# 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, ...
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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://...
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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 ...
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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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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(... 1 vote 0 answers 66 views ### 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 ... 0 votes 1 answer 58 views ### 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 (... • 8,051 0 votes 1 answer 110 views ### 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}^{\... • 8,051 0 votes 1 answer 111 views ### 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 ... • 362 1 vote 0 answers 77 views ### 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 ... • 373 1 vote 0 answers 62 views ### 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 ... • 1,877 1 vote 0 answers 53 views ### 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 ... • 1,877 2 votes 0 answers 58 views ### 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... 1 vote 0 answers 95 views ### 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 (... • 425 1 vote 0 answers 83 views ### 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(... • 405 2 votes 1 answer 210 views ### 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 ... • 1,106 2 votes 1 answer 581 views ### 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 ... 2 votes 0 answers 98 views ### 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 ... • 959 4 votes 1 answer 195 views ### 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 ... • 43.6k 1 vote 1 answer 202 views ### 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? ... • 438 2 votes 2 answers 45 views ### 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}... 0 votes 0 answers 283 views ### 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 ... • 119 2 votes 1 answer 155 views ### 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) ... • 43.6k 1 vote 0 answers 143 views ### 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 ... • 421 6 votes 0 answers 150 views ### 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 ... • 43.6k 1 vote 0 answers 53 views ### 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 ... • 733 2 votes 1 answer 159 views ### 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}). $$But I am not sure how to start. It seems correct and I ... • 536 2 votes 0 answers 42 views ### Maps from a G-set and graph subgroups I am reading now Mike Hill's paper "On the algebras over equivariant little disks" and I have a problem with one (probably) simple equivalence from the proof of the Theorem 2.12. So let G be a ... • 536 33 votes 1 answer 831 views ### Are there intersection theoretic proofs for Ham-Sandwich type theorems? Concrete Question: Let f:\mathbb S^n \to \mathbb R^n be a \mathbb Z/2 smooth equivariant map where the action on the sphere is antipodal, and \mathbb R^n is multiplication of co-ordinates. Can ... • 21k 1 vote 1 answer 61 views ### Fixed point free G-simplicial complex Let G be a finite group, and K be a finite abstract G-simplicial complex. We say K is fixed point free if for each x in geometric realization of K, ||K||, there exist a g\in G such ... • 959 1 vote 1 answer 468 views ### Is a fixed point set of a compact group action on a compact space also compact? [closed] Let G be a compact group acting on a compact topological space X, that is the function$$G\times X\rightarrow X(g,x)\mapsto g\cdot x$$is continuous and satisfies$$g_1\cdot(g_2\cdot x )=(...
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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 $H$...