# Questions tagged [equivariant-homotopy]

Equivariant homotopy theory is the study of how homotopy theory behaves when spaces are considered together with a group action on them.

151 questions
Filter by
Sorted by
Tagged with
97 views

### Torus equivariant Morava K-theory

Let $X$ be a CW complex with a torus action $T$. Is there an established definition in equivariant stable homotopy theory of $T$-equivariant Morava K-theory, $K_p(n)^*_T(X)$? Any explicit references ...
280 views

### $\operatorname{Spaces}/BG$ $\sim$ $\operatorname{Spaces}^G$ $\sim$ $??(\Omega G)$

This is a crosspost (with minor alterations). For a topological group $G$, assigning to a $G$-space $X$ the (canonical) map $EG\times_GX\to BG$ establishes an equivalence between the homotopy category ...
• 17.5k
79 views

### Replacing a $G$-CW-complex with a $G$-homotopy equivalent $G$-simplicial complex - can anyone supply a reference?

Let $G$ be a group (not a topological group, just a group). By a $G$-complex I mean a CW-complex with an action of $G$ that takes cells to cells so that the pointwise and setwise stabilizer of each ...
• 1,446
72 views

### Recover the $C_k$-action of a cyclic object as from the $S^1$-action on Hochschild chain

$\newcommand{\Fun}{\operatorname{Fun}}$Let $X\in\Fun(\Lambda^\mathrm{op}, \mathcal{C})$ be a cyclic object in the ($\infty$-)symmetric monoidal category $\mathcal{C}$, where $\Lambda$ is the cyclic ...
63 views

757 views

### The adjoint representation of a Lie group

Let $G$ be a Lie group and $\text{Ad}(G)$ denote its adjoint representation i.e. the adjoint action of the group $G$ on its Lie algebra $\mathfrak{g}$. The adjoint representation is a real $G$-...
• 131
1 vote
142 views

### Relative $G$-equivariant homology groups

Let $X$ be a free $G$-CW-complex with $G$-equivariant cell filtration by $n$-skeleta $X_0 \subset \dots \subset X_n \subset \dots \subset X$ (for rigorous definition see Chap. II, p. 98 in linked ...
• 6,000
81 views

### Explicit computation of the transfer in the representation ring for unitary groups

For a compact Lie group $G$ we let $R(G)$ be the ring of finite dimensional complex $G$-representations studied by Segal in http://www.numdam.org/item/PMIHES_1968__34__113_0.pdf. This comes with extra ...
• 73
436 views

208 views

### Equivariant complex $K$-theory of a real representation sphere

Consider the one-point compactification of a $U(n)$-representation $V$, denoted by $S^V$. I want to caclulate $\tilde{K}_\ast^{U(n)}(S^V)$. When $V$ is a complex $U(n)$-representation, we can use the ...
135 views

### Equivariant classifying space and manifold models

The classifying space $BS^1$ for $S^1$-bundles can be taken to be the colimit of $\mathbb{CP}^n$ which are smooth manifolds and the inclusions $\mathbb{CP}^n \hookrightarrow \mathbb{CP}^{n+1}$ are ...
• 971
155 views

### Terminology for equivariant homology

The usual $G$-equivariant homology and cohomology groups of a space $X$ with $G$-action are given by the Borel construction: $$H_\ast^G(X)=H_\ast((X\times EG)/G),$$ H^\ast_G(X)=H^\ast((X\times EG)/G)...
• 18.3k
1 vote
78 views

### Equivariant spectrum with coefficients

I am curious to know whether spectra with coefficients as defined in Adams's Blue book be defined to an equivariant setting. In the non-equivariant case, for a spectrum $E$ and an abelian group $A$, ...
• 19
195 views

### Rational G-spectrum and geometric fixed points

For a finite group $G$, how is a rational $G$-spectrum $X$ detected by the geometric fixed point functor $\phi^H$ where we consider the conjugacy class of $H\leq G$? I tried finding a reference for ...
1 vote
78 views

### Do the covariant maps of a sheaf with transfer automatically satisfy a dual gluing axiom?

I'm interested in describing a notion of equivariant sheaves that uses Mackey functors on topoi. We can describe a genuine G-spectrum as a spectral Mackey functor on the topos of finite G-sets. If we ...
99 views

### Mostow-Palais equivariant embedding for manifolds with corners

Let $M$ be a compact smooth manifold and let $G$ be a connected compact Lie group acting on $M$. According to an old theorem of Mostow and Palais, there exists a $G$-equivariant embedding of $M$ into ...
• 373
686 views

### Applications of equivariant homotopy theory to representation theory

Equivariant homotopy theory focuses on spaces together with some group action on them. Jeroen van der Meer and Richard Wong have a paper where they use equivariant methods to compute the Picard group ...
• 404
95 views

### When can the trace on cohomology be computed as the Euler characteristic of fixed points?

In this question all groups are finite, and all spaces are nice (eg, simplicial sets). Given a $G$ space $X$, which we assume has finitely many nonzero cohomology groups, we can compute the trace of ...
• 1,854
163 views

### $E^G_\ast(E)$ tensored with the rationals

Lemma 17.19 of Switzer's "Algebraic topology - Homology and Homotopy" states that $E_\ast(F)\otimes\mathbb{Q}$ is isomorphic to $\pi_\ast(E)\otimes\pi_\ast(F)\otimes\mathbb{Q}$. I wanted to ...
119 views

### Is there a framed nullbordism of $T^4$ with an action of $T^4$ that extends the self-action?

Under the identification of the stable homotopy groups with the (stably) framed bordism groups, it is well known that $\eta\in\pi_1\mathbb{S}$ is represented by $S^1$ with its Lie group framing. ...
• 1,982
124 views

### Equivariant spectra with coefficients

In “The localization of spectra with respect to homology”, Bousfield describes localizations with respect to Moore Spectra. Given a spectrum $E$, and a group $M$, he describes the spectrum with ...
323 views

### Is this class of groups already in the literature or specified by standard conditions?

In recent work Lifting $N_\infty$ operads from conjugacy data on homotopical combinatorics / $N_\infty$ operads in equivariant homotopy theory, collaborators Scott Balchin, Ethan MacBrough, and I ...
• 141
155 views

### Splitting of $BGL_1(KR)$

There are infinite loop space splittings $BGL_1(KO)\simeq BGL_1(KO)[0,2]\times Z$ and $BGL_1(KU)\simeq BGL_1(KU)[0,3]\times Z'$ where $Z$ and $Z'$ are 2 and 3 connected, respectively (i.e. they have ...
• 10.1k
207 views

### Is the equivariant Steenrod algebra useful?

I am a newbie to the field, so please excuse any potential obvious gaps in knowledge. I have been wondering of late about the equivariant (dual) Steenrod algebra in the context of genuine $G = C_p$ ...
422 views

### What are the naive fixed points of a non-naive smash product of a spectrum with itself?

Let $X$ be an object in one of the well-known symmetric monoidal model categories of spectra. E.g., an $\mathtt S$-module in the sense of EKMM, or an orthogonal spectrum, or a $\Gamma$-space, etc. One ...
• 10.7k
86 views

### The slice filtration does not arise from a $t$-structure

I've heard this argument quite a bit that the slice filtration does not arise from a $t$-structure on the category of genuine $G$-spectra. Mike Hill points out in Remark 3.12 of The Equivariant Slice ...
• 121
154 views

### Uniqueness of normal microbundle of a smooth embedding

Suppose $M$ is a topological manifold and $\iota: N\hookrightarrow M$ be a submanifold. A normal microbundle of $N$ consists of an open neighborhood $U$ of $N$ and a retraction $\pi: U \to N$ such ...
• 971
139 views

### Geometric fixed points of induction spectrum

I was reading the paper "The Balmer spectrum of rational equivariant cohomology theories" of J.P.C. Greenlees and I found the following interesting fact, expressed in Lemma 4.2 and Remark 4....
• 757
1 vote
45 views

### A question related to injective envelope for a system of DGA's

I was trying to read Fine and Triantafillou's paper "On the equivariant formality of Kahler manifolds with finite group action". They have defined the enlargement at $H$ of a system of DGA's ...
• 1,059
91 views

### Equivariant imbedding of compact manifold

Let $G$ be a compact Lie group smoothly acting on a smooth compact manifold $X$. Is it true that there exists a smooth $G$-equivariant imbedding of $X$ into a Euclidean space acted linearly (and ...
• 21.1k
178 views

### Double coset decomposition for compact Lie groups

The starting point of my question is the following fact: suppose $G$ is a finite group and let $H,K \leq G$ be arbitrary subgroups, then there exists an isomorphism of $G$-sets as follows \begin{...
• 757
246 views

### Model structure on dg-algebras over an "equivariant fundamental category"?

For purposes of $G$-equivariant rational homotopy theory one wants a Quillen adjunction which generalizes the classical one of Bousfield-Gugenheim from plain dg-algebras/simplicial-sets to (co-)...
• 19.6k
178 views

### Decomposition of fiber product of $G$-sets in $G$-orbits

I have posted an identical question in MSE few days ago, but maybe this site is a better adress to discuss this problem: Let $G$ be a finite group and $K, H \leq G$ two subgroups. Then the right ...
• 6,000
61 views

### Endomorphism in the rational stable $O(2)$-equivariant category of the universal space of the family of finite dihedral subgroups

Let $G$ be a compact Lie group. We can define $\mathfrak{F}G$ to be the collection of conjugacy classes of closed subgroups of $G$ whose Weyl group is finite, a bi-invariant metric on $G$ induces a ...
• 757
121 views

### Equivariant phantom maps

In the stable homotopy category a map of spectra $f\colon X \rightarrow Y$ is called phantom is the induced map between the associated homology theories $X_* \rightarrow Y_*$ is zero, it is know that ...
• 757
153 views

• 795
196 views

### Reduction to graph subgroups for Bredon homology when the $G_1\times G_2$ is $G_2$-free

I have the following problem. Let $\Gamma_{G_1\times G_2}$ be a full subcategory of the orbit category $\mathcal{O}_{G_1\times G_2}$ consisting of graph subgroups of $G_1\times G_2$. Further, let $N$ ...
• 1,759
189 views

### Construction of equivariant Steenrod algebra

I am reading through the calculations in Hu-Kriz "Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence" and I've got a small problem in understanding the ...
• 1,759
483 views

### Are finite $G$-spectra idempotent complete?

Question: Let $G$ be a compact Lie group (you can assume that $G$ is finite if you like). Is the category of finite $G$-spectra idempotent complete? Here, by "finite $G$-spectra", I mean ...
• 61.6k
426 views

### Applications of equivariant homotopy theory in chromatic homotopy theory

I usually do computations in equivariant homotopy theory, but I would like to learn chromatic homotopy theory where one may use the equivariant techniques, e.g., slice spectral sequences, etc. For ...
188 views

### Understanding equivariance of the Tate construction $(-)^{tC_P}$

$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\Fun{Fun}\newcommand\Cat{\text{Cat}}\DeclareMathOperator\CoInd{CoInd}\newcommand\Spaces{\text{Spaces}}$It is stated in line 10, p76, Thomas Nikolaus, ...
• 661
361 views

### Is there a 2-categorical, equivariant version of Quillen's Theorem A?

Quillen's Theorem A says that a functor $F:C \to D$ (between 1-categories) induces a homotopy equivalence of classifying spaces $BC \simeq BD$ if for every object $d$ in $D$ the fiber category $F/d$ ...
• 15.5k
1 vote
225 views

### A $d_1$-differential in the homotopy fixed points spectral sequence

I have the following problem. Let $Q$ denote $\mathbb{Z}/2$ as an abelian group and let $X$ be a $Q$-spectrum. If I want to compute the homotopy groups of $X^{hQ}$, homotopy fixed points spectrum, I ...
• 1,759