All Questions
Tagged with equivariant-homotopy homotopy-theory
19 questions with no upvoted or accepted answers
11
votes
0
answers
264
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
9
votes
0
answers
636
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, ...
- 6,069
8
votes
0
answers
166
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
500
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 $\...
- 25.6k
6
votes
0
answers
162
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 ...
- 803
5
votes
0
answers
120
views
Torus equivariant Morava K-theory
Let $X$ be a CW complex with a torus action $T$. Is there an established definition in equivariant stable homotopy theory of $T$-equivariant Morava K-theory, $K_p(n)^*_T(X)$? Any explicit references ...
- 959
5
votes
0
answers
161
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 ...
- 10.5k
5
votes
0
answers
131
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 $\...
- 121
5
votes
0
answers
218
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 ...
- 7,637
5
votes
0
answers
442
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 ...
- 12.5k
4
votes
0
answers
128
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),...
- 2,630
3
votes
0
answers
85
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
127
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
85
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 ...
- 2,630
3
votes
0
answers
145
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
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
0
answers
244
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 ...
- 1,759
1
vote
0
answers
118
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
200
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\...
- 396