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Given a map $\omega: A\to \Omega X$, one can set up the diagram and construct the map $\sigma : \Sigma A\to X$. It's pretty easy to check that the homotopy classes of $\omega$ and $\sigma$ correspond …

It’s not true. The Poincare sphere $P$ is a manifold, and its suspension is not. But its double suspension is homeomorphic to $S^5$ by Cannon’s “Double Suspension Theorem”. I learned about this from …

My interest is in topological monoids, but I think the question may make sense (in some fashion) for monoids of sets. Let $M$ be a topological monoid, and let $X$ be a pointed space that $M$ acts on, …

This question takes place in the category $\mathrm{CGWH}$ of compactly generated weak Hausdorff spaces. Let $\lambda$ be a limit ordinal, and suppose we have a diagram $\Phi: \lambda \to \mathrm{CGWH} …

No. Take any finite simplicial complex $K$, find an $n$ for which $K$ embeds (piecewise linearly, even!) in $\mathbb{R}^n$. Then for sufficiently small $\varepsilon > 0$, the $\varepsilon$-neighborho …

I've just stumbled on something that seems either too good to be true, or else too good for me not to have heard of it before. It has to do with the basepoint forgetting map $$ u: [A, M] \to \langle A …

The join of $A$ and $B$ is the pushout of the diagram $$ CA \times B \gets A\times B \to A\times CB, $$ which can be formulated in either the pointed or unpointed topological category. This pushout is …

It is natural to ask if it is possible for the mapping cone $X\cup_\alpha CA$ to be homeomorphic to the mapping cone $X\cup_\beta CB$ with $A$ and $B$ nonhomeomorphic. Is there a standard go-to examp …

This is a pretty well-known phenomenon, linked with phantom maps. One of the first existence results was Brayton Gray's paper Spaces of the same $n$-type, for all $n$, Topology 5 (1966) 241--243 Cla …

Let's consider a diagram $\Phi: \lambda \to \mathcal{T}_*$ $$ X_0 \to X_1 \to \cdots \to X_\xi \to X_{\xi+1} \to \cdots $$ of pointed spaces, indexed by some ordinal $\lambda$, in which each $X_\xi$ i …

Yes. In fact $\bigcup_{n\geq 1} P_n $ is the set of all primes. Serre proved that, for each odd prime $p$, there is some very predictable $p$-torsion in $\pi_{k+(p-1)}(S^k)$, for example. There is …

Write $CX$ for the (pointed, or reduced) cone on $X$, and $C^\circ X$ for the open cone inside of it. Let's say a cone map is a map $g:CX\to CY$ such that $g(C^\circ X) \subseteq C^\circ Y$ and $g(X) …

(The discussion below is for pointed spaces.) I'll use $\mathcal{F}_*$ for the pointed version of your $\mathcal{F}$. As Nicholas Kuhn says, this is related to the closed classes studied by E. Dror …

No. Consider the Hopf map $\eta:S^3\to S^2$. If there were such a space $Z$, it would have $\widetilde H^*(Z)=0$, so at the very least $Z$ would be stably trivial, forcing $\eta$ to be stably trivial; …

See this question: Model Structure/Homotopy Pushouts in topological monoids? especially the answer by John Francis. First of all, monoids have classifying spaces, and so a pushout diagram $M_1\g …