An introduction to calculus of functors Ismar Volić 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: (k 0) π k (X), the kth homotopy group of X; H k (X), the kth (singular) homology group of X; H k (X), the kth (singular) cohomology group of X; 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: π k (X) = {maps S k 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), Čech (or sheaf) cohomology (for manifolds), etc.
1. Algebraic topology Example (Euclidean space R n ) H k (R n ) = H k (R n ) = π k (R n ) = 0 { Z, k = 0 0, k 0 k Example (Sphere S n ) H k (S n ) = H k (S n ) = { Z, k = 0,n 0, k 0,n π k (S n ) = unknown in general
1. Algebraic topology Example (Torus T 2 = S 1 S 1 ) { H k (T 2 ) = H k (T 2 Z, k = 0,k = 2 ) = Z Z, k = 1 { π k (T 2 0, k = 0,k > 2 ) = Z Z, k = 1 Example (Real projective space RP n ) Z, k = 0,k = n and n odd H k (RP n ) = Z/2Z, 0 < k < n,k odd 0, otherwise 0, k = 0, π k (RP n Z, k = 1,n = 1 ) = Z/2Z, k = 1,n > 1 π k (S n ), k > 1,n > 0 (can get cohomology using Poincaré duality)
2. Categories and functors π k, H k, and H k 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 X f Y π k π k π k (X) π k(f) π k (Y) Same for H k, but for H k, the bottom arrow is reversed: X f Y H k H k (X) H k (f) H k H k (Y)
2. Categories and functors A way to rephrase the previous slide is to say that π k, H k, and H k 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) Hom C (X,Y). Write f : X Y for an element f Hom C (X,Y). Further, there is a composition function : Hom C (X,Y) Hom C (Y,Z) Hom C (X,Z) (f,g) g f satisfying associativity: h (g f) = (h g) f, and; identity: For each X, there exists an element Id X Hom C (X,X) satisfying, for each f : X Y, f Id X = f = Id Y f.
2. Categories and functors Examples Top: Ob(Top) = topological spaces Hom Top = continuous functions Set: Grp: Ring: Mfld: Vect/F: Ob(Set) = sets Hom Set = functions Ob(Grp) = groups (or abelian groups) Hom Grp = homomorphisms Ob(Ring) = rings Hom Ring = ring homomorphisms Ob(Mfld) = (differentiable) manifolds Hom Mfld = (differentiable) maps Ob(Vect/F) = vector spaces over a field F Hom Vect/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(Id X ) = Id F(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 π k,h k,h k : Top Grp X π k (X),H k (X),H k (X) (H k is contravariant) F: Set Set S P(S), the power set of S F : Grp Set G G, as the underlying set F: Set Grp S Z[S], the free group on S F : Vect/F Vect/F V 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: F T 0 F T k F T k+1 F 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 Emb(O,N) This is contravariant since, given an inclusion O 1 O 2 of open subsets of M, there is a restriction Emb(O 2,N) Emb(O 1,N). Manifold calculus thus applies to the functor Emb(, N) and we get a Taylor tower Emb(, N) T 0 Emb(,N) T k Emb(,N) T k+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 induces isomorphisms Emb(O,N) T Emb(O,N) on π k, k 0, if dim(m)+3 dim(n), and on H k and H k, 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 K n = {embeddings K: R R n fixed outside a compact set} = space of long knots K K n When n = 3, get classical knot theory, which cares about H 0 (K 3 ) = {connected components of the space of knots} = {knot types} = {isotopy classes of knots} H 0 (K 3 ) = {knot invariants f : H 0 (K 3 ) R}, However, higher (co)homology and homotopy are also interesting, even when n > 3 (even though H 0 and H 0 are trivial in this case). Note that the Taylor tower does not converge for K 3, but it still contains lots of information:
4. Applications to knots and links If one knot can be deformed (isotoped) into another, an invariant f H 0 (K 3 ) takes on the same value on those knots. But an invariant does not have to take on different values for different knots. In fact, we do not know if such an invariant or a class of invariants a complete set of invariants that can tell all knots apart exists. Conjecture The set of finite type k invariants, k 0, is a complete set of invariants. Finite type invariants have received much attention in the last 15 years: Motivated by physics (Chern-Simons Theory); Connected to Lie algebras, three-manifold topology, etc.; They have a combinatorial interpretation via chord diagrams (Kontsevich Integral).
4. Applications to knots and links Theorem (V.) The Taylor tower for K 3 classifies finite type knot invariants. More precisely, for each k 0, there is an isomorphism (over R) H 0 (T 2k K 3 ) = {finite type k invariants} H 0 (K 3 ). (And H 0 (T 2k K 3 ) = H 0 (T 2k+1 K 3 ).) Main ingredient in the proof: Configuration space integrals. This theorem puts finite type theory into a more topological setting (prior to this, finite type invariants were studied using physics-like techniques); gives a new explanation of appearance of chord diagrams in the theory via cohomology of configuration spaces (prior to this, chord diagrams were thought of as Feynman diagrams from physics).
4. Applications to knots and links Manifold calculus of functors can also be used for studying the topology of K n, n > 3. This is where the convergence theorem holds. We have Theorem (Lambrechts-Turchin-V.) For n > 3, the rational (co)homology of K n can be completely described using (co)homology of configuration spaces in R n (spaces of distinct points in R n ). Main ingredients in the proof: Calculus of functors; Kontsevich s rational formality of the little n-cubes operad. (Co)homology of configuration spaces can be expressed with chord diagrams, so we get a combinatorial description of H k (K n ) and H k (K n ) (over Q) for n > 3. Have similar results for the rational homotopy of K n, n > 3.
4. Applications to knots and links Can generalize to link spaces. Let n 3 and m 1. Define L n m = {embeddings m R R n } = space of long (string) links Hm n = {link maps m R R n } = space of homotopy long (string) links Bm n = {embeddings with positive derivative m R R n } = space of pure braids All maps are standard outside a compact set; A link map is a smooth map with images of the copies of R disjoint. Can use multivariable manifold calculus of functors (Munson-V.) to study these spaces.
4. Applications to knots and links Example B n m Ln m Hn m ; On the level of components of Hm n, can pass a strand through itself so this can be thought of as space of links without knotting. H H n 3 K K n L B n 3 Ln 3 Hn 3
4. Applications to knots and links Theorem (Munson-V.) Taylor towers for L >3 m and B m >3 converge Taylor towers for L >3 m, H m >3 and B m >3 classify finite type invariants. Conjecture Taylor tower for H m >3 converges; Taylor towers lead to a combinatorial description of rational (co)homology and homotopy of L >3 m, H m >3, and B m >3 (last one is already understood); Can reprove, in the setting of Taylor towers, that finite type invariants form a complete set of invariants for braids (Kohno, Bar-Natan) and homotopy string links (Habegger-Lin); Can use the above in proving the same result for knots and links.
Thank you!