§ PREREQUISITES

§ HOMOTOPY COHERENT DIAGRAMS

Homotopy coherent diagrams represent a relaxed notion of a diagram of topological spaces, one that is required to commute only up to coherent homotopies. In this section we present one approach of introducing group actions on homotopy coherent diagrams.

§ FLOW CATEGORIES

Flow categories have been introduced by Cohen, Jones and Segal as a way to define stable homotopy types associated to Floer homology (see <cit.>, <cit.>, <cit.>). They were used by Lipshitz and Sarkar <cit.> to construct a spatial refinement of Khovanov homology. In this section we describe equivariant cubical flow categories after <cit.> while supplying an alternative description of the (equivariant) cube flow category using the 'free topological category' construction of the previous section.

§.§ Permutohedra and group actions

Let \(n \geq 1\) be a natural number. The symmetric group \(\Sigma_n\) acts on \(\real^n\) by

\[\sigma \acts (x_1, \dotsc, x_n) = (x_{\sigma^{-1}(1)}, \dotsc, x_{\sigma^{-1}(n)}).\]

Suppose \(S =(s_1, \dotsc, s_n) \in \real^n\) is an increasing sequence of real numbers. The \(S\)-permutohedron \(\permutohedron_S\) is the convex hull of \(\Sigma_n\)-translations of \(S\);
if \([n] = (1, \dotsc, n)\), we write \(\permutohedron_{n-1} = \permutohedron_{[n]}\).

For \(i = 1, \dotsc, n,\) let \(\tau_i = \sum_{j=1}^i s_j.\) If \(P \subseteq S\) is a non-empty subset, consider the half-space

\[H_P = \left\{(x_1, \dotsc, x_n) \in \real^n \suchthat \sum_{i \in P} x_i \geq \tau_{\card{P}}\right\}.\]

Then the permutohedron \(\permutohedron_S\) can equivalently be defined as

\[\permutohedron_S = (\setintersection_{\emptyset \neq P \subsetneq S} H_P) \cap \partial H_S\]

(see e.g. <cit.>).
In fact, for any ordered partition \(P_1 \cup \dotsb \cup P_r\) of \(S\), the subset

\[\permutohedron_{P_1, \dotsc, P_r} = \permutohedron_S \cap \setintersection_{i = 1, \dotsc, r-1} \partial H_{P_1 \cup \dotsb \cup P_i}\]

is an \((n-r+1)\)-dimensional face of \(\permutohedron_S.\)
In particular, each \(\permutohedron_{P, S \setminus P}\) is a facet in the sense of <ref>. Moreover, \(\permutohedron_{P_1, \dotsc, P_r}\) is a facet of each of the facets \(\permutohedron_{P_1, \dotsc, P_{i-1} \cup P_i, P_{i+1}, \dotsc, P_r}\), \(i=1, \dotsc, r-1.\) It is the intersection of any two of them; indeed all intersections of the sets \(\permutohedron_{P_1, \dotsc, P_r}\) arise in this way, and the following holds.

The polyhedron \(\permutohedron_S\) becomes an \((n-1)\)-dimensional \(\cornered{n}\)-manifold by letting

\[\partial_i \permutohedron_S = \coprod_{\card{P} = i} \permutohedron_{P, S \setminus P}.\]

For any face \(\permutohedron_{P_1, \dotsc, P_{k+1}}\), there are \(2^k\) faces that contain it, all of the form \(\permutohedron_{P_1, \dotsc, P_i \cup \dotsb \cup P_j, \dotsc, P_{k+1}}\); for the sake of this statement, we treat \(\permutohedron_S\) as the 'face' corresponding to the trivial partition. Denote by \(C_{P_1, \dotsc, P_{k+1}}\) the convex hull of barycentra of all the faces containing \(\permutohedron_{P_1, \dotsc, P_{k+1}}\)

Each of the \(C_{P_1, \dotsc, P_{k+1}}\) is combinatorially equivalent to a \(k\)-dimensional cube, and these cubes form a cubical subdivision of \(\permutohedron_S.\)

The poset of faces of \(\permutohedron_S\) is isomorphic to the poset of internal chains in \(\{0, 1\}^n\): to the face \(\permutohedron_{P_1, \dotsc, P_{k+1}}\) one associates the chain \(u^1 > \dotsb > u^k\) with

\[u^j_i =
\begin{cases}
1, & s_i \in P_1 \cup \dotsb \cup P_j, \\
0, & s_i \in P_{j+1} \cup \dotsb \cup P_{k+1}.
\end{cases}
\]

Consider now the action of a cyclic subgroup \(\langle \sigma \rangle \subseteq \Sigma_n\) on \(\real^n\). The sets \(\permutohedron_S\) and \(\{0, 1\}^n\) are invariant, and so carry the induced action. The poset of faces of \(\permutohedron_S\) also carries this action,
via the corresponding action on the poset of internal chains in \(\{0, 1\}^n\):
\[g \acts (u^1 > \dotsb > u^k) = (g \acts u^1 > \dotsb > g \acts u^k).\]

Now, the fixed point poset of the latter action is isomorphic to the poset of internal chains in some other \(\{0, 1\}^{n'}.\) As the \(\langle \sigma \rangle\)-action descends also to the set of barycentra of faces of \(\permutohedron_S\), the following holds.

The fixed-point set \(\permutohedron_S^{\langle \sigma \rangle}\) is a cubical subcomplex, combinatorially equivalent to a lower-dimensional permutohedron.

Indeed, <cit.> establishes that this can be realized as a diffeomorphism of \((n'-1)\)-dimensional \(\cornered{n'-1}\)-manifolds.

§.§ The cube flow category

The free topological category construction applied to a poset \(\mathcal C\) yields a topological category \(\free \mathcal C\) whose morphism spaces are naturally decomposed as cubical complexes. Abouzaid and Blumberg <cit.> used this in the case that \(\mathcal C= \mathcal P\) is a finite-dimensional poset in the sense that between for any two elements \(p, q \in \mathcal P\), lengths of chains in \(\mathcal P\) starting at \(p\) and ending at \(q\) form a finite set. The upshot is that moduli spaces of \(\free \mathcal P\) are finite-dimensional and have a well-defined boundary.

Consider the topological category \(\cube(n) \coloneqq \free \two{n}\) as defined in <Ref>. We require a description of morphisms spaces of \(\cube(n)\); one such description appears as early as <cit.>, see also <cit.>.

Following the definitions in <Ref>, given \(u, v \in \two{n}\), the space of morphisms \(\cube(n)(u, v)\) is

\[\disjoint_{m \geq 0} \two{n}(u, v)_m \times [0, 1]^m/{\sim},\]

where \(\two{n}(u, v)_m\) is the set of chains in \(\two{n}\) of length \(m\), lying entirely between \(u\) and \(v\). Here, for \(P \subseteq S\) two chains between \(u\) and \(v\), \(\sim\) identifies the cube \([0, 1]^P\) corresponding to \(P\) to the subset of \([0, 1]^S\) obtained by inserting \(1\)s at \(S \setminus P\)-coordinates.
Thus, \(\cube(n)(u, v)\) is isomorphic as a cubical complex to the permutohedron \(\permutohedron^{u \Delta v}\) as described in <Ref>; here,

\[u \Delta v = \{i \in \{1, \dotsc, n\} \suchthat u_i = 1, v_i = 0\}.\]

Likewise, the composition maps in \(\cube(n)\) recover the multifacets \(\partial \cube(n)(u, v)\) making up the boundary of \(\cube(n)(u, v)\) via

\[\partial_i C(n)(u, v) = \coprod_{\abs{v} - \abs{w} = i} \circ (\cube(n)(v, w) \times \cube(n)(u, v)) \subseteq \partial C(n)(u, v).\]

The cube flow category is the topological category \(\cube(n) = \free \two{n}\) equipped with the grading function \(\ob(\cube(n)) = \two{n} \to \mathbb Z\) defined by \((u_1, \dotsc, u_n) \mapsto \abs{u} = u_1 + \dotsb + u_n\).

Consider again the \(\mathbb Z_m\)-action on the category \(\two{n'} = \two{nm} \cong (\two{n})^m\), as in <Ref>. The topological category \(\cube(nm)\) carries an action of \(\mathbb Z_m\), by functors \(\gamma_g \colon \cube(nm) \to \cube(nm)\), \(g \in \mathbb Z_m\), and the maps of moduli spaces \((\gamma_g)_{u, v}\) are isomorphisms of cubical complexes. For \(u \in \cube(nm)\), denote by \((\integer_m)_u\) the isotropy group of \(u\) and consider the \((\integer_m)_u\)-representation \(V_u = \real^{u \Delta 0}\).

The \(\mathbb Z_m\)-action defines a structure of \(\integer_m\)-flow category on \(\cube(n)\), equipped with the grading function \(\gr_{\integer_m}(u) = V_u\).

In keeping with the conventions of <cit.>, we denote this category by \(\cube_\sigma(n')\), \(\sigma\) referring to the permutation of \([nm]\) of order \(m\) which defines the action.

A \(\integer_m\)-equivariant cubical flow category is a \(\integer_m\)-equivariant flow category equipped with a \(\integer_m\)-cover \(f \colon \susp^V \mathcal C \to C_{\sigma}(n'),\) for \(\sigma\) of order \(m\) in \(\Sigma_{n'},\) and for some \(\integer_m\)-virtual representation \(V.\)

§ BURNSIDE FUNCTORS

A Burnside functor is a functor into the Burnside \(2\)-category. After <cit.>, we define a way in which a particular type of homotopy coherent diagram can be described as subordinate to a Burnside functor, in the presence of external group actions on both.

§.§ External actions on Burnside functors

Fix a Burnside functor \(F \colon \mathcal \mathcal C \to \mathcal B.\) Say there exists an action of \(G\) by \(\psi\) on \(\mathcal C.\) An external action on \(F\) compatible with \(\psi\) consists of the following data:

*  a collection of \(1\)-isomorphisms

\[\{\psi_{g, v} \colon F(v) \to F(gv) \suchthat g \in G, \enspace v \in \mathcal C\},\]

*  a collection of \(2\)-isomorphisms

\[\psi_{g, h, v} \colon \psi_{gh, v} \to \psi_{g, hv} \circ \psi_{h, v}\]

(note: should such exist, they are unique),

*  for every morphism \(A \colon x \to y\) in \(\mathcal C\) and every \(g \in G,\) a \(2\)-morphism

\[\psi_{g, A} \colon \psi_{g, y} \circ F(A) \to F(gA) \circ \psi_{g, x}.\]

These data are subject to the following conditions:

*  for objects \(u, v \in \mathcal C\) and a morphism \(A \colon u \to v\), the \(2\)-morphism \(\psi_{gh, A}\) is equal to the composite

\begin{equation*}
\begin{aligned}
\psi_{gh, v} \circ F(A) & \xto{\psi_{g, h, v} \circ_1 \id} \psi_{g, hv} \circ \psi_{h, v} \circ F(A) \xto{\id \circ_1 \psi_{h, A}} \psi_{g, hv} \circ F(hA) \circ \psi_{h, u} \\
& \xto{\psi_{g, hA} \circ_1 \id} F(ghA) \circ \psi_{g, hu} \circ \psi_{h, u} \xto{\id \circ_1 \psi_{g, h, u}} F(ghA) \circ \psi_{gh, u}.
\end{aligned}
\end{equation*}

*  given a composable pair \(u \xto{A} v \xto{B} w\) in \(\mathcal C\), the \(2\)-morphisms

\begin{align*}
\psi_{g, w} \circ F(B) \circ F(A) & \xto{\psi_{g, B} \circ_1 \id} F(gB) \circ \psi_{g, v} \circ F(A) \xto{\id \circ_1 \psi_{g, A}} F(gB) \circ F(gA) \circ \psi_{g, u} \\
& \xto{F_{gA, gB} \circ_1 \id} F(gB \circ gA) \circ \psi_{g, u}
\end{align*}

\[\psi_{g, w} \circ F(B) \circ F(A) \xto{\id \circ_1 F_{A, B}} \psi_{g, w} \circ F(B \circ A) \xto{\psi_{g, B \circ A}} F(gB \circ gA) \circ \psi_{g, u}\]

are equal.

Suppose \(\mathcal C\) carries a \(G\)-action and \(G\) is understood to be acting trivially on \(\two{}\). Then \(\mathcal C \times \{i\}\), \(i = 1, 0\), are \(G\)-invariant subcategories of \(\mathcal C \times \two{}\). Given a Burnside functor \(J \colon \mathcal C \times \two{} \to \burnside\) with an external \(G\)-action by \(\psi\), the subfunctors \(F_i \colon \mathcal C \times \{i\} \to \burnside\) carry external actions induced from \(\psi\) by restriction. This informs the following.

Let \(\mathcal C\) be a small category, acted upon by a group \(G\), and let \(F_1, F_0 \colon \mathcal C \to \burnside\) be functors equipped with external actions of \(G\) by \(\psi_1, \psi_0\). We say that \(F_1\) and \(F_0\) are equivariantly naturally isomorphic if there is a natural isomorphism \(J \colon \mathcal C \times \two{} \to \burnside\) between \(F_1\) and \(F_0\), equipped with an external action of \(G\) extending \(\psi_1\) and \(\psi_0\), respectively.

Let \(F \colon \two{} \to \burnside\) be a Burnside functor and \(G\) a group, understood to be acting trivially on \(\two{}\); denote the single correspondence in \(F(1, 0)\) by \((A, \enspace s \colon A \to X = F(1), \enspace t \colon A \to Y = F(0))\). Then an external action of \(G\) on \(F\) consists of:

* invertible correspondences \(\psi_{g, 1} \colon X \to X\), \(\psi_{g, 0} \colon Y \to Y\), for all \(g \in G\),

* isomorphisms of correspondences \(\psi_{g, h, v} \colon \psi_{gh, v} \to \psi_{g, v} \circ \psi_{h, v}\) for \(v = 1, 0\) and \(g, h \in G\), carrying no additional information beyond their existence,

* isomorphisms of correspondences \(\psi_{g, A} \colon \psi_{g, 0} \circ  A \to A \circ \psi_{g, 1}\) for all \(g \in G\).

These are required to satisfy, for all \(g, h \in G\),

\[\psi_{gh, A} = \psi_{g, h, 0}\circ \psi_{g, A} \circ \psi_{h, A} \circ \psi_{g, h, 1}.\]

By an extension of the <ref>, external actions on Burnside functors from the cube are determined by lower-dimensional data.

Consider the cyclic action of \(\mathbb Z_m\) on \((\two{n})^m\). Suppose \(F \colon (\two{n})^m \to \burnside\) is defined as in <ref>, and that in addition we are given:

* for \(v \in (\two{n})^m\), a \(1\)-isomorphism \(\psi_{g, v} \colon F(v) \to F(gv)\),

* for \(g, h \in G\) and \(v \in (\two{n})^m\), a \(2\)-morphism

\[\alpha_{g, h, v} \colon \psi_{g, h, v} \to \psi_{g, hv} \circ \psi_{h, v},\]

* for each \(g \in \mathbb Z_p\) and \(u \geq_1 v \in (\two{n})^m,\) a \(2\)-morphism

\[\psi_{g, u, v} \colon \psi_{g, v} \circ F(u, v) \to F(gu, gv) \circ \psi_{g, u}.\]

Suppose moreover that this data satisfies the following:

* For any \(u \geq_1 v\) and all \(g, h \in G\),

\[\psi_{gh, u, v} = \alpha^{-1}_{g, h, u} \circ_2 (\psi_{g, hu, hv} \circ \id) \circ_2 (\id \circ \psi_{h, u, v}) \circ_2 \alpha_{g, h, v},\]

* for any \(u \geq_1 v, v' \geq_1 w\) and any \(g \in G\), there is a commutativity hexagon yielding

\[(F(gu, gv, gv', gw) \circ_2 \id) \circ (\id \circ \psi{g, u, v}) \circ (\psi_{g, v, w} \circ \id) = (\id \circ \psi_{g, u, v'}) \circ (\psi_{g, v', w} \circ \id) \circ (\id \circ F(u, v, v', w)).\]

Then there exists a Burnside functor \(F \colon (\two{n})^m \to \burnside\) admitting an external \(\mathbb Z_m\)-action, uniquely up to \(\mathbb Z_m\)-equivariant isomorphism.

§ SPATIAL REFINEMENTS

The aim of this chapter is to, given a Burnside functor (with external action), produce homotopy coherent diagrams with the property that in the homotopy colimit, vertices of the diagram correspond to cells and the Burnside functor describes degrees of attaching maps.

§.§ Stars and star maps

The constructions presented in this sections realise a version of the “charge map”, associating to a configuration of points in \(\real^n\) a map of spheres \(S^n \to S^n\) (see <cit.>). The approach taken here (after <cit.>) allows for composing such maps between (wedges of) spheres along a Burnside functor, in the end furnishing a homotopy coherent diagram. This was already done equivariantly in <cit.>, using spaces of little disks; for our purposes, a wider family of shapes must be used, contatining both disks and products of disks.

In the scope of this section, \(V\) will denote an orthogonal representation of a group \(G\).

For \(X\) a finite \(G\)-set, the configuration space of points of \(X\) in a \(G\)-space \(Y\) is the space

\[\conf_X(Y) = \{\{p_x\}_{x \in X} \in Y^k \suchthat \forall x \neq y \in X \enspace p_x \neq p_y\}.\]

Equivalently, a configuration can be seen as an embedding \(f \colon X \to Y\), whereby \(\conf_X(Y)\) carries an action of \(G\) by

\[(g \acts f)(x) = g \acts f(g^{-1} \acts x).\]

We aim to describe one of the possible equivariant versions of the Pontryagin-Thom collapse map, associating a map between spheres to a configuration. This entails replacing the points of a configuration by “little stars”, as expanded upon below.

Let \(\sphere(V)\) denote the unit sphere in \(V\). Let \(f \colon \sphere(V) \to \real_+\) be a continuous map. By a star in \(V\) we will mean the set

\[B(p, f) = p + \{\alpha \cdot v \in V \suchthat v \in V, \enspace \abs{\alpha} \leq (v)\} \subseteq V\]

for some \(f\) as above and \(p \in V\). The point \(p\) is then called the center point of \(B(p, f)\), and a star is understood to come with a distinguished center point. If \(f\) is \(G\)-invariant, we  call \(B(0, f)\) an invariant star in \(V\). Any star is a star-shaped subset of \(V\); an invariant star in \(V\) is a \(G\)-invariant subset of \(V\), and as such becomes a \(G\)-space.

Let \(\mathbb A \colon \two{} \to \burnside\) be a Burnside functor with external action by \(G\), subordinate to the trivial action of \(G\) on \(\two{}\). Per <ref>, this consists of a correspondence \(X \xot{s} A \xto{t} Y\) along with actions \(\phi_{g, X} \colon X \to X\), \(\phi_{g, Y} \colon Y \to Y\) and \(\phi_{g, A} \colon A \to A\), which commute with the source- and target-maps.

Suppose \(\{B_x\}_{x \in X}\) is a set of \(G\)-invariant stars in \(V\). We fix the radial homeomorphisms \({\varphi_{x, y} \colon B_x \to B_y}\) for all \(x, y\); these satisfy \(\varphi_{x, z} = \varphi_{y, z} \circ \varphi_{x, y}\). The space \(B(X, V) \coloneqq \disjoint_{x \in X} B_x\) carries an action of \(G\) by \(g \acts (x, v) = (g \acts x, g \acts \phi_{x, g \acts x}(v)).\)
The space \(\conf(\{B_x\}, s)\) is defined as the space of embeddings \(\gamma \colon A \to B(X, V)\) satisfying \(\gamma(a) \in B_{s(a)}.\) Topologically, this is the same as \(\disjoint_{x \in X} \conf_{s^{-1}(x)}(B_x).\) The space \(\conf(\{B_x\}, s)\) carries an action of \(G\) by

\[(g \acts \gamma)(a) = g \acts \varphi_{g, g \acts x} \gamma(g^{-1} \acts a).\]

Replacing the points of \(A\) by little stars, we consider the space \(\stars(\{B_x\}, s)\) of embeddings of \(A\)-labeled stars in \(B(X, V)\), again with \(B_a \subseteq B_{s(a)}\). This is topologized as a subset of \({\conf(\{B_x\}, s) \times \maps(X \times S(V), \real^+)}\), and again carries an action by

\[(g \acts \gamma)(a, v) = g \acts \varphi_{g, g \acts x} \gamma(g^{-1} \acts a, v).\]

A configuration of stars with centers \(\{p_x\}_{x \in X}\) can be deformed to one whose stars are all spheres with radius

\[\frac{1}{3}\min_{x, y \in X}(d(p_x, p_y), d(p_x, V \setminus B)).\]

This establishes a strong equivariant deformation retraction from \(\stars(\{B_x\}, s)\) to a bundle of points over \(\conf(\{B_x\}, s)\). The map is also equivariant and induces homotopy equivalences of fixed-point sets, implying the following.

The spaces \(\stars(\{B_x\}, s)\) and \(\conf(\{B_x\}, s)\) are \(G\)-homotopy equivalent.

Note that the target map of the correspondence played no role in the definition of \(\stars(\{B_x\}_X, s)\); rather, it becomes relevant in the definition of the associated map of spheres. Let \(S^V\) denote the one-point compactification \(V \cup \{\infty\}\), considered as a \(G\)-space.
Let \(\mathbb A = (A, s \colon A \to X, t \colon A \to Y)\) be a correspondence and

\[e = \{B_a \subseteq B_{s(a)} \suchthat a \in A\} \in \stars(\{B_x\}, s)\]

a collection of substars in \(V\).
Define a map \(\Phi(e, \mathbb A) \colon \vee_{x \in X} S^V \to \vee_{y \in Y} S^V\) on the summand \(S_x^V\) corresponding to \(x \in X\) to be the map of spheres

\[\restrict{\Phi(e, \mathbb A)}{S_x^V} \colon S_x^V = B_x/{\partial B_x} \to B_x / (B_x \setminus (\bigcup_{\substack{a \in A \\ s(a) = x}} \interior{B}_a)) = \bigvee_{\substack{a \in A \\ s(a) = x}} B_a/{\partial B_a} = \bigvee_{\substack{a \in A \\ s(a) = x}} S_a^V \to \bigvee_{y \in Y} S_y^V,\]

the last map sending \(S_a^V\) by the identity map to \(S_{t(a)}^V.\)
Any map \(\vee_{x \in X} S^V \to \vee_{y \in Y} S^V\) that is of the form \(\Phi(e, \mathbb A)\) for some \(e \in \stars(\{B_x\}, s)\), is called a (\(V\)-dimensional) star map refining the correspondence \(\mathbb A\).

We end this section by stating some facts about star maps without proof.

The map

\[\Phi(-, A) \colon \stars(\{B_x\}, s) \to \maps(\vee_{x \in X} S^k, \vee_{y \in Y} S^V)\]

is continuous.

The star map \(\Phi(e, \mathbb A)\) associated to an element \(e \in \stars(\{B_x\}, s)^H\) fixed by subgroup \(H \subseteq G\), is \(H\)-equivariant.

Given a correspondence \(\mathbb A \colon X \to Y\) as above, \(G\)-representations \(V\) and \(W\), a family of invariant stars in \(\{B_x\}_{x \in X} \subseteq V\) and an invariant star \(B' \subseteq W\), consider the map \(\psi_{B'} \colon \stars(\{B_x\}, s) \to \stars(\{B_x \times B'\}_X, s)\) obtained by taking products of all stars with \(B'\). Then for any \(e \in \stars(\{B_x\}, s)\), the assignment

\[\Phi(e, \mathbb A) \circ \psi_{B'} \in \maps(\vee_{x \in X} S^{V \oplus W}, \vee_{y \in Y} S^{V \oplus W})\]

is a \((V \oplus W)\)-dimensional star map refining the correspondence \(\mathbb A\).

Let \(X \xto{\mathbb A} Y \xto{\mathbb B} Z\) be finite correspondences. Given \(e \in \stars(\{B_x\}, s_A)\) and \(f \in \stars(\{B_y\}, s_B)\), there is a unique arrangement of stars

\[f \circ e \in \stars(\{B_x\}, s_{B \circ A}, \mathbb B \circ \mathbb A, X) = \stars(\{B_x\}, s_{\mathbb B \circ \mathbb A})\]

such that there is an equality of star maps
\(\Phi(f \circ e, \mathbb B \circ \mathbb A) = \Phi(f, \mathbb B) \circ \Phi(e, \mathbb A).\)
Moreover, the assignment

\[\circ \colon \stars(\{B_y\}, s_B) \times \stars(\{B_x\}, s_A) \to \stars(\{B_x\}, s_{B \circ A}, B \circ A, X)\]

is continuous and surjective.

For \((b, a) \in B \times_Y A\), \(b \in B\), \(a \in A\), consider the corresponding stars \(e_b \colon B_b \to B_{s_B(b)}\), \(e_a \colon B_a = B_{s_B(b)}\to B_{s_A(a)}\). The substar \(e_{b, a} \colon B_b \to B_{s_A(a)} = B_{s_{B \circ A}(b)}\) is \(e_a \circ e_b.\)

§ \(G\)-CUBICAL CATEGORIES ARE EXTERNAL ACTIONS ON BURNSIDE FUNCTORS

Independently of <cit.>, a notion of external action on a Burnside functor was introduced by Musyt <cit.>. We use this as a stepping stone in order to pronounce the comparison map establishing equivalence between the notions of equivariant cubical flow categories and external actions on Burnside functors (in the sense of <cit.>).

§ EQUIVALENCE OF REALIZATIONS

Building on the comparison map of the previous section, the aim of the following is to prove that the geometric realizations of an equivariant cubical flow category and that of its associated Burnside functor with external action are equivariantly stably homotopy equivalent.

§ KHOVANOV SPECTRA OF PERIODIC LINKS

We recall the constructions of Khovanov homology and Khovanov spectra, as well as their equivariant extensions due to <cit.>, respectively <cit.> and <cit.>. We present an application of <ref> to this case.

§.§ Khovanov spectra

Given a link diagram \(D\) with \(N\) crossings (numbered \(1\) to \(N\)), the Kauffman cube of resolutions is defined as follows. For \(v = (v_1, \dotsc, v_n) \in \ob(\two{N})\), change the \(i\)-th crossing \((\tikz[baseline=-.8ex] \node[knot under cross, knot, draw, double=Gray] {};)\) to \((\tikz[baseline=-.8ex] \node[knot vert, knot, draw, double=Gray] {};)\) if \(v_i = 0\) and to \((\tikz[baseline=-.8ex] \node[knot horiz, knot, draw, double = Gray] {};)\) if \(v_i = 1.\)

Consider now the Frobenius algebra
\(\mathcal A = \integer[X]/{(X^2)}\)
with comultiplication \(\Delta \colon \mathcal A \to \mathcal A \tensor \mathcal A\) defined by
\(\Delta(1) = 1 \tensor x + x \tensor 1,\)
\(\Delta(x) = x \tensor x.\)
The Khovanov-Burnside functor \(F =F_{Kh} \colon \two{N} \to \burnside\) is defined as follows:

* for \(v \in \ob(\two{N})\),
\(F(v) = \{1, x\}^{\text{circles in the $v$-resolution of D}}\),

* a morphism \(u \to v\) in \(\two{N}\) corresponds to a circle being split into two or two circles being merged into one; \(F(u, v)\) is the correspondence applying the comultiplication, respectively multiplication, rule of \(\mathcal A\) to the labellings,

* for any two chains \(u \to v \to w\), \(u \to v' \to w\) with \(u \geq_2 w\), the \(2\)-morphism

\[F_{u, v, v', w} \colon F(v, w) \circ F(u, v) \to F(v', w) \circ F(u, v')\]

consists of bijections

\[A_{a, b} \coloneqq s^{-1}_{F(v, w) \circ F(u, v)}(x) \cap t_{F(v, w) \circ F(u, v)}^{-1}(z) \to s_{F(v', w) \circ F(u, v')}^{-1}(x) \cap t_{F(v', w) \circ F(u, v')}^{-1}(z) \eqqcolon A'_{a, b}\]

for \(a \in F(u)\), \(b \in F(w)\).
The sets \(A_{a, b}\) and \(A'_{a, b}\) both have \(1\) or \(0\) elements in all but one case. If \(\card A_{x, z} = 2\), then the resolutions along \(u \to v \to w\) split one circle (labeled \(1\) by \(a\)) into two and then merge it back to one (labeled \(x\) by \(b\)); and necessarily, the same can be said about \(u \to v' \to w\). Namely, the morphisms \(u \to w\) correspond to surgery along two edges with endpoints alternating on a single circle \(C_u\). The endpoints cut \(C_u\) into four arcs, among which we distinguish two by the following property: you walk onto them by traveling along one of the surgery edges and turning right. Th two distinghuished arcs are labeled arbitrarily by \(1\) and \(2\), and then the two relevant circles in the \(v\)- and \(v'\)-resolutions are labeled \(C_1, C_2\), respectively \(C'_1, C'_2\). The elements of \(A_{a, b}\) and \(A'_{a, b}\) can then be identified as

\[\alpha = (C_u \mapsto 1) \mapsto ((C_1, C_2) \mapsto (1, x)) \mapsto (C_w \mapsto x),\]

\[\beta = (C_u \mapsto 1)\mapsto ((C_1, C_2) \mapsto (x, 1)) \mapsto (C_w \mapsto x),\]

\[\alpha' = (C_u \mapsto 1) \mapsto ((C'_1, C'_2) \mapsto (1, x)) \mapsto (C_w \mapsto x),\]

\[\beta' = (C_u \mapsto 1)\mapsto ((C'_1, C'_2) \mapsto (x, 1)) \mapsto (C_w \mapsto x),\]

so that \(\restrict{F_{u, v, v', w}}{A_{a, b}}\) can be defined by \(\alpha \mapsto \alpha'\), \(\beta \mapsto \beta'\).

By <ref>, the remaining values of \(F\) are determined up to natural isomorphism. For proof that <ref> of <ref> holds, see <cit.>.

There is a functor \(\integer \langle - \rangle \colon \burnside \to \abelian\), defined as follows: to a set \(X \in \burnside\) associate the free abelian group \(\integer \langle X \rangle\), and to a correspondence \((A, s, t) \colon X \to Y\) the map \(\integer \langle X \rangle \to \integer \langle Y \rangle:\)

\[x \mapsto \sum_{x \in X} \# \{a \in A \suchthat s(a) = x, t(a) = y\} \cdot y.\]

The classical Khovanov homology functor \(\kh \colon \left(\two{n}\right)^{op} \to \abelian\) is the composition \(\kh(D) = \integer \langle - \rangle \circ F_{Kh}\); note that the ladybug matching data encoded by 2-morphisms is forgotten in this composition.

The Khovanov chain complex \(\ckh(D)_*\) is defined as the shift of the totalization of the functor; namely,

\[\ckh_n = \directsum_{\abs{v} = n} \kh(v)[n_-].\]

The differential carries the component \(\kh(u)\) to \(\kh(v)\) by the map
\((-1)^{s_{u, v}} \kh(u, v)\)
if \(u \geq_1 v\), and by the zero map otherwise. The integer \(s_{u, v}\) is defined as
\(\sum_{i=1}^{k-1} u_i,\) where \(u_k\) is the single element in \(\{1, \dotsc, n\}\) with \(u_k = 0\) and \(v_k = 1\).

The complex \(\ckh(D)\) is doubly graded. In addition to the homological grading \(\abs{v} - n_-\), a summand coming from \(\kh(v)\) carries also the quantum grading

\[n - 3n_- + \abs{v} + \card\{\text{circles labeled by $1$}\} - \card\{\text{circles labeled by x}\}.\]

Just the same, the Burnside functor \(F\) can be seen as the sum of direct summands \(F_j\) corresponding to quantum gradings.

A result of <cit.> is that the stable Burnside functor \(\susp^{n_-} F\) has a well-defined realization as a spectrum \(\khovanov(D)\), whose homology is the Khovanov homology. This is the same as our <ref> with \(G\) the trivial group.

§.§ Periodic links

An \(m\)-periodic link is one invariant under a rotation of the sphere of order \(m\), and disjoint from the axis of that rotation. We will give a digest of the constructions of equivariant Khovanov homotopy types due to <cit.> and <cit.>. For a given link, there may be more than one such rotation, defining to distinct equivariant spectra; hence, the rotation is fixed at the outset.

For the following, let \(D\) be a link diagram with \(N = nm\) crossings, invariant under a rotation of the plane \(\rho\) of order \(m\), such that \(\rho(D) = D.\) Consequently, there is an action on the cube of resolutions, which upon numbering the crossings takes the form of the natural \(\mathbb Z_m\)-action on \((\two{n})^m \cong \two{nm}\) as in <ref>.

[Proposition 6.2]stoffregen2018localization construct an external action of \(\mathbb Z_m\) on the Khovanov-Burnside functor \(F_{Kh} \colon \two{n} \to \burnside\) using <ref>. The construction is forced in almost all cases by the group action on \((\two{n})^m\) and the non-equivariant \(F\) itself. The exceptional case is that of ladybug configurations, and the well-definedness of the action follows from the fact that ladybug configurations are invariant under planar isotopy. In parallel, <cit.> use a simplification result <cit.> and construct moduli spaces inductively, with all steps but the one pertaining ladybug configurations already forced.

It is clear that the two constructions are related by the equivalences presented here in <ref>. The results of <Ref> imply the following.

The equivariant stable homotopy types \(\cfcrealization{\mathcal C}\) and \(\hocolim \tilde{F}_{V}^+\) associated to a periodic link by <cit.> and <cit.>, respectively, are equivariantly stably homotopy equivalent. The equivalence can be realised as \(\Sigma^\infty \Psi\), where \(\Psi\) is a cellular map depending on a choice of extended equivariant cubical framed embedding \(\tilde{\iota}\) of \(\mathcal C\).

Let \(\mathbb F\) be a field. Up to chain homotopy, the Khovanov complex \(\ckh(D; \mathbb{F})\) can be equipped with an action of \(\mathbb Z_m\), whereby it can be seen as a \(\mathbb{F}[\mathbb{Z}_m]\)-module. In <cit.>, Politarczyk defined equivariant Khovanov homology with coefficients in a \(\mathbb{F}[\mathbb{Z}_m]\)-module \(M\) by

\[\ekh^{j, q}(D; M) = \ext^j_{\mathbb{F}[\mathbb{Z}_m]}(M; \ckh^{\bullet, q}(D; \mathbb{F})).\]

In <cit.> it is proved that \(\cfcrealization{\mathcal C}\) realizes this notion of equivariant Khovanov homology via Borel cohomology.

Let \(D\) be an \(m\)-periodic link diagram and \(F \colon (\two{n})^m \to \burnside\) the associated Burnside functor with external group action of \(\mathbb{Z}_m\), admitting an equivariant spatial refinement with respect to representation \(V\). For any \(\mathbb{F}[\mathbb{Z}_m]\)-module \(M\), the equivariant Khovanov homology \(\ekh^{j, q}(D; M)\) is isomorphic to the reduced Borel cohomology of \(\hocolim \tilde{F}_V^+\):

\[\ekh^{j, q}(D; M) \cong \mathrm{H}_{\mathbb{Z}_m}^*(\hocolim \tilde{F}_V+, \homs_{\mathbb{F}}(M, \mathbb{F})).\]

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

titleArnold Conjecture and Morava K-theory

Springer-Verlag, Berlin-New York

seriesLecture Notes in Mathematics
titleHomotopy limits, completions and localizations
volumeVol. 304

titleA link surgery spectral sequence in monopole Floer homology
journaltitleAdvances in Mathematics

American Mathematical Society, Providence, RI

seriesMathematical Surveys and Monographs
titleToric topology

titleKhovanov homotopy type, periodic links and localizations
journaltitleMath. Ann.

Birkhäuser, Basel

booktitleThe Floer memorial volume
seriesProgr. Math.
titleFloer's infinite-dimensional Morse theory and homotopy

titleMorse Theory and Classifying Spaces

CRC Press, Boca Raton, FL

booktitleHandbook of homotopy theory
seriesCRC Press/Chapman Hall Handb. Math. Ser.
titleFloer homotopy theory, revisited
year[2020] 2020

equivariant homotopy, excision
pages325 395
titleHomotopy theory of $G$–diagrams and equivariant excision
journaltitleAlgebraic & Geometric Topology

Mathematical Sciences Publishers

titleEquivariant diagrams of spaces
journaltitleAlgebraic & Geometric Topology

* Invalid format of field 'month'

titleG-functors, G-posets and homotopy decompositions of G-spaces
journaltitleFundamenta Mathematicae

titleA categorification of the Jones polynomial
journaltitleDuke Math. J.

titleThe Homotopy Commutative Cube
journaltitleJournal of the London Mathematical Society

Amer. Math. Soc., Providence, RI

booktitleCategorification in geometry, topology, and physics
seriesContemp. Math.
titleThe cube and the Burnside category

Mathematical Sciences Publishers

titleKhovanov homotopy type, Burnside category and products
journaltitleGeometry & Topology

titleA Khovanov stable homotopy type
journaltitleJournal of the American Mathematical Society

titleA refinement of Rasmussen's $S$-invariant
journaltitleDuke Math. J.

titleEquivariant Khovanov Homotopy Type and Periodic Links
University of Oregon

issn0026-2285, 1945-2365
titleEquivariant Khovanov Homology of Periodic Links
journaltitleThe Michigan Mathematical Journal

titleClassifying spaces and spectral sequences
journaltitleInst. Hautes Études Sci. Publ. Math.

titleConfiguration-spaces and iterated loop-spaces
journaltitleInvent. Math.

titleHomotopy coherent diagrams and approximate fibrations

titleLocalization in Khovanov homology
journaltitleGeometry & Topology

titleEquivariant homotopy equivalence of homotopy colimits of
journaltitleArab. J. Math. (Springer)

titleHomotopy Limits and Colimits.
journaltitleMathematische Zeitschrift