Equivariant singular homology and cohomology. I

Topological Abelian Groups and Equivariant Homology

“…The boundary operator is given as the restriction of the boundary operator of the singular chain complex of X . We show below (Theorem 2.9) that this chain complex is naturally isomorphic to Illman's chain complex [13], which defines the Bredon-Illman equivariant homology of X with coefficients in L, H G q (X; L). The main theorem of the paper is the following.…”

“…As shown by Illman [13], given any (covariant) coefficient system, there is an ordinary equivariant homology theory H G * (−; M ) with M as coefficients. Moreover, given any natural transformation µ : M −→ N of coefficient systems, there is an extension of it to a natural transformation µ : H G * (−; M ) −→ H G * (−; N ) of (ordinary) homology theories.…”

Mapping spaces of compact Lie groups and 𝑝-adic completion

“…Now let C = z£. The first map induces an //*(-; ¥p)-isomorphism by [NI], and so a homotopy equivalence after completion (see [BK,I,5.5]), while the second is shown in [JMO, loc. cit.] to be a homotopy equivalence.…”

“…The p-adic completion of a space X that we refer to is the (F^^X of [BK,I,§4.2], which we denote by Xp . However, unless X is nilpotent (e.g., simply-connected), X£ need not be p-complete in the sense of [BK,I,§5 & VII,§2], and so it enjoys few of the properties associated with completion.…”

“…However, unless X is nilpotent (e.g., simply-connected), X£ need not be p-complete in the sense of [BK,I,§5 & VII,§2], and so it enjoys few of the properties associated with completion. In particular, unless X£ is /7-complete, the natural map /: X -► X£ will not induce an isomorphism in Fp-homology, so Xp will not be the fl*(-; Fp ^localization of X (cf.…”