Simple homotopy type of some combinatorially defined complexes

In this paper we provide concrete combinatorial formal deformation algorithms, namely sequences of elementary collapses and expansions, which relate various previously extensively studied families of combinatorially defined polyhedral complexes. To start with, we give a sequence of elementary collapses leading from the barycentric subdivision of the neighborhood complex to the Lov\'asz complex of a graph. Then, for an arbitrary lattice ${\mathcal L}$ we describe a formal deformation of the barycentric subdivision of the atom crosscut complex $\Gamma({\mathcal L})$ to its order complex $\Delta(\bar{\mathcal L})$. We proceed by proving that the complex of sets bounded from below ${\mathcal J}({\mathcal L})$ can also be collapsed to $\Delta(\bar{\mathcal L})$. Finally, as a pinnacle of our project, we apply all these results to certain graph complexes. Namely, by describing an explicit formal deformation, we prove that, for any graph $G$, the neighborhood complex ${\mathcal N}(G)$ and the polyhedral complex $\text{\tt Hom}(K_2,G)$ have the same simple homotopy type in the sense of Whitehead.