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Maybe this is my algebraic topology bias but I'm not sure there's anything one can say about this question in general--there are just too many topological spaces to try to classify them in any sense. …

I hope what that comment on Wikipedia means is that the homotopy category of CW complexes is the "right" homotopy category up to equivalence. There are many other ways to produce the homotopy categor …

I want to use the hairy ball theorem in the following way. Assume for simplicity that we start at time 0 and each ant returns to its original location at time 1. Suppose no two ants ever meet; then …

I'll show that the pushout that glues two copies of $\mathbb{R}$ at the origin does not exist in Man. Suppose for the sake of contradiction that it did; call the resulting manifold $M$, and the commo …

It sounds like maybe you can prove your first statement for simply connected spaces. In that case, you can use the homotopy orbit spectral sequence (the Serre spectral sequence associated to the fibr …

No. For example, let $g : [0, 1] \to [0, 1] \times [0, 1]$ be a continuous surjective map (space-filling curve) and let $p : [0, 1] \times [0, 1] \to [0, 1]$ be the projection onto the first coordina …

To answer your last question, the least number of charts needed to cover any orientable 2-manifold is 2. Consider the usual embedding of an orientable surface Σ in R3 which is symmetric across the pl …

Let P be a polynomial functor from Top to Top, by which I mean a functor of the form P(X) = ∐i ≥ 0 Si × Xi where the Si are finite sets, all but finitely many of which are empty. Corresponding to P t …

No. Here are two reasons: The coproduct of two pointed Kan complexes is usually not a Kan complex (say, if the spaces are connected): we can map $\Lambda^2_1$ into "both summands" and the map will …

There is a theorem of Schwede and Shipley which classifies categories of modules over an A∞ ring spectrum as those stable presentable (∞,1)-categories with a compact generator. Suppose I allow my A∞ …

Why isn't there an intrinsic topological description, or perhaps manifold-theoretic description? At least in some cases, the combinatorial Euler characteristic of X is equal to the homotopy Euler …