The topology of independence complexes of square grids

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.