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Update: A bit of a digital paper chase led me, via David Roberts' thesis (note that in the latest version, it is Chapter 5, section 2 that is most relevant), to this paper on the arXiv. The last sent …

The Eilenberg-Zilber theorem says that for singular homology there is a natural chain homotopy equivalence: $$S_*(X)\otimes S_*(Y) \cong S_*(X\times Y)$$ The map in the reverse direction is the Alex …

Use the source, Luke. Specifically, chapters 14 (Smooth Bump Functions) to 16 (Smooth Partitions of Unity and Smooth Normality). You may be particularly interested in: Theorem 16.10 If $X$ is Li …

The Steenrod square is an example of a cohomology operation. Cohomology operations are natural transformations from the cohomology functor to itself. There are a few different types, but the most ge …

Here's a picture of a smash product that I drew for this talk, as far as I can tell it's what "Qfwfq" is describing in the middle paragraph.

This is the swindle, isn't it? There's an elegant way to phrase this with lots of sines and cosines, but working it all out is too much like hard work. Here's the quick and dirty way. Let $T: S^\in …

It may seem hard to add a new answer to all this, but here's mine. How to motivate the open set garbage of topological spaces: Answer: Don't. There are many ideas in mathematics that can be eas …

I'm going to answer (for my own limited version of the word "answer") the actual question and ignore the motivation (essentially for the reasons given in the comments). And to explain my limited vers …

My instinct (need to sit down with a piece of paper to confirm it) is No and No. For the first, I'm pretty sure that any compactly generated non-compact Hausdorff space will provide a counter-example …

First off, you want your source space, $X$, to be compact (technically, sequentially compact will do). If you don't have that, $C(X,Y)$ need not be locally contractible so no hope of a manifold struc …

Note: whilst typing this, Martin posted his answer. As I come to a completely different conclusion, I'd be very interested in knowing who's right! False. Let $M = [0,1]$ and $K_i = \{0\} \cup [\f …

(I was going to leave this as a comment but decided that it's a bit long for that) A couple of remarks: You express an aversion to Riemannian metrics because you want to be able to apply this in th …

Picking up on Gerald's interpretation of the question (namely, that it really focusses on infinite dimensional vector spaces) then I say: absolutely not! For example, piecewise-smooth paths in some E …

I'd like to recast Reid's (excellent) answer slightly. The essence of it is the following principle: To show that a limit or colimit doesn't exist in some category, embed your category in one whe …

Every real bundle can be complexified so there's a natural transformation $KO(X) \to K(X)$. Going all the way around multiplies by $2$ each time so if you localise at the prime $2$, you get a nice sp …