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[[ Sorry I missed that the question was also concerned with the question in an algebraic topological context. This answer is only concerned with algebraic geometry.]] I think the first question is mu …

I think the easiest way to understand the Bockstein spectral sequence is through the exact couple coming from the long exact sequence of cohomology associated to $0\to\mathbb Z\to\mathbb Z\to \mathbb …

As the cohomology of $(S^1)^n$ is torsion free every stable bundle on $(S^1)^n$ is determined by Chern classes (this also follows from the $K$-theory Künneth formula) so just as for the spheres it is …

I don't know the answer to the actual question but here is a situation which should be similar but simpler. Consider an integral polynomial group law $G$, i.e., a group scheme structure on the affine …

This is an attempt to complete Tyler's argument. We first note that $KO^0(S^5)=\mathbb Z$ (note this true for all spheres of dimension $\equiv 5,6,7 \bmod 8$). This means that every topological vector …

I do not know if this provides more details: I assume the closed ($4$-)manifold $M$ to be oriented (we need the submanifold to be oriented to have an integral homology class and the construction I a …

[[ I have added a discussion of when $p$ is smooth or has quotient singularities. ]] [[ I added a discussion on the cohomology of $[X/G]$. ]] The étale case follows in a way that is altogether analog …

A well-written discussion of the group completion can be found on pp. 89--95 of J.F. Adam: Infinite loop spaces, Ann. of Math. studies 90 (even though he only discusses a particular group completion o …

Well, I think it depends on which dimension you mean and which cohomology. The best fit I think is covering dimension and Čech cohomology. The Čech cohomological dimension is indeed bounded (more or l …

Clearly looking at sheaves on the geometric realisation gives something too far removed from the simplicial picture. This is essentially because there are too many sheaves on a simplex have (most of w …

The reduction mod $2$ of $e(X)$ is the second Stiefel-Whitney class of $X$ which by Wu's formula can be recovered from the homotopy type of $X$ (Steenrod operations and the Poincaré duality for the mo …

Also for the étale fundamental group there is in fact always some universal cover. However, in the abstract way that Grothendieck formulated the theory of coverings a universal cover would only exist …

I would claim that the splitting (and indeed the whole universal coefficient theorem) is not really a topological theorem. If we take the homological version one really works with the chain complex $C …

There is a precise relation at the level of complexes: $C^\ast(X,\mathbb Z)$ is a $G$-complex and as such it is perfect (that is quasi-isomorphic to a finite complex consisting of projective modules) …

The mod $2$ cohomology of $\mathrm{SO}_n$ is not an exterior algebra as soon as $n\geq3$ (there is a unique non-zero element in degree $1$ whose square is non-zero, think of the case $\mathrm{SO}_{3}$ …