This is a repository copy of The homotopy theory of Khovanov homology.

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1 This is a repository copy o The homotopy theory o Khovanov homology. White Rose Research Online URL or this paper: Version: Published Version Article: Everitt, Brent and Turner, Paul (204) The homotopy theory o Khovanov homology. Algebraic and Geometric Topology. pp ISSN Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed or private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow urther reproduction and re-use o the ull text version. This is indicated by the licence inormation on the White Rose Research Online record or the item. Takedown I you consider content in White Rose Research Online to be in breach o UK law, please notiy us by ing eprints@whiterose.ac.uk including the URL o the record and the reason or the withdrawal request. eprints@whiterose.ac.uk

2 The homotopy theory o Khovanov homology BRENT EVERITT PAUL TURNER We show that the unnormalised Khovanov homology o an oriented link can be identiied with the derived unctors o the inverse limit. This leads to a homotopy theoretic interpretation o Khovanov homology. Motivation and introduction In order to apply the methods o homotopy theory to Khovanov homology there are several natural approaches. One is to build a space or spectrum whose classical invariants give Khovanov homology, then show its homotopy type is a link invariant, and inally study this space using homotopy theory. Ideally this approach would begin with some interesting geometry and lead naturally to Khovanov homology. One also might hope to construct something more reined than Khovanov homology in this way (see Lipshitz-Sarkar [2] or a combinatorial approach to this). Another approach is to interpret the existing constructions o Khovanov homology in homotopy theoretic terms. By placing the constructions into a homotopy setting one makes Khovanov homology amenable to the methods and techniques o homotopy theory. In this paper our interest is with the second o these approaches. Our aim is to show that Khovanov homology can be interpreted in a homotopy theoretic way using homotopy limits and to subsequently develop a number o results about the speciic type o homotopy limit arising. The latter will provide homotopy tools appropriate or studying Khovanov homology. Recall that the central combinatorial input or Khovanov homology is the decorated cube o resolutions based on a link diagram D (see Section.). As we explain later, it is convenient to view this cube as a preshea o abelian groups over a certain poset Q, that is, as a unctor F KH : Q op Ab. In the irst section we show that Khovanov homology can be described in terms o the right derived unctors o the inverse limit o this preshea. Published: xxember 20 DOI: 0.240/agt

3 002 Brent Everitt and Paul Turner Theorem.3. Let D be a link diagram and let F KH : Q op Ab be the Khovanov preshea deined in.. Then, KH i (D) = lim Q op i F KH On the let we have singly graded unnormalised Khovanov homology (see Section.) while on the right we have the i-th derived unctor o the inverse limit (see Section.2). This result is central to the homotopy theoretic interpretation o Khovanov homology but is also o independent interest: many cohomology theories are deined as the right derived unctors o some interesting partially exact unctor, or at least can be described in such terms. Examples include group cohomology, shea cohomology and Hochschild cohomology. Obtaining a description in these terms or Khovanov homology reveals its similarity to existing theories not apparent rom the original deinition. Moreover it opens up Khovanov homology to the many techniques available to cohomology theories deined as right derived unctors. Also the construction given in this paper is unctorial with respect to morphisms o presheaves, which being more general, may oer calculational advantage. By connecting with a more amiliar description o higher derived unctors we also obtain a description o Khovanov homology as the cohomology o the classiying space equipped with a system o local coeicients as described in Proposition.3. Right derived unctors o a preshea o abelian groups can be interpreted in homotopy theoretic terms by way o the homotopy limit o the corresponding diagram o Eilenberg-Mac Lane spaces. In the second section we recall basic acts about homotopy limits beore returning to Khovanov homology. We compose the Eilenberg-Mac Lane space unctor K(, n) with the Khovanov preshea F KH o a link diagram to obtain a diagram o spaces F n : Q op Sp whose homotopy limit Y n D = holim Q op F n has homotopy groups described in the ollowing proposition. Proposition 2.8. π i (Y n D) = { KH n i (D) 0 i n, 0 else. For rather elementary reasons the space Y n D is seen to be a product o Eilenberg-Mac Lane spaces and thus determined by the Khovanov homology. Thus the problem o deining an invariant space or spectrum (a homotopy type) is solved by the above as well, but in an uninteresting way. Nevertheless we now ind ourselves within a homotopy theory context so can apply its methods and techniques to Khovanov homology.

4 The homotopy theory o Khovanov homology 003 In the third section we develop this perspective urther by isolating a result about holim and homotopy ibres in this speciic situation which may be useul in the study o Khovanov homology. One central point is that in the preshea setting (or using chain complexes) one has long exact sequences in homology arising rom short exact sequences o presheaves. Typically the latter arise rom a given injection or surjection and one requires some luck or this to be the case. In the homotopy setting, by contrast, any map o spaces has a homotopy ibre and an attendant long exact sequence in homotopy groups. We illustrate the use o this calculus in the last section where we discuss the skein relation as the homotopy long exact sequence o the smoothing change map, reprove Reidemeister invariance rom the homotopy perspective and make an explicit computation. We have tried as ar as possible to make this article readable both by knot theorists interested in Khovanov homology and by homotopy theorists with a passing interest in knot theory. Acknowledgements We thank Hans-Werner Henn, Kathryn Hess, Robert Lipshitz, Sucharit Sarkar, Jérôme Scherer and the reeree or helpul remarks. Khovanov homology and higher inverse limits The main result o this section is a reinterpretation o the (unnormalised) Khovanov homology o a link as the derived unctors o lim over a certain small category.. A modiied Boolean lattice and the inverse limit Let B = B A be the Boolean lattice on a set A: the poset o subsets o A ordered by reverse inclusion. We write or the partial order and or the covering relation, i.e.: x y when subsets x y and x y when x is obtained rom y by adding a single element. Now let D be a link diagram and B the Boolean lattice on the set o crossings o D. Each crossing o D can be 0- or -resolved 0

5 004 Brent Everitt and Paul Turner and i x is some subset o the crossings, then the complete resolution D(x) is what results rom -resolving the crossings in x and 0-resolving the crossings not in x. It is a collection o planar circles. Let V = Z[, u] where Z[S] is the ree abelian group on the set S. This rank two abelian group becomes a Frobenius algebra using the maps m: V V V, ǫ: V Z and : V V V deined by m :, u and u u, u u 0 ǫ : 0, u : u+u, u u u. The Khovanov cube is obtained by assigning abelian groups to the elements o B and homomorphisms between the groups associated to comparable elements. One says cube as the Hasse diagram o the poset B A is the A -dimensional cube, with edges given by the covering relations. For x B let F KH (x) = V k, with a tensor actor corresponding to each connected component o D(x). I x y in B then D(x) results rom -resolving a crossing that was 0-resolved in D(y), with the qualitative eect that two o the circles in D(y) use into one in D(x), or one o the circles in D(y) biurcates into two in D(x). In the irst case F KH (x y) : F KH (y) = V k V k = F KH (x) is the map using m on the tensor actors corresponding to the used circles, and the identity on the others. In the second, F KH (x y) : F KH (y) = V k V k+ = F KH (x) is the map using on the tensor actor corresponding to the biurcating circles, and the identity on the others. All o this is most concisely expressed by regarding B as a category with objects the elements o B and with a unique morphism x y whenever x y. The decoration by abelian groups is then nothing other than a covariant unctor, or preshea, F KH : B op Ab where Ab is the category o abelian groups. The diagram D is suppressed rom the notation. Each square ace o the cube B is sent by the unctor F KH to a commutative diagram o abelian groups. To extract a cochain complex rom the decorated cube these squares must anticommute, and this is achieved by adding ± signs to the edges o the cube so that each square ace has an odd number o signs on its edges. We write [x, y] or the sign associated to the edge x y o B. The Khovanov complex K has n-cochains K n = x =n F KH(x) the direct sum over the subsets o size n (or rows o the cube), and dierential d : K n K n given by d = [x, y]f KH (x y), the sum over all

6 The homotopy theory o Khovanov homology 005 pairs x y with x o size n (or sum o all signed maps between rows n and n). That d is a dierential ollows immediately rom the anti-commuting o the signage. Deinition. The unnormalised Khovanov homology o a link diagram D is deined as the homology o the Khovanov cochain complex: KH (D) = H(K, d). The normalised Khovanov homology o an oriented link diagram D with c negative crossings is a shited version o the above: KH (D) = KH +c (D) The normalised Khovanov homology is a link invariant. All o the above is standard and there are several reviews o this material available (see or example Bar-Natan [], Turner [7] and Khovanov []). A note on the q-grading. Usually there is an internal grading on Khovanov homology making it a bigraded theory. This q-grading is important in recovering the Jones polynomial. A huge amount o inormation is retained however even i this grading is completely ignored. For example Khovanov homology detects the unknot with or without the q-grading. In this paper the q-grading plays no role and we consider the Frobenius algebra V above as ungraded, resulting in a singly graded theory. For what ollows we need to modiy the poset B in a seemingly innocuous way, but one which has considerable consequences (see also the remarks at the end o Section.3). There is a unique maximal element B (corresponding to the empty subset o A) with x or all x B. Now ormally adjoin to B an additional maximal element such that x or all x B with x, and denote the resulting poset (category) by Q = Q A. Extend F KH to a (covariant) unctor F KH : Q op Ab by setting F KH ( ) = 0 and F KH (x ) : F KH ( ) F KH (x) to be the only possible homomorphism. The construction o K extends verbatim to Q: the chains are the direct sum over the rows o Q (identical to B except or the top row where the zero group is added) and the dierential is the sum o signed maps between consecutive rows again identical except between the irst and second rows; we adopt the convention [x, ] = or an

7 006 Brent Everitt and Paul Turner Figure : Regular CW complex (let) with cell poset Q A (right) or A = 3. x with x. The resulting homology is easily seen to be the unnormalised Khovanov homology again. It will be convenient later to identiy Q with the poset o cells o a certain CW complex. Recall that a CW complex is regular i or any cell x the characteristic map Φ x : (B k, S k ) ( k x, k ) is a homeomorphism o B k onto its image. We can then deine a partial order on the cells o by x y exactly when x y, where x is the (CW-)closure o the cell. To realise Q A as such a thing suppose that A = n and let n be an (n )-simplex. Let be the suspension S n, an n-ball, and take the obvious CW decomposition o with two 0-cells (the suspension points) and all other cells the suspensions Sx o the cells x o n. As the suspension o cells preserves the inclusions x y and the two 0-cells are maximal with respect to this we get has cell poset Q. An x Q corresponds to an x -dimensional cell o ; the case n = 3 is in Figure. Using the signage introduced above, i x is a -cell we have [x, ] + [x, ] = 0; i dim x dim y = 2 and z, z 2 are the unique cells with x z i y, then [x, z ][z, y]+ [x, z 2 ][z 2, y] = 0. These properties then ensure that there are orientations or the cells o so that [x, y] is the incidence number o the cells x and y (see Massey [3, Chapter I, Theorem 7.2]). We inish this introductory subsection by recalling the deinition o the inverse limit o abelian groups. Let C be a small category and F : C Ab a unctor. Then the inverse limit lim F is an abelian group that is universal with respect to the property C that or all x C there are homomorphisms lim F F(x) that commute with the C homomorphisms F(x) F(x ) or all morphisms x x in C. The limit is constructed by taking the subgroup o the product Π x C F(x) consisting o those C-tuples (α x ) x C such that or all morphisms x x, the induced map F(x) F(x ) sends α x to α x.

8 The homotopy theory o Khovanov homology 007 It is an easy exercise to see that lim Q op F KH = ker d 0, the degree zero dierential o the cochain complex K, and so () lim F KH 0 = KH (D) Q op.2 Derived unctors o the inverse limit We have seen that presheaves o abelian groups provide a convenient language or the construction o Khovanov homology, and that the inverse limit o the preshea F KH : Q op Ab captures this homology in degree zero. In this subsection we review general acts about the category o presheaves, the inverse limit unctor and its derived unctors. These higher limits give, by deinition, the cohomology o a small category C with coeicients in a preshea. The moral is that they are computed using projective resolutions or the trivial (or constant) preshea, just as group cohomology, say, is computed using projective resolutions or the trivial G-module. The material here is standard (see e.g. Weibel [9, Chapter 2]) and obviously holds in greater generality; rather than working in the category R Mod o modules over a commutative ring R, we content ourselves with Ab := Z Mod. In the ollowing subsection we will show that the higher limits capture Khovanov homology in all degrees, not just degree zero. Recall that a preshea on a small category C is a (covariant) unctor F : C op Ab. The category PreSh(C) = Ab Cop has objects the presheaves F : C op Ab and morphisms the natural transormations τ : F G. For x C we write F(x) or its image in Ab and τ x or the map F(x) G(x) making up the component at x o the natural transormation τ. PreSh(C) is an abelian category having enough projective and injective objects. Many basic constructions in PreSh(C), such as kernels, cokernels, decisions about exactness, etc, can be constructed locally, or pointwise, e.g. the value o the preshea ker (τ : F G) at x C is ker (τ x : F(x) G(x)), and similarly or images. In particular, a sequence o presheaves F G H is exact i and only i or all x C the local sequence F(x) G(x) H(x) is exact. The simplest preshea is the constant one: i A Ab deine A : C op Ab by A(x) = A or all x and or all morphisms x y in C let A(x y) = : A(y) A(x). I : A B is a map o abelian groups then there is a natural transormation τ : A B with τ x : A(x) B(x) the map. Thus we have the constant shea unctor : Ab PreSh(C) which is easily seen to be exact.

9 008 Brent Everitt and Paul Turner We saw at the end o. that the inverse limit lim F exists in Ab or any preshea F PreSh(C). Indeed, we have a (covariant) unctor lim : PreSh(C) Ab by universality. For any A Ab and any F PreSh(C) there are natural bijections (2) Hom PreSh(C) ( A, F) = Hom Z (A, lim F), so that lim is right adjoint to. In particular lim is let exact, and we have the right derived unctors lim i := R i lim : PreSh(C) Ab (i 0) with lim 0 naturally isomorphic to lim. A special case o the adjointness (2) is the ollowing: or any preshea F over C the universality o the limit gives a homomorphism Hom PreSh(C) ( Z, F) lim F that sends a natural transormation τ Hom PreSh(C) ( Z, F) to the tuple (τ x ()) x C lim F. This is in act a natural isomorphism, so we have a natural isomorphism o unctors lim = Hom PreSh(C) ( Z, ) and thus (3) lim i = R i Hom PreSh(C) ( Z, ) or all i 0. I 0 F G H 0 is a short exact sequence in PreSh(C) then there is a long exact sequence in Ab: (4) 0 lim F lim G lim H lim i F lim i G lim i H It turns out that the derived unctors o the covariant Hom unctor in (3) can be replaced by the derived unctors o the contravariant Hom unctor. Let F, G be presheaves over the small category C. Then R i Hom PreSh(C) (F, )(G) = R i Hom PreSh(C) (, G)(F) or all i 0. One thinks o this as a balancing Ext result or presheaves. The corresponding result in R Mod is [9, Theorem 2.7.6], and the reader can check that the proo given there goes straight through in PreSh(C). Summarizing, or F PreSh(C) (5) lim i (F) = (R i Hom PreSh(C) ( Z, ))(F) = (R i Hom PreSh(C) (, F))( Z). To compute the right derived unctors o a contravariant unctor like Hom PreSh(C) (, F), we use a projective resolution. Let P Z be a projective resolution or Z, i.e.: an exact sequence (6) δ P 2 δ P δ P 0 ε Z 0

10 The homotopy theory o Khovanov homology 009 with the P i projective presheaves. Then the inal term in (5) is the degree i cohomology o the cochain complex Hom PreSh(C) (P, F): (7) δ δ Hom PreSh(C) (P, F) Hom PreSh(C) (P 0, F) 0.3 A projective resolution o Z and the Khovanov complex We return now to the particulars o. and compute the cochain complex (7) when F = F KH, the Khovanov preshea in PreSh(Q) where Q is the poset o.. To do this we present a particular projective resolution or the constant preshea Z on Q. We start by constructing a preshea P n in PreSh(Q) or each integer n > 0. Remembering that Q is the cell poset o the regular CW complex o., or x Q set P n (x) := Z[n-cells o contained in the closure o the cell x]. Thus i dim x < n then P n (x) = 0; i dim x = n then P n (x) = Z[x] = Z; and i dim x > n then P n (x) is a direct sum o copies o Z, one copy or each n-cell in the boundary o x. I x y in Q then we take P n (x y): P n (y) P n (x) to be the obvious inclusion. For a given preshea F PreSh(Q) there is a nice characterization o the group o preshea morphisms P n F: Proposition. For F PreSh(Q) the map n : Hom PreSh(Q) (P n, F) dim x=n F(x) deined by n (τ) = dim x=n τ x(x), is an isomorphism o abelian groups. Proo That n is a homomorphism is clear since (τ + σ) x = τ x + σ x. To show injectivity, suppose that n (τ) = 0 rom which it ollows that τ x (x) = 0 or all n-cells x Q. To show that τ = 0 we must prove that τ y : P n (y) F(y) is zero or all y Q. For dim y < n there is nothing to prove since P n (y) = 0. For dim y = n we have P n (y) = Z[y] and τ y (y) = 0 since y is an n-cell. For dim y > n and we have P n (y) = Z[y α dim y α = n and y α in the closure o y] τ y (y α ) = τ y (P n (y y α )(y α )) = F(y y α )(τ yα (y α )) = 0. The irst equality since P n (y y α ): P n (y α ) P n (y) is an inclusion, and the second by naturality o τ and the third since y α is an n-cell. Finally, n is surjective because P n (x) is ree and so there is no restriction on the images τ x (x) F(x).

11 00 Brent Everitt and Paul Turner The isomorphism given in Proposition. allows us to deine a morphism τ : P n F by speciying a tuple λ x F(x), where the sum is over the n-cells x. It is easy to see that the P n are projective presheaves. Given the ollowing diagram o presheaves and morphisms (with solid arrows) and exact row: τ P n G σ F 0 σ then the local maps G(x) x F(x) are surjections. Thus i λx F(x) speciies the map τ then or each x there is a µ x G(x) with σ x (µ x ) = λ x. Hence there exists a morphism τ : P n G speciied by µ x, which clearly makes the diagram commute. The P n are thus projective presheaves. We now assemble the P n s into a resolution o Z by deining maps δ n : P n P n. For x Q let δ n,x : P n (x) P n (x) be the homomorphism deined by δ n,x (y) = y z[y, z] z or y an n-cell x and the sum is over the (n )-cells z y. Here, [y, z] = ± is the incidence number o y and z given by the orientations chosen at the end o.. It is easy to check that these homomorphisms assemble into a morphism o presheaves δ n : P n P n. The sequence δ P n+ τ δ P n δ P n δ is exact at P n i and only i each o the local sequences P (x) is exact at P n (x). But P (x) is nothing other than the cellular chain complex o the dim(x)-dimensional ball corresponding to the closure o x with the induced CW decomposition. In particular { Z, n = 0 H n P (x) = 0, n > 0 so that P (x), and hence P, is exact in degree n > 0. ε To deine an augmentation P 0 Z 0 take ε to be the canonical surjection onto coker (δ): δ ε P P0 coker (δ) 0. The computation o P (x) above immediately shows that coker (δ) = Z. We now have our projective resolution (6) or Z and hence a cochain complex (7) that computes the derived unctors lim i F KH. Proposition. gives an isomorphism

12 The homotopy theory o Khovanov homology 0 o graded abelian groups : Hom PreSh(Q) (P, F KH ) K where K is the Khovanov cochain complex o.. As the ollowing lemma shows, is in act a chain map and thus there is an isomorphism o cochain complexes Hom PreSh(Q) (P, F KH ) = K. Lemma.2 is a chain map Hom PreSh(Q) (P, F KH ) K. Proo We must show that the ollowing diagram commutes. Hom PreSh(C) (P n+, F KH ) n+ K n+ δ d Hom PreSh(C) (P n, F KH ) K n n Let τ Hom PreSh(Q) (P n, F KH ) and write F or F KH. I x is an n-cell, write λ x := τ x (x) F(x) so that n sends τ to the tuple x λ x, the sum over the n-cells o. Applying the Khovanov dierential d we get d( n (τ)) = [x, y] F(y x)(λ x ). x y x Consider now δ(τ) = τδ Hom PreSh(C) (P n+, F KH ). For y an (n+)-cell we have δ y (y) = x y [x, y]x and by an argument similar to that in the proo o Proposition., or x an n-cell we have τ y (x) = F(y x)(λ x ). Thus n+ (δ(τ)) is equal to (τ y δ y )(y) = y]τ y (x) = y x y[x, [x, y]f(y x)(λ x ) = d( n (τ)). y x y dim y=n+ Summarizing: to compute the higher limits o the Khovanov preshea we use the complex Hom PreSh(Q) (P, F KH ), which is isomorphic to K, and this in turn computes the unnormalised Khovanov homology. We have thereore proved the irst theorem: Theorem.3 Let D be a link diagram and let F KH : Q op Ab be the Khovanov preshea deined in.. Then KH i (D) = lim Q op i F KH Remark It is essential that we use the modiied Boolean lattice Q rather than just B: i we work with the Khovanov preshea over B then the higher limits all vanish. This ollows rom the general act that or a preshea over a inite poset with unique maximal element the higher limits all vanish see Mitchell [5].

13 02 Brent Everitt and Paul Turner.4 Aside on the cohomology o classiying spaces with coeicients in a preshea Although not central to what ollows it is worthwhile making the connection with a more topological description o higher limits in which lim i F KH is identiied with the cohomology o a classiying space equipped with a system o local coeicients. We recall that the classiying space BC is the geometric realization o the nerve o the small category C. This point o view is novel in the context o Khovanov homology, so we give a brie presentation o it, but otherwise we make no particular claim to originality here. Starting with a preshea F PreSh(C), the cochain complex C (BC, F) is deined on the nerve o C to have cochains C n (BC, F) = x 0 x n F(x 0 ), n the product over sequences o morphisms x 0 xn in C. I λ C n write λ (x 0 x n ) or the component o λ in the copy o F(x 0 ) indexed by the sequence x 0 x n. The coboundary map d : C n (BC, F) C n+ (BC, F) is given by n+ 2 n+ dλ (x 0 x n+ ) = F(x 0 x )(λ (x x n+ )) n + ( ) i i i+ λ (x 0 xi x i+ n+ x n+ )+( ) n+ n λ (x 0 xn ). i= Write H (BC, F) or the cohomology o C (BC, F). The ollowing result o Moerdijk [6, Proposition II.6.] shows that this cochain complex computes the higher limits: Proposition.2 Let F PreSh(C). Then H (BC, F) = lim C F. From Theorem.3 we immediately get the ollowing description o unnormalised Khovanov homology in terms o the cohomology o the classiying space BQ with a system o local coeicients induced by the Khovanov preshea: Proposition.3 Let D be a link diagram and let F KH : Q op Ab be the Khovanov preshea deined in.. Then, KH (D) = H (BQ; F KH ).

14 The homotopy theory o Khovanov homology 03 Remark Proposition.3 is very similar in spirit to the main result (Theorem 24) o the authors [6] which gives an isomorphism between a homological version o Khovanov homology and a slight variation on the homology o a poset with coeicients in a preshea (termed coloured poset homology in [6]; see also [7]). 2 Interpreting higher limits in homotopy theoretic terms 2. Homotopy limits Limits and colimits exist in the category o spaces but are problematic in the homotopy category: deorming the input data up to homotopy may not result in the same homotopy type. This problem is resolved by the use o homotopy limits and homotopy colimits, which are now standard constructions in homotopy theory. In this section we will use homotopy limits to build spaces whose homotopy groups are Khovanov homology. We begin by recalling the key properties o homotopy limits, and while we will adopt a blackbox approach to the actual construction (leaving the inner workings irmly inside the box), we will provide reerences to the classic text by Bousield and Kan [3]. We briely return to the generality o a small category, but later will again specialise to posets. Let Sp denote the category o pointed spaces. All spaces rom now on will be pointed. Let C be a small category and let Sp C be the category o diagrams o spaces o shape C: an object is a (covariant) unctor : C Sp and a morphism : Y is a natural transormation. Thus a diagram o spaces associates to each object o C a (pointed) space and to each morphism o C a (pointed) continuous unction such that these it together in a coherent way. Given a morphism : Y we will use the notation x or the component at x. The trivial diagram takes value the one-point space or all objects o C and the identity map or all morphisms. For our purposes holim is a covariant unctor holim C : Sp C Sp whose main properties are recalled below in Propositions For a morphism : Y we denote by the induced map holim holim Y. The holim construction is natural with respect to change o underlying category: a unctor F : C C induces a map holim C holim C F. Remark We adopt the convention o Bousield and Kan [3] where i pressed on the matter, space means simplicial set. Furthermore, i thus pressed, we will also

15 04 Brent Everitt and Paul Turner assume that diagrams take as values ibrant simplicial sets [3, VIII, 3.8]. Indeed there are models o Eilenberg-Mac Lane spaces that are simplicial groups, and hence ibrant. The reader should be aware however that in the proper generality the propositions below require ibrant objects. The irst important property o holim is its well-deinedness in the homotopy category; it is robust with respect to deormation by homotopy [3, I, 5.6]: Proposition 2. (Homotopy) Let : Y be a morphism in Sp C such that or all x the map x : (x) Y (x) is a homotopy equivalence. Then : holim holim Y is a homotopy equivalence. Next, a morphism o diagrams which is locally a ibration induces a ibration on holim [3, I, 5.5]: Proposition 2.2 (Fibration) Let : Y be a morphism in Sp C such that or all x the map x : (x) Y (x) is a ibration. Then : holim holim Y is a ibration. There is also a nice description o holim or diagrams over a product o categories [3, I, 4.3]: Proposition 2.3 (Product) Let : C D Sp be a diagram o spaces over the product category C D. Then holim C holim D holim C D holim D holim C We also need to be able to compare diagrams o dierent shape, i.e.: where the base categories are dierent. The result turns out to be easier to state in the context o posets than or small categories, and this suices or us [3, I, 9.2]: Proposition 2.4 (Coinality) Let : P 2 P be a map o posets. (i) Let : P Sp be a diagram o spaces and suppose that or any x P the poset {y P y x} P 2 is contractible. Then holim P2 holim P. (ii) Let : P op Sp be a diagram o spaces and suppose that or any x P the poset {y P y x} P 2 is contractible. Then holim P2 op holim P op.

16 The homotopy theory o Khovanov homology 05 Here a poset is contractible i its geometric realisation BP is contractible, so in particular BP, and hence P, is non-empty. For example i P has an extremal (i.e.: maximal or minimal) element then BP is a cone. Statement (ii) above is simply a restatement o (i), but the potential conusion in taking opposites makes it worth while stating both. For a simple application o Proposition 2.4 let P 2 be a contractible poset and the constant diagram over P 2 having value the space at each x and the identity map at each morphism x y. Let P be the single element poset and Y the diagram having value at this single element. I : P 2 P is the only possible map, then = Y and the conditions o Proposition 2.4 are satisied. Thus holim holim Y. The inal basic property o holim is that it commutes with mapping spaces (o pointed maps between pointed spaces) see [3, I, 7.6]: Proposition 2.5 (Mapping) Let be a diagram o spaces in Sp C and let Y be a (pointed) space. Then Map(Y, holim ) holim Map(Y, ). Here Map(Y, ) is the unctor that takes a pointed space Z to the space o pointed maps rom Y to Z and Map(Y, ) Sp C is the composition Map(Y, ). 2.2 Spaces or Khovanov homology Bousield and Kan give an interpretation o derived unctors o the inverse limit as ollows. Consider the Eilenberg-Mac Lane unctor K(, n) : Ab Sp or which we adopt the construction given by Weibel [9, 8.4.4] where there is an obvious choice o basepoint or K(A, n). For more details on Eilenberg-Mac Lane spaces see May [4, Chapter V] or Hatcher [9, Chapter 4]. The ollowing proposition [3, I, 7.2] gives an interpretation o lim i F in homotopy theoretic terms where C is a small category. Proposition 2.6 isomorphisms: Let F : C Ab be a (covariant) unctor. Then there are natural { π i (holim C K(, n) F) lim n i = C F 0 i n, 0 else. The spaces holim C K(, n) F contain no more inormation than higher derived unctors o F. Indeed, as a consequence o the Dold-Kan theorem (see [9, Section 8.4] or Curtis [4, Section 5]) we have:

17 06 Brent Everitt and Paul Turner Proposition 2.7 For n big enough the space holim C K(, n) F has the homotopy type o a product o Eilenberg-Mac Lane spaces: holim C K(, n) F m K(lim n m C F, m). For a sel contained and elementary argument proving the appropriate result needed here we reer the reader to [5]. Ater these preliminaries on homotopy limits we return to Khovanov homology. Associated to a link diagram D we have the Khovanov preshea F KH : Q op Ab o.. Let n N and let F n : Q op Sp be the diagram o spaces deined by F n = K(, n) F KH, the composition o F KH with the Eilenberg-Mac Lane space unctor K(, n). We can now deine a space Y n D as the homotopy limit o this diagram: Y n D = holim Q op F n = holim Q op K(, n) F KH. Remark The homotopy limit above, taken over the augmented Boolean lattice Q, is what Goodwillie [8], in his theory o calculus o unctors, calls the total iber o the (decorated) Boolean lattice B. From Theorem.3 and Proposition 2.6 we see that Y n D is a space whose homotopy groups are isomorphic to the unnormalised Khovanov homology o D: Proposition 2.8 (8) π i (Y n D) = Indeed by Proposition 2.7 we have { KH n i (D) 0 i n, 0 else. Y n D m K(KH n m (D), m). In order to normalise Khovanov homology a global degree shit is applied. As π i Ω = π i+ or a pointed space, we see that degree shits are implemented at the space level by taking loop spaces. Suppose now D is oriented and has c negative crossings. The collection o spaces Y ={Y n D} is an Ω-spectrum which may be delooped c times to obtain a new Ω-spectrum D = Ω c Y D whose homotopy groups are normalized Khovanov homology: π i ( D) = KH i (D).

18 The homotopy theory o Khovanov homology 07 3 Diagrams over Boolean lattices and homotopy limits This section develops some results on the homotopy limits o diagrams deined over (modiied) Boolean lattices. These will then provide tools applicable to Khovanov homology, and in the next section we illustrate this with homotopy theoretic proos o some Khovanov homology results. In light o the Remark ater Prosposition 2.7 some o the conclusions o this section are consequences o Goodwillie s calculus o unctors, but we preer to (re)prove the results we need in a sel-contained manner. We make extensive use o homotopy ibres and so record here some o their properties. Given a map (o pointed spaces) : Y we deine the homotopy ibre o as a homotopy limit by (9) hoibre( Y) = holim( Y ). By liting the lid o the black box only a raction (see [3, Chapter I]) one sees that this has the homotopy type o the usual homotopy ibre: namely deining (0) E = {(x,α) x,α: [0, ] Y a continuous map such that α(0) = (x)}, then this is a space homotopy equivalent to and the map E Y sending (x,α) α() is a ibration whose ibre is homotopy equivalent to the hoibre (9). Relevant examples o homotopy ibres are () (2) hoibre( ) hoibre( Y) ΩY. Using the long exact homotopy sequence or a ibration and the Whitehead theorem one immediately gets: Lemma 3. For Y connected, i hoibre( Y) then Y. I : Y is a map o pointed spaces with Y contractible, then () extends to (3) hoibre( Y), and similarly i is contractible then (2) extends to (4) hoibre( Y) ΩY. From now on we assume that C is a connected category. Given diagrams, Y Sp C (o pointed spaces) and a morphism : Y one may orm the homotopy ibre

19 08 Brent Everitt and Paul Turner diagram ho( ) by (locally) deining ho( )(x) = hoibre( x : (x) Y (x)) and ho( )(x y) : hoibre( x ) hoibre( y ) the map induced by taking homotopy limits o the two rows o the diagram (x) Y (x) (y) Y (y) with the lethand square commuting courtesy o. It is a standard trick in homotopy theory to compute the homotopy limit o a diagram o homotopy ibres as a homotopy ibre. In the interest o completeness we have included the details, but the main point is that the homotopy ibre is an example o a homotopy limit and homotopy limits enjoy the product property o Proposition 2.3: Proposition 3. Let : Y be a morphism in Sp C. Then holim(ho( )) hoibre(holim holim Y ). Proo Let D be the three element category a α c β b. Deine Z : C D Sp by Z (x, a) = Z (x, b) = (x) Z (x, c) = Y (x). On morphisms let Z ((id,α): (x, a) (x, c)) = Y (x), Z ((id,β): (x, b) (x, c)) = x and, z = a, Z ((θ, z z): (x, z) (x, z)) = (θ), z = b, Y (θ), z = c. We then have holim D Z (x, ) = holim( (x) x Y (x) ) = hoibre( x : (x) Y (x)) [by (9)] = ho( )(x) rom which we get holim C holim D Z holim ho( ). Going the other way we have holim C Z (, a), holim C Z (, b) = holim C, and holim C Z (, c) = holim C Y, so holim D holim C Z = holim(holim holim Y ) hoibre(holim holim Y ). The result now ollows rom Proposition 2.3.

20 The homotopy theory o Khovanov homology 09 Later we will use this result in the orm: i : Y is a map o diagrams, the space holim Y is connected and holim ho( ) contractible, then the induced map : holim holim Y is a homotopy equivalence. Notation or diagrams o spaces. We introduce a convenient notation that we will use extensively. We recall that space means pointed space and diagrams o spaces take values in pointed spaces. A Boolean lattice B will be represented by the circle below let and a diagram : B op Sp by the pictogram below right: Extending this to Q op by deining ( ) = we obtain a diagram o spaces, with Q and : Q op Sp represented as The trivial diagram will be denoted. Let B = B A be Boolean o rank r, i.e.: the lattice o subsets o {,...,r}. For each k r there is a splitting o B into two subposets, both isomorphic to Boolean lattices o rank r : one consists o those subsets containing k and the other o those not containing k. Below we see the splittings or r = 3, with (rom let to right) k =, 2 and 3: A diagram : B op Sp determines (and is determined by) two diagrams o spaces and 2 over these rank r Boolean lattices along with a morphism o diagrams : 2. We denote this situation (and the obvious extension to Q op ) by the pictograms: 2 and 2

21 020 Brent Everitt and Paul Turner This process can be iterated with each o the smaller Boolean lattices to give pictures that are square, cubical, etc. Lemma 3.2 Let : B op Sp be a diagram o spaces. Then, holim holim Proo I B has rank r then the diagram on the let-hand side is over a Boolean lattice o rank r+. Let Q be the extended version o this Boolean lattice and suppose that it has been split as above. Collapsing the bottom Boolean lattice to a point we obtain a new poset P: Q = = P The poset map : Q P which collapses the lower Boolean lattice to a single point satisies the hypotheses o Proposition 2.4 (ii). Moreover we have the ollowing equality o diagrams o spaces (o shape Q op ): = Let Q be the poset obtained rom Q by removing the lower Boolean lattice: then the obvious inclusion i: Q P satisies the hypotheses o Proposition 2.4. Hence we have holim holim holim using Proposition 2.4 twice (with and i).

22 The homotopy theory o Khovanov homology 02 A somewhat more general version o this result is the ollowing. Lemma 3.3 Let, Y : B op Sp be diagrams o spaces such that Y (x) is contractible or all x B op. Then, holim holim Y Proo Let τ be the morphism o diagrams deined by Y As the map Y (x) is a homotopy equivalence or all x, the result ollows rom Proposition 2. and Lemma 3.2. Lemma 3.4 Let : B op Sp be a diagram o spaces. Then, holim Proo Let Q be an extended Boolean lattice o rank one bigger than the rank o B and split as above. Let P be obtained rom B by adding an element which is greater than all other elements (including the existing maximal element in B). Pictorially: Q = = P

23 022 Brent Everitt and Paul Turner The poset map : Q P which identiies elements o the Boolean lattices and sends satisies the hypotheses o Proposition 2.4. Moreover we have the ollowing equality o diagrams o spaces (o shape Q op ). = Since P op has a minimal element it ollows rom [3, I, 4.(iii)] that holim whence the result on applying Proposition 2.4. Proposition 3.2 Let : Y be a morphism o diagrams o spaces over a Boolean lattice. Then, holim hoibre(holim holim Y Y ) Proo Let g be the ollowing morphism o diagrams o spaces Y Y Y We have holim(ho(g)) is homotopy equivalent to Y g hoibre(holim holim Y Y ) holim Y

24 The homotopy theory o Khovanov homology 023 by Proposition 3., Lemma 3.4 and (3). On the other hand writing ho(g) = we see that Z 2 (x) is contractible or all x. Thus, by Lemma 3.3 and Proposition 3. we have holim(ho(g)) holim Z hoibre(holim holim Y ) Z Z 2 Corollary 3. Let : Y be a morphism o diagrams o spaces over a Boolean lattice. Then, holim holim(ho( Y )) Y Lemma 3.2 can be generalized as in the irst part o the ollowing: Proposition 3.3 Let Y : B op Sp be a diagram o spaces with holim Y. Then Y (i). holim holim (ii). holim Ω holim Y Proo For (ii) we have Y holim g hoibre(holim Y ḡ holim ) Ω holim by Proposition 3.2 and Equation (4). Part (i) is similar.

25 024 Brent Everitt and Paul Turner Corollary 3.2 Let, Y : B op Sp be diagrams o spaces such that Y (x) is contractible or all x B op. Then, Y holim Ω holim Proo Let τ be the morphism o diagrams deined by Y As the map Y (x) is a homotopy equivalence or all x, the result ollows rom Propositions 2. and 3.3(ii). Notation or presheaves. We adopt a similar notation or presheaves to that or diagrams o spaces, with the dierence that the circles are white rather than shaded. Thus a preshea F : B op Ab and its extension F : Q op Ab (with F( ) = 0) will be represented by: F and F Given a preshea F we will denote the diagram o spaces K(, n) F by F. Remark For n suiciently large the space holim F is path connected. To see this, one may use Proposition 2.6 to calculate π 0 (holim F ) = lim n F. So long as n is chosen to be greater than the rank o Q the right-hand side is trivial by Theorem.3 (recall that the underlying poset is always assumed connected). We will always assume that n is large enough in this sense.

26 The homotopy theory o Khovanov homology 025 Proposition 3.4 Let Then, F G g H be a short exact sequence in PreSh(Q). (i). hoibre(holim (ii). hoibre(holim G F ḡ holim H ) holim F holim G ) Ω holim H Proo For part (i) the let-hand side is homotopy equivalent to holim(ho(g)) by Proposition 3.. Here ho(g) is the diagram obtained by taking homotopy ibres ater applying K(, n) to the surjection g in the statement o the proposition, so ho(g)(x) = hoibre(k(g(x), n) K(H(x), n)). We can identiy this homotopy ibre with K(ker (g x ), n) = K(F(x), n). The maps in the diagram ho(g) under this identiication correspond to the maps induced by F; this ollows since maps on such Eilenberg-Mac Lane spaces are completely determined by their eect on π n and by the naturality o the hoibre construction. Thus the diagram ho(g) is equivalent to and the result ollows by Proposition 2.. Part (ii) is similar. F Combining Proposition 3.4, Lemma 3. and the Remark preceeding Proposition 3.4 gives the ollowing very useul lemma. Corollary 3.3 Let Then, F G g H be a short exact sequence in PreSh(Q). (i). I holim F then g : holim G holim H is a homotopy equivalence. (ii). I holim H then : holim F holim G is a homotopy equivalence. Another result that will prove useul comes rom combining Propositions 3.4 and 3.2: Corollary 3.4 Let Then, G (i). holim H g F G g H be a short exact sequence in PreSh(Q). holim F (ii). holim F G Ω holim H

27 026 Brent Everitt and Paul Turner 4 Applications to Khovanov homology The results o the previous section give a collection o tools or Khovanov homology, and in this section we illustrate with a ew simple examples. First we isolate the two most useul results. One is Corollary 3.3, which is just a preshea theoretic reormulation o standard arguments involving chain complexes: preshea computational tool. Let H F G be a short exact sequence. I holim H (resp. holim G ) is contractible then holim F holim G (resp. holim H holim F ) An arbitrary morphism o presheaves (not necessarily injective or surjective) cannot be slotted into a short exact sequence. Our second result, which ollows rom Proposition 3. and the discussion preceeding it along with Remark 3, gets around this by using the homotopy theory in a more essential way: homotopy computational tool. Let F G be a morphism in PreSh(Q) with H = ho( F G ). I then holim H (resp. holim G ) is contractible holim F holim G (resp. holim H holim F ). Notation or the Khovanov preshea o a link diagram. We extend our notation to the speciic case o the Khovanov preshea F KH associated to a link diagram. It is convenient to have the link diagram back in the picture: given a link diagam D we denote the associated Khovanov preshea F KH and diagram o spaces K(, n) F KH by: D and D I, as is oten the case, we are interested in a link diagram with a speciied local piece we simply display it inside the circle. Thus or example the unit map ι: Z V, which is deined by ι() =, extends to an injective morphism o presheaves over a Boolean lattice o appropriate rank which we denote by: ι

28 The homotopy theory o Khovanov homology 027 We have link diagrams D and D that are identical outside one part where they dier by the local piece shown. On the let we have the preshea F KH : B op Ab, the cube or D, and on the right F KH : Bop Ab, the cube or D. For x B we have F KH (x) = F KH(x) V and the map ι x : F KH (x) = F KH (x) Z F KH (x) V = F KH (x) is the map ι. These local ι x stitch together to orm a morphism o presheaves ι : F KH F KH. This is what we mean by the picture above. Similarly the counit map ǫ: V Z, deined by ǫ() = 0,ǫ(u) =, extends to a surjective morphism o presheaves over B ǫ and there is a short exact sequence o presheaves (5) ι ǫ The multiplication m: V V V is surjective with ker (m) = V, and this analysis similarly extends to give a short exact sequence o presheaves (6) m The composition m ι is the identity map. Finally, the comultiplication : V V V is injective, coker ( ) = V, and this extends to a short exact sequence o presheaves (7) The composition ǫ is the identity map. Occasionally the link diagram D will be too large or the circle notation above (e.g: in 4.3), and so we will just write D, or a shaded version. For example i D, D 2 are unoriented link diagrams then (5) extends to ι ε and there are similar sequences or m and.

29 028 Brent Everitt and Paul Turner D D (I+) D D (I ) D D (II) D D (III) Figure 2: Reidemeister moves 4. The skein relation By choosing a crossing there is an evident smoothing change morphism o presheaves: ϕ In general this is neither surjective nor injective. There is however an induced map o spaces holim ϕ holim and we can easily describe its homotopy ibre to give a homotopy theoretic incarnation o the skein relation: Proposition 4. hoibre(holim holim ) holim Proo We have holim holim ϕ and the result ollows immediately rom Proposition 3.2. The associated long exact homotopy sequence can be identiied with the usual long exact skein sequence in Khovanov homology (see [8] and [7]).

30 The homotopy theory o Khovanov homology Reidemeister invariance. We now give a homotopy theoretic proo o the invariance o Khovanov homology by Reidemeister moves (Figure 2). The original proos can be ound in [0] (see also []) and more a geometrical argument can be ound in [2]. We recall that the rank o the underlying Boolean lattice is the number o crossings in the given diagram, thus moves (I±) and (II) alter the underlying Boolean lattice. Recalling rom the remarks at the end o Section 2.2 that a negative degree shit in Khovanov homology is equivalent to taking the loop space, we see that we must prove (I+).Y n D Y n D, (I ).Y n D ΩY n D, (II).Y n D ΩY n D and (III).Y n D Y n D Reidemeister moves (I±). Let D and D be two (unoriented) link diagrams locally described as in (I ) in Figure 2. The short exact sequence (7) and Corollary 3.4(ii) give Y n D = holim Ω holim = ΩY nd. A completely analogous argument, using (6) and Corollary 3.4(i), gives Reidemeister (I+) Reidemeister move (II). Let D and D be two link diagrams locally described as in (II) and let F KH be the Khovanov preshea or D. There is a short exact sequence H F KH G: 0 ι = d d 2 d 2 0 d 3 d 4 ε 0

31 030 Brent Everitt and Paul Turner We leave it to the reader to check that G and H are indeed presheaves. All missing horizontal maps are either the identity or zero (it should be clear which is which), ι and ǫ are the unit and counit, and we are using the short exact sequence (5). To check that we have morphisms o presheaves we need to show that εd = and d 3 ι =. The ormer ollows rom ε = and the latter rom mι =. We have holim H = holim Ω holim, with the homotopy equivalences by Propositions 3.4(i), 3.3 and Lemma 3.4 respectively. The map induced by F KH G is thus a homotopy equivalence by the preshea computational tool, and so Y n D equals holim holim Ω holim with the last homotopy equivalent to Ω holim and Lemma 3.2. = ΩY n D by Proposition 3.3(ii) Reidemeister move (III). Let D and D be two (unoriented) link diagrams locally described as in (III) in Figure 2 and let F KH be the Khovanov preshea or D. We start with a short exact sequence

32 The homotopy theory o Khovanov homology 03 G 0 F KH τ G deined by: 0 = d d 2 d 3 d d 2 d d 4 d 5 d 6 d 7 d 8 d 9 d 4 εd 5 d 6 d 7 εd 8 d d 0 d d 2 ι ε 0 One can check that G is indeed a preshea and that τ is surjective. All missing horizontal maps are either the identity or zero. Proposition 3.3(ii) gives holim G 0 Ω 2, hence holim F holim G by the preshea tool (abbreviating F KH to F). We now deine another preshea H and a preshea map σ : G H where all missing maps are either the identity or zero as beore. For this we note that in G we have εd 5 = εd 8 = ε = and d = d 3 : d 9 d d 2 d d d 2 d 9 d d 4 d 6 d 7 d 9 d 4 d 9 d 6 d 7 0 d 9 0 Now, σ is neither injective or surjective so we turn to the homotopy tool. Using the

33 032 Brent Everitt and Paul Turner contractibility o many o the hoibres we have rom Lemma 3.3 and Corollary 3.2 that holim(ho(σ)) Ω holim(ho d 9 d 9 ) Ω with the second to last homotopy equivalence rom Corollary 3., Lemma 3.4 and Proposition 3.3. Thus by Proposition 3. hoibre(holim G σ holim H ) holim ho(σ) It ollows that σ is a homotopy equivalence rom which we obtain Y n D = holim F holim G holim H Now repeat the entire process starting with F KH, the Khovanov preshea or D : a short exact sequence G 0 F KH G and a morphism σ : G H can be deined in a completely analogous way and the homotopy tool invoked to give Y n D = holim F holim G holim H Comparing H and H, it turns out that the vertex groups are visibly identical, as are the edge maps except or an occurrence o d 9 d in H and d 7 d 2 in H. However the ront top ace o F shows that d 9 d = d 7 d 2 and so H and H are in act identical diagrams. Thus, Y n D Y n D, completing the proo o Reidemeister (III). 4.3 An example. We take the technology or a test drive by showing that holim holim Ω 2 holim where D, D 2 are (unoriented) link diagrams and we are simpliying our pictograms as in the remarks immediately beore 4.. The conclusions or Khovanov homology are at the end o this calculation, and although they could be acheived at the level o chain complexes, our purpose here is to illustrate our machinery in action.

34 The homotopy theory o Khovanov homology 033 The Skein relation (Proposition 4.) gives holim hoibre holim holim which in turn is homotopy equivalent to holim holim m holim ( ) Consider the ollowing two diagrams over Q D (with D = ) and the morphism between them: ker m ι m m where is the trivial diagram o spaces. We have the short exact sequence H F G o presheaves: ker m m m 0