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$\DeclareMathOperator\Sym{Sym}$Let $G$ be a compact lie group. Chern–Weil theory tells us that there's a homomorphism: $$H^{*}(BG;\mathbb{R}) \to (\Sym^{\bullet} \mathfrak{g^*})^G$$ which in our case …

After discussing this with Tim we came up with the following answer: The first steifel whiteny class $\omega_1$ of $M$ can be written as the following composition: $$M \to BO(n) \to BO \to BAut(\mathb …

Disclaimer: I don't feel qualified to ask this question and yet it's been troubling me for some time now and I lost my patience and decided to ask to get some kind of answer. If there are any stupid m …

This is not a full answer but it was too long for a comment so I decided to write it as detailed answer (EDIT: I added what I believe to be a full answer at the end resolving both points (1) and (2) …

Let $\mathrm{Diff}_n$, $\mathrm{PL}_n$, $\mathrm{Top}_n$ denote the $\infty$-categories of $n$-manifolds which are respectively smooth/PL/topological, and open embeddings (for instance by taking the h …

A (weak) factorization system on a category $\mathcal{C}$ consists of a pair of classes of morphisms in the category $(L,R)$ satisfying Every morphism $f:x \to y \in \mathcal{C}$ can be factored (n …

Consider a fiber sequence of spaces $$F \overset{i}{\to} E \to B$$ The cofiber $C(i)$ of the inclusion of the fiber comes with a canonical map $C(i) \to B$. Its possible to show (using some point se …

Let $V$ be an object of some stable infinity category (nothing is lost by taking spectra but I see no reason to state the question in this way as it is irrelevant) and suppose we have a two step filtr …

Let $ f: X \to Y$ be a continuous map between connected manifolds s.t. for all $y \in Y$ the fiber $f^{-1}(y)$ is homeomorphic to some fixed connected manifold $Z$. Let $k$ be a ring and for every $ …

Let $M$ be an $n$ dimensional smooth manifold and let $j: M \to \mathbb{R}^{m}$ be an embedding. Associated to this embedding we can form the "collapse map" which is a pointed map from a sphere to the …

(The question will be interesting for topological/Pl as well but in order to not be too vague I will restrict the meaning of manifold to smooth manifold without boundary). I'm interested in the funct …

I'll be using homological grading throughout this question. Let $G$ be a compact connected lie group. The following isomorphisms are classical and can be proven using several methods: $$H^{\bullet}( …

Let $U \subset A \subset X$ be spaces (in the sense of homotopy theory). For every pointed space $Y$ restriction maps induce the following canonical map between mapping spaces: $$fiber(Map(X,Y) \to …

I'm asking this question in the most model-ambiguous way I can since this is the kind of answer i'm looking for. There are various explicit constructions of the Whitehead and Postnikov towers. I'm try …

It seems that the theory of constructible sheaves (in particular anything that goes into proving that they form an abelian category) requires some technical statements about existence of certain strat …