The equivariant-homotopy tag has no usage guidance.

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**1**answer

270 views

### 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 $F: \mathcal{A} \to \mathcal{B}$ ...

**5**

votes

**1**answer

351 views

### 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 ...

**2**

votes

**1**answer

111 views

### 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 $G$-...

**14**

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**2**answers

436 views

### $RO(G)$-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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votes

**1**answer

163 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 $...

**6**

votes

**2**answers

555 views

### 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 ...

**4**

votes

**2**answers

256 views

### 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)^{...

**2**

votes

**1**answer

206 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]\...

**19**

votes

**1**answer

370 views

### 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 ...

**14**

votes

**3**answers

931 views

### 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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**1**answer

216 views

### 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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**0**answers

157 views

### 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 $...

**6**

votes

**1**answer

367 views

### 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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116 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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vote

**1**answer

89 views

### $G$-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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**1**answer

260 views

### 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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**0**answers

100 views

### 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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**0**answers

108 views

### 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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90 views

### 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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177 views

### 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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**0**answers

86 views

### 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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167 views

### 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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votes

**2**answers

332 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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130 views

### 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 ...

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417 views

### Failure of “equivariant triangulation” for finite complexes equipped with a $G$-action

Let $\mathcal{S}$ be the $\infty$-category of spaces, and let $G$ be a finite group, and let $BG$ be the groupoid with one object and automorphisms given by $G$.
Consider the $\infty$-category $\...

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votes

**2**answers

319 views

### Is the category of $G$-spaces a model category?

Let $G$ be a compact Lie group and $\mathcal{C}_G$ the category of $G$-spaces (ie. topological spaces endowed with continuous left $G$-actions). Is there a model category structure on $\mathcal{C}_G$ ...

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913 views

### Adams Operations on $K$-theory and $R(G)$

One of the slickest things to happen to topology was the proof of the Hopf invariant one using Adams operations in $K$-theory. The general idea is that the ring $K(X)$ admits operations $\psi^k$ that ...

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**0**answers

110 views

### Why “non-linear similarity” is the same as equivalence of representations for connected Lie groups?

Let $G$ be a compact Lie group and $V$ a finite-dimensional orthogonal $G$-representation. Write $S^V$ for the quotient $D(V)/S(V)$, where $D(V)$ and $S(V)$ are the unit disk and sphere in $V$, ...

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**2**answers

601 views

### “abstract” description of geometric fixed points functor

I'm sure this must be well known, but I could not find any references.
My basic question is: Are there "abstract" descriptions of the geometric fixed point functors in equivariant stable homotopy ...

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**2**answers

757 views

### Equivariant classifying spaces from classifying spaces

Given compact Lie groups $G$ and $\Pi$, there is a notion of "$G$-equivariant principal $\Pi$-bundle", and a corresponding notion of classifying space, often denoted $B_G\Pi$, so that $G$-equivariant ...

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**1**answer

360 views

### Equivariant homotopy, simplicially

It is a classic result of Kan that the homotopy categories (with appropriate model structures) of simplicial sets and of topological spaces (in fact, one could only care about CW-complexes) are ...

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**0**answers

77 views

### Zeroth G-equivariant Stable Stem [duplicate]

Let G be a finite group. Can anyone give me a motivation and rigorous proof of the Burnside ring A(G) is isomorphic to the zeroth G-equivariant stable stem ?

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305 views

### Is the Milnor construction contractible

Let $G$ be any topological group. Then we can form the infinite join $E_G$ of $G$, i.e. the colimit $G*G*G\cdots$.
Is $E_G$ contractible?
I mean it is clear that $E_G$ is weakly contractible, ...

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162 views

### Fibrations of orthogonal G-spectra and fixed points

There are at least two fixed point functors that characterize stable equivalences of orthogonal G-spectra: the geometric fixed points and the naive fixed points of a fibrant replacement.
Is this true ...

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172 views

### Extensions of discrete groups by spectra

If $G$ is a discrete group, recall that a (naive) $G$-spectrum consists of based $G$-spaces $E_n$ together with based $G$-maps $\Sigma E_n \to E_{n+1}$, where we give the suspension coordinate the ...

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257 views

### Need M combinatorial for existence of injective model structure on $M^G$?

I'm doing some work with model categories and operads, and to check a certain hypothesis I've had to learn a bit of equivariant homotopy theory. Let $M$ be a model category and $G$ be a finite group. ...

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**1**answer

455 views

### Equivariant colimits and homotopy colimits

Suppose we are given a diagram of topological spaces. We can restrict ourselves to the diagrams over finite partially ordered sets and let all spaces be good enough (e.g. CW-complexes). One can take ...

**4**

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**1**answer

218 views

### Can the set of iso classes of G-equivariant H-bundles be given by ordinary homotopy classes of non-equivariant maps?

Let $G$ be a (nice enough) topological group (actually a filtered colimit of compact Lie groups), and let $X$ be a manifold with an action (a proper one in fact) by a Lie group $H$. Let $X//H := (X\...

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**1**answer

148 views

### cofibrations in $O_G$-spaces

For a finite group $G$, let $O_G$ denote the orbit category of $G$. Is there a explicit/nice description of cofibrations in the functor category $Top^{O_G^{op}}$ where the weak equivalences and ...

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2k views

### Reference request: Equivariant Topology

I am teaching a graduate seminar in equivariant topology. The format of the course is that I will give 2-3 weeks of background lectures, then each week a student will present a topic. The students ...

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**0**answers

302 views

### Do non-ordinary Bredon cohomology theories extend?

As shown by Lewis, May, and McClure (MR0598689), the ordinary equivariant Bredon cohomology theory $H^*_G(-; M)$ extends to an $RO(G)$-graded cohomology theory precisely when the coefficient system $M$...

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1k views

### Is the equivariant cohomology an equivariant cohomology?

Suppose a finite group $G$ acts piecewise linearly on a polyhedron $X$. Then there are two kinds of equivariant cohomology (or homology).
$\bullet$ With coefficients in a $\Bbb Z G$-module $M$. A ...

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1k views

### Why are equivariant homotopy groups not RO(G)-graded?

I know very little about the fancy equivariant stable homotopy category, so I apologize if this question is silly for one reason or another, but:
I think that stable homotopy, in the non-equivariant ...

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**1**answer

461 views

### equivariant cohomology with respect to a loop group

Let $G$ be a compact connected simply connected Lie group. Let $LG$ be the corresponding
loop group. What is the cohomology of its classifying space (i.e. what is the equivariant
cohomology of a point ...

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**0**answers

263 views

### Reference for homotopy orbits of pointed spaces

Can someone point me to a good (hopefully simple and brief) place to read about the basics
of homotopy orbits for pointed spaces?
More detail:
As I understand it, in the unpointed case,
we use the ...

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**2**answers

721 views

### Simple examples of equivariant homology and bordism

I'm looking for simple examples of calculations of equivariant homology and of equivariant bordism.
I have a finite group G acting on an CW-complex X. I would like to calculate the equivariant ...

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556 views

### toy examples of equivariant homotopy theory

I've heard a little recently about equivariant homotopy theory, and so I decided to try out some baby examples just to get a feel for it. I'm not even sure if these are the right thing to look at, ...

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**2**answers

608 views

### homotopy invariant and coinvariant

Let $V$ be a chain complex, which is either $Z$ or $Z/2$ graded. A circle action on $V$ is
by definition an action of the dga $H_\ast(S^1)$. This consists of a map $D : V → V$ , which is of square ...

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**1**answer

875 views

### A heart for stable equivariant homotopy theory

Let $G$ be a finite group. I wonder whether the following statement is true, known and written down:
There is a t-structure on the stable $G$-equivariant homotopy category such that the associated ...

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860 views

### What is the current knowledge of equivariant cohomology operations?

In Caruso's paper, "Operations in equivariant $Z/p$-cohomology," http://www.ams.org/mathscinet-getitem?mr=1684248, he shows that the integer-graded stable cohomology operations in $RO(\mathbb{Z}/p)$-...