function λ : I → X˜ is continuous if and only if there is a (continuous) map Λ : I × I → X with λ(s) = [Λ(s, ·)] and Λ(s, 0) = x0 for s ∈ I. Letx ˜0 be the class of the constant path at x0. Then the path λ begins and ends atx ˜0 if and only if there is a map Λ with Λ(0, t) = Λ(1, t) = x0 for t ∈ I. The set {0, 1} × I ∪ I × {0} is a deformation retract of the square I × I. Composing Λ with this deformation gives a homotopy from λ to a constant loop. Hence π1(X,˜ x˜0) is trivial. For details see [3] page 64. Exercise 7.16. Show that if two locally path connected spaces are homotopy equivalent, then so are their universal covers. (Assume that the spaces are semi locally simply connected so that they have universal covers.)

8 Group Actions

8.1. A left action (resp. right action) of a group G on a set X is a homomorphism (resp. anti-homomorphism) G → Perm(X) where Perm(X) denotes the group of permutations of X, i.e. the group of all bijections g : X → X. For a left action it is customary to write the evaluation map as G × X → X :(g, x) 7→ gx whereas for a right action we write G × X → X :(g, x) 7→ xg. This notation makes the homomorphism property look like the associative law. An action is called effective iff the homomorphism G → Perm(X) is injective; for an effective action we view G as a subgroup of Perm(X). An action is called free iff no element of G other than the identity has a fixed point, i.e. iff gx = x =⇒ g = id. When X is a topological space and G is a topological group (meaning that G is a topological space and the group operations are continuous) it is required that the evaluation map be continuous; it then follows the action takes values in the subgroup homeomorphism subgroup Homeo(X) ⊂ Perm(X) of homeomorphisms from X onto itself. In this section the group G will always have the discrete topology, i.e. every subset is open. 8.2. An action of G on X is called transitive iff for every pair of points x, y ∈ X there exists g ∈ G with gx = y (or xg = y for a right action). For each x ∈ X the subgroup of all g ∈ G which satisfy gx = x is called the stabilizer group (some authors say isotropy group) of x and is denoted Gx. When the action is transitive the map G/Gx → X : gGx 7→ gx is well defined and bijective. Here G/H denotes the set of left cosets gH of the subgroup H ⊂ G: the group G acts on the set G/H on the left. The map G/Gx → X intertwines the two actions. As a right action determines a left action and vice versa (precompose with the anti-automorphism g 7→ g−1), all this holds for right actions as well. We denote the set of right cosets of H by H\G. 8.3. Let p : Y → X be a covering space and assume that Y and X are path connected. For x0 ∈ X the fundamental group π1(X, x0) acts on the fiber −1 p (x0) on the right by the rule y0[γ] = y1 iff the liftγ ˜ of γ withγ ˜(0) = y0

21 satisfiesγ ˜(1) = y1. As Y is path connected this action is transitive. The −1 stabilizer group of the point y0 ∈ p (x0) is precisely p∗π1(Y, y0) so there is a bijection −1 p∗π1(Y, y0)\π1(X, x0) → π1 (x0) −1 between the set of right cosets of p∗π1(Y, y0) in π1(X, x0) and the fiber π1 (x0). We may summarize this as: the cardinality of the fiber is the index of the subgroup. 8.4. An automorphism of a covering space p : Y → X is called a deck trans- formation; we denote the group of all deck transformations of the cover by Aut(p). Thus an element of Aut(p) is a homeomorphism f : Y → Y such that p ◦ f = f. 8.5. A group G acts on the set Sub(G) of it subgroups via the adjoint action,

Sub(G) → Sub(G) : H 7→ gHg−1 for g ∈ G. The stabilizer of a subgroup H under this action is called the normalizer of H in G and denoted

N(H,G) := {g ∈ G : gHg−1 = H}.

It is the largest subgroup of G containing H as a normal subgroup; the subgroup H is a normal subgroup of G if and only if N(H,G) = G. Remark 8.6. If G acts on X, H is a subgroup of G, and

XH := {x ∈ X : Gx = H}. denotes the set of points in X having stabilizer group H, then

N(H,G) = {g ∈ G : g(XH ) = XH } and the formula

(G/N) × XH → XH :(gH, x) 7→ gx, N := N(H,G) gives a well deﬁned action of G/N on XH . Theorem 8.7. Let p : Y → X be covering space with X and Y path connected −1 and locally path connected and assume x0 ∈ X and y0 ∈ p (x0). Abbreviate N := N(p∗π1(Y, y0), π1(X, x0)). Then the groups Aut(p) and N/p∗π1(Y, y0) are anti isomorphic. Corollary 8.8. The following are equivalent:

−1 (i) The group Aut(p) acts transitively on the ﬁber p (x0).

(ii) The subgroup p∗π1(Y, y0) is a normal subgroup of π1(X, x0).

22 (iii) For every element [γ] ∈ π1(X, x0) either every lift of γ is closed or no lift is closed. When these equivalent conditions hold the covering is called normal (regular by some authors).

−1 Remark 8.9. For a normal covering p : Y → X and y0 ∈ p (x0), the groups Aut(p) and π1(X, x0)/p∗π1(Y, y0) are anti isomorphic. In particular, for the universal cover p the groups Aut(p) and π1(X, x0) are anti isomorphic. 8.10. Let a discrete group G act on the right on a topological space X and X/G denote the orbit space, i.e.

X/G := {xG : x ∈ X}, xG := {xg : g ∈ G}.

Then the projection X → X/G (with the quotient topology) is a covering space if and only if every point x ∈ X has a neighborhood U such that (Ug1)∩(Ug2) = ∅ for g1 6= g2. When X → X/G is a covering, the group G is (anti isomorphic to) the group of deck transformations so the covering is normal. Conversely, if p : X → Y is a normal covering and G = Aut(p) then Y =∼ X/Gop. (Note. The group Aut(p) is a subgroup of the homeomorphism group and acts on the left. To get a right action we replace G by Gop where the opposite group Gop of G has the same underlying set as G and the product of a and b in Gop is product of b and a in G. Thus the map G → Gop : g 7→ g−1 is an isomorphism.)

Theorem 8.11 (Classiﬁcation Theorem). Let (X, x0) be a path connected pointed space which admits a universal cover. Then the functor described in 7.11 is an equivalence of categories as explained in the proof below. Proof. This means that

(i) For every subgroup H ⊂ π1(X, x0) there is a covering p :(Y, y0) → (X, x0) with H = p∗π1(Y, y0).

(ii) If p :(Y, y0) → (X, x0) and q :(Z, z0) → (X, x0) are connected covering spaces of (X, x0) satisfying p∗π1(Y, y0) ⊂ q∗π1(Z, z0), then there exists a unique map φ :(Y, y0) → (Z, z0) with q ◦ φ = p.

(iii) If p∗π1(Y, y0) = q∗π1(Z, z0), then φ is a homeomorphism and hence an isomorphism of pointed coverings. Part (ii) is a restatement of Corollary 7.12 and part (iii) follows immediately from the uniqueness. For part (i) we construct the inverse functor. Let u : X˜ → −1 X be a universal cover. Abbreviate G := π1(X, x0) and choosex ˜0 ∈ u (x0). The group G acts on X˜ on the right by path lifting starting atx ˜0. The projection u induces a homeomorphism from the orbit space X/G˜ to X. A subgroup H ⊂ G determines a covering ∼ pH : X/H˜ → X/G˜ = X.

23 An inclusion K ⊂ H ⊂ G of subgroups determines a commutative diagram

fK.H X/K˜ / X/H˜ JJ t JJ tt JJ tt pK JJ tt pH J% ytt X/G˜ = X

Example 8.12. The universal cover of a wedge X = S1 ∨ S1 of circles is an inﬁnite tree with four edges at each vertex. Various other coverings of X are shown in [3] on page 58. Example 8.13. A graph is a one dimensional cell complex. A contractible graph is called a tree. A connected graph X contains a maximal tree A and for any maximal tree A in X we have that X/A is a wedge of circles. Thus, by 5.2 the fundamental group π1(X) is free for any graph X. This implies that a subgroup of a free group is free as follows. Let F be a free group and X be a wedge of circles with one circle for every generator of X so that F = π1(X). For any subgroup G of F there is a connected cover p : Y → X with p∗π1(Y ) = G. As p∗ is injective it follows that G is free. 8.14. Now assume that X is connected and locally path connected and that p : (X,˜ x˜0) → (X, x0) is a universal cover. Then X = X/G˜ where G = π1(X, x0) = Aut(p). Given any discrete space F and any left action ρ : G → Perm(F ) deﬁne

−1 X˜ ×ρ F := {[˜x, v] :x ˜ ∈ X,˜ v ∈ F }, [˜x, v] := {(˜xg, ρ(g) v): g ∈ G}.

It is not hard to prove that X˜ ×ρ F → X : [˜x, v] 7→ p(˜x) is a covering space, connected if and only if the action on F is transitive, and that this operation determines a bijective correspondence between isomorphism classes of left actions and isomorphism classes of coverings. (See [3] page 68.) Remark 8.15. This construction is a special case of a more general construction in the theory of fiber bundles as follows. A principal fiber bundle is a fiber bundle π : P → B such that the total space P is equipped with a free transitive right action action of a topological group G whose orbits are the fibers of π. Given any left action ρ : G → Homeo(F ) on a topological space F the associated fiber bundle P ×ρ F → B :[u, ξ] → π(u) is defined by

−1 P ×ρ F := {[u, ξ]: u ∈ P, ξ ∈ F }, [u, ξ] := {(ug, ρ(g) ξ): g ∈ G}.

Exercise 8.16. The projective plane is the quotient P 2 = S2/G where G = 2 2 {±idS2 }. Find all coverings of P × P . Exercise 8.17. The commutator subgroup of a group G is the subgroup [G, G] of G generated by all commutators [a, b] := aba−ab−1 with a, b ∈ G. It is a normal subgroup; the quotient group G/[G, G] is called the Abelianization

24 of G. Show that the Abelianiazation is characterized by the following universal mapping property: For every homomorphism φ : G → A where A is an Abelian group, there is a unique homomorphism ψ : G/[G, G] → A such that ψ ◦ π = φ where π : G → G/[G, G] is the projection.

2 2 m Exercise 8.18. Deﬁne gm,n : R → R by gm,n(x, y) = (x + m, (−1) y + n) 2 and let G = {gm,n : m, n ∈ Z}. The quotient R /G is called the Klein bottle. Determine all double coverings (i.e. the cardinality of the ﬁber is two) of the Klein bottle. Hint: Compute the Abelianization of G.

9 Van Kampen’s Theorem

Deﬁnition 9.1. Let {Gα}α∈Λ be an indexed collection of groups. The free product of this collection is the set of all equivalence classes of words (ﬁnite F sequences) in the disjoint union α∈Λ Gα where the equivalence relation is generated by the following operations: (1) adding or deleting an identity element so that

w1 ··· wkeαwk+1 ··· wn ∼ w1 ··· wkwk+1 ··· wn

where eα is the identity element of Gα; and (2) multiplying adjacent elements if they belong in the same group so that

w1 ··· wkwk+1 ··· wn ∼ w1 ··· (wkwk+1) ··· wn

if wk, wk+1 ∈ Gα.

The free product G of the collection {Gα}α∈Λ is often denoted by

∗ Y G = Gα α∈Λ

(Hatcher [3] uses the notation G = ∗αGα) and when Λ = {1, 2, . . . , n} is ﬁnite the free product is often denoted by

G = G1 ∗ G2 ∗ · · · ∗ Gn.

The free product G is a group: the group operation is induced by catenation and the identity element is the equivalence class of the empty word. When each Gα is isomorphic to Z, the free product is the free group with one generator for each index α ∈ Λ.

9.2. Below we will consider an indexed collection of groups {Gα}α∈Λ and a doubly indexed collection

{iαβ : Gαβ → Gα}α,β∈Λ

25 of group homomorphisms such that Gαα = Gα, iαα = idGα , and Gαβ = Gβα. To such a system we associate a subgroup N of the free product G; it is the −1 normal subgroup generated by all elements of form iαβ(ω)iβα(ω) . We call the normal subgroup N of G the Van Kampen subgroup of the system. (When all the Gαβ for α 6= β are the same the quotient G/N is commonly called the amalgamated product.)

Theorem 9.3 (Van Kampen). Assume X is a space, x0 ∈ X, and [ X = Aα α∈Λ where each Aα is open and path connected and contains the base point x0. Let G be the free product of the indexed collection {π1(Aα, x0)}α∈Λ of groups and N be the Van Kampen subgroup determined by the doubly indexed collection

iαβ : π1(Aα ∩ Aβ, x0) → π1(Aα, x0) of homomorphisms induced from the inclusions (Aα ∩Aβ, x0) → (Aα, x0). Then

(i) If the double intersections Aα ∩ Aβ (α, β ∈ Λ) are path connected, then the natural homomorphism G → π1(X, x0) is surjective.

(ii) If the triple intersections Aα ∩ Aβ ∩ Aγ (α, β, γ ∈ Λ) are path connected. then the kernel of this homomorphism is N so that π1(X, x0) is isomorphic to G/N.

Proof. Let f :(I, ∂I) → (X, x0). Choose a partition 0 = a0 < a1 < ··· < an = 1 so ﬁne that for each k there is an index αk ∈ Λ with f([ak−1, ak]) ⊂ Aαk . Let fk = f|[ak−1, ak]. For k = 1, . . . , n − 1 choose a path gk from x0 to f(ak) lying in Aαk ∩ Aαk+1 and take g0 and gn to be the constant loop at x0. Then

[f] = [g0f1g¯1][g1f2g¯2] ··· [gn−2fn−1g¯n−1][gn−1fngn]

where [gk−1fkgk] is the image in π1(X, x0) of an element of π1(Aαk , x0). This proves (i).

To prove (ii) choose an element [f1]α1 [f2]α2 ··· [fm]αm ∈ G with [fj]αj ∈ 2 πj(Aαj , x0). Let F : I → X be a homotopy from the loop f = f1f2 ··· fm to the constant loop at x0, i.e. F (0, t) = f(t) and F (s, 0) = F (s, 1) = F (1, t) = x0.

We must show that [f1]α1 [f2]α2 ··· [fm]αm ∈ N. 2 Write I as a union of closed rectangles R so small that f(R) ⊂ Aα for some α ∈ Λ. Arrange that the interiors of these rectangles are disjoint and that any vertex v of one of the rectangles lies in at most two other rectangles. (The rectangles are jiggled a bit to achieve this.) By suitably subdividing and reindexing we may assume that each fi is (up to reparameterization) the restriction of F to a union of consecutive intervals in 0 × I each of which is the intersection of 0 × I with one of these rectangles R. Let V be the set of all vertices of these rectangles. For each vertex v ∈ V choose a path gv from x0 to F (v) lying in Aα(R1) ∩ Aα(R2) ∩ Aα(R3) where

26 R1,R2,R3 are the rectangles which contain v. Call two vertices u, v ∈ V adjacent if the line segment [u, v] from u to v lies in an edge of one of the rectangles. For each pair u, v of adjacent vertices let eu,v = F |[u, v]. Then the loop

fu,v := gueu,vg¯v lies in Aα ∩ Aβ where R1 and R2 are the two rectangles adjacent to [u, v] and f(R1) ⊂ Aα and f(R2) ⊂ Aβ. Denote the corresponding homotopy classes as

[fu,v]α ∈ π1(Aα, x0), [fu,v]β ∈ π1(Aβ, x0).

These are distinct elements of the free product G but project to the same element of G/N. Consider an element h of the free product G of form

h = [fv0,v1 ]α1 [fv1,v2 ]α1 ··· [fvn−1,vn ]αn .

Replacing a subscript αk by the another subscript βk as above changes the element h ∈ G but leaves its image in G/N unchanged. If two consecutive factors [fvk−1,vk ]αk [fvk,vk+1 ]αk+1 have the same subscript, i.e. if αk = αk+1 =: α, then vk−1, vk, vk+1 are three of the four vertices of a rectangle R and replacing 0 vk by the fourth vertex vk leaves h unchanged (as an element of G). Thus each 0 of these two operations (replacing αk by βk and replacing vk by vk) leaves the image of h in G/N unchanged. But by means of these operations we can move the constant class in G/N to the class of f in G/N. (Represent f as a product h where the vj lie in (0 × I) ∪ (I × 1).) Thus a preimage of [f] in G lies in N as claimed.

Example 9.4. By Van Kampen, π1(X ∨ Y ) = π1(X) ∗ π1(Y ). In particular, the fundamental group of a wedge of circles is a free group with one generator for each circle. Example 9.5. Any group G is the fundamental group of of a two dimensional cell complex X as follows. Represent G as a quotient G = F/H where F is the free group on the generators {xi}i∈I and H is the normal subgroup generated by (1) the words {rα}α∈Λ. Assume that the 1-skeleton X of X is is a wedge of circles (1) W 1 X := i∈I Si with one circle for each generator xi. Form X by attaching 2 1 1 each a 2-cell eα for each relation rα via a map φα : S → X representing (1) 2 rα ∈ F = π1(X ). For each α ∈ Λ choose zα ∈ eα. Let U be a contractible neighborhood of the wedge point not containing any zα and

(1) A = X \{zα : α ∈ Λ},B = U ∪ (X \ X ).

Then A and B are open sets, A deformation retracts onto X(1), B is contractible, and A ∩ B is homotopy equivalent to a wedge of circles with one circle for each relation rα. Since π1(B) is trivial the free product π1(A) ∗ π1(B) is π1(A) and the Van Kampen subgroup N is the normal subgroup generated by {rα}α∈Λ.

27 Remark 9.6. Shelah [8] proves that the fundamental group of a compact metric space is either finitely generated or uncountable. 9.7. A link is the image K = f(S) of an embedding f : S → R3 of a finite disjoint union of circles. A link with one component (i.e. S = S1) is called a 3 knot. The fundamental group π1(R \ K) is called the group of the link or knot. One generally assumes that the embedding is smooth or piecewise linear (the precise terminology is tame) to avoid pathologies. There is an algorithm which computes the group of a knot or more precisely produces a presentation of the group known as the Wirtinger representation. It is described on page 55 of [3]. Exercise 9.8. Use the Wirtinger representation to prove that the Abelianiza- tion of the group of a knot is Z. Example 9.9. The complement R3 \ S1 of the unknot deformation retracts 1 2 3 1 onto a space homeomorphic to S ∨ S so π1(R \ S ) = Z. Exercise 9.10. Show that if S3 \ K ' X then R3 \ K ' S2 ∨ X. Hence 3 3 π1(S \ K) = π1(R \ K). Example 9.11. The complement S3 \ L of two linked circles 1 1 3 2 L = S × 0 ∪ 0 × S ⊂ S ⊂ C is homeomorphic to T 2 × R via the map t+iα −t+iβ 1 1 iα iβ e e S × S × R → L :(e , e , t) 7→ √ , √ e2t + e−2t e2t + e−2t 3 2 2 2 2 so (under a suitable embedding) π1(R \ L) = π1(S ∨ T ) = π1(T ) = Z . A suitable stereographic projection sends L to the disjoint union of the circle x2 + y2 = 1, z = 0 with the z-axis. Example 9.12. The complement S3 \ L0 of two unlinked circles deformation 1 2 1 2 3 0 retracts X ' S ∨ S ∨ S ∨ S so π1(R \ L ) is a free group on two generators. Example 9.13. Let p, q ∈ Z. The torus knot of type (p, q) is the knot √ 1 1 p q 3 K := {(u, v) ∈ S × S : u = v } ⊂ S ( 2) √ √ where S3( 2) := {(u, v) ∈ C2 : |u|2 + |v|2 = 2} denotes the sphere of radius 2. Exercise 9.14. The one point compactification of R3 is (homeomorphic to) S3. Show that if K ⊂ R3 is compact and S3 \ K ' Y , then R3 \ K ' S2 ∨ Y . Exercise 9.15. Let Y result from the cube I3 by identifying opposite faces with a 90◦ right hand twist. Use Van Kampen’s Theorem to show that the fundamental group of Y is the eight element quaternion group G = {±1, ±i, ±j, ±k}. Then give another proof by showing that the universal cover is S3 → S3/G =∼ Y . Hint: S3 =∼ ∂[−1, 1]4 and each face of [−1, 1]4 is a cube.

28 Exercise 9.16. Conley [1] page 26 studies a ﬂow entering a cylinder X = D2 ×I at the top D2 ×1, exiting at the bottom D2 ×0, with the orbits running vertically downward on the side S1×I ⊂ ∂X, and with a knotted orbit segment A inside the cylinder running from the point p = (0, 0, 1) ∈ D2 × 1 in the top of the cylinder to a point q = (0, 0, 0) ∈ D2 × 0 in bottom as in Exercise 5.11. He argues that there must be a nonempty invariant set inside the cylinder since otherwise the complement of the knotted orbit in the cylinder would deformation retract onto the punctured disk at the bottom. Prove that (D2 × 0) \ q is not a deformation retract of X \ A. In this context the condition that A is knotted means that 3 π1(X \A) is not Z. To see that this can occur, show that π1(X \A) = π1(R \K) where K is constructed from A by adjoining the polygonal arc with successive vertices p0 = 0, p1 = (0, 0, 2), p2 = (0, 2, 2), p3 = (0, 2, −2) , p4 = (0, 0, −2), and p5 = q.

10 Abelian Groups

10.1. Let {Gα}α∈Λ be an indexed family of Abelian groups. The direct prod- Q uct α∈Λ Gα of this family is the set of all functions g defined on the index set Λ such that g(α) ∈ Gα for α ∈ Λ; the direct sum is the subgroup M Y Gα ⊂ Gα α∈Λ α∈Λ of those g of finite support, i.e. g(α) = 0 for all but finitely many α.A free L Abelian group is a group which is isomorphic to G = Z for some index L α∈Λ set Λ. The elements eα ∈ α∈Λ Gα defined by eα(α) = 1 and αα(β) = 0 for β 6= α have the property that every element g ∈ G is uniquely expressible as a finite sum X g = nαeα α where the coefficients nα are integers; such a system of elements of a free Abelian group is call a free basis. It is easy to see that the cardinality a free basis is independent of the choice of the basis; it is called the rank of the free group. An Abelian group G is said to be finitely generated iff there is a surjective homomorphism h : Zn → G. 10.2. A graded Abelian group is an Abelian group C equipped with a with a direct sum decomposition M C = Cn. n∈Z Unless otherwise specified we assume the grading is nonnegative meaning that Cn = 0 for n < 0. A subgroup A of C is called graded iff M A = An where An := A ∩ Cn. n∈Z

29 It is easy to see that the quotient is then also graded, i.e. M C/A = (Cn/An). n∈Z The usage of the direct sum is a notational convenience only; it is better to think of a graded Abelian group as a sequence {Cn}n of Abelian groups. 10.3. homomorphism h : A → B between graded Abelian groups is said to shift the grading by r iﬀ h(An) ⊂ Bn+r for all n; h is said to preserve the grading iﬀ it shifts the grading by 0. The kernel, image, and hence also cokernel of h is graded. The graded Abelian groups and grade shifting homomorphisms form a category as do the graded Abelian groups and grade preserving homomorphisms, 10.4. Suppose that α : A → B and β : B → C are homomorphisms of Abelian groups. We say that the sequence

α β A / B / C is exact at B iﬀ the kernel of β is the image of α, i.e. α(A) = β−1(0). A sequence

··· / Cn+1 / Cn / Cn−1 / ··· is called exact iﬀ it is exact at each Cn. When the sequence terminates (at either end), no condition is placed on the group at the end. To impose a condition an extra zero is added. Thus a short exact sequence is an exact sequence

α β 0 / A / B / C / 0, i.e. it is exact at A, B, and C. For a short exact sequence the map α is injective, the map β is surjective, and C is isomorphic to the quotient B/α(A). Often A is a subgroup of B and α is the inclusion so C ≈ B/A. More generally, an exact sequence α 0 / K / A / B, gives an isomorphism from K to the kernel α−1(0) of α, and an exact sequence

α A / B / C / 0, gives an isomorphism from C to the cokernel B/α(A) of α. Exercise 10.5. Show that for a short exact sequence as in 10.4 the following are equivalent: (1) There is an isomorphism φ : A ⊕ C → B with with φ|A = α and β ◦ φ|C = idC .

(2) There is a homomorphism λ : B → A with λ ◦ α = idA.

30 (3) There is a homomorphism ρ : C → B with β ◦ ρ = idC . When these three equivalent conditions hold we say that the exact sequence splits or that the subgroup α(A) ⊂ B splits in B. Exercise 10.6. Show that a short exact sequence as in 10.4 always splits when C is free but give an example of a short exact sequence which doesn’t split even though A and B are free. Exercise 10.7. The concepts of 10.4 remain meaningful for non Abelian groups although it is customary to use multiplicative notation (1, ab, A × B) rather than additive notation (0, a+b, A⊕B). Show that the implications (1) ⇐⇒ (2) and (1) =⇒ (3) remain true in the non Abelian case but give an example where (3) =⇒ (1) fails.

Lemma 10.8 (Smith Normal Form). For any integer matrix A ∈ Zm×n there are square integer matrices P ∈ Zm×m and Q ∈ Zn×n of determinant ±1 (so P −1 and Q−1 are integer matrices by Cramer’s rule) such that D 0 P AQ−1 = r×(n−r) 0(m−r)×r 0(m−r)×(n−r) where D is a diagonal integer matrix of form   d1 0  ..  D =  .  , 0 dr di > 0, and di divides di+1. Proof. Use row and column operations to transform to a matrix where the least common denominator of the entries is in the (1, 1) position and then use row and column operations to transform so that the other entries in the first row and the first column to vanish. Then use induction on the remaining (m − 1) × (n − 1) matrix. As in elementary linear algebra, the row operations give P and the column operations give Q. See Theorem 11.3 Page 55 of [7] for more details. Corollary 10.9. A subgroup of a free Abelian group is free Abelian of lower or equal rank. Proof. It follows from Smith Normal Form that the range of A is a free subgroup. It is easy to see that any subgroup of Zm is the range of A for some integer matrix A. Hence Smith Normal Form implies that See Lemma 11.1 page 53 of [7] for a more direct argument. Lemma 11.2 page 54 of [7] shows that a subgroup of a free Abelian group is free Abelian even without the hypothesis that the ambient group is finitely generated. 10.10. The torsion subgroup T (G) of an Abelian group G is the subgroup of elements of finite order. It is not hard to see that if G is finitely generated the quotient G/T (G) is free and hence splits by Exercise 10.6. Lemma 10.8 yields the following stronger

31 Corollary 10.11 (Fundamental Theorem of Abelian Groups). Let A ﬁnitely generated Abelian group G has a direct sum decomposition

G = F ⊕ T (G),T (G) = Z/d1 ⊕ · · · Z/dr where F is free, di > 0, and di divides di+1. Remark 10.12. The rank of the free group G/T (G) is also called the rank of G itself. The tensor product Q⊗G of G with the rational numbers Q is a vector space over Q. It is easy to see that the dimension of the vector space Q ⊗ G is the rank of G.

Exercise 10.13. The torus T 2 may be viewed as the quotient R2/Z2 of the group R2 by the subgroup Z2. Consider the linear map R2 → R2 and its inverse represented by the matrices

3 5 7 −5 A = ,A−1 = . 4 7 −4 3

As A has integer entries it deﬁnes a map f : T 2 → T 2 by

2 2 f(x + Z ) = Ax + Z for x ∈ R2. As A−1 also has integer entries, this map is a homeomorphism. How many ﬁxed points does it have? (A ﬁxed point of f is a point p ∈ T 2 such that f(p) = p.) Lemma 10.14 (Five Lemma). Consider a commutative diagram

i j j k A / B / C / D / E

α β γ δ ε 0 i0 j k0 `0 A0 / B0 / C0 / D0 / E0 of Abelian groups and homomorphisms. Assume that the rows are exact and that α, β, δ, and ε are isomorphisms. Then γ is an isomorphism.

11 Abstract Homology

11.1. A chain complex is a pair (C, ∂) consisting of a M C = Cn, n∈Z and a homomorphism ∂ : C → C called the boundary operator such that 2 ∂(Cn+1) ⊂ Cn and ∂ = 0. The chain complex (C, ∂) will be denoted simply by C when no confusion can result. If there are several chain complexes in the discussion we write ∂C for ∂.A chain map from a chain complex (A, ∂A) to

32 a chain complex (B, ∂B) is a group homomorphism φ : A → B which preserves the grading and satisﬁes ∂B ◦ φ = φ ◦ ∂A.

Chain complexes and chain maps form a category, i.e. the identity idC : C → C is a chain map and the composition of chain maps is a chain map. Remark 11.2. Unless otherwise speciﬁed we assume that chain complexes are nonnegative meaning that Cn = 0 for n < 0. 11.3. Let (C, ∂) be a chain complex, Z(C) = ∂−1(0) denote the kernel of ∂, and B(C) = ∂(C) denote the image of ∂. As ∂ shifts the grading by −1 these L L are graded subgroups, i.e. B(C) = n Bn(C) and Z(C) = n Zn(C) where

Bn(C) := ∂(Cn+1) = B(C) ∩ Cn and Zn(C) := Z(C) ∩ Cn. The condition ∂2 = 0 is equivalent to the condition B(C) ⊂ Z(C). The quotient H(C) := Z(C)/B(C) is called the homology group of C. This is also graded, L namely H(C) = n Hn(C) where

Hn(C) := Zn(C)/Bn(C).

By Remark 11.2 Z0(C) = C0. An elements of B(C) is called a boundary, an element of Z(C) is called a cycle, and an element of H(C) is called a homology class. A chain map φ : A → B sends boundaries to boundaries and cycles to cycles and hence induces a graded homomorphism

φ∗ : Hn(A) → Hn(B) for each n. The operation which sends the chain complex C to the graded group H(C) and sends the chain map φ : A → B to the graded homomorphism φ∗ : H(A) → H(B) is a functor, i.e. idC ∗ = idH(C) and (ψ ◦ φ)∗ = ψ∗ ◦ φ∗. Remark 11.4. A chain complex

∂ ∂ ∂ ∂ ··· / Cn+1 / Cn / Cn−1 / ··· is exact at Cn (where n > 0) if and only if Hn(C) = 0. A chain complex whose homology vanishes is sometimes called acyclic. 11.5. An augmented chain complex is a nonnegative chain complex C equipped with an augmentation, i.e. a surjective homomorphism ε : C0 → Z such that ε ◦ ∂|C1 = 0. We use the augmentation to produce a modiﬁed nonnegative chain complex C˜ deﬁned by

−1 C˜n = Cn for n > 0 and C˜0 = ε (0). For an augmented chain complex, the reduced homology group H˜ (C) is the homology of the modiﬁed complex, i.e.

Hñ(C) := Zñ(C)/Bñ(C)

33 −1 where Zñ(C) := Zn(C) for n > 0, Z˜0(C) := ε (0), and Bñ(C) := Bn(C) for n ≥ 0. Obviously Hñ(C) = Hn(C) for n > 0. A chain map φ : A → B between two augmented chain complexes is said to be augmentation preserving iff εB ◦ (φ|A0) = εA. Just as for chain maps, an augmentation preserving chain map induces a graded homomorphism

φ∗ : H˜ (A) → H˜ (B) on reduced homology. Remark 11.6. An exact sequence 0 → A → B → C → 0 of augmented chain complexes gives rise to an exact sequence 0 → A˜ → B˜ → C˜ → 0 of the modified chain complexes. An augmented chain complex can be viewed as chain complex with C−1 = Z and ∂|C0 = ε. This construction gives a chain complex with the same homology as C˜ but is inconvenient because it does not preserves exact sequence of chain complexes in the aforementioned sense: a sequence 0 → Z → Z → Z → 0 is never exact. 11.7. Two chain maps φ, ψ : A → B are called chain homotopic iff there is a chain homotopy between them, i.e. a homomorphism P : A → B such that P (An) ⊂ Bn+1 and ψ − φ = ∂B ◦ P − P ◦ ∂A. Chain homotopy is an equivalence relation and compositions of chain homotopic maps are chain homotopic so the chain complexes and chain homotopy classes form a category. An isomorphism of this category is called a chain homotopy equivalence. A chain complex C is called chain contractible iff the the identity map of C is chain homotopy equivalent to the zero map. Exercise 11.8. Show that a nonnegative free chain complex C is chain contractible if and only f H(C) = 0. Hint: Construct P : Cn → Cn+1 satisfying ∂P = idC − P ∂ by induction on n. Exercise 11.9. Let A and B be nonnegative free chain complexes. Show that two chain maps φ, ψ : A → B are chain homotopic if and only if the induce the same map H(A) → H(B) on homology, i.e. if and only if ψ∗ = φ∗. 11.10. Consider short exact sequence

i j 0 / A / B / C / 0, of chain complexes, i.e. the homomorphisms i and j are chain maps. It is not hard to show that there is a unique homomorphism ∂∗ : H(C) → H(A) called the boundary homomorphism such that for c ∈ Z(C) and a ∈ Z(A) we have

∂∗[c] = [a] ⇐⇒ ∃b ∈ B such that i(a) = ∂B(b) and j(b) = c. where the square brackets signify the homology class of the cycle it surrounds.

34 Theorem 11.11 (Long Exact Homology Sequence). The homology sequence

∂∗ i∗ j∗ ∂∗ i∗ ··· / Hn(A) / Hn(B) / Hn(C) / Hn−1(A) / ··· associated to the short exact sequence of chain complexes of 11.10. The sequence is also natural meaning that a commutative diagram

i j 0 / A / B / C / 0

α β γ 0 i0 j 0 / A / B / C / 0 of chain complexes and chain maps gives rise to a commutative diagram

∂∗ i∗ j∗ ∂∗ i∗ ··· / Hn(A) / Hn(B) / Hn(C) / Hn−1(A) / ···

α∗ β∗ γ∗ α∗ i0 j0 i0 ∂∗ 0 ∗ 0 ∗ 0 ∂∗ 0 ∗ ··· / Hn(A ) / Hn(B ) / Hn(C ) / Hn−1(A ) / ··· of long exact sequences. Theorem 11.12 (Standard Basis Theorem). Assume that C is a chain complex such that each group Cn is free and of ﬁnite rank. Then there is a direct sum decomposition

Ck = Uk ⊕ Vk ⊕ Wk such that ∂(Uk) ⊂ Wk−1 and ∂(Vk) = ∂(Wk) = 0. Moreover, there are bases for Uk and Wk−1 relative to which ∂ : Uk → Wk−1 is represented by a diagonal integer matrix   d1 0  ..  D =  .  0 dr where D is as in Lemma 10.8. Hence Zk = Vk ⊕ Wk and

Hk−1(C) ≈ Vk−1 ⊕ Z/d1 ⊕ · · · ⊕ Z/dr where Z/d = 0 if d = 1.

Proof. Let Zk ⊂ Ck denote the cycles and Bk ⊂ Ck denote the boundaries as usual. Let Wk denote the weak boundaries, i.e. those elements w ∈ Ck such that mw ∈ Bk for some m ∈ Z. Then

Bk ⊂ Wk ⊂ Zk ⊂ Ck.

By Smith Normal Form (Lemma 10.8) there is a basis e1, . . . , en for Ck, a basis f1, . . . , fm for Ck−1, and integers di as in the theorem such that ∂ei = difi for i = 1, . . . , r and ∂ei = 0 for i = r + 1, . . . , n. It follows that

35 1. er+1, . . . , en is a basis for Zk.

2. f1, . . . , fr is a basis for Wk−1.

3. d1f1, . . . , dkfr is a basis for Bk−1.

Then one shows Wk splits in Vk (i.e. that there is a subgroup Vk of Zk with Zk = Vk ⊕ Wk) and takes Uk to be spanned by e1, . . . , er. See Theorem 11.4 page 58 of [7] for more details. Remark 11.13. Smith Normal Form is a special case of Theorem 11.12. The sequence A 0 / Zn / Zm / 0 is a chain complex. 11.14. Here is an algorithm for computing homology for a free ﬁnitely p n m generated chain complex. Assume that Ck+1 = Z , Ck = Z , and Ck−1 = Z n×p so that ∂ : Ck+1 → Ck is represented by a matrix B ∈ Z and ∂ : Ck → Ck−1 m×n is represented by a matrix A ∈ Z where AB = 0. The homology group Hk is the quotient of the kernel of A by the image of B. Assume w.l.o.g. that B is in Smith Normal Form  d 0  1 D 0r×(p−r) . B = ,D =  ..  . 0(n−r)×r 0(n−r)×(p−r)   0 dr

Then as D is invertible over Q and AB = 0 the matrix A must have the form

0 0 m×(n−r) A = 0m×r A ,A ∈ Z . The rank of A equals the rank of A0 so the nullity of A0 is ν := n − r − rank(A). The homology is Ker(A) ≈ ν ⊕ /d ⊕ · · · ⊕ /d . Im(B) Z Z 1 Z r 11.15. The algorithm in 11.14 can easily be modiﬁed to produce the direct sum decomposition Ck = Uk ⊕ Vk ⊕ Wk of Theorem 11.12, i.e. given a sequence n ×n A1,A2,...,An of matrices with Ak ∈ Z k k−1 and Ak−1Ak = 0, the modiﬁed n ×n algorithm constructs matrices Pk ∈ Z k k of determinant ±1 such that   0uk−1×uk 0uk−1×vk 0uk−1×wk −1 Pk−1AkPk =  0vk−1×uk 0vk−1×vk 0vk−1×wk 

D 0wk−1×vk 0wk−1×wk where uk + vk + wk = nk, uk = wk−1, where D is as in Lemma 10.8. Assume inductively that Pn,...,Pk−1 have been constructed. Then, as in 11.14,

−1 0 Ak−1Pk−1 = A 0nk−1×uk .

36 −1 Apply the Smith Normal form algorithm to Ak−1Pk−1 avoiding column operations which modify the last wk−1 columns. This will modify Pk−1 but will not −1 change Pk−1AkPk . Then apply row operations to move the nonzero block in the upper left hand corner to the lower left hand corner. This modiﬁes Pk−2 but not Pk−1.

Exercise 11.16. Assume A ∈ Rm×n and B ∈ Rn×p satisfy AB = 0. Show that Rn = U ⊕ V ⊕ W where U = Im(A∗), W = Im(B), and V = ker(A∗A + BB∗).

12 Singular Homology

12.1. Let X be a topological space. A map σ : ∆n → X is called a singular n-simplex in X. The free Abelian group generated by the singular n-simplices is denoted by Cn(X) and its elements are called singular n-chains. A singular n-chain c is a ﬁnite formal sum of singular n-simplices, i.e.

r X c ∈ Cn(X) ⇐⇒ c = ckσk k=1 where each σk is a singular n-simplex and ck ∈ Z. The boundary operator ∂ : Cn(X) → Cn−1(Y ) is deﬁned by

n X k ∂σ = (−1) σ ◦ ιk i=0

n−1 n where ιk : ∆ → ∆ is deﬁned by

ιk(x1, . . . , xn) = (x1, . . . , xk−1, 0, xk, . . . , xn).

It is easy to see that the sequence

∂ ∂ ∂ ∂ ··· / Cn+1 / Cn / Cn−1 / ··· is a chain complex, i.e. that ∂2 = 0. We write M C(X) := Cn(X) n∈Z where Cn(X) := 0 for n < 0. The cycles of this complex are denoted by Z(X) and the boundaries by B(X). As usual Zn(X) := Z(X) ∩ Cn(X) and Bn(X) := B(X) ∩ Cn(X). The quotient M H(X) := Hn(X),Hn(X) := Zn(X)/Bn(X) n is called the singular homology group of the space X.

37 12.2. Let A ⊂ X be a subspace of X. Then C(A) is a subcomplex of C(X). The quotient complex is denoted M C(X,A) := Cn(X,A),Cn(X,A) := Cn(X)/Cn(A). n Elements of the groups

−1 Zn(X,A) := ∂ Cn(A),Bn(X,A) := Bn(X) + Cn(A) are called relative singular cycles and relative singular boundaries respectively. The homology M H(X,A) := Hn(Z,A),Hn(X,A) := Zn(X,A)/Bn(X,A) n is called the relative singular homology of the pair (X,A).

12.3. The standard augmentation ε : C0(X) → Z of the singular chain complex C(X) is deﬁned by ! X X ε cipi = ci. i i

Here we identify the point p ∈ X with the singular simplex ∆0 → X : 1 7→ p. (Recall that ∆0 = {1}.) The corresponding reduced homology group is denoted H˜ (X) and is called the reduced singular homology group of X. Thus M H˜ (X) = H˜n(X) n where H˜n(X) = Hn(X) for n > 0 and

ε−1(0) H˜0(X) = ⊂ H0(X). B0(X) 12.4. A map f : X → Y induces a homomorphism

f# : Cn(X) → Cn(Y ) deﬁned by f#(σ) := f ◦σ for each singular n-simplex in X. This homomorphism is a chain map, i.e. f# ◦ ∂ = ∂ ◦ f#. This implies that f#(Bn(X)) ⊂ Bn(Y ) and f#(Zn(X)) ⊂ Zn(Y ). Hence f induces a map

f∗ : H(X) → H(Y ).

The induced chain map f# preserves the standard augmentation so f∗(H˜n(X)) ⊂ H˜n(Y ). Moreover, if f :(X,A) → (Y,B) then f#(Cn(A)) ⊂ Cn(B) so f induces a map f∗ : Hn(X,A) → Hn(Y,B) on relative homology.

38 Remark 12.5. The three constructions H(X), H(X,A), and H˜ (X) determine one another as follows. The unique map q : X → {∗} from the space X to the one point space gives a set theoretic equality

H˜ (X) = Ker(q∗) ⊂ H(X). For p ∈ X the sequence {p} → X → {p} gives a splitting H(X) = H˜ (X) ⊕ H({p}). The composition C˜(X) → C(X) → C(x, {p}) induces an isomorphism H˜ (X) ≈ H(X, {p}). In Corollary 15.7 below we will prove that for “nice pairs” (X,A) the projection (X,A) → (X/A, A/A) induces an isomorphism H(X,A) ≈ H(X/A, A/A) ≈ H˜ (X/A). In Remark 1.16 we noted the obvious identiﬁcation X → X/∅. The empty sum is zero so C(∅) = {0}. The map C(X) → C(X, ∅): c 7→ c + {0} = {c} gives a corresponding isomorphism H(X) = H(X, ∅).

Theorem 12.6 (Dimension Axiom). If X is a point p, then Hn(X) = 0 for n > 0 and H0(X) = Z.

Proof. Cn(p) = Z for n ≥ 0 and ∂|Cn(p) = 0 or the identity according as n is odd or even.

Proposition 12.7. If X is pathwise connected, then H0(X) = Z.

Proposition 12.8. Let {Xα}α∈Λ be the path components of X. Then Hn(X) = L α∈Λ Hn(Xα). L L Proof. Zn(X) = α∈Λ Zn(Xα) and Bn(X) = α∈Λ Bn(Xα). Theorem 12.9 (Homotopy Axiom). If f, g : X → Y are homotopic, then the chain maps f# and g# are chain homotopic so f∗ = g∗. Proof. Let F : X × I → Y satisfy F (x, 0) = f(x) and F (x, 1) = g(x). Deﬁne the prism operator P : Cn(X) → Cn+1(Y ) by

n+1 X k P (σ)(x0, . . . , xn) = (−1) F (σ x0, . . . , xn), x0 + ··· + xk−1 k=0 for each singular n-simplex σ in X. The prism operator is a chain homotopy. The geometric interpretation is that the prism ∆n × I is written as a union of (n + 1)-simplices n n [ ∆ × I = [v0, . . . , vk, wk, . . . , wn] k=0

39 n where vi = (ui, 0), wi = (ui, 1), ui are the vertices ∆ , and [...] denotes convex hull. The formula ∂P = g# − f# + P ∂ expresses the boundary of the prism as the top minus the bottom minus the prism on the faces of ∆n. The internal faces of the decomposition cancel. See [3] page 112.

13 ∆-Complexes and Simplicial Complexes

Definition 13.1. Resume the notation of 4.1 and Definition 4.2. A cell complex {Φα : Dα → X}α∈Λ is called a ∆-complex iff each component Dα is a standard n simplex

n n+1 ∆ := {(x0, x1, . . . , xn) ∈ I : x0 + x1 + ··· + xn = 1}

n−1 n (as before, n = nα depends on α) and for each face ι : ∆ → ∆ and each characteristic map Φα the composition Φα ◦ ι is also a characteristic map. The term face here means

ι(x1, . . . , xn) = (x1, . . . , xk−1, 0, xk, . . . , xn) for some k = 1, . . . , n, i.e. ι = ιk as in 1.3. The points of the 0-skeleton of n n a ∆-complex are called vertices and the closed cellse ¯α := Φα(∆ ) are called simplices. A ∆-complex is called a simplicial complex iﬀ the characteristic maps are determined by the vertices, i.e. whenever α, β ∈ Λ satisfy (nα = nβ and) Φα(v) = Φβ(v) for each of the n + 1 vertices v = (0,..., 0, 1, 0,..., 0) of n ∆ , we have α = β. A simplicial complex {Φα : Dα → X}α∈Λ is called a triangulation of X. As was the case with cell complexes, it is customary to denote a ∆-complex by the same letter X as its underlying space rather than by the more correct notation {Φα : Dα → X}α∈Λ. Example 13.2. The simplex ∆n is a simplicial complex with a characteristic m n map Φα : ∆ → ∆ for every subset α ⊂ {0, 1, . . . , n}. The map Φα is the map

m X Φα(x) = xjvij j=0

m for x = (x0, x1, . . . , xm) ∈ ∆ , α = {i0, i1, . . . , im} with i0 < i1 < ··· < im and n where vi is the vertex of ∆ with 1 in the ith place and 0 elsewhere.

Example 13.3. Let T 2 = S1 ×S1 = R2/Z2 be the torus. There is a cell complex Φ: D → T 2 with one 0-cell (the origin (0, 0) in R2), two 1-cells (the intervals I × {0} and {0} × I), and one 2-cell (the unit square I × I); the characteristic map is Φ(x, y) = (x, y) mod Z2. If we add the diagonal we get the ∆-complex of Example 14.2. There is an easy triangulation with vertices (p/3, q/3) mod Z2 for p, q = 0, 1, 2 and edges from (p/3, q/3) to (p + 1)/3, q/3, p/3, (q + 1)/3, and (p + 1)/3, (q + 1)/3. This triangulation has 9 vertices, 27 edges, and 18 faces. By judiciously discarding a few vertices and edges we can achieve a triangulation with 7 vertices. (See Exercise 13.5.)

40 13.4. It is easy to construct simplicial complexes whose geometric realizations are homeomorphic to the sphere S2, the cylinder S1 × I, the M¨obius band, the torus, the Klein bottle, and the projective plane. (See [7] pages 16-19.)

Exercise 13.5. Show that for any triangulation√ of a compact surface, we have 3f = 2e, e = 3(v − χ), and v ≥ (7 + 49 − 24χ )/2. In the case of the sphere, projective plane and torus, what are the minimum values of the numbers v, e and t? (Here v,e and f denote the number of vertices, edges and triangles respectively; χ := v − e + f denotes the Euler characteristic.) Solution. Let P be the set of pairs (E,F ) such that F is a face and E is an edge F in the boundary of F , PE = {F :(E,F ) ∈ P }, and P = {E :(E,F ) ∈ P }. F Then #(PE) = 2 for each E so #(P ) = 2e. Also #(P ) = 3 for each F so #(P ) = 3f. Hence 2e = 3f. From v − e + f = χ we get e = 3v − 3χ. But √ v v(v−1) 2 7+ 49−24χ clearly e ≤ 2 = 2 so 0 ≤ v − 7v + 6χ so v ≥ 2 . For the sphere S2 we have χ = 2 and hence v ≥ 4 with equality for the tetrahedron. For the projective plane we have χ = 1 so v ≥ 6 with equality for the triangulation shown on page 15 of [6]. (This example is obtained by identifying opposite edges of a hexagon and adding a triangle in the interior.) For the torus T 2 we have χ = 0 so v ≥ 7 with equality in the following triangulation: v v v1 4 5 v1 A ¡@ A v7¡ @ v3 A¡ @ v3 QQ Q Q v2 Q v2 @ ¡vA6 @ ¡ A v1 @¡ A v1 v4 v5

Note that in all cases the lower bound is achieved when the 1-skeleton is the complete graph on v vertices. It is a consequence of the proof that this complete graph is the 1-skeleton of a surface of Euler characteristic χ only when the lower bound is an integer.

13.6. Let {Φα : Dα → X}α∈Λ and {Ψβ : Dβ → Y }β∈Λ0 be ∆-complexes. A map f : X → Y is called ∆-map iﬀ for every α ∈ Λ there is a (necessarily 0 unique) β ∈ Λ and a simplicial map fα : Dα → Dβ such that

Ψβ ◦ fα = f ◦ Φα.

n m That a map g : ∆ → ∆ is simplicial means that it sends each vertex vi of n m ∆ to a vertex g(vi) of ∆ and that it is aﬃne, i.e.

n ! n X X g xivi = xig(vi). i=0 i=0

41 With respect to the simplicial structure of Example 13.2 a map g : ∆n → ∆m is simplicial if and only if it is a ∆-map. A ∆-map between simplicial complexes is also called a simplicial map. 13.7. The ∆-complexes and ∆-maps between them form the objects and morphisms of a category, i,e, the identity map of of a ∆-complex is a ∆-map and the composition of two ∆-maps is a ∆-map. The simplicial complexes form a (full) subcategory. 13.8. If (X,A) is a cellular pair and X is a ∆-complex, then the subcomplex A is also a ∆-complex; in this case we call (X,A) a ∆-pair. The cell structure on X/A of 4.18 is then a ∆-complex and the projection X → X/A is a ∆- map. (This construction does not give a simplicial complex even when X is a simplicial complex.) As in 4.18 the operations of suspension, cone, join, and product also yield ∆-complexes (resp. simplicial complexes) when applied to ∆-complexes (resp. simplicial complexes). The characteristic maps of the join are determined by composition with the inverse of the canonical homomorphism

∆m ∗ ∆n → ∆m+n+1 :(x, y, t) 7→ tx + (1 − t)y.

The characteristic maps of the product ∆-complex are given by composition with the maps m+n m n `σ : ∆ → ∆ × ∆ described on page 277 of [3]. In the special case n = 1 this reduces to the prism structure on ∆m × I described in the proof of Theorem 12.9. As is the case for cell complexes, the product topology is the quotient topology for countable complexes but this can fail if one of the factors is uncountable.

14 Simplicial Homology

In this section we deﬁne two subcomplexes ∆(X) and ∆0(X) of the singular chain complex C(X) of a ∆-complex X and state analogs of the basic theorems in homology theory. The analogs are easier to understand than the corresponding theorems on singular theory since the subcomplexes are ﬁnitely generated. In section 16 we present a more general theory which makes singular homology even easier to compute. The inclusions

∆(X) → ∆0(X) → C(X) induce isomorphisms in homology.

14.1. Let {Φα : Dα → X}α∈Λ be a ∆-complex. Each characteristic map n Φα : ∆ → X may be viewed as a singular n-simplex: denote by ∆n(X) the subgroup of the singular chain group Cn(X) generated by the the characteristic maps. By the deﬁnition of ∆-complex there is a function β = β(α, k) which

42 assigns to the index α the (unique) index β such that Φβ = Φα ◦ ιk where n−1 n ιk : ∆ → ∆ is inclusion into the kth face. Hence

n X k ∂Φα = (−1) Φβ(α,k) k=0 which shows that M ∆(X) := ∆n(X) n∈N is a subcomplex of the singular chain complex C(X). The homology

∆ M ∆ H (X) = Hn (X) n∈N of this subcomplex is called the simplicial homology of the ∆-complex X. Thus ∆ ∆ ∆ Hn (X) := Zn (X)/Bn (X) ∆ ∆ where Zn (X) := ∆n(X) ∩ Zn(X) and Bn (X) := ∂(∆n+1(X)). The inclusion ∆(X) → C(X) is a chain map and induces a homomorphism H∆(X) → H(X). Theorem 14.21 below asserts that this homomorphism is an isomorphism.

Example 14.2. The torus T 2 is the surface obtained from the unit square I2 with the identifications (x, 0) ∼ (x, 1) and (0, y) ∼ (1, y). The Klein bottle K2 is the surface obtained from the unit square I2 with the identifications (x, 0) ∼ (x, 1) and (0, y) ∼ (1, 1 − y). The projective plane P 2 is the surface obtained from the unit square I2 with the identifications (x, 0) ∼ (1 − x, 1) and (0, y) ∼ (1, 1 − y). For X = T 2,K2,P 2 there is a ∆-complex Φ : D → X as indicated in Figure 1. For X = T 2 and X = K2 the spaces D is the disjoint union of one 0-simplex v, three 1-simplices a, b, c, and two 2-simplices U and L. For X = P 2 we must adjoin another 0-simplex w to D In each case the bijections Φα is determined by the diagram in the only way possible so that the corresponding maps to I2 are affine and the arrows on the edges match up. Note that had we reversed the arrows marked a in P 2 the vertices of triangle U would be cyclically (not linearly) ordered. The figure would still represent the projective plane but not a ∆-complex. Proposition 14.3. The simplicial homology of the ∆-complex for the torus T 2 is ∆ 2 ∆ 2 2 ∆ 2 H2 (T ) = Z,H1 (T ) = Z ,H0 (T ) = Z ∆ 2 with Hn (T ) = 0 for n > 2. 2 2 2 Proof. The chain groups are ∆2(T ) = Z with generators U, L, ∆1(T ) = 3 2 Z with generators a, b, c, and ∆0(T ) = Z with generator v. The boundary operator is given by

∂U = b − c + a, ∂L = a − c + b, ∂a = ∂b = ∂c = 0.

43 v -b v v -b v w b v

U U U a a a ? a a ? a 6 c L 6 6 c L 6 c L v - v v - v v - w b b b T 2 K2 P 2

Figure 1: Three ∆-complexes

Using row and column operations we ﬁnd the Smith Normal Form for the matrix 2 2 representing ∂ : ∆2(T ) → ∆1(T ) is

 1 1   1 0  −1 B := P  1 1  Q =  0 0  −1 −1 0 0

2 2 and the matrix representing ∂ : ∆1(T ) → ∆0(T ) is of course

A := 0 0 0 .

∆ 2 In the notation of 11.14 ν := n − r − rank(A) = 3 − 1 − 0 = 2 so H1 (T ) = ν 2 ∆ 2 ∆ 2 Z ⊕ Z/1 = Z . From B2 = 0 and rank(B) = 1 we get H2 (T ) = Z2 (T ) = Z ∆ 2 and from A = 0 we get H0 (T2) = ∆0(T ) = Z. Proposition 14.4. The simplicial homology of the ∆-complex for the Klein bottle K2 is

∆ 2 ∆ 2 ∆ 2 H2 (K ) = 0,H1 (K ) = Z ⊕ (Z/2),H0 (K ) = Z

∆ 2 with Hn (P ) = 0 for n > 2. Proof. The chain groups are as for T 2 but the boundary operator is given by

∂U = b − c + a, ∂L = a − b + c, ∂a = ∂b = ∂c = 0.

Using row and column operations we ﬁnd the Smith Normal Form for the matrix 2 2 representing ∂ : ∆2(K ) → ∆1(K ) is

 1 1   1 0  −1 B := P  1 −1  Q =  0 2  −1 1 0 0

2 2 and the matrix representing ∂ : ∆1(K ) → ∆0(K ) is again A = 01×3. In ∆ 2 the notation of 11.14 ν := n − r − rank(A) = 3 − 2 − 0 = 1 so H1 (K ) = ν Z ⊕ (Z/1) ⊕ (Z/2) = Z ⊕ (Z/2). From B2 = 0 and rank(B) = 2 we get ∆ 2 ∆ 2 ∆ 2 2 H2 (K ) = Z2 (K ) = 0 and H0 (K ) = Z as for T .

44 Proposition 14.5. The simplicial homology of the ∆-complex for the projective plane P 2 is

∆ 2 ∆ 2 ∆ 2 H2 (P ) = 0,H1 (P ) = Z/2,H0 (P ) = Z ∆ 2 with Hn (P ) = 0 for n > 2. 2 2 3 Proof. The chain groups are ∆2(P ) = Z and ∆1(P2) = Z as before but 2 2 ∆0(P ) = Z with generators v and w. The boundary operator is given by ∂U = b − a + c, ∂L = a − b + c, ∂a = ∂b = w − v, ∂c = 0.

Using row and column operations we ﬁnd the Smith Normal Form for the matrix 2 2 representing ∂ : ∆2(T ) → ∆1(T ) is  1 1   1 0  −1 B := P  1 −1  Q =  0 2  −1 1 0 0

2 2 and the matrix representing ∂ : ∆1(P ) → ∆0(P ) is 1 1 0 A = . −1 −1 0

∆ 2 In the notation of 11.14 ν := n − r − rank(A) = 3 − 2 − 1 = 0 so H1 (P ) ≈ ν 2 ∆ 2 ∆ 2 Z ⊕ (Z/1) ⊕ (Z/2) = Z/2. As for K we have H2 (P ) = Z2 (P ) = 0. The Smith normal form for A is 1 1 0 1 0 0 M N −1 = −1 −1 0 0 0 0

∆ 2 2 2 ∆ 2 ∆ 2 so B0 (P ) ≈ Z × 0 ⊂ Z = ∆0(P ) =: Z0 (P ) and hence H0 (P ) = Z. 14.6. The standard n-simplex ∆n is itself a simplicial complex with a charac- k n teristic map Φα : ∆ → ∆ for each subset α = {α0, α1, . . . , αk} ⊂ {0, 1, . . . , n},

(α0 < α1 < ··· < αk), namely φα(x) = y where xi = yαi for i = 0, 1, . . . , k and n n yj = 0 for j∈ / α. The space ∆ is homeomorphic to the disk D , its boundary n−1 n Σ := {x ∈ ∆ : some xi 6= 0} is a subcomplex of ∆n homeomorphic to the sphere Sn−1. Proposition 14.7. The ∆-homology of ∆n is given by

∆ n ∆ n H0 (∆ ) = Z,Hk (∆ ) = 0 for k 6= 0. The ∆-homology of Σn−1 is given by

∆ n−1 ∆ n−1 ∆ n H0 (Σ ) = Hn−1(Σ ) = Z,Hk (∆ ) = 0 for k 6= 0, n. n n−1 The boundary ∂id∆n of the identity map of ∆ (viewed as an element of ∆n−1(Σ )) ∆ n−1 gives a generator of Hn−1(Σ ).

45 n n Proof. Deﬁne a chain homotopy Kk : ∆k(∆ ) → ∆k+1(∆ ) by ( Φ{0}∪α if 0 ∈/ α, Kk(Φα) = 0 if 0 ∈ α,

n−1 n−1 so Kk∂ +∂Kk+1 = id and ∂K0 = id−Φ{0}. Also K : ∆k(Σ ) → ∆k+1(Σ ) for k < n and the complexes ∆(∆n) and ∆(Σn−1) are the same except that n n−1 ∆n(∆ ) = Z and ∆n(Σ ) = 0. 0 14.8. Let {Φα : Dα → X}α∈Λ be a ∆-complex and ∆k(X) ⊂ Ck(X) be the subgroup of the nth singular chain group of X generated by all maps of form n Φα ◦ g where Φα : ∆ → X is a characteristic map of the ∆-complex X and k g : ∆ → ∆n is simplicial as in 13.6. Then

0 M 0 ∆ (X) := ∆n(X) n∈N is a subcomplex of the singular chain complex. We denote by

∆0 M ∆0 H (X) = Hn (X) n∈N the homology of this subcomplex. Thus

∆0 ∆0 ∆0 Hn (X) := Zn (X)/Bn (X)

∆0 0 ∆0 0 where Zn (X) := ∆n(X)∩Zn(X) and Bn (X) := ∂(∆n+1(X)). The chain map f# : C(X) → C(Y ) induced by a ∆-map f : X → Y satisﬁes

0 0 f#(∆ (X)) ⊂ ∆ (Y ).

0 Thus H∆ deﬁnes a functor from the category of ∆-complexes to the category of chain groups. By contrast, usually f# will not map ∆(X) to ∆(Y ) an so the operation H∆ is not obviously functorial. However H∆ has the advantage that it is much easier to compute since the group ∆n(X) has much lower rank than 0 the group ∆n(X).

14.9. For a ∆-complex the standard augmentation ε : C0(X) → Z of 12.3 restricts to augmentations of ∆(X) and ∆0(X). The corresponding reduced 0 chain groups are denoted by ∆(˜ X) and ∆˜ (X) so that ∆˜ n(X) = ∆n(X) and ˜ 0 0 ∆n(X) = ∆n(X) for n > 0 and ( ) ˜ ˜ 0 X X ∆0(X) = ∆0(X) = cxx : cx = 0

x∈X0 x∈X0 where the sums are understood to be ﬁnite even if X0 is inﬁnite. The corre- 0 sponding reduced homology groups are denoted H˜ ∆(X) and H˜ ∆ (X).

46 14.10. Let (X,A) be a ∆-pair and deﬁne the quotient chain complexes

0 0 0 C∆(X,A) := C∆(X)/C∆(A),C∆ (X,A) := C∆ (X)/C∆ (A). The homology groups of these complexes are denoted respectively H∆(X,A) 0 and H∆ (X,A). Theorem 14.11. Let X be a ∆-complex. Then the inclusion φ : ∆(X) → ∆0(X) is a chain homotopy equivalence. Proof. Equip the vertices of the standard simplex ∆n with the lexicographical ordering i.e. where vi denotes the vertex with 1 in the ith place and 0 elsewhere k n we have vi < vj ⇐⇒ i < j. An injective simplicial map g : ∆ → ∆ determines a unique simplicial automorphism σ of ∆k such that g ◦ σ is an order preserving embedding from the vertices of ∆k to the vertices of ∆n. Let sgn(σ) denote the sign of the corresponding permutation of the vertices. Deﬁne ψ : ∆0(X) → ∆(X) by ( sgn(σ)Φα(g ◦ σ) if g is injective ψ(Φα ◦ g) = 0 otherwise.

Then ψ is a chain map, ψ ◦ φ is chain homotopic to the identity of ∆(X), and φ ◦ ψ is chain homotopic to the identity of ∆0(X). See [7] page 77. Exercise 14.12. Let p be a point viewed as a ∆-complex in the only way 0 0 possible. Show that H∆(p) = H∆ (p) = 0 for n > 0 and H∆(p) = H∆ (p) = . 0 n n 0 0 Z Hence H˜ ∆(p) = H˜ ∆ (p) = 0. Exercise 14.13. Two ∆-maps f, g : X → Y are said to be ∆-homotopic iﬀ there is a ∆-map F : X×I → Y such that F (·, 0) = f and F (·, 1) = g. Here X×I is the product ∆-complex of 13.8. Show that If two ∆-maps f, g : X → Y are ∆- 0 0 homotopic, then the induced maps f#, g# : ∆ (X) → ∆ (Y ) are ∆-homotopic. Exercise 14.14. Let (X,A) be a ∆-pair, i : A → X denote the inclusion, and p : X → X/A denote the projection. Show that the sequence

i# p# 0 / ∆(˜ A) / ∆(˜ X) / ∆(˜ X/A) / 0 is exact. This gives a long exact sequence

i p i ∂ ˜ ∆ # ˜ ∆ # ˜ ∆ ∂ ˜ ∆ # ··· / Hk (A) / Hk (X) / Hk (X/A) / Hk−1(A) / ··· Exercise 14.15. Continue the notation of 14.14. The exact sequence

i# p# 0 / ∆(A) / ∆(X) / ∆(X)/∆(A) / 0 is exact and gives a long exact sequence

i p i ∂ ∆ # ∆ # ∆ ∂ ∆ # ··· / Hk (A) / Hk (X) / Hk (X,A) / Hk−1(A) / ···

47 Remark 14.16. For a ﬁnite zero-dimensional ∆-complex (i.e. a ﬁnite set) X the exact sequence of 14.15 is

#(A) #(X) #(X)−#(A) 0 → Z → Z → Z → 0 and the exact sequence of Theorem 14.14 is

#(A)−1 #(X)−1 #(X)−#(A) 0 → Z → Z → Z → 0. Exercise 14.17. Let (X,A) be a ∆-pair and Z ⊂ A be an open subset of X such that A \ Z is a subcomplex of A. Then X \ Z is a subcomplex of X and the inclusion (X \ Z,A \ Z) ⊂ (X,A) induces an isomorphism H∆(X \ Z,A \ Z) ≈ H∆(X,A). This might be called the Excision Theorem for simplicial homology. Exercise 14.18. In the situation of 14.17 the map (X \ Z)/(A \ Z) → X/A induced by the inclusion is a ∆-isomorphism and thus induces an isomorphism H˜ ∆(X \ Z)/(A \ Z) → H˜ ∆(X/A). Show that this is the same as the isomorphism obtained by combining the isomorphisms of 14.17 and 14.15. Exercise 14.19. Let A and B be subcomplexes of a ∆-complex X. Then A∩B is also a subcomplex. Show that there is an exact sequence

∆ ∆ ∆ ∆ ∆ · · · → Hn (A ∩ B) → Hn (A) ⊕ Hn (B) → Hn (X) → Hn−1(A ∩ B) → · · · This might be called the Mayer Vietoris Theorem for simplicial homology. Hint: The is sequence 0 → ∆(A) ∩ ∆(B) → ∆(A) ⊕ ∆(B) → ∆(X) → 0 is exact. The map ∆(A) ∩ ∆(B) = ∆(A ∪ B) → ∆(A) ⊕ ∆(B) is the direct sum of the inclusions and the map ∆(A) ⊕ ∆(B) → ∆(X) is the difference of the inclusions of the summands. Remark 14.20. For a finite zero-dimensional ∆-complex (i.e. a finite set) X the exact sequence of 14.19 is

#(A∩B) #(A) #(A) #(A∪B) 0 → Z → Z ⊕ Z → Z → 0 which shows that the Mayer Vietoris Theorem is a generalization of the Inclusion- Exclusion Principle of combinatorics. Theorem 14.21. Let X be a ∆-complex. Then the inclusion ∆(X) → C(X) induces an isomorphism H∆(X) → H(X) between simplicial homology and singular homology. Proof. This is proved for the k-skeleton X(k) by induction on k, the long exact sequences (in both homology theories) for the pair (X(k),X(k−1)), and the Five Lemma 10.14. See [3] Page 128.

48 15 Subdivision

15.1. For a ∆-pair (X,A) the sequence

0 → ∆(A) → ∆(X) → ∆(˜ X/A) → 0 is exact and induces a long exact sequence in simplicial homology. The analogous sequence 0 → C(A) → C(X) → C˜(X/A) → 0 2 1 in singular theory is not exact. For example, if X = I , A = { 2 } × I, and 1 1 σ ∈ C1(X/A) is deﬁned by σ(t) = (0, t) for t < 2 and σ(t) = (1, t) for t > 2 then σ does not lift to X. This nonexactness is the main reason why the relative homology groups H(X,A) are used in place of the reduced homology groups H˜ (X/A). Corollary 15.7 asserts that H(X,A) and H˜ (X/A) are isomorphic. The proof requires the following

Theorem 15.2. Let X be a topological space and U = {Uλ}λ∈Λ be a collection of subsets of X whose interiors cover X. Let CU (X) be the subcomplex of the singular chain complex of X generated by the singular simplices σ : ∆n → X n U such that σ(∆ ) ⊂ Uλ for some λ ∈ Λ and let H (X) denote the homology of this subcomplex. Then the inclusion CU (X) → C(X) induces an isomorphism HU (X) → H(X).

15.3. Fix a convex subset K of Rm. Denote by LC(K) the subcomplex of the singular chain complex C(K) generated by all singular simplices σ : ∆n → Rm of form n X σ(t0, t1, . . . , tn) = tiwi i=0 where w0, w1, . . . , wn ∈ K. The singular simplex σ is called the linear simplex with vertices w0, w1, . . . , wn. The elements of LC(K) are called linear singular chains. Each point b ∈ K determines a cone operator b : Cn(K) → Cn+1(K) denoted by the same symbol via the formula

n X b(σ)(s−1, s0, s1, . . . , sn) = s−1b + siwi. i=0 For each linear n-simplex σ the point

n 1 X b := w σ n + 1 i i=0 is called the barycenter of σ. The map S : LCn(K) → LCn(K) deﬁned inductively by S([w]) = [w],S(σ) = bσ(S∂σ) is called linear barycentric subdivision.

49 Lemma 15.4. The map S is a chain map and is chain homotopic to the identity, i.e. there are maps T : LCn(K) → LCn+1(K) such that ∂T + T ∂ = id − S.

Proof. See [3] page 121-2. 15.5. We now specialize to the K = ∆n the standard n-simplex. Denote the n I vertices of ∆ of v0, v1, . . . , vn. For a set I ⊂ {0, 1, . . . , n} of indices let ∆ denote the convex hull of {vi : i ∈ I}. Each permutation α of {0, 1, . . . , n} de- n termines a linear simplex σα ∈ LCn(∆ ) whose kth vertex wk is the barycenter I n of ∆ k where Ik = {α(0), . . . , α(k)}. Let idn denote the identity map of ∆ viewed as a linear simplex. Then [ idn = εασα α where εα = ±1; the choice of these signs assures that the internal faces cancel. In n S n n n particular, ∆ = α σα(∆ ) and the interiors (in ∆ ) of the simplices σα(∆ ) are pairwise disjoint. The chain chain homotopy T is determined by the formula ! X T (idn) = p# τ + εατα α where p : ∆n × I → ∆n is projection on the ﬁrst factor and τ : ∆n+1 → ∆n × I are linear simplex whose vertices are (b, 1) [here b is the barycenter of ∆n] and n+1 n (vi, 0) for i = 0, . . . , n and where τα : ∆ → ∆ × I is the linear simplex whose vertices are (vα(0), 0) and (vα(i), 1) for i = 0, . . . , n. See [3] page 122. Proof of Theorem 15.2. Let X be any topological space. Deﬁne a chain map S : Cn(X) → Cn(X) and a chain homotopy T : Cn(X) → Cn+1(X) by

S(σ) = σ#(S(idn)),T (σ) = σ#(T (idn)). The formula ∂T + T ∂ = id − S holds so S is chain homotopic to the identity id. By induction and the fact that compositions of chain homotopic maps are chain homotopic we have that the mth iterate Sm of S is chain homotopic to the the identity, in fact

m ∂Dm + Dm∂ = id − S

P i where Dm = i

H(X \ Z,A \ Z) ≈ H(X,A) of the relative homology groups.

50 Proof. Let U = {X \ Z,A}. The map C(X \ Z)/C(A \ Z) → CU (X)/C(A) is an isomorphism and the map CU (X)/C(A) → C(X)/C(A) induces an isomorphisms in homology by Theorem 15.2. Hence

H(X \ Z,A \ Z) ≈ HU (X,A) ≈ H(X,A) where HU (X,A) denotes the homology of CU (X)/C(A). See page 124 of [3]. Corollary 15.7. Assume that (X,A) is a nice pair, i.e. that A has a neighborhood V in X which deformation retracts onto A. Then the projection

(X,A) → (X/A, A/A) induces an isomorphism is relative homology and hence

H(X,A) ≈ H(X/A, A/A) ≈ H˜ (X/A).

Proof. In the commutative diagram

H(X,A) / H(X,V ) o H(X \ A, V \ A)

H(X/A, A/A) / H(X/A, V/A) o H(X/A \ A/A, V/A \ A/A) the horizontal maps on the left are isomorphisms by homotopy, the horizontal maps on the right are isomorphisms by excision, the vertical map on the right is an isomorphism as it is induced by a homeomorphism, and hence the other two vertical maps are isomorphisms. See Proposition 2.22 page 124 of [3]. Corollary 15.8 (Mayer Vietoris). Let X be any topological space and A, B ⊂ X be two subsets whose interiors cover X. Then there is an exact sequence

∂∗ · · · → Hn(A ∩ B) → Hn(A) ⊕ Hn(B) → Hn(X) / Hn−1(A ∩ B) → · · · where the map H(A∩B) → H(A)⊕H(B) is the direct sum of the maps induced by the inclusions A ∩ B → A and A ∩ B → A, the map H(A) ⊕ H(B) → H(X) is the difference of the maps induced by the inclusions A → X and B → X, and the boundary operator ∂∗ is defined in the proof. Proof. The collection U = {A, B} satisfies the hypothesis of Theorem 15.2, and

0 → C(A ∩ B) → C(A) ⊕ C(B) → CU (X) → 0 is an exact sequence of chain complexes.

51 16 Cellular Homology

n Deﬁnition 16.1. By Proposition 14.7 and Theorem 14.21 we have Hn(S ) = Z. Hence each continuous map f : Sn → Sn determines an integer deg(f) called the degree of f via the equation

n n f∗[S ] = deg(f)[S ]

n n where [S ] is a generator of Hn(S ).

16.2. Now let Φ = {Φα : Dα → X}α∈Λ be a cell complex, nα be the dimension of the disk Dα, Let M Cn(Φ) := ZΦα nα=n be the free Abelian group generated by the n-dimensional cells of the complex. For each pair of indices α, β ∈ Λ with nβ = nα − 1 = n − 1 deﬁne

n−1 n−1 n−2 n−1 φαβ : ∂Dα = S → Dβ/∂Dβ = D /S ≈ S

−1 by φαβ = Φβ ◦ qβ ◦ φα where φα = Φα|∂Dα is the attaching map, qβ : (n−1) (n−2) (n−2) X /X → e¯β/X is the identity on eβ and sends the complement −1 of eβ to the wedge point, and Φβ is induced by the inverse of the characteristic map. Deﬁne ∂ : Cn(Φ) → Cn−1(Φ) by X ∂Φα = deg(φαβ)Φβ β

Theorem 16.3. The homomorphism ∂ : C(Φ) → C(Φ) is a chain complex, i.e. ∂2 = 0. The homology H(Φ) of this chain complex is isomorphic to the singular homology H(X) of the space X. Proof. See [3] pages 137-141. An important point is that for each n characteristic map Φ induces a homeomorphism

_ (n) (n−1) Sα ≈ X /X ,Sα := Dα/∂Dα

nα=n from a wedge of n-spheres to the quotient of the n skeleton by the (n − 1)- skeleton. and that homeomorphism in turn induces an isomorphism

(n) (n−1) Cn(Φ) ' Hn(X /X ) of Abelian groups. Example 16.4. The standard representation P/ ∼ of the compact orientable surface Mg of genus g (see part II of Remark 3.3) gives a cell complex structure Φ whose cellular chain complex is

0 0 0 / Z / Z2g / Z / 0

52 so the homology is

2g H0(Mg) = Z,H1(Mg) = Z ,H1(Mg) = Z. See Example 2.36 page 141 of [3]. Example 16.5. The standard representation P/ ∼ of the compact nonori- entable surface Ng of genus g (see part III of Remark 3.3) gives a cell complex structure Φ whose cellular chain complex is

∂ 0 0 / Z / Z2g / Z / 0 where ∂(1) = (2, 2,..., 2) so the homology is

g−1 H0(Ng) = Z,H1(Ng) = Z ⊕ Z/2,H2(Ng) = 0. See Example 2.37 page 141 of [3].

Example 16.6. The 3-torus T 3 = S1 × S1 × S1 may be viewed as a cube with opposite faces identiﬁed. This gives a cell complex structure Φ whose cellular chain complex is

0 0 0 0 / Z / Z3 / Z3 / Z / 0 so the homology is

3 3 3 3 3 3 H0(T ) = Z,H1(T ) = Z ,H3(T ) = Z ,H3(T ) = Z. See Example 2.39 page 142 of [3]. Example 16.7. The Moore Space See Example 2.40 page 143 of [3]. Example 16.8. The real projective space RP n See Example 2.41 page 144 of [3]. Example 16.9. The complex projective space CP n

Example 16.10. The lens space Lm(p, q) See Example 2.42 page 144 of [3].

A Abstract Simplicial Complexes

A.1. An abstract simplicial complex is a collection K of nonempty ﬁnite sets called simplices such that every nonempty subset of a simplex is a simplex, i.e. ∅= 6 τ ⊂ σ ∈ K =⇒ τ ∈ K. A subset of a simplex is called a face of the simplex. The elements of the set [ V := σ σ∈K

53 are called vertices. Since v ∈ V ⇐⇒ {v} ∈ K It is customary not to distinguish between v ∈ V and {v} ∈ K. A subset of K that is itself a simplicial complex is called a subcomplex. If K1 and K2 are simplicial complexes with vertex sets V1 and V2, a map f : K1 → K2 is called a simplicial map if there is a (necessarily unique) map f : V1 → V2 denoted by the same letter such that

f({v0, v1, . . . , vk}) = {f(v0), f(v1), . . . , f(vk)} for {v0, v1, . . . , vk} ∈ K1. Abstract simplicial complexes and simplicial maps form the objects and morphisms of a category. A.2. The dimension dim(σ) of a simplex σ is one less than its cardinality. A simplex of dimension k is also called a k-simplex. The subcomplex K(k) := {σ ∈ K : dim(σ) ≤ k} is called the k-skeleton of the simplicial complex K. As noted above we may denote the set of vertices by V = K(0). A simplicial complex is called n-dimensional iff K(n) = K and K(n−1) 6= K; it is called finite dimensional iff it is n-dimensional for some n ∈ N. A simplicial complex is called finite if the set K is itself finite. A.3. The geometric realization of an abstract simplicial complex K is the set ( ) V X |K| := x ∈ [0, 1] x(v) = 1 and supp(x) ∈ K . v∈V Here V is the vertex set of K and supp(x) := {v ∈ V : x(v) 6= 0} is the support of the element x ∈ |K|. A simplicial map f : K → L induces a map f : |K| → |L| called the geometric realization of f via the formula X f(x)(w) = x(v) f(v)=w for x ∈ |K|. Geometric realization is functorial. Theorem A.4. Assume that K is a finite abstract simplicial complex of dimension n. Let E = R2n+1 denote the Euclidean space of dimension 2n + 1 and EV denote the space of all maps from f : V → E. (The space EV has dimension V #(V )·(2n+1).) Call an element f ∈ E generic iff Whenever v0, v1, . . . , vk are distinct points of V and k ≤ 2n + 1 the vectors f(v1) − f(v0), . . . , f(vk) − f(v0) are linearly independent. Then (i) The set of generic elements is open and dense in EV . (ii) If f is generic, the induced map f : |K| → E defined by X f(x) := x(v)f(v) v∈V is injective.

54 (iii) The topology on |K| induced by this embedding is independent of the choice of the generic element f used to deﬁne it. (iv) In this topology the geometric realization of a simplicial map between ﬁnite complexes is continuous.

Proof. For a sequence W = (v0, v1, . . . , vk) of distinct elements of V of form the (2n+1)×k matrix AW (f) ∈ R whose columns are the k vectors f(vi) − f(v0) for R i = 1, . . . , k. For each k element subset R ⊂ {1,..., 2n + 1} let aW (f) be the determinant of the square matrix that results from AW (f) by deleting the rows whose index is not in R. Then f is generic if and only if for every W there is R R an R such that aW (f) 6= 0. Since aW (f) is a polynomial in the entries of f it cannot vanish identically on an open subset of f’s. This proves (i). Since |K| is compact, the map f : |K| → E is and embedding if and only if it is injective. But a generic map must be injective: if f(x) = f(y) then (as the supports of x and y are of cardinality at most n + 1) the union W of the supports of x and y has cardinality at most 2n+2. But the equation f(x) = f(y) is a linear relation between the vectors {f(v): v ∈ W }. As these vectors are independent we must have x = y as required. Remark A.5. The induced map f : |K| → E can be injective even when f is not generic. For countable K of dimension n, item (i) of Theorem A.4 continues to hold provided open and dense is weakened to residual (countable intersection of open dense sets) so generic maps f exist by the Baire Category Theorem. Item (ii) will hold but item (iv) is false. As a subset of [0, 1]V the geometric realization inherits a topology from the product topology of [0, 1]V . In this topology a set A ⊂ |K| is closed if and only if A ∩ |σ| is closed for every σ ∈ K, but (in the inﬁnite case) this topology can be diﬀerent from the one induced by an embedding in E. A.6. An ordered simplicial complex is a simplicial complex equipped with a linear ordering for each simplex in such a way that if τ is a face of σ then the ordering assigned to τ is the restriction of the ordering assigned to σ.A simplicial complex may be ordered in many ways. Generally we will choose an ordering for book keeping purposes and then prove that the choice doesn’t matter. Remark A.7. The geometric realization |K| of an ordered simplicial complex is a ∆-complex as follows. Let

G n D := ∆σ σ∈K

n n be a disjoint union of standard simplices with one copy of ∆σ := ∆ for every n n-simplex σ ∈ K and let Φσ : ∆ → |σ| be deﬁned by

Φσ(x) = x0δv0 + x1δv1 + ··· xnδvn

n for x = (x0, x1, . . . , xn) ∈ ∆ and σ = {v0, v1, . . . , vn} where v0 < v1 < ··· vn.

55 Problem A.8. An abstract unordered ∆-complex is a disjoint union G W = σα, α∈Λ of nonempty ﬁnite subsets together with a function which assigns to each pair (w, α) with #(σα) > 1 w ∈ σα a bijection ι : σβ → σ \{w}. (The index β ∈ Λ is also a function of (α, w).) The elements of W are called vertices, the sets σα are called simplices, and the maps ι are called attaching maps. An abstract ordered ∆-complex is an abstract unordered ∆-complex together with a linear ordering for each simplex σα such that each attaching map ι is the unique order isomorphism from its domain to its target. Does every abstract unordered ∆- complex arise from an abstract ordered ∆-complex? (In other words given an abstract unordered ∆-complex can we order its simplices so that for each (w, α) the bijection ι is the unique order isomorphism?) Remark A.9. An abstract ∆-complex determines a cell complex Φ : D → X where G D := |σα| α∈Λ is the disjoint union of the geometric realizations of the simplices, X = D/∼, and Φ : D → X is the projection into the identiﬁcation space. The equivalence relation is generated by X X x = xvδv ∈ |σβ| y = xvδι(v) ∈ |σα| =⇒ x ∼ y. v∈β v∈β

This cell complex is called the geometric realization of the abstract ∆- complex.

References

[1] Charles Conley: Isolated Inveariant Sets and the Morse Index, CBMS Con- ferences in Math 38, AMS 1976. [2] William Fulton: Algebraic Topology, A First Course Springer GTM 153, 1995. [3] Allen Hatcher: Algebraic Topology, Cambridge U. Press, 2002. [4] John G. Hocking & Gail S. Young: Topology, Addisow Wesley, 1961. [5] Morris W. Hirsch: Diﬀerential Topology, Springer GTM 33, 1976. [6] William S. Massey: A Basic Course in Algebraic Topology, Springer GTM 127, 1991. [7] James R. Munkres: Elements of Algebraic Topology, Addison Wesley, 1984.

56 [8] S. Shelah: Can the fundamental group of a space be the rationals? Proc. Amer. Math. Soc. 103 (1988) 627–632. [9] Stephen Smale: A Vietoris mapping theorem for homotopy. Proc. Amer. Math. Soc. 8 (1957), 604–610 [10] J. W. Vick: Homology Theory (2nd edition), Graduate Tests in Math 145, Springer 1994. [11] J.H.C. Whitehead: Combinatorial homotopy II, Bull. A.M.S 55 (1949) 453-496. [12] George W. Whitehead: Elements of Homotopy Theory Graduate Tests in Math 61, Springer 1978.