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Edit: Modified in accordance with Tom Leinster's entirely reasonable objections. Sorry to exhume this question from 5+ years ago. In case someone is still looking for an answer, note that a very spec …

One set of sufficient conditions may be obtained if your map plays nicely with respect to subspaces fixed by closed subgroups. Let $X$ and $Y$ be $G$-spaces for any $G$ (not only the torus) and assume …

It is somewhat jarring to hear of people who "know nothing about the homology theories of topological spaces and their applications" but are "familiar with homological algebra, category theory, spectr …

Peter May has been working on an entire book (or maybe just a comprehensive set of lecture notes?) addressing your exact question (and much more). The preprint version which he shared with me is calle …

Let $X$ be a reasonable topological space (say one that has the homotopy type of a finite CW complex) and consider a subset $\Gamma$ of $X \times X$ so that the projection $p:\Gamma \to X$ onto the fi …

Question: Given a finite simplicial complex $K$, what general techniques allow one to efficiently compute (a presentation of) the group $\text{Aut}(K)$ of $K$'s automorphisms? Since this is str …

Let $(X,\leq)$ be a poset with distinguished element $p$, and let $P'$ be the poset of "finite chains which weakly descend to $p$" given by all $\sigma = (x_0 \geq x_1 \geq \cdots \geq x_k \geq p)$ or …

At least in the degree-theoretic world, the index notation appears to be dominant. When we write the Lefschetz-Hopf theorem $$L(f) = \sum_{x \in \text{Fix}(f)} \text{stuff}_x(f),$$ the $\text{stuff} …

A brief account of PL Morse theory has been requested by Andras in the comments, so I am writing it down here. Note that this does not address the main question on PL singularity theory. Note also tha …

It seems really hard to impose combinatorial Sperner-like conditions which would guarantee the nontriviality of the Hopf invariant. But if you allow things to get slightly more algebraic by constructi …

A simplicial complex with partially ordered vertices such that the vertex set of each simplex is a chain of the poset is called an ordered simplicial complex. This avoids the confusion with orientabil …

Your question is precisely the subject of Justin Curry's recent preprint. Bottom line: if you agree to identify functions $f,g:[0,1] \to \mathbb{R}$ whenever they have the same merge-tree, then ther …

One way to make things "minimal" (given the lack of any further information) is to construct a simplicial complex whose cup products are all trivial, so the (co)homology generators don't interact with …

Mark Grant's answer provides a good class of examples, but a slightly more general class can be found after (many, many hours) of reading Whitehead's "Combinatorial Homotopy II" available here. I th …

Background Let $\mathcal{A} = \lbrace A_0, \ldots, A_M \rbrace$ be an arbitrary sequence of finitely generated Abelian groups. It is well-known that a finite CW complex $X_\mathcal{A}$ may be construc …