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Such questions are typically framed in terms of Classifying Spaces, but the answer is yes. It follows from, for instance, from Proposition 2.1 in Graeme Segal's article Classifying spaces and spectra …

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 …

Since there were no answers for a few months, I asked this question to my colleague and triangulation expert Frank Lutz. Since his response was wonderful and exhaustive, I am reproducing it here for t …

Question Are there efficient algorithms to check if a finite simplicial complex defined in terms of its maximal facets is shellable? By efficient here I am willing to consider anything with smaller …

With apologies to fellow algebraic topologists, I confess that I have no idea how to answer this innocent-looking question: (1) Let's say we know that a finite simplicial complex $S$ is the baryce …

I would like to compute the homology of certain low dimensional CW complexes and I am hoping to take advantage of software that handles simplicial sets as input. Thus, I would like to convert a CW com …

Fix a number $n$, and define $\gamma(n)$ to be the number of simplicial complexes on $n$ unlabeled vertices up to homotopy equivalence. It is unlikely that an explicit formula exists, but what is know …

Let $\mathcal{C}$ be a small category. Then, its Segal subdivision $\text{sd }\mathcal{C}$ is a new category whose objects are morphisms of $\mathcal{C}$, and a morphism from $f:x \to y$ to $g: w \to …

I have a really tame category $C$: there are only finitely many objects $C_0$, each hom-set $C(x,y)$ for $x,y \in C_0$ has at most one element, and (aside from identity morphisms) if $C(x,y)$ is non-e …

Let's say we are given two finite simplicial complexes, which I will suggestively call $E$ and $B$. We'd like an algorithm for the following decision problem: Does there exist a simplicial map $p: …

The answer to the question as stated seems to be "no". Consider a three-element Markov partition $\mathcal{M} = \{A, B, C\}$ with directed edges $(A,B)$, $(B,C)$ and $(C,A)$. There is an obvious peri …

Let $K$ and $L$ be finite simplicial complexes that remain fixed throughout. How does one efficiently sample (according to the uniform distribution) elements from the finite set of simplicial map …

I realize that I am several months late to the Cerf theory party, but this paper of Chari and Joswig might be of interest to the original poster and certainly deserves a mention in the context of this …

It's always nice to see people working on discrete Morse theory. Answer 1 It is an "if and only if". Meaning: the partial order $\prec$ is defined by $\alpha \prec \gamma$ if and only if $\gamma$ pr …

This got too long for a comment, so I am placing it here. I don't think there is a theory already out there, but that should not be too surprising. After all, the "combinatorial distance" between a s …