Is there an algorithm that can decide, wether the polyhedron of a neighborhood complex (as defined here) of a given finite graph is simply connected? There is no such algorithm for arbitrary simplicial complexes, see k-connectedness of simplicial complexes.

I recently saw a proof of this using the fact that the star of a vertex $v$ of a simplicial complex is open. However, this does not hold if $st(v) = v$ where $st(v)$ means the star of $v$ (i.e. $v$ lies only in a zero-simplex). Is there any reason as to why $ K […]

I refer to example 4, fig.3.6, p.17 of Munkres’ Algebraic Topology. He says the given triangulation scheme “does more than paste opposite edges together”. Not clear to me. For those who don’t have the book to hand, a rectangle is divided into 6 equal squares by a horizontal midline and two verticals; each square has […]

In a $1$-simplex, it’s clear that the boundary is the set of the two $0$-simplices and that the interior is all the points in between them. But what about in a $0$-simplex? I’m asking because I know that the topological realisation of a simplex is the union of the interiors of simplices. This would suggest […]

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