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### Group action induced on homology under change of coefficients

Let $M$ be a closed manifold equipped with a (continuous or smooth) $\mathbb{Z}_2$-action such that - for simplicity - both $H_{k-1}(M;\mathbb{Z})$ and $H_{k+1}(M;\mathbb{Z})$ are zero for some ...

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### Rational homology and finite group actions

I'm looking for examples of the following phenomena. Let $X$ be a reasonable space (say, a CW complex) and $G$ be a finite group acting on $X$. For all $k \geq 1$, the projection map $X \rightarrow ...

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### reference needed for some well know results on cohomology of the orbit spaces

The following results are well known
If the group $\mathbb Z_2$ acts freely on a mod $2$ cohomology $n$-sphere $X$, then the orbit space
$X/\mathbb Z_2$ is a cohomology real projective $n$-space.
...

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### Gysin sequence for $\mathbb S^3$ bundle

Let the group $G=\mathbb S^3$ act semi freely on a paracompact space $X$. Then exercise 12 of G.E. Breadon's book , Introduction to compact transformation groups pg 169 asks to derive the following ...

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### free action on mod p cohomology sphere

It is well known that the group $G=\mathbb Z_2\oplus\mathbb Z_2$ cannot act freely on mod 2 cohomology n-sphere.
Is it also true that this group $G$ cannot act freely on any mod p cohomology ...

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### Why do we need a $G$-universe?

Let $G$ be a compact Lie group. Before defining $G$-prespectra, we have to define a $G$-universe $\mathcal U$.
Question: Why do we need a $G$-universe?
A $G$-universe is defined to be a countably ...

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### Which topological spaces are coset spaces of locally compact groups?

What is a topological characterization of the class of spaces that have the form $G/H$ for a locally compact, Hausdorff group $G$ and a closed subgroup $H$ ?
Such a space $X=G/H$ necessarily ...

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### What, precisely, does Klein's Erlangen Program state?

People write that the Erlangen Program is a "program" (like the "Langlands Program"), i.e. a series of related conjectures, which in this case were all solved. There are various intuitive accounts, ...

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### Collapsing of Riemannian manifolds with a group action

Let $M$ be a complete Riemannian manifold with bounded sectional curvature and $G$ a compact connected Lie group acts smoothly on $M$. Consider the fixed point set $F$, it is of course a submanifold ...

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### Free involutions and equivariant maps

The following paper of Conner and Floyd does not include proofs of many theorems/ results.
'Fixed point free involution and equivariant maps , Bull. Amer. Math. Society vol 66, no. 6( 1960)'.
I would ...

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### 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 ...

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### Intersections of conjugates of the icosahedral group in SO(3)

(Related question)
Let $I$ be the group of orientation preserving symmetries of a regular icosahedron. This is a $60$ element subgroup of $SO(3)$, isomorphic with the alternating group $A_5$. It is ...

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### Dimensions of orbit spaces

Let $G$ be a compact Lie group acting effectively on a compact, Hausdorff topological space $X$. I am looking for results of the type
If $X$ is a ... and the action is ... then $\dim(X/G)\leq ...

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### Does the Borel functor take equivariant fibrations to fibrations?

Let $p\colon X\to B$ be a fibration. Let $G$ be a topological group acting continuously on $X$ and $B$, and assume that the map $p$ is $G$-equivariant.
We can apply the Borel functor $EG\times_G-$ ...

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### Is there a smooth free circle action on the Klein bottle?

Can the circle group $S^1$ act smoothly and freely on the Klein bottle? I'm sure there is some obvious reason why the answer is no, which eludes me right now.
We can view $K$ as the quotient of ...