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.