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Any two isomorphic simplicial complexes are simple-homotopy-equivalent. This is a fairly simple result, but it is not obvious. Yet I have been surprisingly unable to find it in the literature on ...

Let's say I have a finite simplicial complex $X$ with a finite covering map $\pi: \widetilde{X} \rightarrow X$ and a discrete gradient vector field $V$ on $X$ (which for my purposes I prefer to its ...

In "vanilla" Morse theory, you can construct cycles representing integral homology classes of a smooth manifold from a Morse function on the manifold (by looking at the flow into/out of the critical ...