All Questions
Tagged with equivariant-homotopy homotopy-theory
50
questions
27
votes
4
answers
3k
views
(∞, 1)-categorical description of equivariant homotopy theory
I'm trying to learn a bit about equivariant homotopy theory. Let G be a compact Lie group. I guess there is a cofibrantly generated model category whose objects are (compactly generated weak ...
- 24.7k
25
votes
2
answers
2k
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 ...
- 2,247
22
votes
1
answer
1k
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. ...
- 59k
15
votes
3
answers
2k
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 ...
15
votes
2
answers
1k
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 ...
- 6,029
14
votes
1
answer
472
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 ...
- 403
13
votes
1
answer
603
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 ...
- 372
12
votes
1
answer
436
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{...
- 34.9k
12
votes
2
answers
2k
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 ...
- 1,951
11
votes
0
answers
250
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 ...
- 111
10
votes
2
answers
463
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 ...
- 59k
9
votes
1
answer
500
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 ...
9
votes
1
answer
703
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 ...
- 4,730
9
votes
0
answers
631
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, ...
- 6,029
8
votes
1
answer
913
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-...
- 7,469
8
votes
1
answer
262
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 ...
- 34.9k
8
votes
0
answers
154
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 ...
- 745
8
votes
0
answers
498
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 $\...
- 25.1k
6
votes
1
answer
716
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 ...
6
votes
1
answer
244
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 ...
- 997
6
votes
1
answer
421
views
An exact sequence involving THH
Disclaimer: I'm a relative beginner in this area. I'm trying to prove that if one has a commutative ring $R$ and a prime number $p$, then there is an exact sequence of the form
$$\DeclareMathOperator\...
- 63
6
votes
1
answer
663
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 ...
6
votes
1
answer
332
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 ...
- 59k
6
votes
0
answers
151
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 ...
- 933
5
votes
1
answer
493
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 ...
- 73
5
votes
1
answer
182
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 ...
5
votes
1
answer
170
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.$
...
- 745
5
votes
0
answers
149
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 ...
5
votes
0
answers
122
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 $\...
- 111
5
votes
0
answers
202
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 ...
- 7,447
4
votes
2
answers
1k
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 ...
- 113
4
votes
2
answers
353
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)^{...
- 1,970
4
votes
1
answer
208
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 ...
- 661
4
votes
1
answer
262
views
Calculate homotopy groups of $\mathbb{Z}_2$-equivariant loop spaces of "complex" topological spaces
Let $X$ be a topological space such that complex conjugation is defined (e.g. $\mathbb{C}^n$) and let us define the set of maps $$S_d:= \left\{f: (I^d,\partial I^d)\to (X,x_0)\mid \overline{f(k)} = f(...
4
votes
0
answers
127
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),...
- 2,590
4
votes
0
answers
414
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 ...
- 12.4k
3
votes
3
answers
398
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 $\mathcal{G}^0$...
- 3,031
3
votes
1
answer
278
views
What is the pointed Borel construction of the $0$-sphere?
From what I understand, the Borel construction takes a $G$-space $X$ and produces a topological space $X\times_{G}\mathbf{E}G$―the homotopy quotient $X/\!\!/G$ of $X$ by $G$ in the $\infty$-category ...
- 9,727
3
votes
0
answers
79
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
3
votes
0
answers
117
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 ...
user340793
3
votes
0
answers
122
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 ...
- 139
3
votes
0
answers
82
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 ...
- 2,590
3
votes
0
answers
144
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 ...
user2664
2
votes
1
answer
540
views
characterization of cofibrations in CW-complexes with G-action
Is there a condition for a $G$-equivariant map $X \to Y$ to be a cofibration of $G$-spaces? Here $X$ and $Y$ are CW complexes, the group $G$ is finite, and acts by cellular maps.
I am using the model ...
- 367
2
votes
1
answer
175
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 $...
- 661
2
votes
0
answers
99
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 $G$...
- 21
1
vote
1
answer
299
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]\...
- 1,970
1
vote
0
answers
196
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 ...
- 1,679
1
vote
0
answers
116
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 ...
- 745
1
vote
0
answers
199
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\...
- 386