  The independence complex of a graph G is a simplicial complex whose simplices
are the independent sets in G. In the last couple of decades, the independence
complexes of square grids (with various boundary conditions) have gained much
attention because of their connections with the hard square model from
statistical physics. In this article, we prove that if G is an $m\times n$ grid
with open or cylindrical boundary condition then its independence complex is
homotopy equivalent to a wedge of spheres. A part of this result settles a
conjecture of Iriye.

Combinatorics;Algebraic Topology