# Questions tagged [equivariant-homotopy]

The equivariant-homotopy tag has no usage guidance.

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### Definition of $Fun^G( \mathcal C, \mathcal D)$ in the setting of quasicategories

Our research group is currently going through the paper by Thomas Nikolaus and Peter Scholze On Topological Cyclic Homology and in the Appendix B, Proposition B.5., they use the notation $Fun^{B \...

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### Is a $G$-cell complex always a $G$-CW complex?

I recall vaguely once reading that a cell complex—constructed like a CW-complex but without assuming the cells are appended in order of increasing dimension —"is" actually a CW-complex. I ...

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### Homotopy fixed points of complex conjugation on $BU(n)$

Stably it is known that $(\mathbb Z\times BU)^{hC_2}\simeq \mathbb Z\times BO$ holds. The homotopy fixed point spectral sequence for KU with complex conjugation action can be completely calculated and ...

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### Did Peter May's “The homotopical foundations of algebraic topology” ever appear?

In the monograph Equivariant Stable Homotopy Theory, Lewis, May, and Steinberger cite a monograph "The homotopical foundations of algebraic topology" by Peter May, as "in preparation." It's their [107]...

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### Does there exist a “Margolis-type” definition of equivariant cellular towers?

I am interested in cellular towers in the equivariant stable homotopy category $SH_G$ corresponding to a compact Lie group along with a complete universe for it.
Note here that a cellular tower for ...

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### Reference Request: Equivariant Symplectic bordism

Non-equivariantly, symplectic bordism has been developed extensively by Ray, Gorbunov, and specially S. Kochman in this memoir: http://dx.doi.org/10.1090/memo/0496 Yet the coefficients ...

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172 views

### Do Mackey (co)homology functors factor through derived categories? References with details?

Let $G$ be a compact Lie group; I will write $SH_G$ for the (equivariant) stable homotopy category of $G$-spectra (say, with respect to a complete universe; does its choice affect the homotopy ...

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### What is the relationship between $O_G$-parameterized $\infty$-categories and $\infty$-categories enriched in $Top_G$?

Let $G$ be a finite group. Barwick et al define a $G-\infty$-category to be a fibration over the orbit category $O_G$ of transitive $G$-sets. But in the non-$\infty$-land, the natural guess at where I ...

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### “Oriented representation” sphere

I am trying to understand basic notions from Hill-Hopkins-Ravenel paper: https://arxiv.org/abs/0908.3724
In the Example 3.10 we are considering equviariant cellular chain complex for $n$-dimensional ...

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### Extending a weak version of Sullivan's generalized conjecture

Apologies for the title.
Miller's theorem (formerly Sullivan's conjecture) gives that for a finite group $G$ and a finite dimensional connected CW complex $X$, the based mapping space $\operatorname{...

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### $RO(Q)$-graded homotopy fixed point spectral sequence

I am trying to understand some part of J. Greenlees's "Four approaches to cohomology theories with reality": https://arxiv.org/abs/1705.09365
I have a problem with understanding $RO(Q)$-graded ...

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137 views

### Bredon cohomology of a sign representation for a cyclic group of order 4

Yet another question "I compute Bredon cohomology of something and I am not sure, whether it is correct".
So I am taking a sign representation $\sigma$ of cyclic group of order 4, $C_4$. Then I ...

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162 views

### “Strict” homotopy theory of topological stacks/orbifolds

If we fix a finite group $G$, there are two different useful homotopy theories on the set of $G$-equivariant topological spaces (which are CW complexes, say). One, the "weak" homotopy theory, is given ...

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173 views

### Relation between the category of orthogonal G-spectra and the category of orthogonal H-spectra [closed]

I just read some parts of the book "Equivariant orthogonal spectra and S-modules" by Mandell and May. I wonder whether there is any description of the relation between the categories of orthogonal G-...

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### Homotopy theoretic description of homotopy fixed points (and obstructions) for an action of group $G$ on a groupoid $X$

There are several scattered statements about fixed points and obstructions which I'd very much like to see unified in some framework.
To state them let $G$ be a group acting on a connected (1-...

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### What's with equivariant homotopy theory over a compact Lie group?

For some reason -- I'm not quite sure why -- I've developed the impression that I'm supposed to "tiptoe" around equivariant homotopy theory over groups that are not finite. Should I?
Let me explain. ...

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### Naive equivariant transfer

Given a group $G$, a $\mathbb Z$-graded cohomology theory $E^*_G$, and a $n$-sheeted covering $p\colon X \to Y$, I would like a transfer map $$p_!\colon E^*_G X \to E^*_G Y$$ satisfying $$\require{...

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### Orbit decomposition of the restriction of an equivariant sheaf?

All sets and groups in the question are finite.
In order to understand equivariant sheaves better I'm trying to prove some basic facts from Mackey theory using equivariant sheaves. The main obstacle ...

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### Question about finite G-sets [closed]

Let G be a finite group with subgroups H and K. Then the set of not necessarily equivariant maps from G/H to G/K is itself a finite G-set under the conjugation action. Is there a good description of ...

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### When is $X_{hG} \to X/G$ a weak equivalence for $X$ a free $G$-space, $G$ compact Lie?

Two questions (more details below):
Let $G$ be a compact Lie group and $X$ a $G$-space such that all stabilizer
subgroups are conjugate to a fixed $H \leq G$. Denote by $\pi: X
\to X/G$ the quotient ...

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110 views

### Spin bordism with non free involution

Is there a comprehensive account of GEOMETRIC equivariant spin bordism groups with respect to the group $ \mathbb{Z}/2$ (instead of homotopy theoretical trough equivariant Thom Spectra),...

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435 views

### Models for equivariant genuine commutative ring spectra

The following question is an attempt at understanding various flavours of equivariant commutative ring spectra; it may not be suitable level for this forum.
Let $\mathcal{C}(G)$ be a symmetric ...

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256 views

### Equivariant K-theory of projective representation on complex projective space

Let $G$ be a group and let $V$ be a complex projective representation of $G$, so that $G$ acts on the projectivization $\mathbb{P}(V)$. Is there any way to calculate the $G$-equivariant complex $K$-...

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### Trouble with Stable Equivariant Profinite Homotopy Theory

I've heard that there are some problems in developing a good formalism for stable equivariant homotopy theory (either from the spectral mackey functors perspective or from the orthogonal spectra ...

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### Where can I find basic “computations” of equivariant stable homotopy groups?

I am new to this subject; so please correct me if I will say something wrong or if you don't like my notation. In particular, I don't know whether it is reasonable to consider an infinite group $G$ (...

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### Twists of equivariant spectra

Let $A$ be a spectrum, defined by deloopings $A_n$ (n an integer). Then the identity $A = S^1\wedge A_1$ together with antipodal equivariant spectrum structure on $S^1$ gives genuine $\mathbb{Z}/2$-...

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### When do non-exact functors induce morphisms on $K$-theory?

Let $\mathcal{A}$ and $\mathcal{B}$ be Waldhausen or exact categories, so that we can take the $K$-theory spectrum of $\mathcal{A}$ and $\mathcal{B}$. An exact functor $F: \mathcal{A} \to \mathcal{B}$ ...

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### Are two equivariant maps between aspherical topological spaces homotopic?

Let $f: X \rightarrow Y$ be continuous, X,Y pathwise connected and aspherical (i.e. trivial higher homotopy groups). Then $\pi_1(X)$ acts on the universal cover of $X$ via deck transformations, and on ...

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### Universal space for the family of subgroups of a finite cyclic group

Let $G$ be a compact Lie group and let $\mathcal{P}_G$ denote the family of proper subgroups of $G$. The universal space for the family $\mathcal{P}_G$ is a cofibrant $G$-space which does not have $G$-...

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### $RO(G)$-graded homotopy groups vs. Mackey functors

Everything here is model-independent: either take co/fibrant replacements wherever appropriate, or work $\infty$-categorically.
Also, I've looked through other similar MO questions, but I didn't find ...

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272 views

### (Non)-equivariant equivalence in $G$-spectra

In HHR, an important part is the periodicity theorem. For proving the theorem, they invert a carefully defined class $D \in \pi^{C_8}_{19\rho_8}(N^8_2MU_{\mathbb{R}})$ and they can find an element in $...

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### Genuine equivariant ambidexterity

A particular case of Lurie and Hopkins' ambidexterity theory is that if $G$ is a finite group acting on a $K(n)$-local spectrum $X$ then the norm map
$$ X_{hG} \to X^{hG} $$
is a $K(n)$-local ...

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### homotopy equivalence between configuration spaces

Let $M$ be a $m$-dimensional compact manifold without boundary and $W(M)$ the non-compact $CW$-complex obtained by glueing $[0,1)\times (0,1)^{m-1}$ to $M$, identifying the boundary $\{0\}\times(0,1)^{...

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256 views

### isotopy equivalence (topological meaning) between $CW$-complexes

Let $M$ and $N$ be $CW$-complexes.
Definition. (different from the isotopy notion in geometry of submanifolds). A (topological) isotopy is a fibre-wise continuous map
$$
F: M\times [0,1]\...

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### What is higher equivariant homotopy?

In Lurie's "Survey of elliptic cohomology" it is claimed that there exists some mystical "2-equivariant homotopy theory" for elliptic cohomology. The classical equivariant elliptic cohomology is ...

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### Motivation for equivariant homotopy theory?

I'm in the process of learning equivariant homotopy theory, so I was wondering: what is the importance of equivariant homotopy theory, and what has it been applied to so far? I know of HHR's solution ...

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### Equivariant Fredholm operators classify equivariant K-theory

Let $\mathcal{F}$ be the space of Fredholm operators on a separable Hilbert space $H$ with the topology induced by the operator norm.
If $X$ is compact,
Atiyah-Jänich proved that
$$[X,\mathcal{F}]\...

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### How does the HHR Norm functor interact with the cotensor over $G$-spaces?

Let $N_H^G$ be the norm functor from orthogonal $H$-spectra to orthogonal $G$-spectra. We know the category of orthogonal $G$-spectra $\mathcal{S}_G$ is enriched over the category of based $G$-spaces $...

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### Iterated Homotopy Quotient

If one has a normal Lie group inclusion $H\to G$, with quotient $G/H$, and a $G$-manifold $X$, one can take the quotient $X/H$. Then the $G$-action on $X/H$ factors thru a $G/H$-action, so one can ...

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241 views

### $G$-CW complex structure of certain G-space

Let $G$ be a finite group and $H$ be a subgroup of $G$ . Let us denote $X= G/H \ast G/H \ast \cdots \ast G/H $($k$ times).Where $\ast$ denotes the topological join operation. My question are as ...

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### $G$-CW complex structure of universal a $\mathcal{F}$-space

Let $G$ be a finite group and $H$ be an abelian subgroup of $G$. Let $\mathcal{F}$ be a family of all subgroups of $H$ , i.e. $\mathcal{F}= \{K : K \leq H\}$ Define universal $\mathcal{F}$-space $E\...

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### Naive G-spectrum representing geometric equivariant cobordism

Let $G$ be a finite group. By the transversality results of Wasserman $G$-equivariant bordism (say real) should be a naive homology theory, and as such it should be represented by a naive G-spectrum.
...

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### Maps between equivariant loop spaces

I have an elementary question about equivariant loop spaces that I feel it should be well known.
Given a finite group $G$ and a finite $G$-set $J$ let $S^J=\mathbb{R}[J]^+$ be the permutation ...

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### Explicit calculation of G-CW(V) structure of a G-space

I know explicitly the $Z/6$-CW($ξ^2$)-complex structure of $D(ξ^2)$, where $ξ$ is the non-trivial irreducible representation of $Z/6$ without fixed points. I am looking for an explicit calculation of ...

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### Why is the oriented $G$-homotopy type of a $G$-complex uniquely determined by the periodicity generator?

Say we have a periodicity generator $e \in H^k(BG)$. I can show that we then have a $(k-1)$-dimensional $G$-complex $X$ with free $G$-action. It's also not that difficult to see that it has trivial $G$...

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### Equivariant Homotopy

Let $G=\mathbb{Z}/2\mathbb{Z}$ be $\{\pm1\}$ and let there be two $G$-spaces given: $X=$ The surface of a cylinder including its boundary circles and $S^4$. That means we two G-actions $f_1:G\times X\...

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### Equivariant model structure on $G-\mathrm{Gpd}$

Let's denote $G\text{-}\mathrm{Gpd}$ the presheaf category $[\mathbf{B}G, \mathrm{Gpd}]$. Now assume that $\mathrm{Gpd}$ is endowed with its natural model structure where weak equivalences are ...

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### G-spaces and SG-module spectra

This question is related to the one here, but has a slightly different angle.
Let $G$ be a topological group and let $X$ be a $G$-space. Taking the suspension spectrum $\Sigma^{\infty}_+ X$ (in my ...

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431 views

### Isomorphism between the Burnside ring $A(G)$ and the zeroth $G$-equivariant stable homotopy $\pi^{G}_0(S^0)$

Let $G$ be a compact Lie group. I know that the Burnside ring $A(G)$ is isomorphic to the zeroth $G$-equivariant stable homotopy $\pi^{G}_0(S^0)$. What is the isomorphism between $A(G)$ and $\pi^{G}_0(...

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### When is the diagonal inclusion a $\Sigma_2$-cofibration?

Recall that a space $X$ is called locally equiconnected or LEC if the diagonal map $d:X\hookrightarrow X\times X$ is a cofibration. For example, CW-complexes are LEC. There is some discussion of this ...