Homotopy type of skeleta of the flag complex over a finite vector space and generalized Galois numbers. (English) Zbl 1539.57026
Summary: Let \({\mathscr{F}}\) stand for the flag complex associated to the lattice of proper subspaces of a finite-dimensional vector space \(V\). This paper aims at giving a (discrete) Morse theoretical proof of the fact that the \(k\)-th skeleton of \({\mathscr{F}}\) is homotopy equivalent to a wedge of spheres of dimension \(\min \{k,\dim ({\mathscr{F}})\} \). The tight control provided by Morse theoretic methods (through an explicit discrete gradient field) allows us to give a formula for the number of spheres appearing in each of these wedge summands. As an application, we derive an explicit formula for the number of flags on \(V\) of a given dimension, i.e., the number of simplices in \({\mathscr{F}}\) of the given dimension. Rather than depending on generalized Galois numbers, our formula for flags is given in terms of weighted inversion statistics of the symmetric group.
MSC:
| 57Q70 | Discrete Morse theory and related ideas in manifold topology |
| 55P15 | Classification of homotopy type |
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