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9
votes
1answer
249 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
1answer
349 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
1answer
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
votes
2answers
434 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 ...
7
votes
1answer
162 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
2answers
539 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
2answers
255 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
1answer
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
1answer
369 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
3answers
927 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 ...
12
votes
1answer
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}]\...
10
votes
0answers
156 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
1answer
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 ...
0
votes
0answers
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 ...
1
vote
1answer
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\...
6
votes
1answer
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. ...
3
votes
0answers
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 ...
1
vote
0answers
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 ...
2
votes
0answers
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 $G$...
1
vote
0answers
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\...
3
votes
0answers
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 ...
5
votes
0answers
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 ...
4
votes
2answers
331 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(...
7
votes
0answers
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 ...
7
votes
0answers
416 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 $\...
5
votes
2answers
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 $\mathcal{C}_G$ ...
20
votes
2answers
911 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 ...
3
votes
0answers
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 $V$, ...
11
votes
2answers
599 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 ...
22
votes
2answers
748 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 ...
9
votes
1answer
359 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 ...
2
votes
0answers
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 ?
5
votes
0answers
301 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, ...
8
votes
0answers
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 ...
5
votes
0answers
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 ...
4
votes
2answers
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. ...
6
votes
1answer
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
votes
1answer
217 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\...
0
votes
1answer
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 ...
14
votes
8answers
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 ...
2
votes
0answers
301 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 $M$...
17
votes
0answers
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 ...
12
votes
6answers
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 ...
5
votes
1answer
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 ...
2
votes
0answers
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 ...
6
votes
2answers
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 ...
8
votes
0answers
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, ...
3
votes
2answers
607 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 ...
8
votes
1answer
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 ...
17
votes
0answers
858 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 $RO(\mathbb{Z}/p)$-...