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By a version of Poincaré duality, $\mathrm H^n(M, \mathbb Z)$ is isomorphic to the $0$-th Borel-Moore homology group $\mathrm H_0^{\mathrm{BM}}(M, \mathbb Z)$, (see, for example, IX-4 in Iversen's boo …

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 …

One can define singular homology directly for an orbifold, mimicking the standard constructions, but I don't know if this has been written up. The alternative is to take a space that is homotopy equiv …

Suppose that $\overline G$ is a Lie group such that the connected component of $1$ is $\mathbb C^*$. Assume that $\mathbb C^*$ is central in $\overline{G}$, and set $G := \overline G/\mathbb C^*$. The …

I don't think there is a canonical construction of the associated line bundle, this will be only defined up to homotopy, in some sense. Here is a slightly more geometric construction. Let $G'$ be the …

The answer is positive. Let $P$ be the principal $\mathrm{GL}_n$-bundle associated with $E$; then the space of flags is the quotient $P/B$, where $B$ is the Borel subgroup of $\mathrm{GL}_n$ consistin …

For 1): take a double covering $E\to B$, where $E$ and $B$ are compact oriented surface of genus 3 and 2 respectively, and give $E$ a structure of Riemann surface with trivial automorphism group. Abo …

Gabriele Vezzosi and I have been musing on the following. Consider the standard double cover $\mathop{\rm Pin}_{n} \to \mathrm{O}_{n}$, whose kernel is $\mathbb Z/2\mathbb Z$. This allows to associate …

If $V$ is a finite closed cover of a space $X$, with the property that the higher cohomology of all finite intersections vanishes, then the Čech cohomology of $V$ coincides with that of $X$. This is b …

The Mayer-Vietoris sequence is an upshot of the relationship between sheaf cohomology and presheaf cohomology (a.k.a. Cech cohomology). Let $X$ be a topological space (or any topos), $\mathcal U$ a c …

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

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 …