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In the same vein of this MO question, one can ask: If two spaces $X$, $Y$ have isomorphic generalized cohomology rings $\mathrm{h}^{\bullet}(X)\cong \mathrm{h}^{\bullet}(Y)$ for every multiplicative …

Abstractly, on the topological circle $S^1$ there are only two real line bundles, up to isomorphism: the trivial one $\mathcal{O}$ and the Moebius strip $\mathcal{O}(1)$ (thinking of $S^1$ as $\mathbb …

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 …

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 …

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).

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 …

My question can be simply (and loosely) stated as follows: Is there a general (but not too general) construction that captures, as specializations, both Weil cohomologies in algebraic geometry and …

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 …

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 …

Consider the real algebraic line bundle $\mathcal{O}(-1)$ over the real algebraic variety $\mathbb{R}\mathbb{P}^1$. It is nontrivial hence continuously isomorphic to the "Moebius" line bundle (there a …

The answer to the first question is: Because in local chart we want the action to commute with transition functions, and the latter are traditionally assumed to be acting on the left. I'll explain b …

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 …

Are there accepted/standard definitions of "decomposition", "filtration", and "stratification" of a topological space (or of a manifold, or of an algebraic variety) $X$? I tend to understand "decompo …

You can find something in Allen Hatcher, "Spectral Sequences" .

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 …