Complete homology and related dimensions of groups

Abstract

Abstract. In 1992, Goichot introduced a complete homology theory for arbitrary groups developed by Vogel. In 1998, Triulzi investigated another complete homology theory using the satellite approach invented by Mislin. These developments provide the motivation to find a new invariant ichd G of a group G, which is related to complete homology theories. We investigate its properties and related questions. In particular, using this invariant, we show that Triulzi complete homology is isomorphic to Vogel-Goichot complete homology in many classes of groups. Also we show that hd G ¼ 0 if and only if G is locally finite when G satisfies one of a number of conditions.

On Algebraic Invariants for Free Actions on Homotopy Spheres

“…In [18,Proposition 5.4] the author claimed that for any group G, sfli G = silf G if both are finite. Soon after [18] was published, the author realised that there is an incorrect argument in the proof of [18,Proposition 5.4] even though sfli G ≤ silf G is true. Asadollahi et al showed in [2,Theorem 3.7] that sfli G = silf G provided that ZG is coherent.…”

“…Asadollahi et al showed in [2,Theorem 3.7] that sfli G = silf G provided that ZG is coherent. Thus [18, Question B] cannot be proved by the argument in [18,Question B]. However, Emmanouil showed in [10] that [18,Question B] has a positive answer for any group G.…”

“…Thus [18, Question B] cannot be proved by the argument in [18,Question B]. However, Emmanouil showed in [10] that [18,Question B] has a positive answer for any group G.…”

“…The possible values of pccd G are integers greater than or equal to −1 and ∞. It is known that if G = * n∈N G n and G n is a free abelian group of rank n, then pccd G = ∞, and if G is the Thompson group T , ⊕ ∞ i=1 Z, or GL n (K), where K is a subfield of the algebraic closure of Q, then pccd G = −1 [17,18].…”