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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 …

Assume given a group $G$, a subgroup $H$ and a $g\in G$ such that $gHg^{-1}$ is properly contained in $H$. Let now $Y$ be a topological space with an action of $G$ such that $Y\rightarrow Y/G=:X$ is a …

We have that if $f\colon S^{2k}\to BU$ is an actual map of topological spaces (it is a little bit unclear from your formulation if you assume this) then $\langle c_k,[f]\rangle>$ is a multiple of $(k- …

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 …

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 …

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 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 …

This seems to work for getting a central extension (even though it looks a little bit too simple). To be specific I am using May's notation with $\newcommand{\hw}{\overline{W}}\hw$ for the classifying …

The exact sequence $0\rightarrow\mathrm{Z}/p\rightarrow\mathrm{Z}/p^2\rightarrow\mathrm{Z}/p\rightarrow0$ ($p$ a prime say) gives a map $\mathrm{Z}/p\rightarrow\mathrm{Z}/p[1]$ in the derived category …

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 …

It depends completely on what you mean by cofibrations. The choice is not quite simple to make as the homotopy category of real commutative dga's is anti-equivalent to "real homotopy" which would sugg …

If $P$ and $Q$ are closed subspaces of $Y$ and $Y$ is their union, we have a short exact sequence of sheaves on $Y$ $0\rightarrow\mathbb Z\rightarrow i_\ast\mathbb Z\bigoplus j_\ast\mathbb Z\rightarro …

As Simon points out, the answer is no in a simple case and if you think about his argument the answer should probably be no as soon as $G^p$ is infinite. However, there is a statement that is very clo …

[[ 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 …