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Questions tagged [equivariant-topology]

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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) ...
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0answers
18 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 ...
5
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0answers
71 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 ...
0
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0answers
8 views

relation between drawstring bag and globe

$S^2$ has two antipodal points which remain still when spun. Visually this is clear, but how can you prove, topologically, that a spinning action on $D^2$ leave two points in $D^2 / \sim \partial D^2$ ...
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0answers
19 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 ...
2
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1answer
70 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 ...
2
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0answers
28 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 ...
33
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1answer
622 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 ...
1
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1answer
45 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 ...
1
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1answer
176 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 )=(...
1
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0answers
69 views

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 $H$...