On the homotopy type of CW-complexes with aspherical fundamental group

This paper is concerned with the homotopy type distinction of finite CW-complexes. A (G, n)-complex is a finite n-dimensional CW-complex with fundamental-group G and vanishing higher homotopy-groups up to dimension n - 1. In case G is an n-dimensional group there is a unique (up to homotopy) (G, n)-complex on the minimal Euler-characteristic level χ_min (G, n). For every n we give examples of n-dimensional groups G for which there exist homotopically distinct (G, n)-complexes on the level χ_min (G, n) + 1. In the case where n = 2 these examples are algebraic.