On groups which act freely and properly on finite dimensional homotopy spheres

Periodic Complexes and Group Actions

“…Y is a free Γ-CW complex), then it acts freely and properly on R n × S m for some m, n > 0. Indeed Y /Γ has countable homotopy groups, hence is homotopic to a countable complex which in turn is homotopic to an open submanifold V in some Euclidean space; applying the h-cobordism theorem we can infer that for sufficiently large q we have a diffeomorphism V × R q ∼ = R n × S m for some m, n > 0 (this appears in [22], lemma 5.4; see also [12], page 139). Conversely the existence of a free and proper Γ-action on R n × S m implies that Γ is countable (as R n × S m is a separable metric space) and that Γ has periodic cohomology (via the Gysin sequence).…”

Complete homology and related dimensions of groups

“…In [8], Gedrich and Gruenberg defined invariants spli G and silp G of a group G. These invariants play an important role in complete cohomology theory and finiteness conditions of groups (cf. [11], [15]). We can define new invariants sfli G, silf G and splf G similar to these invariants as follows.…”

“…It follows from Theorem 4.2 that Vogel-Goichot complete homologyH H Ã ðG; ÀÞ and Triulzi complete homologyĤ H Ã ðG; ÀÞ are isomorphic. Since spli G < y, G admits a complete projective resolution and cd G < y by [15, Theorem 2.5] and hence any complete projective resolutions of G are chain homotopy equivalent; see [10], or [15]. Since hd G < y, we have H Ã ðG; I Þ ¼ Tor i ðZ; I Þ ¼ 0 for i > hd G and I G-injective.…”

“…In a similar way as for a complete projective resolution (see [7], [8], and [15]), we can define the notion of a complete injective resolution. A complete injective resolution of a G-module M is an acyclic complex of injective G-modules which agrees with an ordinary injective resolution in a su‰ciently high dimension k. The number k A N is called the coincidence index of the complete injective resolution; see [16].…”

“…It follows from Lemma 5.2 that Vogel-Goichot complete homology can be computed using any complete projective resolution of G. By an argument using a stablizer spectral sequence just as for Farrell homology we have the result (cf. [4], [15] …”