Simple Homotopy Types and Finite Spaces

Abstract

Whitehead’s theory of simple homotopy types is inspired by Tietze’s theorem in combinatorial group theory, which states that any finite presentation of a group could be deformed into any other by a finite sequence of elementary moves, which are now called Tietze transformations. Whitehead translated these algebraic moves into the well-known geometric moves of elementary collapses and expansions of finite simplicial complexes.