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I think the situation rationally is less hopeless than Tyler Lawson's answer indicates. In fact any simply connected suspension is rationally homotopy equivalent to a wedge of spheres (this is part of …

The short answer is "morally, $Y$ should be the loop space of $X$," and we'll see how far this answer gets towards justifying that. First some words that are not about spaces. My go-to example of …

Here's something you can say. Recall (Hatcher 3G.1) that if $G$ is a finite group and $\pi : X \to X/G$ a covering map associated to an action of $G$ on a space $X$, then over a field $F$ of character …

Let me write $C_k$ for unordered configurations and $F_k$ for ordered configurations. The natural map $F_k \to C_k$ is a covering map, so the two spaces have the same higher homotopy groups, hence one …

I do not see why the difference $c_1(E_\mathbb{C})c_1(F_\mathbb{C})$ between these two expressions is necessarily 2-torsion. What am I missing? Real line bundles are classified by $H^1(X, \mathbb …

Assuming that $G$ is discrete, the homotopy quotient $X/G$ fits into a fiber sequence $$X \to X/G \to BG$$ and hence, by the long exact sequence in homotopy, its fundamental group $\widetilde{G} = \ …

This is not really an answer. The inclusion of homotopy $1$-types into homotopy types has a left adjoint $\tau_{\le 1}$ given by truncation, and hence preserves homotopy limits (e.g. products). The qu …

In high dimensions you always have available variants of Dijkgraaf-Witten theory (references at the bottom of the page). In dimension $n$ this is a TFT constructed from a finite group $G$ and a charac …

Consider the variation where we ask for smooth manifolds and smooth fiber bundles. Then I claim that $R$ is not finitely generated. The starting observation is that if $F \to E \xrightarrow{p} B$ is …

One way to think about how to distinguish, if not classify, such spaces is by homotopy operations. In the same way that cohomology operations are natural transformations between cohomology functors, h …

Another class of examples comes from group cohomology: if $G$ is any group with interesting higher cohomology $H^n(G, A), n \ge 2$ then there are non-nullhomotopic maps $BG \to B^n A$ for some $n \ge …

This is the same as the induced map $\pi_4(KO) \to \pi_4(KU)$. Via Bott periodicity (see also this answer), this is the same as the induced map $\pi_0(KSp) \to \pi_0(KU)$ coming from the map sending a …

This is long and may not at all address your question, but here's a claim I'd like to defend: taking pullbacks of universal things is an abstract-nonsense operation, but taking pushforwards is no …

This isn't a complete answer. In general, suppose $X \to BG$ is the classifying map of a principal $G$-bundle and we want to describe the obstructions to reducing it to a principal $H$-bundle, or equi …

Not an answer. $h$ is already universal enveloping provided only that $K$ is the rationalization of a simply connected space; this is, for example, Theorem 21.5 in Felix, Halperin, and Thomas. In par …