Questions tagged [equivariant-homotopy]
Equivariant homotopy theory is the study of how homotopy theory behaves when spaces are considered together with a group action on them.
146
questions
3
votes
2
answers
519
views
The adjoint representation of a Lie group
Let $G$ be a Lie group and $\text{Ad}(G)$ denote its adjoint representation i.e. the adjoint action of the group $G$ on its Lie algebra $\mathfrak{g}$. The adjoint representation is a real $G$-...
3
votes
1
answer
191
views
Reduction to graph subgroups for Bredon homology when the $G_1\times G_2$ is $G_2$-free
I have the following problem. Let $\Gamma_{G_1\times G_2}$ be a full subcategory of the orbit category $\mathcal{O}_{G_1\times G_2}$ consisting of graph subgroups of $G_1\times G_2$. Further, let $N$ ...
- 1,679
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
127
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 ...
- 933
3
votes
0
answers
114
views
Is there a framed nullbordism of $T^4$ with an action of $T^4$ that extends the self-action?
Under the identification of the stable homotopy groups with the (stably) framed bordism groups, it is well known that $\eta\in\pi_1\mathbb{S}$ is represented by $S^1$ with its Lie group framing. ...
- 1,883
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
127
views
Extensive survey of computations of equivariant stable stems
Where can I find a comprehensive survey of computations of equivariant stems?
To my knowledge, the status is:
Classical Work of Araki and Iriye, Osaka J. Math. 19 (1982). ...
- 2,590
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
81
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 $\mathbb{Z}/2$-...
- 7,882
3
votes
0
answers
139
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
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
3
votes
0
answers
117
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 $V$, ...
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
372
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 $G$-...
- 123
2
votes
2
answers
205
views
Bredon cohomology of a sign representation for a cyclic group of order 4
Yet another question "I compute Bredon cohomology of something and I am not sure, whether it is correct".
So I am taking a sign representation $\sigma$ of cyclic group of order 4, $C_4$. Then I ...
- 1,679
2
votes
1
answer
152
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{...
- 757
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
1
answer
171
views
Orbit decomposition of the restriction of an equivariant sheaf?
All sets and groups in the question are finite.
In order to understand equivariant sheaves better I'm trying to prove some basic facts from Mackey theory using equivariant sheaves. The main obstacle ...
- 7,469
2
votes
1
answer
233
views
Do Mackey (co)homology functors factor through derived categories? References with details?
Let $G$ be a compact Lie group; I will write $SH_G$ for the (equivariant) stable homotopy category of $G$-spectra (say, with respect to a complete universe; does its choice affect the homotopy ...
- 16.3k
2
votes
0
answers
26
views
Is anything known about the equivariant homotopy theory of surfaces with the action of a finite subgroup of the mapping class group?
The Nielson realization theorem for a surface says that every finite subgroup of the mapping class group is realized by a finite subgroup of homeomorphisms on the surface. Furthermore, for a genus $g \...
- 51
2
votes
0
answers
156
views
The dimension of the representation ring
Let $G$ be a compact Lie group. I am trying to characterize the algebraic properties of the representation ring $R(G)$ of $G$. In the case of the $n$-torus, the representation ring $R(T)$ is ...
2
votes
0
answers
137
views
Terminology for equivariant homology
The usual $G$-equivariant homology and cohomology groups of a space $X$ with $G$-action are given by the Borel construction:
$$H_\ast^G(X)=H_\ast((X\times EG)/G),$$
$$H^\ast_G(X)=H^\ast((X\times EG)/G)...
- 18.1k
2
votes
0
answers
194
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
2
votes
0
answers
82
views
The slice filtration does not arise from a $t$-structure
I've heard this argument quite a bit that the slice filtration does not arise from a $t$-structure on the category of genuine $G$-spectra. Mike Hill points out in Remark 3.12 of The Equivariant Slice ...
- 121
2
votes
0
answers
128
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....
- 757
2
votes
0
answers
48
views
Projective resolution of a dual coefficient system
I was trying to read the paper "Equivariant minimal models" by G. Triantafillou(1982) and was trying to compute cohomology of a system of DGA with rational coefficient system. Given a finite ...
- 1,039
2
votes
0
answers
179
views
Understanding equivariance of the Tate construction $(-)^{tC_P}$
$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\Fun{Fun}\newcommand\Cat{\text{Cat}}\DeclareMathOperator\CoInd{CoInd}\newcommand\Spaces{\text{Spaces}}$It is stated in line 10, p76, Thomas Nikolaus, ...
- 661
2
votes
0
answers
74
views
Does there exist a "Margolis-type" definition of equivariant cellular towers?
I am interested in cellular towers in the equivariant stable homotopy category $SH_G$ corresponding to a compact Lie group along with a complete universe for it.
Note here that a cellular tower for ...
- 16.3k
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
2
votes
0
answers
84
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 ?
- 745
2
votes
0
answers
335
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$...
- 855
1
vote
2
answers
200
views
Relation between the category of orthogonal G-spectra and the category of orthogonal H-spectra [closed]
I just read some parts of the book "Equivariant orthogonal spectra and S-modules" by Mandell and May. I wonder whether there is any description of the relation between the categories of orthogonal G-...
- 990
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
1
answer
200
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\...
- 35
1
vote
1
answer
135
views
Relative $G$-equivariant homology groups
Let $X$ be a free $G$-CW-complex with $G$-equivariant cell filtration by
$n$-skeleta $X_0 \subset \dots \subset X_n \subset \dots \subset X$ (for
rigorous definition see
Chap. II, p. 98 in linked ...
- 5,716
1
vote
0
answers
78
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
1
vote
0
answers
75
views
Do the covariant maps of a sheaf with transfer automatically satisfy a dual gluing axiom?
I'm interested in describing a notion of equivariant sheaves that uses Mackey functors on topoi. We can describe a genuine G-spectrum as a spectral Mackey functor on the topos of finite G-sets. If we ...
- 1,748
1
vote
0
answers
45
views
A question related to injective envelope for a system of DGA's
I was trying to read Fine and Triantafillou's paper "On the equivariant formality of Kahler manifolds with finite group action".
They have defined the enlargement at $H$ of a system of DGA's ...
- 1,039
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
0
votes
1
answer
195
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 ...
- 1
0
votes
1
answer
149
views
Question about finite G-sets [closed]
Let G be a finite group with subgroups H and K. Then the set of not necessarily equivariant maps from G/H to G/K is itself a finite G-set under the conjugation action. Is there a good description of ...
0
votes
0
answers
330
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 ...
- 35