All Questions
Tagged with equivariant-homotopy stable-homotopy
30 questions
2
votes
1
answer
300
views
G-equivariant homotopy between G-spaces
I apologize for asking too many questions in a single post. I am not very conversant with equivariant homotopy theory. While discussing with some faculty I was told that certain fact is true. All ...
- 139
4
votes
2
answers
459
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(...
2
votes
0
answers
72
views
Equivariant $K$-theory and proper actions of discrete groups
The work of Lück and Oliver describes the generalization of equivariant $K$-theory to infinite discrete groups. When $X$ is a finite proper $G$-CW complex, there exist Bott isomorphisms $K^n_G(X)\cong ...
- 21
4
votes
1
answer
238
views
Equivariant complex $K$-theory of a real representation sphere
Consider the one-point compactification of a $U(n)$-representation $V$, denoted by $S^V$. I want to caclulate $\tilde{K}_\ast^{U(n)}(S^V)$. When $V$ is a complex $U(n)$-representation, we can use the ...
user501794
6
votes
2
answers
450
views
What are the naive fixed points of a non-naive smash product of a spectrum with itself?
Let $X$ be an object in one of the well-known symmetric monoidal model categories of spectra. E.g., an $\mathtt S$-module in the sense of EKMM, or an orthogonal spectrum, or a $\Gamma$-space, etc.
One ...
- 10.9k
5
votes
1
answer
223
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 ...
4
votes
1
answer
165
views
$E^G_\ast(E)$ tensored with the rationals
Lemma 17.19 of Switzer's "Algebraic topology - Homology and Homotopy" states that $E_\ast(F)\otimes\mathbb{Q}$ is isomorphic to $\pi_\ast(E)\otimes\pi_\ast(F)\otimes\mathbb{Q}$. I wanted to ...
user493302
3
votes
0
answers
152
views
Equivariant classifying space and manifold models
The classifying space $BS^1$ for $S^1$-bundles can be taken to be the colimit of $\mathbb{CP}^n$ which are smooth manifolds and the inclusions $\mathbb{CP}^n \hookrightarrow \mathbb{CP}^{n+1}$ are ...
- 803
1
vote
0
answers
84
views
Equivariant spectrum with coefficients
I am curious to know whether spectra with coefficients as defined in Adams's Blue book be defined to an equivariant setting. In the non-equivariant case, for a spectrum $E$ and an abelian group $A$, ...
- 19
10
votes
2
answers
516
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 ...
- 64k
2
votes
0
answers
242
views
Is the equivariant Steenrod algebra useful?
I am a newbie to the field, so please excuse any potential obvious gaps in knowledge. I have been wondering of late about the equivariant (dual) Steenrod algebra in the context of genuine $G = C_p$ ...
- 21
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
2
votes
1
answer
236
views
Double coset decomposition for compact Lie groups
The starting point of my question is the following fact: suppose $G$ is a finite group and let $H,K \leq G$ be arbitrary subgroups, then there exists an isomorphism of $G$-sets as follows
\begin{...
- 767
9
votes
0
answers
434
views
Stable homotopy type of $BG^{\wedge}_p$ in algebraic terms
In the mid 90's, Martino- Priddy proved that given two finite groups $G, H$, the following are equivalent:
$\mathbb{F}_p\mathrm{Inj}(P,G)\cong \mathbb{F}_p\mathrm{Inj}(P,H)$ as $\mathbb{F}_p\mathrm{...
- 795
2
votes
0
answers
170
views
Geometric fixed points of induction spectrum
I was reading the paper "The Balmer spectrum of rational equivariant cohomology theories" of J.P.C. Greenlees and I found the following interesting fact, expressed in Lemma 4.2 and Remark 4....
- 767
4
votes
0
answers
124
views
Equivariant phantom maps
In the stable homotopy category a map of spectra $f\colon X \rightarrow Y$ is called phantom is the induced map between the associated homology theories $X_* \rightarrow Y_*$ is zero, it is know that ...
- 767
16
votes
6
answers
2k
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 ...
- 13.5k
7
votes
0
answers
206
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}^{\...
- 71
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 ...
- 6,069
9
votes
2
answers
1k
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 ...
- 9,491
9
votes
2
answers
821
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 ...
- 1,961
6
votes
1
answer
651
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 $G$ (...
- 16.9k
8
votes
1
answer
361
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 $...
- 583
7
votes
1
answer
482
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.
...
- 425
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
11
votes
0
answers
402
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 $...
- 423
9
votes
1
answer
1k
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 ...
- 1,273
3
votes
0
answers
85
views
Twists of equivariant spectra
Let $A$ be a spectrum, defined by deloopings $A_n$ (n an integer). Then the identity $A = S^1\wedge A_1$ together with antipodal equivariant spectrum structure on $S^1$ gives genuine $\mathbb{Z}/2$-...
- 7,972
3
votes
0
answers
142
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 ...
- 425
8
votes
0
answers
211
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 ...
- 425