Questions tagged [equivariant-topology]

Equivariant topology is the study of topological spaces that possess certain symmetries. In studying topological spaces, one often considers continuous maps, and while equivariant topology also considers such maps, there is the additional constraint that each map "respects symmetry", in the sense of preserving action of a group on the space.

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Example of CW-complex with $G$-action, which is not $G$-CW-complex

Let $G$ be a quasi-compact, Hausdorff topological group and let $G$ act on a CW-complex $X$ such that the $G$-action sends cells to cells and boundaries of cells to boundaries of cells. Further, ...
Fabio Neugebauer's user avatar
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Cofibrant replacement in coarse model structure onG-spaces

This question is about the Borel model category structure on $G$-spaces for a topological group $G$. This is also sometimes called the coarse projective model structure. In this article https://...
Fabio Neugebauer's user avatar
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Find $\mathscr X$ such that $\operatorname{Spaces}/BG$ $\sim$ $\operatorname{Spaces}^G$ $\sim$ $\mathscr X(\Omega G)$ for any $G$

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 of $G$-spaces and the homotopy category of ...
მამუკა ჯიბლაძე's user avatar
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Transfer map for the Mackey functor $\underline{\pi}_n^H$.

Let $G$ be a finite group ad let $X$ be a $G$-space. Consider the following Mackey functor, that I will denote by $\underline{\pi}_n$: $G/H\mapsto \pi_n^H(X)$, where $\pi_n^H(X)$ refers to the stable ...
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Is every CW-complex a $G$-CW-complex for $G$ finite?

Whenever $G$ is a finite group, a $G$-CW-complex structure on $X$ is equivalent to a CW-complex structure on which $G$: (1) The image of an open cell is another open cell (and boundaries are preserved)...
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Transport of group actions via homeomorphisms

Let $G$ be a topological group acting on the topological space $X$ and let $\phi\colon X\xrightarrow{\simeq} Y$ be a homeomorphism between the spaces $X$ and $Y$. Can I induce a $G$-action on $Y$ by $\...
Mathematics enthusiast's user avatar
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Chain complex of symmetric powers

Let $X$ be a pointed topological space, $k$ a field (or I suppose, more generally a commutative ring) and $n$ a positive integer. We know (by a reduced version of the Eilenberg-Zilber theorem) that ...
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When are homeomorphic $G$-spaces isomorphic?

Let $X,Y$ be $G$-spaces ($G$ a topological group). In general $X$ and $Y$ can be homeomorphic as topological spaces without being isomorphic as $G$-spaces. For example $X=Y=S^1$ and $G=\mathbb{Z}_2$ ...
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Definition of Cartan Model - Equivariant forms

Let $G$ be a connected Lie group and let $\mathfrak{g}$ be its Lie algebra. Let $M$ be a $G$-manifold. The Cartan model of $M$ is the $\Omega_G(M) := \{ a \in S(\mathfrak{g}^*) \otimes \Omega(M) | a \...
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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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Equivariant Submersion Theorem

I have a question about the Equivariant Submersion Theorem (Proposition 2.7) in the article Equivariant control data and neighborhood deformation retractions by Markus J. Pflaum and Graeme Wilkin. The ...
Bastiaan Cnossen's user avatar
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Induction preserves weak equivalences

Let $G$ be a finite group and $H \leq G$ be a subgroup. There is an induction functor $G \ltimes_H - : \mathbf{Sp}^H \to \mathbf{Sp}^G$ from the category of $H$-spectra to the category of $G$-spectra (...
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Geometric Fixed Points of Thom Spectrum

Recall that the Thom spectrum is an orthogonal spectrum $\operatorname{mO} \in \mathbf{Sp}$ defined via $\operatorname{mO}(V) = \operatorname{Th}(\operatorname{Gr}_{\dim{V}}(V \oplus \mathbb{R}^{\...
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Is this a torus action?

I was constructing an explicit example of a torus action in order to compute the Sullivan model of the associated Borel fibration, and I came up with the following action: $T^2 \times T^2 \to T^2$ ...
groupoid's user avatar
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Calculating the equivariant K-group $K_G^*(S^1)$ - where's the mistake?

Let a finite group $G$ act on the circle $S^1$ via a group homomorphism $\varphi \colon G \to S^1$. Let $K = \ker \varphi$. I wish to calculate the equivariant K-theory group $K_G^*(S^1)$. One method ...
Motmot's user avatar
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Clarification of G-CW(V)-complex via an example

I am trying to understand the definition of a G-CV(V)-complex given by Costenoble and Warner. It seems to me that there are two different definitions. Let $G$ be a finite group and let V be an ...
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Associated fibre bundle to a $(\Gamma,\alpha)$-equivariant $G$-principal bundle

Let $\Gamma$ and $G$ be compact Lie groups and $\alpha:\Gamma\to Aut(G)$ group homomorphism with the condition that $(\gamma,g)\mapsto \alpha(\gamma)(g)$ is continuous. $(\Gamma,\alpha,G)$-bundles are ...
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Schubert cells in generalized flag manifolds

Let $G$ be a compact Lie group and $T$ be a maximal torus. We call $G/T$ a generalized flag manifold since for $G = U(n)$ this quotient is isomorphic to the manifold of complete flags in $\mathbb{C}^n$...
Lennart Meier's user avatar
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Simplicial resolutions and the homotopy fixed points spectral sequence

According to this set of notes, which says (paraphrasing): "To construct the homotopy fixed points spectral sequence, we use the fact that the bar construction gives a simplicial resolution of $(...
Jordan Levin's user avatar
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Equivariant maps up to homotopy

Let $G$ be a topological group and $X$, $Y$ be $G$-spaces. Suppose there is a continuous map $F:X\to Y$ which is $G$-equivariant up to homotopy, namely the two maps $G\times X\to Y$, $(g,x)\mapsto F(...
Yeah's user avatar
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Quotient of 3-torus by $\mathbb{Z}/3\mathbb{Z}$ action

Consider the quotient of $T^3 = \mathbb{S}^1\times \mathbb{S}^1\times \mathbb{S}^1$ by the $\mathbb{Z}/3\mathbb{Z}$ action $(a,b,c) \mapsto (c,a,b)$, i.e., cycling the coordinates. Is this ...
Ryan's user avatar
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Is Tu's new 'Introductory Lectures on Equivariant Cohomology' an effective introduction to equivariant topology?

I went to a number of lectures this summer introducing ideas in equivariant algebraic topology. I was interested in learning more and I found a book, Tu's Introductory Lectures on Equivariant ...
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Homotopic equivariant maps

Let $G$ be a finite group and $X, Y$ be "nice" free topological G-spaces (such as finite free G-simplicial complexes). Moreover assume $f, g: X\to Y$ are $G$-maps. I have a feeling that the ...
123...'s user avatar
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A misunderstanding about Sullivan's conjecture

In this old blog post, Akhil Mathew describes the Sullivan conjecture and part of Miller's proof of a special case. There's a point in the beginning which is not clear to me, about $p$-completions at ...
Maxime Ramzi's user avatar
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Fundamental Group of a free G-space

It is very well known that any group $H$ can be the fundamental group of a topological space. What happened if we restrict the class of topological spaces to the free equivariant topological spaces? ...
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Proof of Equivalence of Least Squares Problem

Let $b \in \mathbb R^n $ and $A \in \mathbb R^{m\times n}$ with $m \geqslant n$ and $\operatorname{rank}(A)\le n$. Prove that the following statements are equivalent; $\hat{x} = \operatorname{argmin}...
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Proving that two distances are topologically equivalent.

I want to prove that $d(x,y)=|x-y|$ and $d'(x,y)=|\ln(x)-\ln(y)|$ are topologically equivalent. Proof: $\tau_d \subset \tau_{d'} \iff \forall x \in X, \forall \epsilon >0, \exists r>0$ such ...
user144435's user avatar
2 votes
1 answer
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Taking homotopy fixed points preserves fibrations

I'm reading a paper where they have an appendix about homotopy fixed point sets of a $G$-space, and at some point they claim that if $f:X\to Y$ is a $G$-map that is an ordinary (non-equivariant) ...
Maxime Ramzi's user avatar
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Identifying a balanced smash product of G-spaces

Let $X$ be a pointed $G$-space, why is it true that there is a natural isomorphism $G_+\wedge_HX≅G/H_+\wedge X$, where $H$ is a subgroup of $G$? (Maybe some assumptions are needed on $H$, in which ...
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Equivariant homotopy theory, topos theory and intuitionistic algebraic topology

This might be a very naive question, but I don't really see what would go wrong, so I'm wondering if this has already been done. The idea is the following : equivariant homotpy theory as far as I can ...
Maxime Ramzi's user avatar
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Killing homology below middle dimension with equivariant surgery

Assume a finite group acts smoothly on a manifold $M$ of dimension $n$. Suppose $a\in H_i(M)$, where $i=1,\ldots,[n/2]$. Is there a way to kill $a$ with equivariant surgery and keep the same fixed ...
piotrmizerka's user avatar
2 votes
1 answer
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Homology of an equivariant product

I am struggling with checking the following fact: $$ H_*(E\Sigma_n\times_{\Sigma_n}X^{\times n})\cong H_*(\Sigma_n;C_*(X)^{\otimes n}). $$ But I am not sure how to start. It seems correct and I ...
Igor Sikora's user avatar
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Maps from a $G$-set and graph subgroups

I am reading now Mike Hill's paper "On the algebras over equivariant little disks" and I have a problem with one (probably) simple equivalence from the proof of the Theorem 2.12. So let $G$ be a ...
Igor Sikora's user avatar
33 votes
1 answer
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Are there intersection theoretic proofs for Ham-Sandwich type theorems?

Concrete Question: Let $f:\mathbb S^n \to \mathbb R^n$ be a $\mathbb Z/2$ smooth equivariant map where the action on the sphere is antipodal, and $\mathbb R^n$ is multiplication of co-ordinates. Can ...
Andres Mejia's user avatar
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1 answer
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Fixed point free G-simplicial complex

Let $G$ be a finite group, and $K$ be a finite abstract $G$-simplicial complex. We say $K$ is fixed point free if for each $x$ in geometric realization of $K$, $||K||$, there exist a $g\in G$ such ...
123...'s user avatar
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1 answer
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Is a fixed point set of a compact group action on a compact space also compact? [closed]

Let $G$ be a compact group acting on a compact topological space $X$, that is the function $$G\times X\rightarrow X$$ $$(g,x)\mapsto g\cdot x$$ is continuous and satisfies $$g_1\cdot(g_2\cdot x )=(...
piotrmizerka's user avatar
2 votes
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Equivariant homotopy vs homotopy of fixed points

Let $G$ be a Hausdorff group and $X, Y$ two $G$-spaces. Assume that $f,g:X\to Y$ are two $G$-maps which are $G$-homotopic. If we consider the fixed point functor $(\cdot)^H$ for a (closed) subgroup $H$...
freakish's user avatar
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