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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 …

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 $U \subset A \subset X$ 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 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 …

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 …

The question I'm trying to answer is the following: Let $P \to X$ be a principal $G$-bundle (over a connected CW complex) satisfying that all pullbacks to spheres (of arbitrary dimension) are …

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 …

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 …

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 …

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 $ …

(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 …

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 …