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Is there a heuristic reason why principal homogeneous spaces of a group (object) $G$ (in some categories) are called $G$-torsors? Does it have anything to do with the idea of "torsion", somehow? When …

Not that I understand anything of this, but there is the following paper by Peter Scholze that seems to be in-topic here: On torsion in the cohomology of locally symmetric varieties, Annals of Mathem …

I've once been told that "torsion in homology and cohomology is regarded by topologists as a very deep and important phenomenon". I presume an analogous statement could be said in the context of algeb …

Line bundles on a scheme $X$ live in $H^1(X,O_X^*)$, where $O_{X}^{*}$ is the sheaf of invertible functions. If $X$ is noetherian separated, then we can think of this $H^1$ to be Čech cohmology w.r.t …

Let $X$ be a simply connected complex projective surface (hence a real $4$-manifold). Let $(H^2(X,\mathbb Z)/\mathrm{tors}, q_X)$, $(A^1(X)/\mathrm{tors},q_X')$ be the corresponding lattices in cohomo …

Google tells me that there are these Horikawa surfaces (I don't know what they are) which posses infinitely many differentiable structures. The fact that the author says the Horikawa surface makes me …

How to prove that the first Stiefel-Whitney class $w_1 (E)$ of a real rank $n$ vector bundle over a manifold M is equal to $w_1(\det E)$, where $\det E$ is the $n$-th wedge power of $E$? (I want to a …

This (Brown--Cohen, A proof that simple-homotopy equivalent polyhedra are stably homeomorphic) gives a partial answer (maybe subsumed by the answer by Igor Rivin).

If $\mathscr{M}$ is the moduli stack of mathematical objects $X$ of some specified kind such that $\mathrm{Aut}(X)$ always (naturally) contains a group (object) $G\subseteq\mathrm{Aut}(X)$, then $\mat …

I have heard stated the following Theorem. If $\Sigma$ is a (orientable) surface, then $\mathrm b_1(\Sigma)$ counts the maximum number of "circular cuts" (embedded circles $C_1,\ldots,C_m$) that you …

The root gerbes $${^r}\sqrt{\mathscr{L}/X}$$ associated to a line bundle on a scheme (or stack) $X$

Let $X$ be a compact differentiable manifold of dimension $m$ or, if you prefer, a smooth complex projective manifold of complex dimension $n=m/2$. The Euler characteristic $\chi(X):=\Sigma_{i=0}^{m …

In (real world, for $n=3$) crystallography, given a point group $K\subseteq\mathrm{O}(n)$ and a ($K$-invariant) lattice $T\subseteq\mathbb{R}^n$, the set of possible crystallographic classes (where, …

This could as well have been asked in the comments to this question, but I prefer to open a new one for the sake of clarity. Say $G$ is a reductive group over the complex numbers, with compact real f …

What goes wrong in the axiomatic definition of a generalized (co)homology theory if one drops the axiom of homotopy invariance i.e. that homotopic maps should induce the same map in (co)homology? Or …