Khovanov homotopy type, periodic links and localizations

Given an m-periodic link \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L\subset S^3$$\end{document}L⊂S3, we show that the Khovanov spectrum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {X}_L$$\end{document}XL constructed by Lipshitz and Sarkar admits a group action. We relate the Borel cohomology of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {X}_L$$\end{document}XL to the equivariant Khovanov homology of L constructed by the second author. The action of Steenrod algebra on the cohomology of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {X}_L$$\end{document}XL gives an extra structure of the periodic link. Another consequence of our construction is an alternative proof of the localization formula for Khovanov homology, obtained first by Stoffregen and Zhang. By applying the Dwyer–Wilkerson theorem we express Khovanov homology of the quotient link in terms of equivariant Khovanov homology of the original link.

A link L in S 3 is said to be m-periodic if there exists an orientation-preserving action of a cyclic group Z m on S 3 such that L is an invariant subset of S 3 and the fixed point set is an unknot disjoint from L. A diagram D of a link L is called m-periodic if 0 / ∈ D and D is invariant under rotation ρ m of order m of the plane about the point 0 ∈ R 2 . The Khovanov complex of an m-periodic link admits an induced action of Z m [8,38].
Removing a tubular neighborhood of the fixed point axis F of the rotation of S 3 produces an annular link L ⊂ S 1 × D 2 invariant under a fixed point-free rotation of S 1 × D 2 . Such links in S 1 × D 2 are also called m-periodic. The Z m -action on the Khovanov complex preserves the filtration (1.1), hence it descends to a Z m -action on the annular Khovanov chain complex.
The primary purpose of this paper is to study the Khovanov homotopy type and the annular Khovanov homotopy type of periodic links. The following two theorems constitute the central geometric part of the present article.   If we set m = p a prime number, we have the following result.

b) If D p is a p-periodic link diagram and D is the associated quotient diagram. Then
for any q ∈ Z From Theorems 1.2 and 1.3, we can obtain nontrivial relations between the (annular) Khovanov homology of a periodic link and the annular Khovanov homology of the quotient thereof. The simplest forms of the relation are the following versions of the Smith inequality. Theorem 1.4 Let p be a prime and let L p be a p-periodic link with associated quotient link L. Then, for every q, k ∈ Z the following holds i dim F p AKh i, pq−( p−1)k,k (L p ; F p ) ≥ j dim F p AKh j,q,k (L; F p ). Theorem 1.5 For any p-periodic link L p ⊂ S 3 and any q ∈ Z we have Lipshitz and Sarkar [34] showed that the action of stable cohomology operations on the Khovanov homology might lead to substantially stronger link invariants. Similarly, in our case, stable cohomology operations can be used to strengthen Theorems 1.4 and 1.5. As a corollary of our construction, we obtain Theorem 8.10, which gives a functorial way to determine the annular Khovanov homology of the quotient link from the equivariant (annular) Khovanov homology of a periodic link. To be more precise, we show in Theorem 8.3 that equivariant (annular) Khovanov homology is isomorphic to the Borel equivariant cohomology of X Kh (respectively X AKh ). Careful analysis of the action of the Steenrod algebra on appropriately localized Borel cohomology, see [14], recovers the cohomology of the fixed point set X

General context
Since the advent of various homological invariants for three-manifolds or knots in three-manifolds, there has been a question on the behavior of these invariants under passing to the quotient by a group action. One direction of the research was in the Floer theory. Early results in knot Floer homology include Levine's paper [30], which was later used by Hendricks to obtain a rank inequality for knots in double branched covers (see [18]). More recent advances in this direction include another paper of Hendricks [19], and finally, a paper by Lidman and Manolescu [31], where Smithtype inequalities are obtained for monopole Floer and Heegaard Floer homologies.
For Khovanov homology theory, the first localization results were obtained by Seidel and Smith [44], where the authors used their own definition of Khovanov homology based on the Lagrangian Floer theory [43]. Note that the equivalence of the Seidel-Smith Khovanov homology with the original one is still conjectural in positive characteristic. Motivated by their results, Hendricks et al. [20] constructed equivariant Lagrangian Floer theory for more general groups.
In order to study Khovanov homology using techniques from algebraic topology, it is convenient to realize Khovanov homology of a link as the singular homology of a topological space. In a series of papers, Lipshitz and Sarkar, later also with Lawson, defined and studied the so-called Khovanov homotopy type [28,[33][34][35] (see also [9] for a review in a language of algebraic topology) with the property that cohomology of the space is the Khovanov homology of a link. A question remained whether Khovanov homotopy type (sometimes called 'Khomotopy type') that they constructed admits a group action if the underlying link is periodic.
The affirmative answer was given in the first version of this paper, and, independently by Stoffregen and Zhang [46]. The updated version of this paper contains proofs of fixed point results, which were not present in the first version. In particular, Theorems 1.2 and 1.3 were proved first by Stoffregen and Zhang. Note that the two constructions, even though they lead to the same result, are of substantially different nature. Stoffregen and Zhang use the approach to Khovanov homotopy type via the Burnside category [28]. Conceptually, this approach seems to require more case-bycase analysis. On the other hand, Burnside rings have deep connections with ordinary homology theory and Mackey functors; see the book of Costenoble and Waner [11]. Therefore, the construction of equivariant Khovanov homology via Burnside rings has the potential of revealing deeper structure in the equivariant Khovanov homology.
On the contrary, our approach is very concrete and down-to-earth. Most of the arguments reduce either to the Riemann-Hurwitz formula or to Counting Moduli Lemma 6.6, which is a direct application of relations in Cob 3 •/l . Moreover, we give a specific and conceptual reason why the fixed point category of the Khovanov flow category is the annular Khovanov flow category and not just the Khovanov flow category; see Sect. 7, especially Lemma 7.5.
Even more important is that we get an explicit cell decomposition of geometric realizations. Consequently, without much effort we obtain an identification of the chain complex C * (X Kh ) with the Khovanov chain complex CKh as R[Z m ] modules (for some ring R); see Proposition 8.2. It follows that Borel homology of the geometric realization is the equivariant Khovanov homology defined by Politarczyk. Theorem 8.3 might seem to have complicated proof, but this is because we have rather general assumptions on the coefficient module. Finally, methods of algebraic topology, like the Dwyer-Wilkerson theorem, allow us to recover the annular homology of the quotient link in terms of the equivariant Khovanov homology of the original link; see Theorem 8. 10. The latter result is not present in the Stoffregen-Zhang paper. Furthermore, to the best of our understanding, passing from the results of Stoffregen and Zhang to Theorem 8.10 might require a few steps.
We expect that the equivariant homotopy type of Stoffregen-Zhang is equivariantly homotopy equivalent to our construction. We do not have proof of that fact.
The special case ( p = 2) of Theorem 1.4 was proved by Zhang [51]. She also proved Corollary 1.5 for p = 2 and certain classes of periodic links.

Outline of the paper
Our construction of the equivariant Khovanov homotopy type is based on the construction of the Khovanow homotopy type via cubical flow categories [28], which is a simplification of the original construction [33]. We consider an equivariant version of cubical flow category, called equivariant cubical flow category (see Sect. 3.4). A remarkable difference from the non-equivariant definition is that the grading function gr is replaced by an equivariant grading function gr G taking values in the representation ring RO(G). Consequently, the moduli spaces are expected to be of dimension gr G (x) − gr G (y) (refer to Definition A.7 for the definition of "dimension" in this setting). This approach is motivated by the construction of ordinary (Bredon) homology theory [11], and it makes the construction of equivariant Khovanov homotopy type significantly simpler.
After defining equivariant cubical flow categories and a suitable generalization of the notion of a neat embedding to the equivariant case, we construct the equivariant Khovanov homotopy type. Thanks to the choice of the grading function, this part of the construction is straightforward.
To show invariance under the choice of link diagram, we need to do substantially more work. The key tool is, as in [33], the Whitehead theorem, but in the equivariant case, the assumptions of the Whitehead theorem are much harder to verify. In particular, before proving invariance, we have to study fixed points of the equivariant cubical flow category; Sect. 3.8 is devoted to this study. Apart from that, the invariance of the group action on the choice of the diagram is proved analogously as in the non-equivariant case.
The fixed point theorem requires even more work. From Sect. 3.8, we know that the fixed point category is a cubical flow category, but we need to show that this category is the (annular) Khovanov flow category of the associated quotient link. This is the statement of Theorem 7.1. The proof requires a more in-depth understanding of topological and combinatorial properties of the morphism spaces M(x, y). The general idea is to use Bar-Natan's formulation of the Khovanov theory in terms of dotted cobordisms. A moduli space M(x, y) is nontrivial if there exists a suitable cobordism between resolution configurations. Counting Moduli Lemma 6.6 expresses the number of connected components of the moduli spaces in terms of the genera of the components of . If we pass to a cover, we can use the Riemann-Hurwitz Theorem to study the genus of the cover of the cobordism. Then, Bar-Natan's formalism allows us to relate the moduli spaces of the periodic link and the moduli space of its quotient link.
Next, we pass to homological statements. Our primary tool is the BQAS (Borel-Quillen-Atiyah-Segal) Localization Theorem [3,40] and a Smith-type inequality [45] which relates the rank of the homology groups of a periodic knot with the rank of the homology group of the quotient knot. As an immediate corollary of Theorem 1.2 we obtain Smith inequalities for (annular) Khovanov homology.
While analogs of the BQAS Localization Theorem recover only the rank of the homology of the quotient knot, by applying more refined tools from algebraic topology we obtain a significantly stronger result. Indeed, using the result of Dwyer and Wilkerson [14], it is possible to give a complete description of the Khovanov homology of the quotient knot in terms of the equivariant Khovanov homology of a p-periodic knot, for a prime p. By Theorem 8.3, the Borel cohomology of X Kh (D) can be identified with the equivariant Khovanov homology EKh * , * (L; F p ) introduced by the second author [38]. Repeating the construction of [38] one can obtain the equivariant annular Khovanov homology EAKh * , * , * (L; F p ), which, by an analog of Theorem 8.3, is isomorphic to the Borel cohomology of X AKh (L). Therefore, EKh * , * (L; F p ) and EAKh * , * , * (L; F p ) admit an action of the cohomology algebra H * (BZ p ; F p ), of the classifying space of Z p and the action of the mod p Steenrod algebra A p . These two algebraic structures are sufficient to recover the annular Khovanov homology of the quotient knot from equivariant annular Khovanov homology of the periodic knot, as shown in Theorem 8. 10.
The structure of the paper is as follows. Section 2 recalls the construction of Lipshitz and Sarkar. The reader familiar with the construction can skim through this section, maybe except Sect. 2.2, where the degree of the cover map f is expressed in terms of maximal chains in suitably defined posets. Section 3 generalizes the construction of a geometric realization of a cubical flow category to the construction of a geometric realization of an equivariant cubical flow category. The results in this section are stated for general equivariant flow categories and general finite groups. Section 4 deals with Khovanov homotopy type. We construct the equivariant Khovanov flow category as well as its annular analog. We show that passing to geometric realization yields a space that is independent of various choices up to equivariant stable homotopy equivalence. This independence is proved in Sect. 5. In Sect. 6, we make preparatory steps to prove the fixed-point theorems. We recall Bar-Natan's construction of Khovanov homology via Cob 3 •/l -category and use this construction to establish Counting Moduli Lemma 6.6, which computes the number of connected components of the moduli space in terms of the genus of the cobordism in Bar-Natan's setting. Sect. 7 proves Categorical Fixed Point Theorem (Theorem 7.1).
In Sect. 8, we change the setting and deal with homologies of geometric realizations. We show that Borel homology of the equivariant geometric realization of the Khovanov category coincides with Politarczyk's equivariant Khovanov homology of a periodic link (Theorem 8.3). The Dwyer-Wilkerson theory allows us to calculate the Khovanov homology of a quotient link in terms of the equivariant Khovanov homology of the associated periodic link, see Theorem 8.10.
Some technical results are moved to the Appendix. In Appendix A, we review the definitions of manifolds with corners, while in Appendix B, we review the definition and basic properties of permutohedra. We also establish a technical result, Proposition B.11, which essentially says that the intersection of a permutohedron with a hyperplane is a permutohedron of lower dimension. To the best of our knowledge, it is a result not known in the literature. A consequence of this technical fact is Proposition B.18. It states that if a group acts on R n by permuting coordinates, a fixed point set of a permutohedron is again a permutohedron.
Finally, we note that we present detailed examples of computations in a forthcoming paper [4].

Flow categories
In this section we use the notion of an n -manifold introduced in the Appendix A. The necessary background on permutohedra is given in Appendix B.

Definition 2.1
A flow category is a topological category C such that the set of objects is finite, discrete, and is equipped with a grading function gr C : Ob(C) → Z. Morphism spaces satisfy the following three conditions: Moreover, for any x, y ∈ Ob(C) we define the moduli space from x to y as If τ ∈ Z, we define the τ -th suspension of C, τ (C), to be the flow category with the same objects and morphisms and associated grading function Definition 2.2 (see [28,Section 3.1]) The cube flow category Cube(n), for n ∈ Z + , is the flow category such that: (1) Ob(Cube(n)) = {0, 1} n with grading defined by The set of objects of Cube(n) can be partially ordered: For two objects u > v of the flow category with gr(u) − gr(v) = d we define is defined with the aid of identification from Lemma B.4. Namely, for a triple of objects u > w > v such that gr(u) − gr(w) = k, gr(w) − gr(v) = l, there exists a 1 , a 2 , . . . , a k+l ∈ {1, 2 . . . , n} with a 1 < a 2 < · · · < a k+l , such that We use the notation 0 n = (0, . . . , 0) ∈ Ob(Cube(n)) and 1 n = (1, . . . , 1) ∈ Ob(Cube(n)).

Example 2.3
In [33, Definition 3.14] there is described a method to assign a flow category C f to every Morse-Smale function f : M → R, where M is a smooth compact manifold. Objects of C f are critical points of f , the grading of an object is the index of the associated critical point, and the morphism spaces are moduli spaces of non-parametrized gradient flow lines of f .
The n-dimensional cube [0, 1] n can be equipped with the structure of a CW-complex with cells denote the cellular cochain complex of the cube associated to the CW-structure described above.

Posets associated to cubical flow categories
The goal of this subsection is to calculate combinatorially the degree of the map f x,y : M C (x, y) → M Cube (f(x), f(y)). Proposition 2.13 is a step in establishing Counting Moduli Lemma 6.6 below, which is needed to prove the Categorical Fixed Point Theorem (Theorem 7.1).
Let P be a finite poset. A chain in P is a linearly ordered subset of P. A chain is called maximal if it is maximal with respect to the inclusion relation. We denote by max(P) the set of maximal chains of P.
, then we say that c is a full chain if w 1 = u and w k = v. Every maximal chain is necessarily full.
We write P(Cube(n)) for the poset of all objects of Cube(n). While P(Cube(n)) = Ob(Cube(n)), we use the notation P(Cube(n)) whenever we want to emphasize the partial order on the objects of the cube category. Choose More generally, every full chain in P(u, v) determines a face of s−1 . Namely, to a full chain u > w 1 > · · · > w k > v, we associate the face which is the image of the composition map (2.4) A maximal chain in P(u, v) corresponds to a vertex of M Cube(n) (u, v). Conversely, to a vertex z = (z 1 , . . . , z s ) of n−1 we associate a maximal chain u = w 1 > w 2 > · · · > w s = v such that w i+1 differs from w i at the z i -th coordinate. Denote this maximal chain by P z (u, v).
Suppose now C is a cubical flow category and f : C → Cube(n) is the cubical functor. Until the end of this subsection, we will make the following assumption.

Assumption 2.8
For any x, y ∈ Ob(C) such that gr(x) − gr(y) = 1, the moduli space M C (x, y) is either empty or it is a single point.
Note that this assumption is trivially satisfied in the case of the Khovanov, respectively the annular Khovanov flow category, defined in Sects. 4.3 and 4.4.
Under Assumption 2.8 we can define the following relation on objects: we say that x y if gr(x) − gr(y) = 1 and M C (x, y) is non-empty. In general is the transitive closure of this relation. Proof If x y, there exists a chain x = x 0 x 1 · · · x s = y and therefore y) is non-empty, then it is a union of permutohedra s−1 . Choose a vertex of one of these permutohedra, which corresponds to 10 The map f : Ob(C) → Ob(Cube(n)) is order-preserving. Let x, y ∈ Ob(C) and s = gr(x) − gr(y). Assume that M C (x, y) is non-empty.
. is a codimension one face. We recall this distinction (present in LLS papers) in the appendix and we've checked all the instances of face/facet used in the paper. A maximal chain in P C (x, y) corresponds to a single vertex in M C (x, y), because if the chain is maximal, all the moduli spaces M C (x i , x i+1 ) consist of a single element by Assumption 2.8.

Definition 2.11
For a maximal chain m ∈ P C (x, y), the associated vertex v m ∈ M C (x, y) is the vertex associated to m by the above construction.
The correspondence can be reversed. Each face of M C (x, y) determines a chain in P C (x, y) precisely as in the case of the cube flow category. The following result is a special case.

Lemma 2.12 For every vertex
where all moduli spaces are zero-dimensional. In particular, with x 0 = x and x r +1 = y, we have gr(x i ) − gr(x i−1 ) = 1, which implies that the chain x 0 · · · x r +1 is maximal. Clearly, the vertex associated to this chain is v.
For any vertex z ∈ n−1 we define P z (x, y) ⊂ P C (x, y) to be the preimage of P z (u, v) under f P x,y .

Proposition 2.13
For any vertex z of n−1 , Proof Fix a vertex z ∈ n−1 . As the map f is a cover, we infer that #π 0 (M C (x, y)) = #f −1 x,y (z). To show that # max P z (x, y) = #f −1 x,y (z), let first v ∈ f −1 x,y (z), and denote by m the maximal chain m in P(x, y) associated to v. Clearly m ∈ max P z (x, y). On the other hand, every maximal chain m ∈ max P z (x, y) has an associated vertex v m ∈ M C (x, y) such that f(v m ) = z. This shows that there is a bijection between max P z (x, y) and f −1 x,y (z).

Neat embeddings
Recall that Lawson, Lipshitz and Sarkar described in [28, Section 3] a construction that turns a cubical flow category into a CW-complex. The construction is a simplification of the construction of Lipshitz and Sarkar in [33]. In Sects. 2.3, 2.4 and 2.5 we give a brief review.
Let (C, f) be a cubical flow category, and fix d

For any triple of objects
such that (CNE-1) For each x, y ∈ Ob(C) the following diagram commutes (CNE-2) For any u, v ∈ Ob(Cube(n)) the map is a neat embedding (see Definition A.5). (CNE-3) For any triple x > y > z ∈ Ob(C) the following diagram commutes Here the vertical maps are given by ι, the top horizontal map is the composition of morphisms and the bottom horizontal map is as defined in (2.5).

Framed cubical neat embeddings
To perform the construction of Lawson, Lipshitz and Sarkar, we need to construct an extension of ι to a framed cubical neat embeddingῑ, i.e a collection of embeddings ι x,y : for some > 0, in such a way that the commutativity from (CNE-3) is preserved with ι replacing ι. In generalῑ can be constructed as follows: where π R u,v : Ifῑ is a framed neat embedding of the cube flow category, thenῑ determines a sign assignment. Namely, for u, v ∈ {0, 1} n such that gr(u)−gr(v) = 1, we set ν(u, v) = 0 if ι u,v (M Cube(n) (u, v)) is framed positively with respect to the standard framing of [−R, R] d |v| , and ν(u, v) = 1 otherwise. In this case, we say thatῑ refines ν. Lemma 2.14 Any sign assignment ν determines a framed cubical neat embedding of the cube flow category which refines ν.
Any sign assignment for the cube flow category induces a sign assignment for its cover in an obvious way. In particular, a framed neat embedding of a cubical flow category induces a framed neat embedding of the underlying cube flow category, hence a sign assignment on the cube flow category; see [28, Section 3.5] for more details.

Cubical realizations
Let us fix a cubical flow category (C, f), a cubical neat embedding ι of C relative to a tuple d • = (d 0 , d 1 , . . . , d n−1 ) and fix > 0 in such a way that the map (2.6) is an embedding. As in [28,Definition 3.29] we construct a based CW-complex (||C||, x 0 ) in the following way: (1) For any x ∈ Ob(C), if u = f(x), we define the cell associated to x as where M Cube(n) (u, 0 n ) is defined to be {0} if u = 0 n and [0, 1]×M Cube(n) (u, 0 n ) otherwise. (2) The cells X (x) are glued together inductively. First we start with a disjoint union of cells X (y) for {y : f(y) = 0 n ∈ {0, 1} n }. For arbitrary x ∈ Ob(C), the cell X (x) is glued to the union y : f(x)>f(y) X (y). The gluing map is described below. (3) For any x, y ∈ Ob(C) with f(x) = u > v = f(y) the cubical embedding provides an embedding θ y,x : X (y) × M C (x, y) → X (x) given by (2.10) The first inclusion is given by the map ι x,y . The last inclusion comes from the composition map if v = 0, or the inclusion {0} → [0, 1] if v = 0. Denote by X y (x) ⊂ X (x) the image of the above map. (4) The attaching map for X (x) sends X y (x) ∼ = X (y) × M C (x, y) to X (y) via the projection onto the first factor. The complement of ∪ y X y (x) in ∂ X (x) is mapped to the base point.

Remark 2.15
It is proved in [28,Lemma 3.30] that the attaching maps are well-defined. This boils down to showing that if x, y, z ∈ Ob(C) are such that f(x) > f(y) > f(z), then there exists a map κ x,y,z that makes the following diagram commute.

Definition 2.16
The CW-complex ||C|| is called the cubical realization of the cubical category C. The formal desuspension: where τ is as in Definition 2.5, is called the C-homotopy type.
We note that we deviate slightly from [28]. We want the cubical realization to be a CW-complex, i.e. a topological space. After desuspension we obtain an object in the Spanier-Whitehead category, for which we use different notation X (C). Thanks to this distinction, many statements become more transparent, like the statement of Proposition 3.27.

Remark 2.17
It follows directly from the construction that if C is a union of categories C 1 , . . . , C s (in the sense that objects are set sums of objects, and there are no morphisms between objects in different summands), then X (C) is the wedge sum of X (C 1 ), . . . , X (C s ).

Chain complex associated with a cubical flow category
For completeness of the exposition, we recall how to compute the singular cohomology of the cubical realization. A detailed account is given in [33,Section 3] and [28,Section 3.2].
Let (C, f) be a cubical flow category with f : τ C → Cube(n). Choose a sign assignment ν for Cube(n). Define a cochain complex C * (C, f) in the following way: • The group C k (C) is freely generated over Z by the objects of C whose grading is equal to k; • If x ∈ Ob(C) has grading k, then we define where n x,y is the signed count of points in M C (x, y). In particular, if we choose a framed cubical neat embedding which refines a sign assignment ν, then The following result follows immediately from the construction of ||C||.
Lemma 2.18 C * (C, f) is a cochain complex, that is, ∂ 2 = 0, and its associated cohomology is equal to the cohomology of X (C), the C-homotopy type.

Equivariant flow categories
In this section we adapt the construction from Sect. 2 to the equivariant setting. First, we will introduce some terminology from equivariant differential topology. General references include [37,47,49].

Terminology
Let G be a finite group. An orthogonal representation of G is a homomorphism ρ : where O(V ) denotes the group of orthogonal automorphisms of some inner product space V . In particular, V is implicitly equipped with an inner product which is preserved by G. In the present article, we consider only finite-dimensional representations.
If it does not lead to confusion, we will refer to a representation ρ : G → O(V ) as V . In particular, for a subgroup H ⊂ G, the notation V | H means the representation If W ⊂ V are two G-representations, then by V − W , we denote the orthogonal complement of W in V . This notation is extended to the case when W is not necessarily a subrepresentation of V by introducing a Grothendieck group (actually a ring) of representations. More specifically, the representation ring RO(G) is the ring whose elements are formal differences The direct sum induces the addition, and the tensor product over R induces the multiplication.
We pass to the definition and basic properties of G-manifolds. Some more technical results are deferred to the Appendix. General references for group actions on manifolds include [26,47,49].
We say that M is a G-manifold, if it is a manifold (possibly with boundary) equipped with a smooth action of G. Observe that for any x ∈ M, the isotropy group G x acts on the tangent space T x M. By abuse of notation, we will denote by T x M the tangent representation of G x . For any subgroup H ⊂ G define Let M be a compact G-manifold and let p : E → M be a vector bundle over M. We say that E is a G-vector bundle if there exists an action of G on E by vector bundle morphisms such that p commutes with the action of G on E and M. If V is a G-representation, then a V -bundle is a G-vector bundle p : E → M such that for any x ∈ M there exists an isomorphism of G x representations between V | G x and p −1 (x).

Equivariant cell complexes
In order to fix the terminology, we recall the notion of a G-cell and a G-cell complex.

Definition 3.2 Let H ⊂ G be a subgroup and let
• X 0 is a disjoint union of orbits, • for any n > 0, The G-cell complex is called a Rep(G)-complex in [16,Section 1.13]. If we restrict the class of cells allowed in the construction, we obtain the following special cases.
• If we assume that all representations V n are of the form a V ⊕ R a n , for some fixed representation V and some integers a n , we obtain a G-CW(V ) complex in the sense of e.g. [ Topological spaces we construct are usually G-cell complexes, while in Sect. 8, we apply theorems for G-CW complexes. Therefore we need to translate from one object to another. The following result is well-known to experts.

Proposition 3.3 Any G-cell complex has a G-homotopy type of a G-CW complex.
Proof There are essentially two ways of approaching this result. In [37, Proposition X.2.8] it is proved that a G-CW(V ) complex is G-homotopy equivalent to a G-CW complex, and the proof can be adapted to the case of general G-cell complexes.
Another way is to refine the cell structure, namely to find a triangulation of B R (V ) by cells such that G acts on B R (V ) by permuting cells. This can be done using the results of Illman [24] (if G is a finite group, [23] suffices).

Definition 3.4
The equivariant Spanier-Whitehead category SW G is the category whose objects are the pairs (X , V ), where X is a finite G-CW complex and V is a virtual G-representation. Morphisms are defined by where Z runs through the family of finite-dimensional G-representations such that both V ⊕ Z and W ⊕ Z are G-representations.
The equivariant Spanier-Whitehead category is a full subcategory of the equivariant stable homotopy category, see [37,Proposition XII.7.3] and the preceding discussion.

Group actions on flow categories
We introduce now the definition of a group action on a flow category. To understand the details, it might be helpful the reader to keep in mind that the construction is modeled on the flow category associated with an equivariant Morse function. Definition 3.5 Let G be a finite group and let C be a flow category. We say that C is a G-equivariant flow category (as usual, we will omit G when it is clear from the context) if it is equipped with the following data: (1) for any g ∈ G there exists a grading preserving functor there is an equivariant grading function Moreover, these data must satisfy the following conditions: ) is a diffeomorphism of gr(x) − gr(y) − 1 -manifolds, which satisfies the following property for all z ∈ Ob(C) such that gr(y) < gr(z) < gr(x). Here we identify In particular, for any g 1 , g 2 ∈ G, υ g 1 • υ g 2 = υ g 1 ·g 2 .
(EFC-7) Let x, y ∈ Ob(C) and define G x,y = {g ∈ G : In the non-equivariant setting, it is possible to define the suspension k C of a flow category C by shifting the grading function by k ∈ Z. In the equivariant setting, we define the suspension of a flow category C by any virtual representation V − W ∈ RO(G). The category V −W C has the same objects and morphisms as C but different grading function given by

Definition 3.6
Given two G-equivariant flow categories C 1 and C 2 , a functor f : such that for any g ∈ G, we have is topologically a (trivial) covering map and for any object The notion of a cover will allow us to check easily some of the conditions (EFC-1)-(EFC-7) for C 1 if they are satisfied for C 2 . More precisely, we have the following result.

Lemma 3.8
Suppose C 2 is a G-equivariant flow category, C 1 is flow category and f : C 1 → C 2 is a trivial cover. Assume there is an action of G on C 1 satisfying conditions (EFC-1), (EFC-2) and (EFC- 3), such that f commutes with the action. Then, there is a unique structure of a G-equivariant flow category on C 1 such that f is a trivial G-cover.
Proof For an element x ∈ Ob(C 1 ) we set gr G (x) = gr G (f(x)). Then (EFC-4)-(EFC-6) are satisfied. Condition (EFC-7) follows from the fact that the G-dimension is preserved under maps that are local G-diffeomorphisms.

Equivariant cube flow category
Recall that objects of the cube flow category are elements of {0, 1} n . If σ ∈ Perm n is a permutation of an n-element set such that σ m = id, then σ induces an action of Z m on {0, 1} n . As in Appendix B.3 consider the action of Z m on R n defined by formula (B.3). We will denote this representation by V σ . For x ∈ Ob(Cube(n)) denote by (Z m ) x the isotropy group of x and consider the following Proposition 3.9 Let σ ∈ Perm n satisfy σ m = id. The cube flow category Cube(n) can be equipped with the structure of a Z m -equivariant flow category such that the action on the set of objects is generated by σ . Moreover, for any object x we have Therefore, we can define (G σ ) x,y =σ | M Cube(n) (x,y) . Lemma B.17 implies that conditions (EFC-1), (EFC-2) and (EFC-3) are satisfied.
In order to define the grading function observe that Lemma B.17 implies that for any x ∈ Ob(Cube(n)), the space Therefore, in order to satisfy condition (EFC-7), the only choice for gr Corollary 3.10 Using the notation from Proposition 3.9, suppose that σ is a product of n/m disjoint cycles of lenght m. Then, for any x ∈ Ob(Cube(n)) we have Proof If σ is a product of n/m disjoint cycles of length m,

Definition 3.11
Given σ ∈ Perm n such that σ m = id, we denote by Cube oe (n) the Z m -equivariant cube flow category for which the action on objects is generated by σ . Let C be a Z m -equivariant flow category. We say that C is a Z m -equivariant cubical flow category if it is a cubical flow category and, for some Z m -virtual representation V −W and some σ ∈ Perm n satisfying σ m = id, the functor f :

Remark 3.12
In the construction of the equivariant Khovanov homotopy type it is enough to restrict to categories Cube oe (n), where σ is a product of n/m distinct cycles of length m.

Remark 3.13
Note that the constructions in this section work equally well with any fixed subgroup G ⊂ Perm n . We restrict our attention to cyclic groups because this is the only relevant case for us.

Equivariant neat embedding
Let G = Z m and let (C, f) be a G-equivariant cubical flow category. Fix a sequence e • = (e 0 , e 1 , . . . , e n−1 ) of positive integers. For an orthogonal G-representation V and any u > v ∈ Ob(Cube oe (n)) define where B R (V ) denotes the closed ball in V centered at 0 and of radius R. We abbreviate V u,v = V | G u,v , recalling that the symbol V | H denotes the restriction of the representation to the subgroup H , the underlying linear space is the same.
Definition 3.14 An equivariant cubical neat embedding of a cubical flow category (C, f) relative to e • and relative to the representation V , is a cubical neat embedding (i.e. satisfying axioms (CNE-1), (CNE-2) and (CNE-3)). The maps ι x,y are required to be G x,y := G x ∩ G y -equivariant. Furthermore, any x, y ∈ Ob(C) and any g ∈ G the following diagram is commutative The right vertical arrow is labeled by g · (−), which should be read that the map is induced by the group action. More specifically, G acts on V and g ∈ G takes V u,v to V gu,gv . Moreover, g takes M Cube oe (n) (u, v) to M Cube oe (n) (gu, gv). Combining these actions we have the map that takes . This is the right vertical map in the above diagram.

Remark 3.15
From Definition 3.14 it follows that the diagrams of maps in (CNE-1) and (CNE-3) are diagrams of G x,y maps (in case of (CNE-1)), respectively G x,y,z = G x ∩ G y ∩ G z (in case of (CNE-3)).

Proposition 3.16 Any equivariant cubical flow category admits an equivariant cubical neat embedding.
Proof Consider x, y ∈ Ob(C). The space In particular, by the Mostow-Palais Theorem (see Theorem A.11) there exists a representation W x,y such that M C (x, y) embeds in W x,y . Define V to be the direct sum of W x,y over all pairs x, y ∈ Ob(C). We want to construct embeddings ι x,y : is given by the definition of the cubical flow category (see Definition 2.5 above).
Our task is therefore to construct the map j x,y . We shall proceed by induction is a finite set of points. The construction of j x,y in this case is obvious. Conditions (CNE-1), (CNE-2) are satisfied, while (CNE-3) is empty. The diagram in Definition 3.14 commutes.
Suppose the embedding has been constructed for all x, y with δ < k and we aim to construct a map j x,y for |f(x)| − |f(y)| = k. By the induction assumption, the map j x,y is defined already on the boundary of M C (x, y). We extend this map to a G-equivariant map on the whole of M C (x, y) by Lemma A.10, maybe increasing the values of some of the e i . Conditions (CNE-1) and (CNE-2) for j x,y × f is trivially satisfied. Condition (CNE-3) follows from the construction, because j x,z on the interior of M C (x, z) is an extension of j x,z on the boundary. Commutativity of the diagram in Definition 3.14 follows from equivariance of j x,y . The next step in the construction of Lawson, Lipshitz, and Sarkar is the construction of a framed cubical neat embedding. The notion of an equivariant framed cubical neat embedding is a direct generalization of the notion of a framed cubical neat embedding. Namely, given the set of maps ι x,y : constituting an equivariant cubical neat embedding (see Definition 3.14), an equivariant framed cubical neat embedding is an extension of ι x,y to equivariant maps We require that that (CNE-3) holds for ι x,y replaced by ι x,y and M C (x, y) replaced by the product In the non-equivariant setting, passing from a cubical neat embedding to a framed cubical neat embedding is described in Sect. 2.4. In the equivariant setting, no adjust-ments are needed, because the projections π R u,v and π M u,v considered in Sect. 2.4 are already equivariant by construction.

Equivariant cubical realization
We consider now an analog of the construction of a CW-complex ||C|| given in Sect. 2.5.
For each x ∈ Ob(C) such that u = f(x) we define: In fact, with the choice of d i = e i dim V and an identification EX(x) ∼ = X (x), E`is exactly the same map as θ . Again the key point is that E`x ,y is G x,y -equivariant. Write EX y (x) ⊂ EX(x) for the image of E`(y).
Analogously to the non-equivariant case, the complex ||C|| is constructed inductively by taking the cells EX(x) and the attaching map taking EX y (x) to EX(y) via the projection EX y (x) ∼ = EX(y) × M C (x, y) → EX(y). As in the non-equivariant case, the remaining part ∂ EX(x) \ y EX y (x) is mapped to the base point.

Remark 3.17
The map Eθ y,x gives a well-defined attaching map, see item (4) in Sect. 2.5 and equation (2.11). This is because, as we mentioned above, Eθ is essentially the map θ from Sect. 2.5. Another possibility is to observe that the map κ x,y,z constructed in the proof of [28,Lemma 3.16] is equivariant because of the axioms (EFC-1)-(EFC-3). We omit the details.

Proposition 3.18
The space ||C|| has the structure of a G-cell complex of Definition 3.2.
Proof If x 1 , x 2 , . . . , x k is an orbit of x 1 ∈ Ob(C), then there exists an equivariant homeomorphism It is easy to verify that the gluing maps are compatible with the homeomorphism from (3.5). The following result is a direct consequence of the construction: to explain the notion of a G-CW-complex in more detail.

Proposition 3.21
Let (C, f : W C → Cube oe (n)) be a G-equivariant cubical flow category. Let ι be an equivariant cubical neat embedding relative to e • = (e 0 , e 1 , . . . , e n−1 ) ∈ N n and relative to an orthogonal G-representation V . There exists a G-cell complex ||C||, such that every object x ∈ Ob(C) corresponds to a single cell of ||C|| of dimension gr G (x). Moreover, the forgetful functor (i.e. the one which forgets the action of G) maps ||C|| to the stable homotopy type constructed by Lawson et al. [33].

Fixed points of the cubical realization
The purpose of this subsection is to study the fixed point sets (with respect to a subgroup H ) of the group action on the cubical realization. The results will play an essential role in the proof of the invariance of the equivariant Khovanov homotopy type under Reidemeister moves. Recall that X H denotes the set of fixed points of H , that is, Let C be an equivariant cubical flow category. For any H ⊂ G define the H -fixed subcategory C H in the following way.
• The objects of C H are those objects of C that are fixed under the action of H , that is Ob(C H ) = Ob(C) H ; • The morphisms between objects are given by fixed point submanifolds, that is,

Remark 3.22
If H is a normal subgroup of G (in the paper we work with G cyclic, so any subgroup of G is normal), it is possible to endow C H with the structure of a G/H -equivariant flow category.
We will now give an instance of an H -fixed subcategory that is the most important in our approach.

Proposition 3.23
Let H be a subgroup of Z m and consider Cube oe (n) for σ ∈ Perm n such that σ m = id. Then there is a functor R H : Cube oe (n) H → Cube(n ) that induces an isomorphism of categories.
The integer n is calculated as follows. If σ is a product of p disjoint cycles (a i1 , . . . , a in i ) with n i |m and p i=1 n i = n, then we set i = gcd(n i , m/|H |) and n i = n i / i . We have n = n i .

Proof
The key idea is to use Theorem B.15. There is a technical difficulty namely Theorem B.15 does not give us a canonical diffeomorphism. Therefore we first fix a concrete diffeomorphism between M Cube(n) (1 n , 0 n ) H and M Cube(n ) (1 n , 0 n ), next we show that it can be used to define a map between all moduli spaces of the Cube(n) H category and corresponding moduli spaces of the Cube(n ) category.
To begin with, if (v 1 , . . . , v n ) is an object in Ob(Cube oe (n) H ), then by definition it is an object in Cube oe (n) fixed by the action of H . This amounts to saying that, for i = 1, . . . , p and j = 1, . . . , where to simplify the notation we write v i, j instead of v a i, j . The functor R H on objects is defined as We now define R H on morphisms. Consider first 0 n , 1 n ∈ Cube oe (n). They are fixed under the action of any subgroup where we also used the notation x i, j as a shorthand for x a i, j . We note that the dimension of L is precisely n i = n .
Take now general u, v ∈ Ob(Cube oe (n)) H with u > v. We assume that u = 1 n , v = 0 n . The case where precisely one inequality holds is analogous and it is left to the reader. Consider the product By the axioms of the cube category u,v embeds as a codimension 2 face in the moduli space M Cube oe (n) (1 n , 0 n ) = n−1 .
In fact, consider the partition and The first map is an embedding to a fiber { pt} × M Cube oe (n) (u, v) H × {pt} for two chosen points in M Cube oe (n) (v, 0 n ) H and M Cube oe (n) (1 n , u) H , respectively. R H does not depend on the choice. The last map is the projection onto the second factor. The map ψ was defined above as a map from n−1 ∩ L to n −1 . It takes the face p to the face we omit the details. We sketch the proof of the fact that R H respects the compositions. Suppose that u, w, v ∈ Ob(Cube oe (n)) H with u > w > v. Let R H u, R H w, R H v be the corresponding objects in Cube(n ). We need to show that the following diagram commutes.
Consider the refinement p w of the partition p defined in (3.7) given as where the subsets P ·,· are as above (below (3.7)). Let u,w,v be the face corresponding to this partition. Then M 1 = u,w,v ∩ L. By construction of ψ, it takes A to a face On the other hand, the reduction (p w ) B is easily seen to be the partition But then the corresponding face is (p w ) B = M 2 ., which essentially boils down to the statement that refinements commute with reductions. We can still add some more details, but the proof might eventually become less readable. Making the morphism ψ in Proposition B.11 might be possible, but it would definitely required a much longer proof.
Finally, the equivariant grading on Cube oe (n) described in Proposition 3.9 has the property that if x ∈ Ob(Cube oe (n)) H , then gr G (x) H is equal to the grading of R H (x). This is a straightforward verification. Proof In order to prove that C H is a flow category, we need to verify the axioms (FC-1), (FC-2) and (FC-3). The axiom (FC-1) is obvious. The axiom (FC-2) follows from the axiom (EFC-7) and Proposition B.11. The axiom (FC-3) follows from the axiom (EFC-3). This shows that C H is a flow category. It remains to prove that the functor f H makes C H a cubical flow category.
Since f commutes with the group action, it takes objects in C that are fixed under H to objects of Cube oe (n) that are fixed under H . In particular, f H is well-defined on objects.
To show that it is well-defined on morphisms, observe that for any x, y ∈ Ob(C) H , the map

Lemma 3.25
Let C be a framed cubical flow category and ι a neat embedding of C relative to e • = (e 1 , e 2 , . . . , e n−1 ) and relative to a representation V . Then, for any H ⊂ G, ι yields a neat embedding of C H , denoted by ι H , relative to e H • = (e 1 + · · · + e k−1 , e k + e k+1 + · · · + e 2k−1 , . . . , e n−k + e n−k+1 + · · · + e n−1 ) and V H , where k denotes the order of H . 14. An equivariant neat embedding where x, y ∈ Ob(C) H , yields an embedding Observe that for some R > R. Using the map R H from Proposition 3.23 we obtain a neat embedding x,y follow immediately from analogous properties of the maps ι x,y . To see this recall that by (3.3) we have where e H i = e k·i + e k·i+1 + · · · + e k·(i+1)−1 and k denotes the order of H . Discussion in Proposition 3.23 implies that EX(x) H ∼ = X H (x), for any x ∈ Ob(C H ).
In order to complete the proof of Proposition 3.27, we need to show that the attaching maps coincide. This holds, provided that θ H (y, x) = E`(y,

Equivariant chain complexes
In Sect. 2.6 we constructed a cochain complex C * (C, f), whose cohomology was equal to the cohomology of the cubical realization ||C||. Suppose now that the underlying cubical flow category admits an action of the group G = Z m . In order to describe the induced action of G on the chain complex, notice that, for any g ∈ G, we obtain a homomorphism of abelian groups yielding an action of G. This action, however, does not, in general, commute with the differential on C * (C, f).
The differential of the chain complex (2.12) depends on the sign assignment ν on the cube flow category Cube(n) σ , see (2.13). We will denote, abusing the notation, a generator of G by σ . The symmetry group acts on sign assignments via σ (ν)(x, y) = ν(σ (x), σ (y)).
To remedy this, we recall that the sign assignments form a 1-chain in [0, 1] n with values in F 2 (see Sect. 4.1). The difference between any two sign assignments satisfies a cocycle condition. Therefore, there exists a 0-cochain (3.9) Lemma 3. 28 The map t σ : C * (C) → C * (C) given by x → (−1) c(f(x)) G σ (x) commutes with the differential and therefore it generates the G-action on the chain complex C * (C). is to define the group action on C * (C).
Proof We need to check that the coefficient in ∂t σ (y) at t σ (x) is equal to the coefficient in ∂ y at x. The latter is equal to compare to (2.13). We want to compute now the former. Write y = G σ (y), x = G σ (x) . By (2.13) the coefficient in ∂ y at x is equal to Given the definition of t σ , we have t σ (x) = (−1) c(f(x)) x and t σ (y) = (−1) c(f(y)) y . Thus, in light of (3.11), the coefficient in ∂t σ G(y) at t σ (x) is given by y). (3.12) Finally, to show the equality of (3.10) and (3.12) we need to guarantee that but this follows immediately from (3.9).

Remark 3.29
This sign problem is not uncommon. It appears in the construction of the equivariant Khovanov homology [38, Section 2]. The approach in [38] is essentially the same as the one we use here, but it is expressed in a different language.

Equivariant subcategories
Suppose that C is an equivariant downward closed subcategory of C. Let C be the complementary upward closed subcategory. As C is invariant under the group action, the subcategory C is also an invariant subcategory.
The following result is a direct generalization of [ Suppose C is an equivariant cubical flow category, C is a downward closed subcategory, and C is the complementary upward closed category, and let the maps ι, κ and ρ be as in Proposition 3.30. We ask under which conditions one of these maps is an equivariant homotopy equivalence. This holds under some extra assumptions that we spell in Lemma 3.31. Although these assumptions are harder to verify, the methods developed in Sect. 3.8 simplify the process.  Invariance of the equivariant homotopy types is proved in Sect. 5.

Khovanov chain complex
In this subsection we rely on [33, Section 2]. Let V be a two-dimensional vector space over a field F with + and − as generators. We make it a graded space by assigning a grading q( + ) = 1, q( − ) = −1, called the quantum grading. Given two resolution configurations D 1 and D 2 , we define the resolution configuration D 1 \ D 2 by declaring, see [33]: Another operation that we can perform on a resolution configuration D is taking the For general labeled resolution configurations the partial order is defined as the transitive closure of the above relation.   Fig. 2. The arcs correspond to 0-resolutions; in Fig. 2 the dotted line represents the arc associated to that particular resolution. To avoid cumbersome notation, we will drop the subscript D when it is clear from the context.
Let V (D(v)) be the vector space over F generated by all possible labeled resolution configurations (D(v), x). Define the Khovanov complex of D in homological grading i = |v| − n − as The vector space CKh i inherits the quantum grading from V . To be more precise, to a homogeneous element x = 1 ⊗ · · · ⊗ t ∈ V (D(v)), i ∈ {+, −}, we associate the grading q(x) = q( i ) + n + − 2n − + |v|. Then CKh i splits as a direct sum of spaces CKh i,q , where the second index denotes the quantum grading.
In order to make CKh * into a cochain complex, we need to choose a sign assignment ν. This done, we define the differential of an element (D(v), x) with homological grading i as The cohomology groups of the complex CKh, that is Kh i,q (D) = ker(∂ i )/ Im(∂ i−1 ), are link invariants [27]. They are the Khovanov homology groups of the link represented by D.

Annular Khovanov chain complex
Asaeda, Przytycki, and Sikora [1] gave a construction of a variant of Khovanov homology for a link in the solid torus. This construction was later refined by Roberts [41]. Given L ⊂ S 1 × R 2 , we fix a diagram D of L so that D can be drawn on an annulus S 1 × R 1 . The starting point of the construction of Asaeda, Przytycki, and Sikora is to assign an extra annular grading to each of the generators of the Khovanov complex of D. For any v ∈ {0, 1} n and any labeled resolution configuration D = (D(v), x), the annular grading of D, denoted Ann(D), is defined in the following way. Let Z(D(v)) = {Z 1 , Z 2 , . . . , Z k }. We say that the circle Z i , for 1 ≤ i ≤ k is trivial, if it is null-homotopic in S 1 × D 1 , and nontrivial otherwise. For any 1 ≤ i ≤ k we define It is easy to check that for any decorated resolution configuration (D, x, y) we have Indeed, the above property is trivial for decorated resolution configurations of index one. The general case follows from the transitivity of the relation . Consequently, there exists a filtration of the Khovanov complex where CA k (D) is the subcomplex of CKh * , * (D) generated by those labeled resolution configurations (D(v), x) such that Ann(D(x), x) ≤ k. The annular Khovanov complex of D is the triply-graded cochain complex defined as In this setting, the annular Khovanov homology of D, denoted by AKh * , * , * (D), is defined as the homology of CAKh * , * , * (D). Annular Khovanov homology is an invariant of an annular link.

Khovanov homotopy type
In this subsection we apply constructions described in Sects. 2 and 3 to a specific flow category, the Khovanov flow category, which is at the heart of the Lipshitz-Sarkar construction. Let D be an oriented link diagram with n = n + + n − ordered crossings. The starting point of the construction is to assign to every decorated resolution con-

Annular Khovanov homotopy type
Recall that any labeled resolution configuration D of a link L ⊂ S 1 × D 2 , has an associated annular grading Ann(D). Define the annular Khovanov flow category C AKh (D) to be the subcategory of C Kh (D) with the same set of objects but with morphisms preserving the annular grading. For x, y ∈ Ob(C Kh ) we have M C AKh (x, y) = ∅ unless Ann(x) = Ann(y). In the latter case M C AKh (x, y) = M C Kh (x, y). For a labeled resolution configuration (D, x, y) we also denote by M AKh (D, x, y) the moduli space M C AKh ((s(D), x), (D, y)).

Lemma 4.4 The categories C AKh (D), C ≥k
Kh (D), C ≤k Kh and C k Kh are cubical flow categories.
Proof The cubical functor f : C Kh (D) → Cube(n) restricts to a cubical functor on each of these categories. Verifying the axioms of the cubical flow category is straightforward.

Given Lemma 4.4 we can define ||C AKh ||, ||C ≥k
Kh ||, ||C ≤k Kh || and ||C k Kh || as a cubical realization of the suitable categories and then the corresponding desuspensions X (C AKh ), X (C ≥k Kh ), X (C ≤k Kh ) and X (C k Kh ). Notice that the decomposition of the annular flow category C AKh (D) according to the quantum and annular gradings induces a decomposition . Repeating the proof of the invariance of the stable homotopy type of X Kh (D) under Reidemeister moves, we obtain the invariance of the stable homotopy type of X AKh (D) under Reidemeister moves in the solid torus. Therefore, the stable homotopy type X AKh (D) is an invariant of an underlying link L. The following result relates the cohomology of the cubical realization C AKh with the annular Khovanov homology.

Lemma 4.5 For any k ∈ Z, and any quantum grading q ∈ Z, there exists an isomorphism
Proof By construction of the category C AKh , the cochain complex associated with AKh * ,q,k is precisely the cochain complex for ||C q,k AKh || up to a shift. We conclude by Lemma 2.18.

Equivariant Khovanov flow category
Our goal is to construct a group action on the Khovanov flow category of a periodic link. Let m be an integer. Let D m be a diagram of an m-periodic link and consider G = Z m , which acts effectively on R 2 by rotations, preserving the diagram D m . The action of G permutes the crossings of D m . Let σ be a permutation corresponding to a generator of G. We have σ m = id. The following proposition shows how to extend the action of G on crossings of D m to the action on the Khovanov flow category. Recall that the action of Z m on the cube flow category was linear when restricted to any moduli space M Cube (f(x), f(y)). Therefore, the group action on morphisms in the cube category is completely determined by its restriction to the set of vertices of the respective permutohedra. This indicates that the group action on the moduli spaces M C Kh (x, y) should be built inductively with respect to the dimension of the moduli spaces.
The construction of G σ is straightforward for index 1 decorated configurations. Namely, M Kh (D(v), x, y) is a single point by construction (see the construction of M Kh (D(v), x, y) in [33,Section 5]). This means that if x, y ∈ Ob(C Kh ) have ind(y) = ind(x) − 1 and M C Kh (x, y) is non-empty, the functor f induces a diffeomorphism between the moduli spaces M C Kh (x, y) and M Cube(n) (f(x), f(y)), because each of them consists of a single point. Therefore G σ on zero-dimensional moduli space is uniquely determined by the action of G σ on the cube flow category Cube(n).
We now pass to the construction of G σ for moduli spaces corresponding to index k + 1 decorated configurations and k ≥ 1. The construction is inductive. That is, in the construction we suppose G σ has already been constructed for all moduli spaces corresponding to resolution configurations of index k or less.
. Moreover, the following diagram is commutative The extension of G σ can be defined on M C Kh (D(v), x, y) as follows. Let denote the respective connected components. Without loss of generality, we may and will assume that G σ maps ∂Y i onto ∂Y i . For any 1 ≤ i ≤ k, we define The axioms (EFC-1) and (EFC-2) are trivially satisfied and (EFC-3) is guaranteed by the fact that the construction is performed inductively.
To complete the proof we need to consider the case, when (D(u) \ D(v), x, y) is a ladybug configuration. The action of Z m preserves the ladybug matching by [33,Lemma 5.8]. Therefore, we again obtain a well-defined extension of f to the whole M Kh (D(v), x, y) and the extension of G σ is given again by (4.2). This completes the construction of the group action on the flow category C Kh (D m ). Conditions (EFC-1)-(EFC-3) are trivially satisfied.
We define the grading via Lemma 3.8. As this is an important step of the construction, we unfold the definition of the grading. Namely, for an element y = (D(v), x) ∈ Ob(C Kh (D m )) we define

Proof of Theorem 1.2
We will prove only part (b) of the theorem, namely that X (C Kh ) is a well-defined object in the equivariant Spanier-Whitehead category. The case of annular Khovanov homology is completely analogous.
Suppose that D m is an m-periodic diagram representing an m-periodic link L. By Proposition 4.6, the Khovanov flow category C Kh (D m ) admits a group action. Proposition 3.21 shows that the cubical realization || V C Kh (D m )|| admits a Z m -action, for an appropriate representation V . In particular, it ensures the existence of the Khovanov homotopy type as an object in the equivariant Spanier-Whitehead category, see Sect. 3.1.
To conclude the proof of Theorem 1.2, we need to show that the equivariant stable homotopy type X (D m ) does not depend on the choices made. We prove invariance step by step. • Independence of V . Let us introduce the following notation. Suppose ι V is a cubical neat embedding relative to e • and relative to a representation V . Let V → W be an equivariant embedding. Composing this embedding with ι V we obtain a neat embedding relative to e • and W , which we denote by ι W V . We observe that if W = V ⊕ V , then by construction Suppose ι V and ι V are two cubical neat embeddings relative to e •V and V and to e •V and V , respectively. By increasing the entries of e •V and e •V and using independence on e • discussed above, we may and will assume that e •V = e •V = e • . We will also assume that the entries of e • are sufficiently large. Under the latter assumption, with W = V ⊕ V , the two embeddings ι W V and ι W V a re equivariantly isotopic by the Mostow-Palais Theorem (Theorem A.11). By this we mean that for any x, y ∈ Ob(C), there exists an equivariant isotopy ι t x,y satisfying compatibility relations (EFC-1)-(EFC-3) for all t ∈ [0, 1]. Such isotopy is constructed by defining j t x,y , once j 0 x,y and j 1 x,y have been defined (see proof of Proposition 3.16). The construction of j t x,y is inductive as in Proposition 3.16, using Mostow-Palais Theorem at each stage. We omit straightforward details. Given the isotopy, we obtain that ||C|| ι W V and ||C|| ι W V are equivariantly homotopy equivalent, and therefore ||C|| ι V and ||C|| ι V are equivariantly stably homotopy equivalent, as desired. Proving the independence on the choice of the diagram and on the ladybug matching is more complicated; we prove these results in Sects. 5.1 and 5.2, respectively.

Independence under equivariant Reidemeister moves
Let Consider now the subcategory of C Kh (D m ), denoted C 1 , consisting of those labeled resolution configurations (D (v), x) such that: • either there exists a value of k with v n+2k−1 = 0 and v n+2k = 1, and x assigns a label + to the extra circle created (see Fig. 4b); • or there exists a value of k satisfying v n+2k−1 = v n+2k = 1 (see Fig. 4c).  Fig. 4d, e. We observe that C 3 is an upward closed category.

Lemma 5.3 The subcategory C 3 is G-invariant, and for any subgroup H ⊂ G the complex C * (C H
3 ) is acyclic. We omit the proof of Lemma 5.3, since it is analogous to the proof of Lemma 5.2. Let C 4 be the complementary category of C 3 in C 2 ; that is to say, C 4 is the category such that there exists a value of k satisfying v n+2k−1 = 1 and v n+2k = 0 (see Fig. 4f). Moreover, observe that C 4 is isomorphic to the category C Kh (D m ) corresponding to the original diagram D m .
In this setting, we apply Lemma 3.31 twice to get the desired result. Namely, we first state that ||C Kh (D m )|| is equivariantly stably homotopy equivalent to ||C 2 || and then that ||C 2 || is equivariantly stably homotopy equivalent to ||C 4 || = ||C Kh (D m )||. This concludes the proof of Proposition 5.1.
Since C * (Cube(m)) is acyclic, the Künneth Theorem implies that C * (C 1 ) is also acyclic.
An analogous argument works for the fixed point sets categories. Namely, let H ⊂ G be a subgroup of order k and consider C H 1 . Let σ denote the permutation of crossings of D m . Notice that the subset of crossings {c n+1 , c n+2 , . . . , c n+2m } consists of two orbits of G c n+1 , c n+3 , c n+5 , . . . , c n+2m−1 and c n+2 , c n+4 , . . . , c n+2m .

Independence of the ladybug matching
Before we give the proof of the independence of the ladybug matching, we introduce some notation: given a link diagram D m together with a Z m -action by rotations, we write D m for the link diagram with a Z m action by rotations in the opposite direction. We write C Kh (D m ) for the corresponding equivariant Khovanov flow category; the underlying non-equivariant Khovanov flow category is the same, but the group action is inverted. There is also an equivariant stable homotopy equivalence between ||C # Kh (D m )|| and ||C Kh (D m )||. This is shown using the same argument as in the proof of [33,Proposition 6.5]: the isomorphism of framed flow categories C # Kh (D m ) and C Kh (D m ) is equivariant, if we revert the group action on one side.

Proposition 5.4 Let D m be a periodic diagram and let C
The composition of the two equivariant stable homotopy equivalences yields the desired equivariant stable homotopy equivalence.

Moduli spaces via the Cob 3 •/l category
In order to prove the Categorical Fixed Point Theorem 7.1 and, more generally, in order to understand the fixed point set of X Kh (D) when D is a periodic link diagram, we need a deeper understanding of the structure of moduli spaces M Kh . The key tool is Bar-Natan's cobordism category Cob 3 •/l reviewed in Sect. 6.1. The main result of this section, which is used in the proof of the fixed point theorem, is the Counting Moduli Lemma 6.6, which computes the number of connected components of the moduli space M Kh in terms of the genus of the associated cobordism.

Dotted cobordism category of R 3
As alluded to above, we begin with recalling Bar-Natan's formulation of Khovanov homology; see [2]. Let Cob 3 • denote the graded additive category whose objects set is generated by finite collections of disjoint simple closed curves, i.e., (1) If Z ⊂ R 2 is a finite collection of pairwise disjoint simple closed curves, then , then a formal shift of Z , denoted Z { }, for some ∈ Z, also belongs to Ob(Cob 3 • ), then their formal direct sum Z 1 ⊕ Z 2 also belongs to Ob(Cob 3 • ).

As Cob 3
• is an additive category, it is enough to define morphisms on generators. If  Fig. 6d where χ( ) denotes the Euler characteristic of . Cobordisms are usually drawn from left to right or from bottom to top. Dots can move freely within the connected components of a given cobordism and decrease the degree of the respective map by 2. The category Cob 3 •/l is the quotient of Cob 3 • by local relations depicted in Fig. 6. Particularly useful is the neck cutting relation depicted in Fig. 6d. Indeed, a recursive application of the neck cutting relation quickly reduces any morphism in Cob 3 •/l to a morphism given as a disjoint sum of unknotted surfaces of genus 0 and 1.

Lemma 6.1 [2]
Let be a dotted surface representing a morphism in Hom Cob 3 (1) If any connected component of is of genus greater than 1, then (

2) If any connected component of is a singly-dotted torus, then
= 0 ∈ Hom Cob 3 Proof The lemma is a consequence of the relation drawn in Fig. 7, which follows directly from the neck cutting relation shown in Fig. 6d and the fact that a double dot annihilates every morphism, as shown in Fig. 6c. = 2 Fig. 7 The relation used in the proof of Lemma 6.1
•/l ) have c 1 and c 2 connected components, respectively, then is generated by a disjoint union of c 1 cocaps bounding Z 1 and c 2 caps bounding Z 2 .

Example 6.3 By Lemma 6.2(4), if Z ∈ Ob(Cob 3 •/l ) consists of c connected components, then Hom Cob 3
•/l (∅, Z ) is generated by c disjoint dotted or undotted caps. For any field F, we can identify where V is the vector space used in Sect. 4.1. In order to do that enumerate circles of Z by Z 1 , . . . , Z c . If c = 1, the identification is given in Fig. 8. The case c > 1 is a simple extension. Indeed, if where the sign of the i-th factor is + if C i is undotted and − otherwise, for 1 ≤ i ≤ c.
For any Z 1 , Z 2 ∈ Ob(Cob 3 •/l ) a distinguished generator in Hom Cob 3 a disjoint union of cocaps and caps as described Lemma 6.2 (5). The distinguished basis of Hom Cob 3

Remark 6.5
The isomorphism in Lemma 6.2(5) can be described in the following way. For two objects Z 1 , Z 2 ∈ Ob(Cob 3 •/l ), the composition in Cob 3 •/l induces a trilinear map Z 1 ,Z 2 : T Kh (Z 2 ) * × Hom Cob 3 which yields an isomorphism If ∈ Hom Cob 3 , then it is easy to check that where the summation extends over distinguished generators S 1 and S 2 of T Kh (Z 1 ) and T Kh (Z 2 ), respectively.

Counting moduli lemma
Let (D, x, y) The result we present next is the key tool in the study of the fixed points of moduli spaces.  (D, x, y)).
In particular, if M Kh (D, x, y) = ∅, then where c 1 is the number of genus 1 connected components of (z, D).

Remark 6.7
The lemma can be deduced from the discussion at the end of [29,Section 2.11]. As the precise statement is absent in that paper, and the result is widely used in the present paper, we present a sketch of the proof using Cob 3 •/l -categories and posets. (Z(D)) denote the distinguished generator corresponding to (s A (D), x ). Without loss of generality we can pick z = (1, 2, 3, . . . , n) ∈ n−1 . Proposition 2.13 implies that #π 0 (D, x, y) = # max P z (D, x, y), so it is sufficient to prove that # max P z (D, x, y) = Z(D),Z(s A (D)) (S 1 , (z, D), S 2 ). (6.1)

Proof For a resolution configuration (s
In order to prove (6.1), we proceed by induction on the index of the resolution configuration. Let Z 01 , . . . , Z 0s y be the circles in Z(D) and Z 11 , . . . , Z 1s x be the circles in Z(s(D))).
For an index 1 resolution configuration, the poset P z (x, y) consists of a single chain x y and the surface has genus 0. Then both sides of (6.1) are equal to 1.
Suppose now that (6.1) has been proved for all index n−1 resolution configurations, and let (D, x, y) be a resolution configuration of index n. There are two cases. Either A 1 is a split, or it is a merge. We will deal only with the (harder) case, when A 1 is a split, leaving the other case to the reader., we give only the half of the proof.
Suppose Z 01 splits into two circles Z 011 and Z 012 . If y(Z 01 ) = − , then there is a unique y 1 such that (D, y) ≺ (s {A 1 } (D), y 1 ). We infer that # max P z (x, y) = # max P z 1 (x, y 1 ). The neck-cutting relation shows that Suppose finally that y(Z 01 ) = + . Then there are two different assignments y 1 and y 2 such that for y j = (s {A 1 } (D), y j ) we have y ≺ y j (with j = 1, 2): one assigns + to Z 011 and − to Z 012 , the other one does the opposite. In particular Let S 3 , S 4 be the distinguished generators associated to y 1 and y 2 , respectively. We need to prove that which follows from the neck-cutting relation (see Fig. 10).
where ζ denotes the complex coordinate on C \ {0} = C * . If D is a resolution configuration in C * , then define the p-lift of D to be the p-periodic resolution configuration D p such that Z(D p ) = π −1 p (Z(D)) and A(D p ) = π −1 p (A(D)). Analogously, for a labeled resolution configuration (D, x) and a decorated resolution configuration (D, x, y) we define the p-lift (D p , x p ) and (D p , x p , y p ), where x p = x • π p and y p = y • π p .

Theorem 7.1 (Categorical fixed point theorem) Let D p be a p-periodic annular link diagram and let D denote the quotient diagram. For any q, k ∈ Z there exists an isomorphism of cubical flow categories
which induces the following isomorphism of cubical flow categories, for any q ∈ Z, As a corollary we obtain the statement equivalent to Geometric Fixed Point Theorem 1.3.

Corollary 7.2 For any annular link diagram D we obtain
Proof of Corollary 7.2 Notice that by Proposition 3.27 and Theorem 7.1 we have The case of the Khovanov flow category is analogous.

Proof of Theorem 7.1
The desired isomorphism of cubical flow categories will be first defined on objects, then on morphisms.
The map F p preserves the annular grading.

Proof of Lemma 7.3
The inverse map is given by taking the quotient of a respective labeled resolution configuration. Moreover, invariance of the annular grading under the map F p is evident.
We will now pass to constructing the map on morphisms. The key property that we will require is that for all resolution configurations (D, x, y) the following diagram where u = f (D, y), v = f(s(D), x), u p = f(D p , y p ) and v p = f(s(D p ), x p ). We begin with morphisms in C AKh (D) corresponding to index one configurations. All these configurations are depicted in Fig. 11. We recall now a result of Zhang, which is proved on a detailed case-by-case analysis. Fig. 11 we have

Lemma 7.4 (See [51, Section 5.3]) For all configuration depicted in
Next result connects M AKh with M Kh for index one configurations. (D, x, y) is one of the configurations of Fig. 11. If M Kh (D p , x p , y p ) Z p is non-empty, Ann(D p , y p ) = Ann(s(D p ), x p ).

Proof of Lemma 7.5
The proof is done on a case-by-case analysis. Cases (e) and (f) of Fig. 11 are trivial, because no circles in D p or s(D p ) is non-trivial (in the sense of Sect. 4.2). Thus Ann(D p , y p ) = Ann(s(D p ), x p ) = 0.
Case (b) is dual to (a), and case (d) is dual to (c), so it is enough to prove the lemma for cases (a) and (c) only. We will deal with case (a) only, leaving case (c) (which is easier) Fig. 12 Proof of Lemma 7.5 to the reader. The resolution configurations D p and s(D p ) are depicted in Fig. 12. Note that the annular grading of any resolution configuration on the right is zero, because there are no non-trivial circles. Therefore, it is enough to show that if the resolution configuration on the left has non-trivial annular grading, then M Kh (D p , x p , y p ) Z p = 0. The configuration (D p , y p ) has non-trivial annular grading in precisely two cases: either y p assigns − to both circles on the left, or it assigns + to both circles. In the first case, as the surgery on any of the arcs merges two circles labeled with − , the moduli The other case is that y p assigns + to both circles. After the surgery on one of the arcs connecting the two circles, we obtain a single circle labeled with + . All other p − 1 arcs are splits. Any split of an + labeled circle yields a circle labeled with + and a circle labeled with − , while a split of a circle labeled with − has two circles both labeled with − . It follows that x p assigns + to a positive number of circles and − also to a positive number of circles. Such configuration (the underlying p circles are drawn in Fig. 12 on the right) cannot be Z p -invariant. Hence M Kh (D p , x p , y p ) Z p is empty. This concludes the proof of case (a).

Corollary 7.6 For any index one configuration, there is a bijection between
Proof From Lemma 7.5 we immediately obtain a bijection between M AKh (D p , x p , y p ) Z p and M Kh (D p , x p , y p ) Z p . By Lemma 7.4, it is enough to consider the case, when M AKh (D, x, y) is non-empty. Then it is a zero-dimensional permutohedron 0 , that is, a single point. Call it z. Let (z, D) be the corresponding surface. It has genus zero.
Let z p ∈ Z p p−1 denote the unique fixed point of the Z p action. The surface (z p , D p ) is a p-fold cover of (z, D) and it is easily seen to have genus zero as well. From Counting Moduli Lemma 6.6 we deduce that M AKh (D p , x p , y p ) is connected, hence it is diffeomorphic to p−1 . Therefore M AKh (D p , x p , y p ) Z p is a single point.

Remark 7.7
Since the moduli spaces in Corollary 7.6 are either empty or a single point, the bijection of Corollary 7.6 is uniquely defined. Continuing the proof of Theorem 7.1, we extend F p from objects to morphisms corresponding to index one resolution configurations. We now discuss the index two resolution configurations.
Assume first that (D, x, y) is not a ladybug and M AKh (D, x, y) is non-empty. Then (D, x, y) is a genus zero resolution configuration. In particular, M AKh (D, x, y) = 1 is an interval with two boundary components. By dimension counting argument, M AKh (D p , x p , y p ) Z p is a union of some number of copies of one-dimensional permu- Hence M AKh (D p , x p , y p ) Z p is also an interval. Then f takes it diffeomorphically to If the genus of (D, x, y) is one, the moduli space is not connected. Decorated resolution configurations of index two and genus one are called ladybug configurations, they are depicted in Fig. 13.
We discuss these three cases separately. The two other cases are dealt with in the following lemma, whose proof is deferred to Sect. 7.2. Fig. 13b or Fig. 13c. Then

Lemma 7.9 Suppose (D, x, y) is a ladybug configuration depicted in
The isomorphism makes the diagram (7.1) commutative.

Proof of Lemma 7.9
We will prove Lemma 7.9 only for the resolution configuration depicted in Fig. 13b. The case of Fig. 13c is similar (and easier), we leave it to the reader. Lemma 6.6 implies that M Kh (D, x, y) = 1 1 2 1 has two connected components. Let Z denote the unique circle in D and let A 1 and A 2 denote the arcs, where A 1 is the arc lying inside Z . As x y we must have y(Z ) = + and x(Z ) = − and in this case the poset P(x, y) consists of four elements x x j y, j = 1, . . . , 4, where: Here x 1 A 2 assigns + to the inner circle and + to the other circle and x 2 A 2 the opposite way. The assignments x 1 are paired under right ladybug matching, that is, the vertices corresponding to posets x x 1 y and x x 3 y belong to the same connected component of M Kh (D, x, y); see [33,Section 5.4].
Every maximal chain containing x p 1 is of the form v σ 1 σ 2 := (D p , y) ≺ x 10 σ 1 σ 2 ≺ · · · ≺ x p0 Proof It is enough to prove the result if σ 1 = σ 1 and σ 2 differs from σ 2 by swapping two adjacent elements (or σ 2 = σ 2 and σ 1 differs from σ 1 by a single transposition of elements). The proof in that case is essentially a repetition of the argument used in the proof of Lemma 7.10 so we leave it to the reader.
Let be the connected component of M C Kh (x p , y p ) that contains all of the v σ 1 σ 2 and v σ 1 σ 2 . Note that the group action takes v σ 1 σ 2 to v σ 1 σ 2 for some other permutation σ 1 , σ 2 , therefore the component 1 is preserved. The fixed point set Z p is a one-dimensional permutohedron 2 , which is diffeomorphic with Z p 2 p−1 . The diffeomorphism is realized by the restriction of the cover map f : We define now the isomorphism M C Kh (x, y) → M C Kh (x p , y p ) Z p in such a way that the segment connecting the vertices x x 1 y and x x 3 y in M C Kh (x, y) is mapped to a segment in Z p (which is a disjoint union of segments). The segment connecting the vertices x x 2 y to x x 4 y is mapped to the other connected component of M Kh (x p , y p ) Z p .

Equivariant Khovanov homology
We begin with a brief review of the construction of the equivariant Khovanov homology [38]. Later on, we merge this construction with the construction of the equivariant Khovanov homotopy type that we introduced in Sect. 4.

Review of the construction
Let D be an m-periodic diagram representing an m-periodic link L. The symmetry of L can be realized by a cobordism in S 3 × I in the following way. Suppose the rotation center is at 0 ∈ R 2 . Consider D × I ⊂ R 2 × I and twist it by the diffeomorphism η : R 2 × I → R 2 × I given by (x, t) → ( 2π t/m x, t), where θ is a counterclockwise rotation by the angle θ . The image is a cobordism from D to D. Note that this is a product cobordism, and there are no handle attachments.
In [2] a map φ D of Khovanov chain complexes was associated to each cobordism of diagrams . The chain homotopy class of this map was later shown to be functorial, i.e., not depending on the isotopy type of ; see [48]. is well-defined and can be used to show that the group action on the Khovanov chain complex is well-defined. In fact, defining a group action is relatively easy, but many proofs of invariance can be simplified, once we have a functorial map φ D Notice that only the chain homotopy type of φ D is well-defined. However, since D is a composition of Reidemeister moves, it is possible to choose a representative for φ D , which induces a group action on the Khovanov complex. Khovanov homology is an invariant of periodic links [38]. The construction of equivariant Khovanov homology also works in the annular case. The methods of [38] carry over to the annular case without significant modifications. Namely, we observe that the annular chain complex CAKh * ,q,k (D; R) admits a Z maction, hence it is a -module. Next, we define EAKh j,q,k (L; M) = Ext j (M; CAKh * ,q,k (D; R)).
Essentially the same argument as in [38] can be used to show that EAKh is an invariant of an annular link.

Equivariant Khovanov homology as Borel cohomology
We now have two ways of getting equivariant homology from the Khovanov theory. One way is to use the definition given in (8.2). Another way uses the Borel cohomology of space X Kh (D). We will now show that the two constructions agree. In the rest of this section we denote by C * (X Kh (D); R) the reduced cellular cochain complex of X Kh (D) associated to the CW-structure described in Sect. 3.7. First we state a preparatory result.

Proposition 8.2 Let D m be an m-periodic diagram of a link. There exists an identification of cochain complexes of R[Z m ]-modules
Here it should be understood that the structure of the R[Z m ] cochain complex is given by the Z m -action on C * (X Kh (D m ); R) and on CKh(D m ).
Proof The statement is a consequence of the construction of the cochain complex of C * (X Kh (D m )). The cellular cochain complex C * (X Kh (D m ); R) was constructed in Sect. 2.6. The construction is that the generators of C * (X Kh (D m ); R) correspond to the generators of CKh * (D m ; R). The differential on C * (X Kh (D m ); R) is the same as in CKh(D m ; R). In Sect. 3.9 it was shown that the induced group actions on C * (X Kh (D m ); R) and CKh(D m ; R) coincide.
In order to state and prove the next result, we need to set up some notation and recall some basic facts from homological algebra. If C * is a chain complex, we will associate to it a cochain complex C * r defined by C −n r = C n with the differential d −n r : C −n r → C −n+1 r defined by d −n r = (−1) n d n , where d n : C n → C n−1 is the differential in C * . For two cochain complexes C * and D * we define the Hom cochain complex with the differential d n If P * is a projective resolution of a cochain complex C * and I * is an injective resolution of D * , then (8.4) Recall that to any discrete group G, we can associate a contractible space EG equipped with the free action of G. By BG = EG/G we denote the classifying space of G. For a G-space X and any finitely generated R[G]-module M we define the Borel equivariant cohomology of X where C * r (X ) denotes the cochain complex associated to the cellular cochain complex C * (X ) of X using the convention described above. In particular, we have C * (X ; R) = Hom * R (C * r (X ); R). There is a natural G-map EG × X → EG which, after taking quotient of both sides, yields a map We define the reduced Borel cohomology of X , to be It is easy to check that where g ∈ G acts on Hom F (M, Proof To begin with, observe that Here the first equality is the definition of equivariant Khovanov homology while the second equality follows from Proposition 8.2. Next, let P * M be a projective resolution of M. We have where (1) comes from the definition of C * r (X Kh (D m ) q ; F), (2) is the definition of the Ext functor and the isomorphisms (3) and (4)  We remark that the same argument as in the proof of Theorem 8.3 shows the following result.

Proposition 8.4 Let L m be an m-periodic annular link and let D m be an m-periodic diagram of L m . For a field F and any F[G]-module M, it holds
AKh (D m ), Hom F (M, F)).

Stable cohomology operations
Given two generalized cohomology theories X (·) and Y (·), a stable cohomology operation of degree k is a family of natural transformations between functors X l (·) and Y k+l (·) commuting with suspension. We focus on stable cohomology operations in singular homology over a finite field. These operations form a Steenrod algebra. Standard references include [17,Section 4.L] and [13,Section 10.4]. The Steenrod algebra A 2 over Z 2 is generated by the Steenrod squares Sq i : H * (·, Z 2 ) → H * +i (·, Z 2 ), with Sq 1 being the Bockstein homomorphism corresponding to the short exact sequence 0 → Z 2 → Z 4 → Z 2 → 0.
For a prime p > 2, the Steenrod algebra A p is generated by the Bockstein homomorphism β, and operations P k : H * (·, Z p ) → H * +2k( p−1) (·, Z p ). The homomorphism β is of degree 1, and it is the connecting homomorphism of the long exact sequence of cohomology induced by the short exact sequence of groups Coming back to Khovanov homology we make the following observation, see [34,35].

Proposition 8.5
Let α be a stable cohomology operation of degree k over Z p . Then, given a link L and q ∈ Z, the map α induces a well defined map There appeared several algorithms for computing Steenrod squares in Khovanov homology, so the invariants based on Steenrod squares can be effectively computed (see [35,36]). The knotkit package [42] implements the algorithm of [35]. We remark that the maps Sq 1 and β are determined by the integral Khovanov homology, see [35,Section 2.5].
The next statement shows that Steenrod operations commute with group action.

Fixed Point theorems
Recall that BZ p is the classifying space of the finite cyclic group of order p. The cohomology ring of BZ p is given below, see [17,Example 3E.2], is the exterior algebra over F p generated by Z . Write S p ⊂ H * (BZ p ; F p ) for the multiplicative set generated either by X , if p = 2, or by Y , when p > 2.
Theorem 8.7 [3,40] Let X be a Z p -CW-complex with p a prime. There exists an isomorphism of graded Z p -algebras As an immediate corollary of Theorems 8.7 and 1.2, we get Theorem 8.8 Let L p ⊂ S 1 × D 2 be a p-periodic link with L denoting the quotient link. For any q, k ∈ Z there exists an isomorphism of S −1 p H * (BZ p ; F p )-modules Let A p denote, for any prime p, the mod p Steenrod algebra, i.e. the algebra of stable F p -cohomology operations. It turns out that Theorem 8.8 can be strengthened considerably when we take into account the action of the Steenrod algebra. Before stating the main result, let us introduce the following terminology.

Theorem 8.10
Let p be a prime.
(a) If L p ⊂ S 1 × I is p-periodic link and L is the quotient link, then for any q, k ∈ Z there exists an isomorphism of rings

Consequently
AKh * ,q,k (L; (b) For a p-periodic link L p ⊂ S 3 and for any q ∈ Z it holds: where L denotes the quotient link.
Proof This is an immediate corollary of [14, Corollary 2.5.] and Theorem 8.3.
Smith inequalities given in Theorems 1.4 and 1.5 are corollaries of Theorem 8.10. We prove now Theorem 1.4; an analogous proof works for the case of Theorem 1.5.
Proof of Theorem 1. 4 We have the following chain of inequalities The first inequality is a consequence of the definition of equivariant annular Khovanov homology. The second inequality is a natural consequence of the properties of the localization, and the last equality follows from Theorem 8.10.

Conflicts of interest On behalf of all authors, Maciej Borodzik states that there is no conflict of interest.
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Example A.2
An n-dimensional permutohedron has a structure of an n -manifold, see Subsection B.1 below.
We will need the following construction, see [28,Construction 3.4].

Construction A.3
Suppose X is an n -manifold and Y is an m -manifold. The product X × Y is given the structure of an n + m -manifold by declaring.
In some cases, it is more convenient to view n -manifolds as certain functors to the category of topological spaces. Let 2 1 denote the category consisting of two objects 0 and 1 with a single non-identity morphism 0 → 1. For an integer n > 1 let a 1 , a 2 , . . . , a n ) and b = (b 1 , b 2 , . . . , b n ). An n-diagram, for n ≥ 1, is a functor from the category 2 n to the category of topological spaces.
We can associate an n-diagram to a given n -manifold X by declaring, for every a = (a 1 , a 2 , . . . , a n ) ∈ 2 n : 1, 1, . . . , 1), Moreover, for any b ≤ a in 2 n the map X (b) → X (a) is the inclusion. We point out that X (a) is an |a| -manifold with the corresponding |a|-diagram obtained by restricting X to the full subcategory of objects b such that b < a (recall that |a| = a i ). Let M be a manifold with corners. Choose a Riemannian metric on M. We have the following generalization of a classical result.

Proposition A.4
There is an open tubular neighborhood U of ∂ M, homeomorphic to M × [0, 1) and a subset V ⊂ T M| ∂ M such that the exp map yields a diffeomorphism between U and V .
Proof The proof is analogous to the proof of the collar neighborhood theorem, see, for instance, [21,Section 4.6].
We now recall the concept of a neat embedding, which roughly means an embedding with no pathological behavior near the boundary. Various similar notions are discussed in detail in [25,Section 3].
Definition A.5 Let X and Y be two n -manifolds. A neat embedding is an embedding ι : X → Y such that: (1) ι is an n-map, i.e. ι −1 (∂ i Y ) = ∂ i X , for any 1 ≤ i ≤ n, (2) the intersection of X (a) and Y (b) is perpendicular with respect to some Riemannian metric on Y , for b < a in 2 n .
The following result is a direct consequence of [33,Lemma 3.11].

A.2. Group actions on n -manifolds
Let Diff n (X ) denote the group of diffeomorphisms of the n -manifold X that are also n-maps. If G is a finite group, then a smooth action of G on X is a homomorphism γ : G → Diff n (X ).
An action of G is said to be effective if γ is injective. Throughout this paper, we assume that group actions are effective. Moreover, we will often identify g ∈ G with its image γ (g) ∈ Diff n (X ).
Definition A.7 Let V − W ∈ RO(G). We say that X is of dimension V − W , and denote it by dim X = V − W , if for any interior point x there exists an isomorphism of representations We have the following equivariant analog of Proposition A.4.

Proposition A.8 Let M be an n -manifold with an action of a finite group G. Choose a G-invariant Riemannian metric on M. Then ∂ M admits a G-equivariant tubular neighborhood U such that there exists a G-invariant subset V ⊂ T M| ∂ M such that the exp-map takes V diffeomorphically and G-equivariantly to U .
Proof The proof for standard manifolds with boundary is given in [26,Section 3]. The case of manifolds with corners is analogous.
Definition A. 9 Let M be an n -manifold acted upon by G. Let V be an orthogonal representation of G. The manifold M is said to be subordinate to V if for each x ∈ M there exists an invariant neighborhood U x of x, and an equivariant differentiable embedding of U x in V t \{0} for some t. We write G(V ) for the category whose objects are G-manifolds subordinate to V and whose maps are continuous equivariant maps.
Next result is an equivariant version of [33,Lemma 3.11]. It is needed in the proof of Proposition 3.16. A key ingredient in the proof is the equivariant version of Whitney embedding theorem, due to Mostow and Palais.  Definition B. 6 The facet corresponding to P is denoted by P,S\P (often simply P ) and it is called the facet associated with subset P.
In the following corollary we use the notation of the proof of Lemma B.4.
Remark B.8 For consistency of the notation, we observe that the interior of S corresponds to the trivial partition p of {1, . . . , r } into a single subset.
It is clear that if p and p are two partitions of {1, . . . , r }, then p ⊂ p if and only if p is a refinement of p.

B.2. Intersecting a permutohedron with a hyperplane
We describe the intersection of a permutohedron with hyperplanes given by sets of equations {x a i = x b i }. It turns out that this intersection is a lower-dimensional permutohedron. The key statement in this section is Proposition B.11, which identifies the intersection of a permutohedron S ∩ L with s 1 ,...,s r −1 . The identification of Proposition B.11 is such that the combinatorial structure of the boundary is preserved. In order to spell this control over the combinatorial structure, we need to introduce a simple notion.
i . If B = {b 1 , . . . , b } is a finite subset of {1, . . . , r } and p is a partition such that no P i is a subset of B, the reduction of p with respect to B is a partition p B obtained as a subsequent reduction of p by the elements b i , starting from the largest element, then taking the second largest and so on.
The following result can be deduced from the cubical decomposition of a permutohedron, see [28,Section 3.4]. We give a self-contained proof of that result.  Proof We argue by contradiction. Suppose, a ∈ P i , b ∈ P j with i = j. We have an inclusion p ⊂ P i ,S\P i . By Corollary B.7, if x = (x 1 , . . . , x r ) ∈ P i ,S\P i , then x a ≤ s |P i | and x b ≥ s |P i |+1 . Since s k is a strictly increasing sequence, we conclude that x a < x b , hence P i ,S\P i ∩ L = ∅. In particular p ∩ L = ∅.
Continuation of the proof of Proposition B.11. We will construct the isomorphism by induction, starting with the lowest dimension faces, i.e. vertices. For a partition p of length r , p ∩ L is empty by Lemma B.13. Suppose p is a length r − 1 partition. Unless a and b belong to the same subset of the partition, p ∩ L = ∅. Consider the case, when {a, b} subset P j for some j = 1, . . . , r − 1. Obviously P j = {a, b} and all other subsets P i , consists of single elements, P i = {p i } for some p i different than a and b. By Corollary B.7, p is given by x a , x b ∈ [ j, j + 1]

(B.3)
Since n−1 is an invariant subset, Z m acts on n−1 .
We are now going to show that actionσ endows n−1 with a structure of a Z mmanifold, and we will compute its Z m -dimension. Let V σ denote the Z m -representation induced by the action of σ on R n . If σ is a product of p disjoint cycles of lengths n 1 , n 2 , . . . , n p , then there is an isomorphism of representations where R[Z n i ] denotes the real group algebra of Z n i , for i = 1, 2, . . . , p, and Z m acts on R[Z n i ] via the projection Z m → Z n i . Permutohedron n−1 is contained in the affine hyperplane which is invariant under Z m . Orthogonal projection of L onto the hyperplane L 0 = (x 1 , x 2 , . . . , x n ) : shows that the action ofσ restricted to L yields a representation isomorphic to V σ −R.

Remark B.16
Recall that V σ − R denotes the orthogonal complement of a onedimensional trivial representation R inside V σ .
Lemma B.17 Letσ be as above.
(1) At every interior point x of n−1 the tangent representation T x n−1 is isomorphic to (V σ − R)| (Z m ) x , where (Z m ) x denotes the isotropy group at x ∈ n−1 . (4) Suppose that a 1 < a 2 < · · · < a k are elements of P and b 1 < b 2 < · · · < b n−k are the elements of its complement. Suppose that σ (P) = P and define maps The fact that |σ (P)| = |P| completes the proof of the third statement. In order to prove the last assertion notice that if P is invariant under σ , then its complement is also invariant.