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.
156 questions
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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 ...
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Finitely dominated universal spaces for the family of solvable subgroups
$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Sz{Sz}$In short, I am interested in the question which finite groups $G$ admit a finitely dominated universal space with respect to the family of ...
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Serre spectral sequence of Borel construction
Let $G$ be finite $p$-group, $X$ be path-connected $G$-space, $E=EG\times_{G}X$ be the Borel construction and $BG$ be the classifying space of $G$. Consider Serre spectral sequence of the fibration
$$...
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A $\mathbb{Z}_2$-equivariant map from $n$-torus to $2$-sphere that is null-homotopic is $\mathbb{Z}_2$-homotopic to a non-surjective map?
I have been thinking on the problem below for a while and I am not sure if it is correct or not. I am trying to see if there exists a counter-example for the problem below.
Problem: Let $f: (S^1)^n \...
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Reference request for a theorem of Jaworowski
Jan Jaworowski, in 2000, proved the following theorem (I came to know about it from here)
Jaworowski (2000) : Let $Y$ be a finite simplicial complex of dimension $k$ and let $n\ge 2k$. If $f:S^n\to Y$...
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$\operatorname{Spaces}/BG$ $\sim$ $\operatorname{Spaces}^G$ $\sim$ $??(\Omega G)$
This is a crosspost (with minor alterations).
For a topological group $G$, assigning to a $G$-space $X$ the (canonical) map $EG\times_GX\to BG$ establishes an equivalence between the homotopy category ...
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Calculate homotopy groups of $\mathbb{Z}_2$-equivariant loop spaces of "complex" topological spaces
Let $X$ be a topological space such that complex conjugation is defined (e.g. $\mathbb{C}^n$) and let us define the set of maps $$S_d:= \left\{f: (I^d,\partial I^d)\to (X,x_0)\mid \overline{f(k)} = f(...
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Recover the $C_k$-action of a cyclic object as from the $S^1$-action on Hochschild chain
$\newcommand{\Fun}{\operatorname{Fun}}$Let $X\in\Fun(\Lambda^\mathrm{op}, \mathcal{C})$ be a cyclic object in the ($\infty$-)symmetric monoidal category $\mathcal{C}$, where $\Lambda$ is the cyclic ...
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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 $G$-...
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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 ...
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Replacing a $G$-CW-complex with a $G$-homotopy equivalent $G$-simplicial complex - can anyone supply a reference?
Let $G$ be a group (not a topological group, just a group). By a $G$-complex I mean a CW-complex with an action of $G$ that takes cells to cells so that the pointwise and setwise stabilizer of each ...
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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\...
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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 ...
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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 ...
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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 ...
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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
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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 ...
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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 ...
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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 \...
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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 ...
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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 ...
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$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
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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 ...
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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)...
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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 ...
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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 $A$, ...
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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 ...
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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 ...
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Mostow-Palais equivariant embedding for manifolds with corners
Let $M$ be a compact smooth manifold and let $G$ be a connected compact Lie group acting on $M$. According to an old theorem of Mostow and Palais, there exists a $G$-equivariant embedding of $M$ into ...
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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 $G = C_p$ ...
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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. ...
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When can the trace on cohomology be computed as the Euler characteristic of fixed points?
In this question all groups are finite, and all spaces are nice (eg, simplicial sets).
Given a $G$ space $X$, which we assume has finitely many nonzero cohomology groups, we can compute the trace of ...
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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
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Model structure on dg-algebras over an "equivariant fundamental category"?
For purposes of $G$-equivariant rational homotopy theory one wants a Quillen adjunction which generalizes the classical one of Bousfield-Gugenheim from plain dg-algebras/simplicial-sets to (co-)...
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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]...
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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 ...
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How does one rigorously lift a map $Sp \rightarrow Sp$ of spectra to equivariant spectra?
This is in part motivated from my attempt to understand tate diagonal in III.1 of Thomas Nikolaus, Peter Scholze, On topological cyclic homology, arXiv:1707.01799. I just want to make my ...
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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 ...
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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{...
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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 ...
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Applications of equivariant homotopy theory in chromatic homotopy theory
I usually do computations in equivariant homotopy theory, but I would like to learn chromatic homotopy theory where one may use the equivariant techniques, e.g., slice spectral sequences, etc.
For ...
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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{...
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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 ...
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Decomposition of fiber product of $G$-sets in $G$-orbits
I have posted an identical question in MSE few days ago, but maybe this site is a better adress to discuss this problem:
Let $G$ be a finite group and $K, H \leq G$ two subgroups. Then
the right ...
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Equivariant imbedding of compact manifold
Let $G$ be a compact Lie group smoothly acting on a smooth compact manifold $X$.
Is it true that there exists a smooth $G$-equivariant imbedding of $X$ into a Euclidean space acted linearly (and ...
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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....
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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 $N$ ...
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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 $\mathcal{G}^0$...
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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 ...
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The set of isomorphism classes of Z/nZ-equivariant line bundles over a 2 dimensional Z/nZ-CW complex
Suppose I wish to find the set of isomorphism classes of $\mathbb{Z}/n\mathbb{Z}$-equivariant line bundles over a 2-dimensional, compact $\mathbb{Z}/n\mathbb{Z}$-CW-complex $X$, i.e. $\mathrm{Vect}^{1}...
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