This volume deals with the theory of finite topological spaces and its relationship with the
homotopy and simple homotopy theory of polyhedra. The interaction between their intrinsic
combinatorial and topological structures makes finite spaces a useful tool for studying
problems in Topology Algebra and Geometry from a new perspective. In particular the methods
developed in this manuscript are used to study Quillen's conjecture on the poset of p-subgroups
of a finite group and the Andrews-Curtis conjecture on the 3-deformability of contractible
two-dimensional complexes. This self-contained work constitutes the first detailed exposition
on the algebraic topology of finite spaces. It is intended for topologists and
combinatorialists but it is also recommended for advanced undergraduate students and graduate
students with a modest knowledge of Algebraic Topology.
