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An Introduction to Calculus of Functors

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An Introduction to Calculus of Functors An introduction to calculus of functors Ismar Voli´c Wellesley College International University of Sarajevo May 28, 2012 Plan of talk Main point: One can use calculus of functors to answer questions about embedding spaces, and in particular about knots and links. Outline: 1 Generalities about algebraic topology 2 Categories and functors 3 Calculus of functors and embedding spaces 4 Applications to knots and links 1. Algebraic topology Algebraic topology attempts to classify topological spaces up to some equivalence, such as homeomorphism: Spaces X and Y are homeomorphic if there exists a map (continuous function) f : X → Y which has a continuous inverse; or homotopy equivalence: Spaces X and Y are homotopy equivalent if there exist maps f : X → Y and g : Y → X such that f ◦ g and g ◦ f are homotopic to identity maps. (Essentially, X can be deformed into Y .) homeomorphism =⇒ homotopy equivalence The goal is to find algebraic invariants for spaces, i.e. assign algebraic objects to spaces such that, if two spaces are equivalent, the algebraic objects are isomorphic. 1. Algebraic topology The most basic invariants are: πk (X ), the kth homotopy group of X ; Hk (X ), the kth (singular) homology group of X ; k H (X ), the kth (singular) cohomology group of X ; (k ≥ 0) Homology is hardest to define but easiest to compute. Cohomology is dual to homology, but it has more structure and is hence harder to compute. Homotopy is easy to define: k πk (X )= {maps S → X modulo the relation of homotopy}, but it is the hardest to compute. There are other equivalent versions of singular (co)homology: simplicial homology, cellular homology, deRham cohomology (for manifolds), Cechˇ (or sheaf) cohomology (for manifolds), etc. 1. Algebraic topology Example (Euclidean space Rn) n k n Z, k = 0 Hk (R ) = H (R )= (0, k 6= 0 n πk (R ) = 0 ∀k Example (Sphere Sn) n k n Z, k = 0, n Hk (S ) = H (S )= (0, k 6= 0, n n πk (S ) = unknown in general 1. Algebraic topology Example (Torus T 2 = S 1 × S 1) 2 k 2 Z, k =0, k =2 Hk (T )=H (T )= (Z × Z, k =1 2 0, k =0, k > 2 πk (T )= (Z × Z, k =1 Example (Real projective space RPn) Z, k =0, k = n and n odd n Hk (RP )= Z/2Z, 0 < k < n, k odd 0, otherwise 0, k =0, n Z, k =1, n =1 πk (RP )= Z/2Z, k =1, n > 1 n πk (S ), k > 1, n > 0 (can get cohomology usingPoincar´eduality) 2. Categories and functors k πk , Hk , and H not only assign a group to any space X , but also assign a map of groups (a homomorphism) to each map of spaces. This feature allows us to transfer comparisons between spaces (which is what maps in topology do) to comparisons between groups. More precisely, there is a commutative diagram f X / Y πk πk πk (f ) πk (X ) / πk (Y ) k Same for Hk , but for H , the bottom arrow is reversed: f X / Y k k H H k f k H ( ) k H (X )o H (Y ) 2. Categories and functors k A way to rephrase the previous slide is to say that πk , Hk , and H are functors from the category of topological spaces to the category of groups. More precisely, we have Definition A category C consists of a class of objects Ob(C), and; for any two objects X and Y , a class of morphisms (or maps, or arrows) HomC(X , Y ). Write f : X → Y for an element f ∈ HomC(X , Y ). Further, there is a composition function ◦: HomC(X , Y ) × HomC(Y , Z) −→ HomC (X , Z) (f , g) 7−→ g ◦ f satisfying associativity: h ◦ (g ◦ f ) = (h ◦ g) ◦ f , and; identity: For each X , there exists an element IdX ∈ HomC(X , X ) satisfying, for each f : X → Y , f ◦ IdX = f = IdY ◦ f . 2. Categories and functors Examples • Top: Ob(Top) =topologicalspaces HomTop = continuous functions • Set: Ob(Set)=sets HomSet = functions • Grp: Ob(Grp) =groups(orabeliangroups) HomGrp = homomorphisms • Ring: Ob(Ring) =rings HomRing = ring homomorphisms • Mfld: Ob(Mfld) =(differentiable) manifolds HomMfld = (differentiable) maps • Vect/F: Ob(Vect/F) = vector spaces over a field F HomVect/F = F -linear functions 2. Categories and functors A map between categories is a functor: Definition A covariant functor F : C → D between categories C and D is a mapping that associates an object F (X ) ∈ D to each object X ∈C, and; associates a morphism F (f ): F (X ) → F (Y ) in D to each morphism f : X → Y in C, satisfying F (IdX )= IdF (X ), and; F (g ◦ f )= F (g) ◦ F (f ). A contravariant functor is the same except it reverses arrows, that is, it assigns a morphism F (f ): F (Y ) → F (X ) in D to each morphism f : X → Y in C. 2. Categories and functors Examples k πk , Hk , H : Top −→ Grp k k X 7−→ πk (X ), Hk (X ), H (X ) (H is contravariant) F : Set −→ Set S 7−→ P(S), the power set of S F : Grp −→ Set G 7−→ G, as the underlying set F : Set −→ Grp S 7−→ Z[S], the free group on S F : Vect/F −→ Vect/F ∗ V 7−→ V , the dual vector space (contravariant) (should say where morphisms are sent too, but this is not hard in each case) 2. Categories and functors Various concepts can be studied with or expressed as functors: partially ordered sets, (pre)sheaves, tangent and cotangent bundles, groups actions and representations, Lie algebras, tensor products, limits and colimits, etc. In general, category theory is helpful in two ways: It organizes, frames, and contextualizes information so it can be managed easier, and One can often prove statements in the abstract setting of categories and functors and then apply them to particular situations. One example of where this occurs is calculus of functors. 3. Calculus of functors Calculus of functors is a theory that aims to “approximate” functors in algebra in topology much like the Taylor polynomials approximate ordinary smooth real or complex-valued functions. In general, given a functor F : C −→ D this theory gives a “Taylor tower” of approximating functors: l F IWWW ll II WWWW lll II WWWWW lll II WWWWW lll II WWWWW v lll I$ WWWW ol o o o o WW+ T0F · · · Tk F Tk+1F · · · T∞F Depending on F , this tower might converge, i.e. there is an equivalence, for all X ∈C, X ≃ T∞F (X ). 3. Calculus of functors There are currently three varieties of functor calculus: Homotopy calculus Orthogonal calculus Manifold calculus Each is designed to study different kinds of functors. Here we are interested in manifold calculus: Given a smooth manifold M, its open subsets form a category O(M) with inclusions maps as morphisms. Manifold calculus then studies contravariant functors F : O(M) −→ Top The main example of such a functor is the space of embeddings: 3. Calculus of functors and embeddings Definition Let M and N be smooth manifolds. An embedding of M in N is an injective map f : M ֒→ N whose derivative is injective and which is a homeomorphism onto its image. When M is compact, an embedding is an injective map with the injective derivative. The set of all embeddings of M in N can be topologized so we get the space of embeddings Emb(M, N) (a special case of a mapping space). For many M and N, this is a topologically interesting space, so we want to know ∗ π∗(Emb(M, N)), H∗(Emb(M, N)), H (Emb(M, N)). What is especially interesting is the case of knots and links, as we will see later. 3. Calculus of functors and embeddings But Emb(M, N) can also be thought of as a functor on O(M) given by Emb(−, N): O(M) −→ Top O 7−→ Emb(O, N) This is contravariant since, given an inclusion O1 ֒→ O2 of open subsets of M, there is a restriction Emb(O2, N) → Emb(O1, N). Manifold calculus thus applies to the functor Emb(−, N) and we get a Taylor tower Emb(−, N) ZZZZ hhh RRR ZZZZZZZ hhhh RRR ZZZZZZ hhh RRR ZZZZZZZ hhhh RRR ZZZZZZ ht hh R) ZZZZZZZ, o o o o o T0 Emb(−, N) · · · Tk Emb(−, N) Tk+1 Emb(−, N) · · · T∞ Emb(−, N) 3. Calculus of functors and embeddings Theorem (Goodwillie-Klein-Weiss) The Taylor tower for Emb(−, N) converges under certain dimensional assumptions. Namely, given O ∈ O(M), the map Emb(O, N) −→ T∞ Emb(O, N) induces isomorphisms on πk , k ≥ 0, if dim(M) + 3 ≤ dim(N), and k on Hk and H , k ≥ 0, if 4dim(M) ≤ dim(N). In practice, we set O = M to extract information about the space of embeddings of the entire manifold (so functoriality is used in proving the statement, but we then specialize). Let’s look at knots and links: 4. Applications to knots and links n n {K = {embeddings K : R ֒→ R fixed outside a compact set = space of long knots K ∈Kn When n = 3, get classical knot theory, which cares about 3 H0(K )= {connected components of the space of knots} = {knot types} = {isotopy classes of knots} 0 3 3 H (K )= {knot invariants f : H0(K ) → R}, However, higher (co)homology and homotopy are also interesting, 0 even when n > 3 (even though H and H0 are trivial in this case). Note that the Taylor tower does not converge for K3, but it still contains lots of information: 4.
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