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Is there a sequence of topological spaces $X_n$ (manifolds ideally), where the sum of the Betti numbers of $X_n$ remains bounded but the Lusternik–Schnirelmann category is unbounded, as $n \to \infty$ …

The following seems to be well known to experts, but I would be happy if there is a paper or textbook that I can cite. Note: all of the manifolds are assumed to be without boundary. Suppose that $C$ …

Is it true that if a finite CW complex $X$ is simply connected, and $\tilde{H}_i(X, \mathbb{Q}) =0$ for $i \neq D$, then $X$ is rationally homotopy equivalent to a bouquet of $D$-dimensional spheres? …

Gil Kalai has a beautiful paper from 1983 where he shows that, on average, $\mathbb{Q}$-acyclic $d$-dimensional simplicial complexes $S$ with complete $(d-1)$-skeleton on $n$ vertices have $$| H_{d-1} …

Andrew Newman just posted a preprint to the arXiv, showing that for every prime $p$ and $d \ge 2$, you can get $p$-torsion in homology $H_{d-1}(K)$ with only $O(\log^{1/d} p)$ vertices. (The implied c …

One way to approach this question quantitatively is suggested by probability. One can put various measures on the space of all simplicial complexes on $n$ vertices. One perhaps fairly natural measur …

The classical examples are homology spheres.

What I would like to know is exactly what the title asks: What are the most general classes of simplicial complexes or posets for which the Charney-Davis conjecture is known, and what is the …

Let $G$ be a simple graph on a finite vertex set. The clique complex $X(G)$ is the simplicial complex whose faces are complete subgraphs of $G$, and the independence complex $I(G)$ is the simplicial c …

I have a number of results on hard disks in various types of regions, and preprints are in progress. The terminology "hard spheres" (or "hard disks" in dimension 2) comes from statistical mechanics, a …

One way to think about whether "most" spaces are aspherical is measure-theoretically. Here a few examples and non-examples of random topological spaces being aspherical. Examples Presentation complex …

(1) The Computational Homology Project offers free software CHomP that will compute homology of simplicial complexes, at least with finite field coefficients. (2) Dionysus, from the computational top …

The following does not answer your question, but adding just in case it is helpful. If you weaken "simply connected" to $H_1(\Delta, \mathbb{Q}) = 0$, and weaken "every vertex is in a bounded number o …

Andrew Newman just posted a preprint to the arXiv showing that the answer is doubly exponential in $n$. In particular, he showed that the number of homotopy types of simplicial complexes on $n$ verti …