The space of tropically collinear points is shellable
Given \(d,n\in\mathbb{N}\), let \(T_{d,n}\) be the space of \(n\) tropically collinear points in a tropical projective space \(\mathbb{T}\mathbb{P}^{d-1}\). A point in \(T_{d,n}\) is represented by a \(d\times n\) real matrix of tropical rank no greater than two. Let \(T'_{d,n}\) be the space \(T_{d,n}\), up to translation, and let \(T''_{d,n}\) denote the intersection of \(T'_{d,n}\) with the unit sphere. The authors endow \(T_{d,n}\) and \(T'_{d,n}\) with a simplicial fan structure. Then \(T''_{d,n}\), the link of \(T'_{d,n}\) at the origin, is a simplicial complex of one dimension less. A shelling on a simplicial complex \(S\) is a particular type of ordering on the facets of \(S\). Shellability is a strong combinatorial property which implies, among other things, non-vanishing homology only in the top dimension. Concerning the spaces \(T_{d,n}\), \(T'_{d,n}\), \(T''_{d,n}\), the shellability of one implies the shellability of the other two. The authors identify \(T_{d,n}\) with a subcomplex of the space of phylogenetic trees on \(d+n\) taxa, which was proved to be shellable by [\textit{H. Trappamann} and \textit{G. M. Ziegler}, J. Comb. Theory, Ser. A 82, No. 2, 168--178 (1998; Zbl 0916.06004)]. This way, they construct a shelling of \(T_{d,n}\). Then they use the corresponding shelling of \(T''_{d,n}\) to compute the homology of this space. A particular type of trees, called combs, are crucial at this point. The results of the paper can be summarized as follows: \(T''_{d,n}\) is a shellable simplicial complex having the homotopy type of a wedge of \(h\) \(n+d-4\)--dimensional spheres, where \[ h=\sum_{k=1}^{\min\{n-1,d-1\}}(k!)^2 S(n-1,k)S(d-1,k), \] and \(S(m,k)\) denotes the Stirling number of the second kind (i.e., the number of partitions of \(\{1,2,\dots,m\}\) into \(k\) non-empty subsets). The shellability of \(T_{d,n}\) was proved by [\textit{M. Develin}, Collect. Math. 56, No. 1, 1--19 (2005; Zbl 1069.52014)], for \(d=3\), and conjectured for any \(d,n\). This is a clearly written, interesting paper which adds to the steady and fast growing development of tropical algebraic geometry.
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