Geodesic
On the homotopy and strong homotopy type of complexes of discrete Morse functions
Canadian mathematical bulletin, Tome 66 (2023) no. 1, pp. 19-37

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In this paper, we determine the homotopy types of the Morse complexes of certain collections of simplicial complexes by studying dominating vertices or strong collapses. We show that if K contains two leaves that share a common vertex, then its Morse complex is strongly collapsible and hence has the homotopy type of a point. We also show that the pure Morse complex of a tree is strongly collapsible, thereby recovering as a corollary a result of Ayala et al. (2008, Topology and Its Applications 155, 2084–2089). In addition, we prove that the Morse complex of a disjoint union $K\sqcup L$ is the Morse complex of the join $K*L$. This result is used to compute the homotopy type of the Morse complex of some families of graphs, including Caterpillar graphs, as well as the automorphism group of a disjoint union for a large collection of disjoint complexes.
DOI : 10.4153/S0008439522000121
Mots-clés : Discrete Morse theory, Morse complex, dominated vertex, strong collapsibility, homotopy 
Donovan, Connor; Lin, Maxwell; Scoville, Nicholas A. On the homotopy and strong homotopy type of complexes of discrete Morse functions. Canadian mathematical bulletin, Tome 66 (2023) no. 1, pp. 19-37. doi: 10.4153/S0008439522000121
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     title = {On the homotopy and strong homotopy type of complexes of discrete {Morse} functions},
     journal = {Canadian mathematical bulletin},
     pages = {19--37},
     year = {2023},
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