Simple homotopy of flag simplicial complexes and contractible contractions of graphs
Anton Dochtermann, Takahiro Matsushita
TL;DR
The paper establishes a precise bridge between simple homotopy of flag complexes and the $ $frak{I}$-homotopy type of graphs by proving that $G$ and $H$ are $ $frak{I}$-homotopy equivalent iff $C(G)$ and $C(H)$ are simple homotopy equivalent, and that $C(G)$ contractible implies $G$ is $ $frak{I}$-contractible. It leverages results from BFJ and CYY and extends the characterization to arbitrary simplicial complexes via links, showing that simple homotopy can be realized through vertex insertions/deletions with contractible links. A key consequence is that the converse of the BFJ relation with $s$-homotopy holds, unifying several notions of graph and flag-complex homotopy. The work also provides an explicit structural understanding of simple homotopy in terms of vertex links and a Cyl/barycentric-subdivision framework, broadening the applicability to general simplicial complexes and clarifying redundancy in $ $frak{I}$-contractible moves.
Abstract
In his work on molecular spaces, Ivashchenko introduced the notion of an $\mathfrak{I}$-contractible transformation on a graph $G$, a family of addition/deletion operations on its vertices and edges. Chen, Yau, and Yeh used these operations to define the $\mathfrak{I}$-homotopy type of a graph, and showed that $\mathfrak{I}$-contractible transformations preserve the simple homotopy type of $C(G)$, the clique complex of $G$. In other work, Boulet, Fieux, and Jouve introduced the notion of $s$-homotopy of graphs to characterize the simple homotopy type of a flag simplicial complex. They proved that $s$-homotopy preserves $\mathfrak{I}$-homotopy, and asked whether the converse holds. In this note, we answer their question in the affirmative, concluding that graphs $G$ and $H$ are $\mathfrak{I}$-homotopy equivalent if and only if $C(G)$ and $C(H)$ are simple homotopy equivalent. We also show that a finite graph $G$ is $\mathfrak{I}$-contractible if and only if $C(G)$ is contractible, which answers a question posed by the first author, Espinoza, Frías-Armenta, and Hernández. We use these ideas to give a characterization of simple homotopy for arbitrary simplicial complexes in terms of links of vertices.