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This is essentially a (cw) reference request, because it seems like the sort of thing that should have been looked at already. Setting aside, for now, how to think what the localization of a general …

Quite simply, I'd like to know what is the broadest or most natural context in which either (or both) of Mather's cube theorems hold. If you like, this may mean any of What properties of $Top$ or $T …

Regarding the table of $SO(n)$s-of-origin in Davis+Mahowald (if you can get MathSciNet), is there a good reason that it should take longer for $\pi_{15}(SO)$ to be representable than $\pi_{19}(SO)$, w …

The following runs out of steam towards the end; I may also be making important mistakes, so be on your guard --- but that's half the fun! Anyways, it was too long for a comment. Choose a locally-fi …

Well, the silly answer is that $f:\mathbb{S}^k\to X$ represents a torsion element of order $p$ if $p \cdot f:\mathbb{S}^k\to X$ extends along $\mathbb{S}^k \hookrightarrow D^{k+1}$ to a map $\varphi: …

First, it's important that the infinite connect sum $A\# B\#A\#\cdots$ is not the limit of the finite connect sums $A,A\#B, A\#B\# A,\dots$; in fact, I'm sure the binary connect sum is as wrong a nota …

I'd just like to point out that there is a monad on $Top$, (which in the homotopy category looks rather dull,) assigning to each space $X$ its cone $CX$, the mapping cylinder of $X\to * $. The unit m …

This is probably making a hash of the earlier answers, but bundles are special fibrations; specifically, they are fibrations with (not canonically) isomorphic fibers. And we all like fibrations, righ …

There are a short list of operations described as generating the desired polyhedra: $ X : \mathrm{Prism} \vdash C X : \mathrm{Prism} $ $ l : \mathrm{list}\ \mathrm{Prism} \vdash \Pi l : \mathrm{ …

I have two related sorts of example to suggest, probably exhibiting my categorical bias vs. the analytic/geometric-topology weight of the preceding examples. Galois-theoretic Let $P\in K[x]$ be an i …