All Questions
Tagged with equivariant-homotopy at.algebraic-topology
102
questions
28
votes
2
answers
2k
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 ...
- 26.5k
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
Did Peter May's "The homotopical foundations of algebraic topology" ever appear?
In the monograph Equivariant Stable Homotopy Theory, Lewis, May, and Steinberger cite a monograph "The homotopical foundations of algebraic topology" by Peter May, as "in preparation." It's their [107]...
- 25.6k
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
22
votes
0
answers
2k
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 ...
- 5,515
21
votes
0
answers
1k
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 $RO(\mathbb{Z}/p)$-...
- 855
18
votes
8
answers
3k
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 ...
15
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 ...
- 13.1k
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
1
answer
497
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}]\...
- 653
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
14
votes
0
answers
319
views
Is this class of groups already in the literature or specified by standard conditions?
In recent work
Lifting $N_\infty$ operads from conjugacy data on homotopical combinatorics / $N_\infty$ operads in equivariant homotopy theory, collaborators
Scott Balchin, Ethan MacBrough, and I ...
- 141
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
371
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
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
10
votes
2
answers
1k
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}$ ...
- 25.1k
10
votes
1
answer
912
views
Cyclic spaces and S^1-equivariant homotopy theory
I'm trying to understand the relationship between cyclic spaces and S1-equivariant homotopy theory. More precisely, I only care about S1-spaces up to equivalence of fixed point spaces for the finite ...
- 24.7k
10
votes
0
answers
703
views
Adams Spectral Sequence for Equivariant Cohomology Theories
In ordinary algebraic topology the Adams spectral sequence can be applied for any cohomology theory $E$ and in good cases it converges to the stable homotopy classes of maps (of the E-nilpotent ...
- 1,233
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 ...
- 9,186
9
votes
3
answers
633
views
Homotopy group action and equivariant cohomology theories
Many of the introductory notes on generalized equivariant cohomology theories assume that one is working over the category of $G$-spaces or $G$-spectra. However, one thing that concerns me is that the ...
- 914
9
votes
1
answer
235
views
"Oriented representation" sphere
I am trying to understand basic notions from Hill-Hopkins-Ravenel paper: https://arxiv.org/abs/0908.3724
In the Example 3.10 we are considering equviariant cellular chain complex for $n$-dimensional ...
- 1,679
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
1
answer
205
views
Almost free circle actions on spheres
$\DeclareMathOperator{\Fix}{\operatorname{Fix}}$I am looking for any reference regarding the following problem:
Problem: Consider a smooth almost-free action of $S^1$ on a smooth sphere $S^n$. Then ...
- 1,384
8
votes
2
answers
735
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 ...
- 1,951
8
votes
1
answer
349
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 $...
- 573
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
8
votes
0
answers
207
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
7
votes
2
answers
635
views
Naive Z/2-spectrum structure on E smash E?
Let $E$ be a spectrum. Then $E \wedge E$ is a $\mathbb{Z}/2$-spectrum in the naivest possible sense, i.e., an object with $\mathbb{Z}/2$-action in the (∞,1)-category of spectra. Can I make it ...
- 24.7k
7
votes
1
answer
455
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
7
votes
1
answer
507
views
Naive equivariant transfer
Given a group $G$, a $\mathbb Z$-graded cohomology theory $E^*_G$, and a $n$-sheeted covering $p\colon X \to Y$, I would like a transfer map $$p_!\colon E^*_G X \to E^*_G Y$$ satisfying $$\require{...
- 2,974
7
votes
0
answers
188
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
6
votes
2
answers
407
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 ...
- 10.3k
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
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
2
answers
227
views
Fibre preserving maps of Borel constructions
Let $G$ be a discrete group with universal principal bundle $EG\to BG$, and let $X$ and $Y$ be left $G$-spaces. An equivariant map $\overline{f}:X\to Y$ induces a fibre-preserving map $f:EG\times_G X\...
- 34.9k
6
votes
1
answer
594
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 $G$ (...
- 16.3k
6
votes
2
answers
895
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 ...
- 21.8k
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
6
votes
0
answers
142
views
Homotopy groups of certain geometric fixed point spectrum
Let $G$ be a finite group and $E$ be a genuine $H$-spectrum for $H\leq G.$ Then for any subgroup $K$ of $G$, consider the $K$-spectrum $X=Res^G_K Ind^G_H(E).$
Is there any reference for computing the ...
- 745
5
votes
2
answers
718
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 $\mathcal{C}_G$ ...
- 4,850
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
267
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\...
- 33.2k
5
votes
2
answers
450
views
Burnside ring and zeroth G-equivariant stem for finite G
Let $G$ be a finite group. The theorem that the Burnside ring $A(G)$ is isomorphic to the zeroth stable stem $\pi^{G}_0(S)$ is usually said to originate from Segal. I search for a reference of a proof ...
- 1,233