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Lusternik-Schnirelmann category for simplicial complexes. (English) Zbl 1302.55004

The authors define Lusternik-Schnirelmann category in simple homotopy type and use it obtain a lower bound for the number of discrete Morse functions. For recall, if \(K\) is a simplicial complex, an elementary collapse is an injection \(K\backslash \{\sigma, \tau\}\to K\) where \(\sigma, \tau\) are faces such that \(\sigma\) is a face of \(\tau\) and \(\sigma\) has no other coface. Then two complexes \(K_1\) and \(K_2\) have the same simple homotopy type if there is a sequence of collapses connecting \(K_1\) to \(K_2\). If \(K_2= \{v\}\) a single vertex, then \(K_1\) is called collapsible. A complex \(K\) has discrete geometric pre-category \(\leq m\) if \(K\) can be covered by \(m+1\) closed collapsible subcomplexes. The discrete geometric category of \(K\), \(\mathrm{dgcat}(K)\), is the minimum of the discrete geometric pre-categories of all the complexes in the same simple homotopy type as \(K\). The authors prove that if \(K\) has \(n\) vertices, then dgcat\((K)\leq n-1\). On the other hand if \(G\) is a planar graph, then dgcat\((G)\leq 2\). The main result says that the number of critical values of a discrete Morse function \(f:K\to \mathbb R\) is \(\geq \mathrm{ dgcat}(K)+1\).

MSC:

55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
55U05 Abstract complexes in algebraic topology
57M15 Relations of low-dimensional topology with graph theory
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