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So i turned my comment into an answer after reading [1] again. A Whitney stratification, i.e. a stratification satisfying Whitney's condition b (and so automaticly a), induces a triangulation compati …

Yes, take for example $X=Y=S^1$ and let $\mathbb{Z_2}$ act on $S^1$ via the antipodal action. Then the product action of $\mathbb{Z}_2\times \mathbb{Z}_2$ on $X\times Y$ is free. However, by a theore …

Maybe not research level. Let $Z\cong \mathbb{R}$ be the $z$-axis of $\mathbb{R}^3$. Clearly $\pi_1(\mathbb{R}^3-Z)\cong \mathbb{Z}$. Now if $F\subset Z$ is a closed non-empty subset, then one easily …

I hope this is not trivial. Let $B$ be a nice topological space (paracompact, CW-complex or whatever you think is nice) For $i=1,\ldots,n$ let $x_i \in H^i(B,\mathbb{Z}_2)$ be certain cohomology cla …

Geometric respectively topological methods are widely applied in representation theory. As far as I know mainly cohomological methods are used. I wonder if there are concrete applications of the …

This should only be considered as a comment to the answer above, unfortunately it does not fit in a comment. One can show, that $\mathbb{Z}_p\times \mathbb{Z}_p$ cannot act freely on $S^n$ also via " …

It is known that $M$ has the homology of a sphere, see [1] But you probably knew that. What you also can describe pretty nicely is the cohomology of the orbit space $S^n/G$. To be more precise $S^n/G …

Let $X$ be a nice space (compact $CW$-complex or triangulated space, compact manifold, whatever works), $f:Y\to X$ be a finite covering of degree $n$, and $\chi(X)$ be the euler characteristic. By the …

I hope this question is not too vague. Let $G$ be a complex reductive group, $B$ a Borel subgroup of $G$, and $P$ a parabolic containing $B$. Denote by $\pi:G/B\to G/P$ the canonical map. Consider th …

Maybe I add something to Peter Michor's answer. The slice theorem says, that for $y\in M_{(H)}$ with $G_y=H$ there exists a $G$-invariant open neighbourhood $U_y$, such that $$ U_y\cong G\times_H V …

This is a problem i am thinking for a while but did not find an answer. Maybe one of you knows it. Let $G:= SO(n,\mathbb{R})$, $G\times \mathbb{R}^n\to \mathbb{R}^n$ the standard representation of $G …