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On a smooth manifold $M$ one can define various algebraic structures, natural with respect to diffeomorphisms: the differential graded-commutative algebra $\Omega(M)$ of differential forms on $M$; t …

The constant rank theorem says that if $f\colon M→N$ is a smooth map whose rank equals some fixed $k≥0$ at any point of $M$, then, locally with respect to $M$ and $N$, the map $f$ assumes the easiest …

Connes defined a noncommutative analog of a closed oriented Riemannian spin^c manifold using spectral triples. Using his definition it is unclear how to separate the smooth structure from the metric. …

The category of internal locales in the Grothendieck topos of sheaves on a locale X is equivalent to the slice category over X. In other words, internal locales over X are precisely morphisms of local …

This is a followup to my previous question, which asked whether the category of commutative or noncommutative C*-algebras or von Neumann algebras is equivalent to the category of commutative or noncom …

A folk theorem says that star-shaped open subsets of R^n are diffeomorphic to R^n. Is there a citeable reference for a proof of this result? For the sake of being definite, let's say that “citeable” m …

The fiberwise Stokes theorem says that given a differential form on a smooth fiber bundle whose fibers have boundary, the difference between the fiberwise integral of the differential and the differen …

every group can be declared a Lie group if one allows the more general definition since one is permitted to have an uncountable number of connected components. A manifold is paracompact if and o …

Suppose $d≥0$, $m≥0$, $n≥0$, and $\def\R{{\bf R}} f\colon \R^m→\R^n$ is a smooth map whose rank at any point of $\R^m$ is at most $d$. Here and below, smooth means infinitely differentiable. Can we fi …

The étale space construction produces non-Hausdorff and nonparacompact spaces (e.g., smooth manifolds) in many practical examples that have nothing to do with algebraic geometry. The étale space is …

The proof of this fact is available in modern textbooks. For example, see Theorem 7.16 in Jet Nestruev's Smooth Manifolds and Observables (Second Edition, 2020). In fact, the cited book contains a lot …

For the purposes of this question, let's say that a blurring of a smooth triangulation $T$ of a smooth manifold $X$ is a smooth homotopy $h\colon [0,1] \times X \to X$ such that $h_0=\operatorname{id} …

How can we characterize the algebras (at least within all the C^∞(M)'s), that come from compact manifolds? An algebra of the form C^∞(M) corresponds to a compact manifold if and only if all of it …

Suppose M is an arbitrary smooth manifold and D is its bundle of 1-densities. On the category of finite-dimensional vector bundles over M and linear differential operators between them there is a cont …

A classical result by Brown and Gersten says that to verify the homotopy descent property for the Zariski topology it suffices to verify it for Zariski squares and the empty cover of the empty scheme. …