Questions tagged [equivariant-homotopy]

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5
votes
0answers
135 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 ...
10
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2answers
368 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 ...
8
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1answer
243 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 ...
2
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0answers
129 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, ...
8
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0answers
299 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 $F/d$ ...
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0answers
119 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 ...
6
votes
3answers
293 views

How does one rigorously lift a map $Sp \rightarrow Sp$ of spectra to equivariant spectra?

This is in part motivated from my attempt to understand tate diagonal in III.1 of Thomas Nikolaus, Peter Scholze, On topological cyclic homology, arXiv:1707.01799. I just want to make my ...
2
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1answer
136 views

Equivariant colimit and equivariant functors

This is rather specific B.5 of Thomas Nikolaus, Peter Scholze, On topological cyclic homology, arXiv:1707.01799 (on last line p147), which I am having fundamental confusion. We have the categories $...
4
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1answer
181 views

Computing homotopy colimit of a space with free $S^1$-action

Context. I am trying to understand the argument in B.4 of Thomas Nikolaus, Peter Scholze, On topological cyclic homology, arXiv:1707.01799 (on p147). I am still lost. But from Maxime's helpful ...
9
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1answer
160 views

Almost free circle actions on spheres

$\DeclareMathOperator{\Fix}{\operatorname{Fix}}$I am looking for any reference regarding the following problem: Problem: Consider a smooth almost-free action of $S^1$ on a smooth sphere $S^n$. Then ...
7
votes
1answer
192 views

The set of isomorphism classes of Z/nZ-equivariant line bundles over a 2 dimensional Z/nZ-CW complex

Suppose I wish to find the set of isomorphism classes of $\mathbb{Z}/n\mathbb{Z}$-equivariant line bundles over a 2-dimensional, compact $\mathbb{Z}/n\mathbb{Z}$-CW-complex $X$, i.e. $\mathrm{Vect}^{1}...
5
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1answer
153 views

Slices for certain $C_p$-spectrum

By the work of Hill-Yarnall, for the group $G=C_p,$ all the slices for any spectrum, in particular, for $S^V \wedge H\underline{\mathbb{Z}}$, are classified. Here $V$ is a representation of $C_p.$ ...
7
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0answers
87 views

Existence of relative equivariant minimal models

In equivariant rational homotopy theory the existence of minimal models (i.e. the equivariant generalization of minimal Sullivan models) has been established by Triantafillou (jstor:1999119) and Scull ...
5
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1answer
156 views

An induction formula for spectral Mackey functors, and a fake proof

I'm trying to get a grasp of Barwick's model for genuine $G$-spectra, that is, spectral Mackey functors 1. There's a classical formula about induction, that should be easy to prove, that I was trying ...
6
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0answers
103 views

Homotopy groups of certain geometric fixed point spectrum

Let $G$ be a finite group and $E$ be a genuine $H$-spectrum for $H\leq G.$ Then for any subgroup $K$ of $G$, consider the $K$-spectrum $X=Res^G_K Ind^G_H(E).$ Is there any reference for computing the ...
8
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0answers
132 views

A question on recognition of equivariant loop spaces

I have a question about equivariant loop space that has been bothering me, and that I have not been able to find an answer to in the obvious places. We know from the work of Segal that to give a loop ...
4
votes
1answer
146 views

Bredon cohomology of a permutation action on $S^3$

I've seen a couple of similar questions asking to verify computations of Bredon cohomology here and here, so I will ask one such question myself. Let $\mathbb{Z}/2$ act on $S^3\subset \mathbb{C}^2$ by ...
5
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0answers
104 views

Equivariant splitting of loop space of a suspension

It is well known, e.g. by Cohen's "A model for the free loop space of a suspension", that there is a stable splitting of the free loop space $\mathcal{L} \Sigma X $of the suspension $\...
7
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3answers
364 views

Homotopy group action and equivariant cohomology theories

Many of the introductory notes on generalized equivariant cohomology theories assume that one is working over the category of $G$-spaces or $G$-spectra. However, one thing that concerns me is that the ...
6
votes
2answers
186 views

Fibre preserving maps of Borel constructions

Let $G$ be a discrete group with universal principal bundle $EG\to BG$, and let $X$ and $Y$ be left $G$-spaces. An equivariant map $\overline{f}:X\to Y$ induces a fibre-preserving map $f:EG\times_G X\...
7
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0answers
140 views

A confusion about geometric fixed points via spectral Mackey functors and smashing localisations

Let $G$ be a finite group and $N$ a normal subgroup. One of the modern ways to construct the $\infty$-category of $G$-spectra is as product-preserving spectral presheaves $\text{Sp}^G = \text{Fun}^{\...
3
votes
3answers
311 views

K-theory of free $G$-sets and the classifying space, and generalization

$\newcommand\Sets{\mathrm{Sets}}\DeclareMathOperator\Nerve{Nerve}$Let $G$ be a finite group, $\mathcal{G}^0$ be the category of finite free $G$-sets and isomorphisms between them. Then $\mathcal{G}^0$...
3
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0answers
108 views

What is the definition of $\operatorname{Fun}^{B \mathbb Z}$ used in Nikolaus--Scholze Proposition B.5? [duplicate]

I am trying to understand the relationship between cyclic objects in a quasicategory $\mathcal C$ and $S^1$-equivariant objects in $\mathcal C$ as presented in Nikolaus--Scholze "On Topological Cyclic ...
3
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0answers
97 views

Extensive survey of computations of equivariant stable stems

Where can I find a comprehensive survey of computations of equivariant stems? To my knowledge, the status is: Classical Work of Araki and Iriye, Osaka J. Math. 19 (1982). ...
6
votes
1answer
198 views

$p$-adic equivalence of spectra with $G$-action

In Krause and Nikolaus' "Lectures on topological Hochschild homology and cyclotomic spectra" (which can be found here), there is a lemma I'm trying to understand, but one line in the proof ...
9
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1answer
284 views

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 \...
5
votes
1answer
487 views

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 ...
14
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1answer
384 views

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 ...
24
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2answers
2k views

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]...
2
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0answers
61 views

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 ...
3
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0answers
77 views

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 ...
2
votes
1answer
191 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 ...
6
votes
1answer
255 views

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 ...
8
votes
1answer
184 views

“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 ...
12
votes
1answer
397 views

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{...
5
votes
1answer
188 views

$RO(Q)$-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 ...
2
votes
2answers
177 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 ...
5
votes
0answers
188 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 ...
1
vote
2answers
188 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-...
7
votes
1answer
750 views

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-...
19
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1answer
864 views

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. ...
7
votes
1answer
466 views

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{...
2
votes
1answer
118 views

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 ...
0
votes
1answer
144 views

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 ...
8
votes
1answer
340 views

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 ...
4
votes
0answers
120 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),...
8
votes
2answers
524 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 ...
8
votes
0answers
334 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 $K$-...
11
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0answers
210 views

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 ...
6
votes
1answer
480 views

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 $G$ (...