Abstact: From the early days of rigorous statistical mechanics the study of discrete models (such as Ising Model, Percolation, and the Potts Model) on discrete spaces (such as the discrete lattice, the discrete torus, and cylindrical spaces) relied upon topological tools such as Peierls contours and various level-line preserving maps. To handle these, discrete topological analogues for manifolds, boundaries, and homology & cohomology theories were required and many connectivity and structural theorems were used. These were developed ad-hoc, and analogues of elementary topological results were repeatedly established in large sections and appendices of the various papers. Observing that such ad-hoc topological arguments are burdening the progress of the field, Timár established in the late 2000's a preliminary connectivity toolkit, avoiding direct use of algebraic topology. In this talk, we discuss a new construction of topological level components, which contains Timár's framework and is viable for Peierls contours, using algebraic topology directly, allowing the use of theorems such as the Poincare duality and the Mayer-Vietoris sequence, which were previously unavailable in this discrete context. 

Joint work with Ohad Feldheim.