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$\mathbb C^{n^2} \smallsetminus {0}$ is homotopy equivalent to $S^{2n^2-1}$, not $S^{n^2-1}$, and $\mathrm H^{2n^2-1}(GL_n(\mathbb C)) = 0$.

Linearly reductive groups, that is, linear algebraic groups, say over a field, in which the functor of invariants from finite dimensional representations to vector spaces is exact. I am not sure this …

In the stated generality, it is false; for example, suppose that $G_1$ and $G_2$ are trivial groups, $P = B \times G$ (here $B$ is the base), and the maps $P_1 \to P$ and $P_2 \to P$ are given by two …

When $n$ is a prime, the additive structure of $H^*(BPU_n, \mathbb Z)$ has been computed independently in Kameko, Masaki; Yagita, Nobuaki, The Brown-Peterson cohomology of the classifying spaces of t …

Atiyah's statement makes sense only if one gives $U(1)$ the discrete topology, otherwise it is just plain false (continuous $U(1)$-bundles are classified topologically by their first Chern class, whic …

If $G$ is an algebraic group (say over $\mathbb C$), acting on an algebraic variety $X$, the orbits are always locally closed smooth algebraic subvarieties of $X$. This is standard, and easy to prove. …

Suppose that $\mathbb R^n$ is a module of the Clifford algebra $C_m$, with generators $e_1, \dots, e_m$, and relations $e_i^2 = -1$, and $e_ie_j+e_je_i = 0$. After a base change in $\mathbb R^n$, the …

I think so. This is implied by the fact that the group of outer automorphisms of $H$ is finite. When $H$ is semisimple, then the group of outer automorphisms is contained in the group of automorphisms …

[Edit]: my previous counterexample was irredeemably wrong; hopefully this one works. Suppose that there exists an étale $G$-equivariant map $V' \times^{P}G \to V$; then there exists an invariant neig …