Homotopy types of truncated projective resolutions

Abstract

We work over an arbitrary ring R. Given two truncated projective resolutions of equal length for the same module we consider their underlying chain complexes. We show they may be stabilized by projective modules to obtain a pair of complexes of the same homotopy type.

Realizing algebraic 2-complexes by cell complexes

“…For finite groups it has been shown that the 'geometric' D (2) property is equivalent to the 'algebraic' realization property [1, theorem I]. Indeed in one direction this is extended to all finitely presented groups [1, appendix B]:…”

“…In 1965 Wall introduced the problem and showed that a counter example must have n 2(see [5]). The question of whether or not such a complex exists is known as 'Wall's D (2) problem'. So a counter example must be (up to homotopy equivalence) a finite 3-complex of cohomological dimension 2, which is not homotopy equivalent to any finite 2-complex.…”

“…So a counter example must be (up to homotopy equivalence) a finite 3-complex of cohomological dimension 2, which is not homotopy equivalent to any finite 2-complex. The question of whether or not such a complex exists is known as 'Wall's D (2) problem'.…”

“…Thus our main result is: THEOREM 1•1. A finitely presented group G satisfies the D (2) property if and only if it satisfies the realization property.…”

Construction of Homotopy Equivalence of Truncated Complexes

“…In 2007, Mannan in [3] considered two truncated projective resolutions of equal length for the same module and obtained a pair of complexes of the same homotopy type. This paper generalizes projective resolutions to proper left C-resolutions and similar results are obtained, where C is a subcategory of R-modules.…”