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Home , Morse homology, Morse theory

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R † t homology its : MS-1643401. 61 f u product cup hschain This . – db many by ed C efine- 63 ce. ∗ ssetting ’s them). n ensional, yof gy ,Morse ], ( sociated unlike o singular f f does ) on ho- M M spectrum (or pro-spectrum), and outlined a construction, under restrictive hy- potheses, of such an object. While they suggest that these spectra might be de- termined by the ambient, infinite-dimensional together with its polar- ization (a structure which seems ubiquitous in Floer theory), their construction builds a CW complex cell-by-cell, using the moduli spaces appearing in Floer theory. (We review their construction in §2.4. Steps towards describing in terms of a polarized manifold have been taken by Lipyanskiy [52].) Although the Cohen-Jones-Segal approach has been stymied by analytic difficul- ties, it has inspired other constructions of stable homotopy refinements of various Floer homologies and related invariants [1,6,7,12,13,16,24,33–35,40–42,56,74]. Since its inception, Floer homologies have been used to define invariants of objects in low-dimensional topology—3-manifolds, knots, and so on. At the end of the 1990s, Khovanov defined another knot invariant, which he calls sl2 homol- ogy and everyone else calls Khovanov homology [36], whose graded Euler char- acteristic is the Jones polynomial. (See [4] for a friendly introduction.) While it looks formally similar to Floer-type invariants, Khovanov homology is de- fined combinatorially. No obvious infinite-dimensional manifold or functional is present. Still, Seidel-Smith [77] (inspired by earlier work of Khovanov-Seidel [39] and others) gave a conjectural reformulation of Khovanov homology via Floer homology. Over Q, the isomorphism between Seidel-Smith’s and Khovanov’s invariants was recently proved by Abouzaid-Smith [2]. Manolescu [58] gave an extension of the reformulation to sln Khovanov-Rozansky homology [38]. Inspired by this history, we gave a combinatorial definition of a spectrum refining Khovanov homology, and studied some of its properties [49–51]. This circle of ideas was further developed with Ng [48] and in ongoing work with Lawson [44, 45], and extended in many directions by other authors (see §3.3). Another approach to a homotopy refinement was given by Everitt-Turner [19], though it turns out their invariant is determined by Khovanov homology [18]. Inspired by a different line of inquiry, Hu-Kriz-Kriz also gave a construction of a Khovanov stable homotopy type [26]. The two constructions give homotopy equivalent spectra [44], perhaps suggesting some kind of uniqueness. Most of this note is an outline of a construction of a Khovanov homotopy type, following our work with Lawson [44] and Hu-Kriz-Kriz’s paper [26], with an emphasis on the general question of stable homotopy refinements of chain complexes. In the last two sections, we briefly outline some of the structure and uses of the homotopy type (§3.3) and some questions and speculation (§3.4). Another exposition of some of this material can be found in [46]. Acknowledgments. We are deeply grateful to our collaborator Tyler Lawson, whose contributions and perspective permeate the account below. We also thank Mohammed Abouzaid, Ciprian Manolescu, and John Pardon for comments on a draft of this article.

2 Spatial refinements

The spatial refinement problem can be summarized as follows.

2 Start with a C∗ with a distinguished, finite basis, arising in some interesting setting. Incorporating more information from the setting, con- struct a based CW complex (or spectrum) whose reduced cellular chain complex, after a shift, is isomorphic to C∗ with cells corresponding to the given basis. A result of Carlsson’s [11] implies that there is no universal solution to the spatial refinement problem, i.e., no functor S from chain complexes (supported in large gradings, say) to CW complexes so that the composition of S and the reduced cellular chain complex functor is the identity (cf. [69]). Specifically, for G = Z/2×Z/2 he defines a module P over Z[G] so that there is no G-equivariant Moore space M(P,n) for any n. If C∗ is a free resolution of P over Z[G] then S(C∗) would be such a Moore space, a contradiction. Thus, spatially refining C∗ requires context-specific work. This section gives general frameworks for such spatial refinements, and the next section has an interesting example of one.

2.1 Linear and cubical diagrams

Let C∗ be a freely and finitely generated chain complex with a given basis. After shifting we may assume C∗ is supported in gradings 0,...,n. Let [n + 1] be the category with objects 0, 1,...,n and a unique morphism i → j if i ≥ j. Let B(Z) denote the category of finitely generated free abelian groups, with objects finite sets and HomB(Z)(S,T ) the set of linear maps ZhSi → ZhT i or, equivalently, T × S matrices of integers. Then C∗ may be viewed as a functor F from [n +1] to B(Z) subject to the condition that F sends any length two arrow (that is, a morphism i → j with i − j = 2) to the zero map. Given such a functor F :[n + 1] → B(Z), that is, a linear diagram

ZhF (n)i → ZhF (n − 1)i→···→ ZhF (1)i → ZhF (0)i (2.1) with every composition the zero map, we obtain a chain complex C∗ by shifting the gradings, Ci = ZhF (i)i[i], and letting ∂i = F (i → i − 1). This construction [n+1] is functorial. That is, if B(Z)• denotes the full subcategory of the functor category B(Z)[n+1] generated by those functors which send every length two arrow to the zero map, and if Kom denotes the category of chain complexes, [n+1] then the above construction is a functor ch: B(Z)• → Kom. Indeed, it would [n+1] be reasonable to call an element of B(Z)• a chain complex in B(Z). [n+1] A linear diagram F ∈ B(Z)• may also be viewed as a cubical diagram G: [2]n → B(Z) by setting

F (i) if v = (0,..., 0, 1,..., 1) G(v)=  n−i i (2.2) ∅ otherwise.  | {z } | {z } On morphisms, G is either zero or induced from F , as appropriate. Conversely, [2]n [n+1] a cubical diagram G ∈ B(Z) gives a linear diagram F ∈ B(Z)• by setting

3 F (i) = |v|=i G(v), where |v| denotes the number of 1’s in v. The component of F (i +1 → i) from ZhG(u)i⊂ ZhF (i + 1)i to ZhG(v)i⊂ ZhF (i)i is ` (−1)u1+···+uk−1 G(u → v) if u − v = e , the kth unit vector, k (2.3) (0 if u − v is not a unit vector. b α [n+1] [2]n These give functors B(Z)• B(Z) with β ◦ α = Id. β n The composition ch ◦ β : B(Z)[2] → Kom is the totalization Tot, and may be viewed as an iterated mapping cone. Up to chain homotopy equivalence, one can also construct Tot using homotopy colimits. Define a category [2]+ by n n adjoining a single object ∗ to [2] and a single morphism 1 →∗; let [2]+ = ([2]+) . n Given G ∈ B(Z)[2] , by treating abelian groups as chain complexes supported in homological grading zero, we get an associated cubical diagram A: [2]n → Kom. n Extend A to a diagram A+ : [2]+ → Kom by setting

A(v) if v ∈ [2]n A+(v)= (2.4) (0 otherwise.

Then the totalization of G is the homotopy colimit of A+ (see [10, 76, 82]).

2.2 Spatial refinements of diagrams of abelian groups As a next step, given a finitely generated chain complex represented by a functor F :[n + 1] → B(Z) we wish to construct a based cell complex with cells in dimensions N,...,N + n whose reduced cellular complex—with distinguished basis given by the cells—is isomorphic to the given complex shifted up by N. Let T (SN ) denote the category with objects finite sets and morphisms N N HomT (SN )(S,T ) the set of all based maps S S → T S between wedges of N-dimensional spheres; applying reduced N th homology to the morphisms W N W produces a functor, also denoted HN , from T (S ) to B(Z). A strict N- N dimensional spatial lift of F is a functor P :[n+1] → T (S ) satisfying HN ◦P = F and P is the constant map on anye length two arrow in [n + 1], i.e., a strict chain complex in T (SN ) lifting F . Just as morphisms in B(Z) are matrices,e if we replace SN by the sphere spectrum S, we may view a morphism in T (S) as a matrix of maps S → S by viewing S S as a coproduct and T S as a product. Given such a linear diagram P , we can construct a based cell complex by taking mapping cones and suspendingW sequentially [14, §5].W If CW denotes the category of based cell complexes, then P induces a diagram X :[n + 1] → CW,

f f −1 f2 f1 X(n) −→n X(n − 1) −→n · · · −→ X(1) −→ X(0) (2.5) with every composition the constant map. Since f1 ◦ f2 is the constant map, there is an induced map g1 : ΣX(2) → Cone(f1) from the reduced suspension

4 to the reduced cone. Then we get a diagram Y :[n] → CW,

Σfn Σf3 g1 Y (n − 1)=ΣX(n) −→ · · · −→ Y (1) = ΣX(2) −→ Y (0) = Cone(fn). (2.6)

Take the mapping cone of g1 and suspend to get a diagram Z :[n − 1] → CW and so on. The reduced cellular chain complex of the final CW complex is the original chain complex, shifted up by N. This construction is also functorial: N [n+1] if T (S )• is the full subcategory generated by the functors which send every length two arrow to the constant map, then the construction is a functor N [n+1] T (S )• → CW. The construction can also be carried out in a single step. Construct a category [n + 1]+ by adjoining a single object ∗ and a unique morphism i →∗ for all i 6= 0. Extend X :[n + 1] → CW to X+ :[n + 1]+ → CW by sending ∗ to a and take the homotopy colimit of X+. N [n+1] n A linear diagram P ∈ T (S )• produces a cubical diagram Q: [2] → T (SN ) by the analogue of Equation (2.2). There is a totalization functor Tot: N [2]n N [n+1] T (S ) → CW extending the functor T (S )• → CW so that

N [n+1] N [2]n T (S )• T (S ) CW ecell C∗ [−N] (2.7) [n+1] [2]n B(Z)• B(Z) Kom. commutes. The totalization functor is defined as an iterated mapping cone or as a homotopy colimit of an extension of Q analogous to Equation (2.4).

2.3 Lax spatial refinements Instead of working with strict functors as in the previous section, sometimes is it more convenient to work with lax functors. A lax or homotopy coherent or (∞, 1) functor F : C → Top is a diagram that commutes up to which are specified, and the homotopies themselves commute up to higher homotopies which are also specified, and so on [15,54,82]. More precisely, F consists of based topological spaces F (x) for x ∈ C , and higher homotopy maps F (fn,...,f1): n−1 f1 fn [0, 1] × F (x0) → F (xn) for composable morphisms x0 −→ · · · −→ xn with certain boundary conditions and restrictions involving basepoints and identity morphisms; the case n = 1 is maps corresponding to the arrows in the diagram, n = 2 is homotopies corresponding to pairs of composable arrows, etc. A strict C functor may be viewed as a lax functor. Let hTop denote the category of lax functors C → Top, with morphisms given by lax functors C ×[2] → Top. (There are also higher morphisms corresponding to lax functors C × [n] → Top.) There is a notion of a lax functor to T (SN ) induced from the notion of lax N [n+1] N [n+1] functors to Top. Let hT (S )• be the subcategory of hT (S ) consisting f1 fn of those objects (respectively, morphisms) F such that F x0 −→ · · · −→ xn is the constant map to the basepoint for any string of morphisms in [n + 1]  (respectively, [n + 1] × [2]) with some fi of length ≥ 2 in [n + 1] (respectively,

5 [n + 1] ×{0, 1}). We call such functors chain complexes in T (SN ). (In Cohen- N n Jones-Segal’s language, chain complexes in T (S ) are functors J0 → T∗.) [n+1] If one starts with a chain complex F ∈ B(Z)• and wishes to refine it to a based cell complex, instead of constructing a strict N-dimensional spatial lift N [n+1] in T (S )• , it is enough to construct a lax N-dimensional spatial lift, that N [n+1] is, a functor P ∈ hT (S )• with HN ◦ P = F . Such a P produces a cell complex by adjoining a basepoint to get a lax diagram [n + 1]+ → CW and then taking homotopy colimits. Alternatively,e we may convert P to a lax cubical [2]n diagram Q ∈ hT (SN ) and proceed as before. The iterated mapping cone construction becomes intricate since the associated diagram X : [2]n → CW is n lax. So, extend to a lax diagram X+ : [2]+ → CW as before and then take its homotopy colimit. This generalization to the lax set-up remains functorial and the analogue of Diagram (2.7) still commutes.

2.4 Framed flow categories Cohen-Jones-Segal first proposed lax spatial refinements of diagrams F :[n + 1] → B(Z) via framed flow categories, using the Pontryagin-Thom construc- tion [14]. A framed flow category is an abstraction of the flows of a Morse-Smale function. Concretely, a framed flow category C consists of: 1. A finite set of objects Ob(C ) and a grading gr: Ob(C ) → Z. After translat- ing, we may assume the gradings lie in [0,n]. 2. For x, y ∈ C with gr(x) − gr(y) − 1 = k, a morphism set M(x, y) which is a k-dimensional hki-manifold. A hki-manifold M is a smooth manifold with corners so that each codimension-c corner point lies in exactly c facets (closure of a codimension-1 component), equipped with a decomposition of k its boundary ∂M = ∪i=1∂iM so that each ∂iM is a multifacet of M (union of disjoint facets), and ∂iM ∩ ∂j M is a multifacet of ∂iM and ∂j M [29, 43].

⊃ (An associative composition map M(y,z) ×M(x, y) ֒→ ∂gr(y)−gr(z)M(x, z .3 M(x, z). Setting M(i, j)= M(x, y), (2.8) x,y gr(x)=ai,gr(y)=j the composition is required to induce an isomorphism of hi−j −2i-manifolds

∂j−kM(i, k) =∼ M(j, k) ×M(i, j). (2.9)

i−j−1 D(i−j) Neat ιi,j : M(i, j) ֒→ [0, 1) × (−1, 1) for some large .4 D ∈ N, namely, smooth embeddings satisfying

−1 j−k−1 i−j−1 D(i−k) ιi,k ([0, 1) ×{0}× [0, 1) × (−1, 1) )= ∂j−kM(i, k) (2.10) and certain orthogonality conditions near boundaries. These embeddings are required to be coherent with respect to composition. The space of such collections of neat embeddings is (D − 2)-connected.

6 a b c d • • • • (−1, 1) (−1, 1) (−1, 1)

[0, 1) [0, 1) [0, 1) 3 (−1, 1) (−1, 1) (−1, 1)

J B I • • dc • • (−1, 1) (−1, 1) Ja [0, 1) ba ca • • dI db

Figure 2.1: A framed flow category C . Ob(C ) = {x,y,z,w} in gradings 3, 2, 1, 0, respectively. The 0-dimensional morphism spaces are: M(x, y)= {a}, M(y,z) = {b,c}, and M(z, w) = {d}, each embedded in (−1, 1). The 1- dimensional morphism spaces are: M(x, z) = I (resp. M(y, w) = J) an in- terval, embedded in [0, 1) × (−1, 1)2, with endpoints {ba,ca} (resp. {db, dc}) embedded in {0}× (−1, 1)2 by the product . The 2-dimensional mor- phism space is a B, embedded in [0, 1)2 × (−1, 1)3; it is a h2i-manifold with 3 boundary decomposed as a union of two arcs, ∂1B = dI ⊂{0}× [0, 1) × (−1, 1) 3 and ∂2B = Ja ⊂ [0, 1) ×{0}× (−1, 1) . Coherent framings of all the normal bundles are represented by the tubular neighborhoods ιi,j . In the last subfigure, (−1, 1)3 is drawn as an interval by projecting to the middle (−1, 1).

5. Framings of the normal bundles of ιi,j , also coherent with respect to com- D(i−j) i−j−1 × (position, which give extensions ιi,j : M(i, j) × [−1, 1] ֒→ [0, 1 (−1, 1)D(i−j).

N [n+1] A framed flow category produces a lax linear diagram P ∈ hT (S )• with N = nD. On objects, set P (i) = {x ∈ C | gr(x) = i}. On morphisms, define the map associated to the sequence m0 → m1 → ··· → mk to be the constant map unless all the arrows are length one. To a sequence of length one arrows, i → i − 1 →···→ j, associate a map

[0, 1]i−j−1 × SN = [0, 1]i−j−1 × [−1, 1]nD/∂ x∈P (i) x∈P (i) _ a (2.11) → SN = [−1, 1]nD/∂ y∈_P (j) y∈aP (j) using ιi,j and the Pontryagin-Thom construction [14, §5]. We can then apply the totalization functor to P to get a cell complex with cells in dimensions N,N +1,...,N + n. As Cohen-Jones-Segal sketch, for a flow category coming from a generic gradient flow of a Morse function, the cell complex produced by the totalization functor is the N th reduced suspension of the Morse cell complex built from the unstable disks of the critical points [14]. Much of the above data can also be reformulated in Pardon’s language of

7 S-modules [67, §4.3]. Let S[n + 1] be the (non-symmetric) multicategory with objects pairs (j, i) of integers with 0 ≤ j ≤ i ≤ n, unique multimorphisms (i0,i1), (i1,i2),..., (ik−1,ik) → (i0,ik) when k ≥ 1, and no other multimor- phisms (cf. shape multicategories from [45]). Let Top be the multicategory of based topological spaces whose multimorphisms X1,...,Xk → Y are maps X1 ∧···∧ Xk → Y . An S-module is a multifunctor S[n + 1] → Top. D(i−j) Given C , define S-modules S, J by setting S(j, i) = ∨y∈P (j)S and D(i−j) J(j, i) = (∨x∈P (i)S ) ∧ J (i, j) where J is the category with objects inte- gers and morphisms J (i, j) the one-point compactification of [0, 1)i−j−1 if i ≥ j (which is a point if i = j) and composition J (j, k)∧J (i, j) → J (i, k) induced by the inclusion map [0, 1)j−k−1 ×{0}× [0, 1)i−j−1 ֒→ ∂[0, 1)i−k−1 [14, §5]. On a multimorphism (i0,i1),..., (ik−1,ik) → (i0,ik), S sends the (y0,...,yk−1) ∈ D(i1−i0) D(ik−ik−1) P (i0) ×···× P (ik−1) summand S ∧···∧ S homeomorphically D(ik −i0) to the y0 summand S , and J sends the (x1,...,xk) ∈ P (i1)×···×P (ik) D(i1−i0) D(ik −ik−1) summand (S ∧···∧ S ) ∧ (J (i1,i0) ∧···∧ J (ik,ik−1)) to the D(ik−i0) xk summand S ∧ J (ik,i0), homeomorphically on the first factor, and using the composition in J on the second factor. Given a neat embedding of ν C , we can define another S-module N by setting N(j, i)= ιi,j , the Thom space of the normal bundle of ιi,j , if i > j. (When i = j, N(j, j) is a point.) On multi- morphisms, N is induced from the inclusion maps im(ιi1,i0 )×···×im(ιik ,ik−1 ) → im(ιik,i0 ). The Pontryagin-Thom collapse map is a natural transformation—an S-module map—from J to N, which sends the x ∈ P (i) summand of J(j, i) ν to the Thom-space summand ∪y∈P (j)ιx,y in N(j, i). A framing of C produces ν another S-module map N → S which sends the summand ιx,y in N(j, i) to the y summand of S(j, i). Composing we get an S-module map J → S, which is N [n+1] precisely the data needed to recover a lax diagram in hT (S )• . Finally, as popularized by Abouzaid, note that since the (smooth) framings ν D(i−j) of ιi,j were only used to construct maps ιi,j → ∨y∈P (j)S , a weaker struc- ture on the flow category—namely, coherent trivializations of the Thom spaces ν ιi,j as spherical fibrations—might suffice.

2.5 Speculative digression: matrices of framed Perhaps it would be tidy to reformulate the notion of stably framed flow cat- egories as chain complexes in some category B(Cob) equipped with a functor B(Cob) → T (S). It is clear how such a definition would start. Objects in B(Cob) should be finite sets. By the Pontryagin-Thom construction, a map S → S is determined by a framed 0-manifold; therefore, a morphism in B(Cob) should be a matrix of framed 0-manifolds. To account for the homotopies in T (S), B(Cob) should have higher morphisms. For instance, given two (T × S)- matrices A, B of framed 0-manifolds, a 2-morphism from A to B should be a (T × S)-matrix of framed 1-dimensional cobordisms. Given two such (T × S)- matrices M,N of framed 1-dimensional cobordisms, a 3-morphism from M to N should be a (T ×S)-matrix of framed 2-dimensional cobordisms with corners, and so on. That is, the target category Cob seems to be the extended

8 category, an (∞, ∞)-category studied, for instance, by Lurie [55]. Since matrix multiplication requires only addition and multiplication, the construction B(C ) makes sense for any rig or symmetric bimonoidal category C and, presumably, for a rig (∞, ∞)-category, for some suitable definition; and perhaps the framed cobordism category Cob is an example of a rig (∞, ∞)- category. Maybe the Pontryagin-Thom construction gives a functor B(Cob) → T (S), and that a stably framed flow category is just a functor [n+1] → B(Cob). Rather than pursuing this, we will focus on a tiny piece of Cob, in which all 0-manifolds are framed positively, all 1-dimensional cobordisms are trivially- framed intervals and, more generally, all higher cobordisms are trivially-framed disks. In this case, all of the information is contained in the objects, 1-morphisms, and 2-morphisms, and this tiny piece equals B(Sets) with Sets being viewed as a rig category via disjoint union and Cartesian product.

2.6 The cube and the Burnside category The Burnside category B (associated to the trivial group) is the following weak 2-category. The objects are finite sets. The 1-morphisms Hom(S,T ) are T × S matrices of finite sets; composition is matrix multiplication, using the disjoint union and product of sets in place of + and × of real numbers. The 2-morphisms are matrices of entrywise bijections between matrices of sets. (The category B is denoted S2 by Hu-Kriz-Kriz [26], and is an example of what they call a ⋆-category. The realization procedure below is a concrete analogue of the Elmendorf-Mandell machine [17]; see also [44, §8].) There is an abelianization functor Ab: B → B(Z) which is the identity on objects and sends a morphism (At,s)s∈S,t∈T to the matrix (#At,s)s∈S,t∈T ∈ n ZT ×S. We are given a diagram G ∈ B(Z)[2] which we wish to lift to a diagram [2]n Q ∈ hT (SN ) . As we will see, it suffices to lift G to a diagram D : [2]n → B. Since B is a weak 2-category, we should first clarify what we mean by a diagram in B. A strictly unital lax 2-functor—henceforth just called a lax functor—D : [2]n → B consists of the following data: 1. A finite set F (x) ∈ B for each v ∈ [2]n.

2. An F (v)×F (u)-matrix of finite sets F (u → v) ∈ HomB(F (u), F (v)) for each u > v ∈ [2]n.

3. A 2-isomorphism Fu,v,w : F (v → w) ◦1 F (u → v) → F (u → w) for each n u>v>w ∈ [2] so that for each u>v>w>x, Fu,w,z ◦2 (Id ◦1Fu,v,w) = (Fv,w,z ◦1 Id)◦2 Fu,v,z, where ◦i denotes composition of i-morphisms (i =1, 2). Next we turn such a lax diagram D : [2]n → B into a lax diagram Q ∈ n N [2] hT (S ) , N ≥ n + 1, satisfying HN ◦ Q = Ab ◦ D. Associate a box Bx = [−1, 1]N to each x ∈ D(v), v ∈ [2]n. For each u > v, let D(u → v) = (Ay,x)x∈D(u),y∈D(v) and let E(u →ev) be the space of embeddings ιu,v = {ιu,v,x}x∈D(u) where

(ιu,v,x : Ay,x × By ֒→ Bx (2.12 y a

9 whose restriction to each copy of By is a sub-box inclusion, i.e., composition of a translation and dilation. The space E(A) is (N − 2)-connected. For any such data ιu,v, a collapse map and a fold map give a box map

N N ιu,v : S = Bx/∂ → By/∂ → By/∂ = S . (2.13) x∈D(u) x∈D(u) x∈D(u) y∈D(v) y∈D(v) _ a y∈aD(v) a _ b a∈Ay,x

Given u>v>w and data ιu,v, ιv,w, the composition ιv,w ◦ ιu,v is also a box map corresponding to some induced embedding data. [2]n The construction of the lax diagram Q ∈ hT (SNb) isb inductive. On objects, Q agrees with D. For (non-identity) morphisms u → v, choose a box N N map Q(u → v)= ιu,v : x∈D(u) S → y∈D(v) S refining D(u → v). Staying in the space of box maps, the required homotopies exist and are unique up to W W homotopy becauseb each E(A) is N − 2 ≥ n − 1 connected, and there are no sequences of composable morphisms of length >n − 1. (See [44] for details.) The above construction closely follows the Pontryagin-Thom procedure from §2.4. Indeed, functors from the cube to the Burnside category correspond to certain kinds of flow categories (cubical ones), and the realizations in terms of box maps and cubical flow categories agree [44].

3 Khovanov homology 3.1 The Khovanov cube Khovanov homology [36] is defined using the Frobenius algebra V = H∗(S2). 0 2 2 2 Let x− ∈ H (S ) and x+ ∈ H (S ) be the positive generators. (Our labeling is opposite Khovanov’s convention, as the maps in our cube go from 1 to 0.) Via the equivalence of Frobenius algebras and (1 + 1)-dimensional topological field theories [3], we can reinterpret V as a functor from the (1 + 1)-dimensional 1+1 bordism category Cob to B(Z) that assigns {x+, x−} to circle, and hence

π0(C){x+, x−} to a one-manifold C. For x ∈ V (C), let kxk+ (respectively, kxk ) denote the number of circles in C labeled x (respectively, x ) by x. Q − + − For a cobordism Σ: C1 → C0, the map V (Σ): ZhV (C1)i = ⊗π0(C1)Zhx+, x−i →

ZhV (C0)i = ⊗π0(C0)Zhx+, x−i is the tensor product of the maps induced by the connected components of Σ; and if Σ: C1 → C0 is a connected, genus-g cobordism, then the (y, x)-entry of the matrix representing the map, x ∈ V (C1), y ∈ V (C0), is 1 if g = 0, kxk+ + kyk− = 1, 2 if g = 1, kxk+ = kyk− = 0, (3.1) 0 otherwise

(cf. [5, 26]).  Now, given a link diagram L with n crossings numbered c1,...,cn, Khovanov n n 1+1 constructs a cubical diagram GKh = V ◦ L ∈ B(Z)[2] where L: [2] → Cob

10 is the cube of resolutions (extending Kauffman [32]) defined as follows [36]. For v ∈ [2]n, let L(v) be the complete resolution of the link diagram L formed by th resolving the i crossing ci by the 0-resolution if vi = 0 and by the 1-resolution if vi = 1. For a morphism u → v, L(u → v) is the cobordism which is an elementary saddle from the 1-resolution to the 0-resolution near crossings ci for each i with ui > vi, and is a product cobordism elsewhere. The dual of the resulting total complex, shifted by n−, the number of nega- tives crossings in L, is usually called the Khovanov complex

∗ CKh (L) = Dual(Tot(GKh ))[n−], (3.2) and its cohomology Kh∗(L) the Khovanov homology, which is a link invariant. There is an internal grading, the quantum grading, that comes from placing the two symbols x+ and x− in two different quantum gradings, and the entire com- plex decomposes along this grading, so Khovanov homology inherits a second i i,j grading Kh (L)= ⊕jKh (L), and its quantum-graded

(−1)iqj rank(Khi,j (L)) (3.3) i,j X recovers the unnormalized Jones polynomial of L. The quantum grading persists in the space-level refinement but, for brevity, we suppress it.

3.2 The stable homotopy type Following §2.6, to give a space-level refinement of Khovanov homology it suffices n to lift GKh to a lax functor [2] → B. One can show that the TQFT V : Cob1+1 → B(Z) does not lift to a functor Cob1+1 → B [26, §3.2]. However, we may instead work with the embedded 1+1 cobordism category Cobe , which is a weak 2-category whose objects are closed 1-manifolds embedded in S2, morphisms are compact cobordisms embedded in S2 × [0, 1], and 2-morphisms are isotopy classes of isotopies in S2 × [0, 1] rel n 1+1 n boundary. The cube L: [2] → Cob factors through a functor Le : [2] → 1+1 Cobe . (This functor Le is lax, similar to what we had for functors to the Burnside category except without strict unitarity.) So it remains to lift V to a 1+1 (lax) functor Ve : Cobe → B

V 1+1 e Le Cobe B. [2]n (3.4) L V Cob1+1 B(Z)

On an embedded one-manifold C, we must set

Ve(C)= {x+, x−}. (3.5) πY0(C)

11 2 For an embedded cobordism Σ: C1 → C0 with Ci embedded in S ×{i}, the matrix Ve(Σ) is a tensor product over the connected components of Σ, i.e., m j j if Σ = ∐j=1 Σj : C1,j → C0,j and (y , x ) ∈ Ve(C0,j ) × Ve(C1,j ), then the (Πm yj , Πm xj ) entry of V (Σ) equals j=1 j=1 e 

Ve(Σ1)y1,x1 ×···× Ve(Σm)ym,xm . (3.6)

And finally, if Σ: C1 → C0 is a connected genus-g cobordism, then for x ∈ Ve(C1) and y ∈ Ve(C0), the (y, x)-entry of Ve(Σ) must be a

1-element set if g = 0, kxk+ + kyk− = 1, 2-element set if g = 1, kxk+ = kyk− = 0, (3.7) ∅ otherwise.

One-element sets do not have any non-trivial automorphisms, so we may set all the one-element sets to {pt}. The two-element sets must be chosen carefully: they have to behave naturally under isotopy of cobordisms (the 2-morphisms 1+1 ′ ∼ ′ in Cobe ) and must admit natural isomorphisms Ve(Σ ◦ Σ ) = Ve(Σ) ◦ Ve(Σ ) ′ when composing cobordisms Σ : C2 → C1 and Σ: C1 → C0. Decompose S2 × [0, 1] as a union of two compact 3-manifolds glued along Σ, A ∪Σ B. Set Ve(Σ) to be the (cardinality two) set of unordered bases {α, β} for ker(H1(Σ) → H1(∂Σ)) =∼ Z2 so that α (respectively, β) is the restric- tion of a generator of ker(H1(A) → H1(A ∩ (S2 ×{0, 1}))) =∼ Z (respectively, ker(H1(B) → H1(B ∩ (S2 ×{0, 1}))) =∼ Z), and so that, if we orient Σ as the boundary of A then hα ∪ β, [Σ]i = 1 (or equivalently, if we orient Σ as the boundary of B then hβ ∪ α, [Σ]i = 1). This assignment is clearly natural. ′ Given cobordisms Σ : C2 → C1 and Σ: C1 → C0, we need to construct a ′ ′ natural 2-isomorphism Ve(Σ) ◦ Ve(Σ ) → Ve(Σ ◦ Σ ). The only non-trivial case is when Σ and Σ′ are genus-0 cobordisms gluing to form a connected, genus-1 cobordism. In that case, letting x ∈ Ve(C2) (respectively, y ∈ Ve(C0)) denote the generator that labels all circles of C2 by x− (respectively, all circles of C0 by x+), we need to construct a bijection between the (y, x)-entry My,x of ′ ′ Ve(Σ) ◦ Ve(Σ ) and the (y, x)-entry Ny,x of Ve(Σ ◦ Σ ). Consider an element Z of My,x; Z specifies an element z ∈ V (C1). There is a unique circle C in C1 ′ that is non-separating in Σ ◦ Σ and is labeled x+ by z. Choose an orientation o of Σ ◦ Σ′, orient C as the boundary of Σ, and let [C] denote the of C ′ ′ in H1(Σ ◦ Σ , ∂(Σ ◦ Σ )). Assign to Z the unique basis in Ny,x that contains the Poincar´edual of [C]. It is easy to check that this map is well-defined, independent of the choice of o, natural, and a bijection. 1+1 This concludes the definition of the functor Ve : Cobe → B. The spatial n N [2] n lift QKh ∈ hT (S ) is then induced from the composition Ve ◦Le : [2] → B. Totalization produces a cell complex Tot(QKh ) with

∗ ∗ Ccell(Tot(QKh ))[N + n−]= CKh (L). (3.8) th We define the Khovanove spectrum XKh (L) to be the formal (N + n−) desus- pension of Tot(QKh ). The stable homotopy type of XKh (L) is a link invari-

12 ant [26, 44, 49]. The spectrum decomposes as a wedge sum over quantum grad- j ings, XKh (L) = j XKh (L). There is also a reduced version of the Khovanov stable homotopyW type, XKh (L), refining the reduced Khovanov chain complex. 3.3 Properties ande applications In order to apply the Khovanov homotopy type to knot theory, one needs to extract some concrete information from it beyond Khovanov homology. Doing so, one encounters three difficulties: .. 1. The number of vertices of the Khovanov cube is 2n, where n is the number of crossings of L, so the number of cells in the CW complex XKh (L) grows at least that fast. So, direct computation must be by computer, and for relatively low crossing number links. .. 2. For low crossing number links, Khi,j (L) is supported near the diagonal j 2i − j = σ(L), so each XKh (L) has nontrivial homology only in a small number of adjacent gradings, and these Khi,j (L) have no p-torsion for p> 2. If X is a spectrum so that Hi(X) is nontrivial only for i ∈{k, k+1} then the homotopy type of X is determined by H∗(X), while if H∗(X) is nontrivial in only three adjacente gradings and has no p-torsion (p> 2) then the homotopy type of X is determined by He∗(X) and the Steenrode operations Sq1 and Sq2 [8, Theorems 11.2, 11.7]. e .. 3. There are no known formulas for most algebro-topological invariants of a CW complex. (The situation is a bit better for simplicial complexes.) 2 i,j One can find an explicit formula for the operation Sq : Kh (L; F2) → i+2,j 1 Kh (L; F2)[51]. The operation Sq is the Bockstein, and hence easy to j compute. Using these, one can determine the spectra XKh (L) for all prime links up to 11 crossings. All these spectra are wedge sums of (de)suspensions of 6 ba- sic pieces (cf. .. 2), and all possible basic pieces except CP 2 occur (see . 1). The i,j first knot for which XKh (K) is not a Moore space is also the first non-alternating knot: T (3, 4). Extending these computations, Seed showed: Theorem 1. [75] There are pairs of knots with isomorphic Khovanov coho- mologies but non-homotopy equivalent Khovanov spectra. n n The first such pair is 1170 and 132566. Jones-Lobb-Schuetz introduced moves and simplifications allowing them to give a by-hand computation of Sq2 for T (3, 4) and some other knots [31].

Theorem 2. [44] Given links L, L′, XKh (L ∐ L′) ≃ XKh (L) ∧ XKh (L′) and, if L and L′ are based, XKh (L#L′) ≃ XKh (L) ∧ XKh (L′) and XKh (L#L′) ≃ ′ XKh (L) ∧XKh (U) XKh (L ). Finally, if m(L) is the mirror of L then XKh (m(L)) is the Spanier-Whiteheade dual to XKh (L)e. e Corollary 3.1. For any integer k there is a knot K so that the operation Sqk : Kh∗,∗(K) → Kh∗+k,∗(K) is nontrivial. (Compare . 3.)

13 Proof. Choose a knot K0 so that in some quantum grading, Kh(K0) has 2- i n torsion but Kh(K0; F2) has vanishing Sq for i> 1. (For instance, K0 = 133663 k f k works [79].)f Let K = K0# ··· #K0. By the Cartan formula, Sq (α) 6= 0 for some α ∈ Kh(K; F2). Therez is}| a short{ exact sequence 0 → Kh(K; F ) → Kh(K; F ) → Kh(K; F ) → 0 f 2 2 2 [70, §4.3] induced by a cofiber sequence of Khovanov spectra [49, §8], so if f f k β ∈ Kh(K; F2) is any preimage of α then by naturality, Sq (β) 6= 0, as well. Plamenevskaya defined an invariant of links L in S3 transverse to the stan- dard contact structure, as an element of the Khovanov homology of L [68]. Theorem 3. [48] Given a transverse link L in S3 there is a well-defined coho- motopy class of XKh (L) lifting Plamenevskaya’s invariant. While Plamenevskaya’s class is known to be invariant under flypes [48], the homotopical refinement is not presently known to be. It remains open whether either invariant is effective (i.e., stronger than the self-linking number). The Steenrod squares on Khovanov homology can be used to tweak Ras- mussen’s concordance invariant and slice-genus bound s [71] to give potentially new concordance invariants and slice genus bounds [50]. In the simplest case, Sq2, these concordance invariants are, indeed, different from Rasmussen’s in- variants [50]. They can be used to give some new results on the 4-ball genus for certain families of knots [44]. More striking, Feller-Lewark-Lobb have used these operations to resolve whether certain knots are squeezed, i.e., occur in a minimal-genus cobordism between positive and negative knots [20]. In a different direction, the Khovanov homotopy type admits a number of extensions. Lobb-Orson-Sch¨utz and, independently, Willis proved that the Kho- vanov homotopy type stabilizes under adding twists, and used this to extended it to a colored Khovanov stable homotopy type [53,83]; further stabilization re- sults were proved by Willis [83] and Islambouli-Willis [28]. Jones-Lobb-Sch¨utz proposed a homotopical refinement of the sln Khovanov-Rozansky homology for a large class of knots [30] and there is also work in progress in this direction of Hu-Kriz-Somberg [27]. Sarkar-Scaduto-Stoffregen [72] have given a homotopical refinement of Ozsv´ath-Rasmussen-Szab´o’s odd Khovanov homology [64]. The construction of the functor Ve is natural enough that it can be used [45] to give a space-level refinement of Khovanov’s arc algebras and tangle invari- ants [37]. In the refinement, the arc algebras are replaced by ring spectra (or, if one prefers, spectral categories), and the tangle invariants by module spectra.

3.4 Speculation We conclude with some open questions: . 1. Does CP 2 occur as a wedge summand of the Khovanov spectrum associated to some link? (Cf. §3.3.) More generally, are there non-obvious restrictions on the spectra which occur in the Khovanov homotopy types?

14 . 2. Is the obstruction to amphichirality coming from the Khovanov spectrum stronger than the obstruction coming from Khovanov homology? Presum- ably the answer is “yes,” but verifying this might require interesting new computational techniques. . 3. Are there prime knots with arbitrarily high Steenrod squares? Other power operations? Again, we expect that the answer is “yes.” . 4. How can one compute Steenrod operations, or stable homotopy invariants beyond homology, from a flow category? (Compare [51].) . 5. Is the refined Plamenevskaya invariant [48] effective? Alternatively, is it invariant under negative flypes / SZ moves? . 6. Is there a well-defined homotopy class of maps of Khovanov spectra asso- ciated to an isotopy class of link cobordisms Σ ⊂ [0, 1] × R3? Given such a cobordism Σ in general position with respect to projection to [0, 1], there is an associated map [68], but it is not known if this map is an isotopy invariant. More generally, one could hope to associate an (∞, 1)-functor from a quasicategory of links and embedded cobordisms to a quasicategory of spectra, allowing one to study families of cobordisms. If not, this is a sense in which Khovanov homotopy, or perhaps homology, is unnatural. Applications of these cobordism maps would also be interesting (cf. [81]). . 7. If analytic difficulties are resolved, applying the Cohen-Jones-Segal con- struction to Seidel-Smith’s symplectic Khovanov homology [77] should also give a Khovanov spectrum. Is that symplectic Khovanov spectrum homo- topy equivalent to the combinatorial Khovanov spectrum? (Cf. [2].) . 8. The (symplectic) Khovanov complex admits, in some sense, an O(2)-action [25, 57, 73, 78]. Does the Khovanov stable homotopy type? . 9. Is there a homotopical refinement of the Lee [47] or Bar-Natan [5] deforma- tion of Khovanov homology? Perhaps no genuine spectrum exists, but one can hope to find a lift of the theory to a module over ku or ko or another ring spectrum (cf. [12]). Exactly how far one can lift the complex might be predicted by the polarization class of a partial compactification of the symplectic Khovanov setting [77]. . 10. Can one make the discussion in §2.5 precise? Are there other rig (or ∞-rig) categories, beyond Sets, useful in refining chain complexes in categorifica- tion or Floer theory to get modules over appropriate ring spectra?

. 11. Is there an intrinsic, diagram-free description of XKh (K) or, for that mat- ter, for Khovanov homology or the Jones polynomial?

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