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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 …

I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians. In an effort to ge …

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 …

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

$\DeclareMathOperator\Sym{Sym}$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 …

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 …

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 …

Let $G$ be a topological group, $Vect(BG)$ the monoid of complex vector bundles over its classifying space (not the stack!) and $Rep(G)$ its monoid of complex representations. Generally $Vect(BG) \ne …

Let $X$ be a compact manifold. I'm interested in whether any of the following cases admits a general closed formula for (complex)-$K$-theory. Let $E$ be a complex vector bundle with a given line bundl …

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 …

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) …

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 …

Let $P \to X$ be a principal $G-$bundle and let $f: X \to BG$ be its classifying map. As I understand there's some way to associate a monodromy representation $\pi_1(X) \to G$ to it. I know how to con …

I'm trying to piece together a proof of the projective bundle formula from several incomplete sources. Here's the statement I'd like to prove: Projective bundle formula: Let $\pi: E \to X$ be a ve …