Homology of spaces with operators. I

Abstract

The object of this paper is to study simultaneous invariants of a topological space X and of an abstract group W acting as a group of transformations on X. We assume that W also acts as a group of operators on the coefficient group G. In addition to the ordinary homology groups of X over G we define two ones that we call the equivariant and the residual homology groups. The three kinds of homology groups are connected by homomorphisms which form an exact sequence, that is, a sequence in which each homomorphism has as kernel the image of the preceding homomorphism. The same is done with cohomologies. We use singular homology theory [31(1) throughout; this permits us to drop all finiteness assumptions on W. In this respect our approach seems to differ from the one very successfully used by P. A. Smith [16]. In Chapter I we develop the theory of abstract complexes with operators. In Chapter II we define the special groups for a topological space X and show that if X is a simplicial complex and if W operates on X simplicially, then the special groups can be computed from the simplicial scheme of X. In the last part of Chapter II we define for each (discrete) group W an abstract complex Kw which has Was a group of operators. The equivarient groups of Kw are shown to be isomorphic with the homology and cohomology groups of the group W studied by Eilenberg and MacLane [5, 6, 7] and Hopf [10]. In Chapter III we assume that W operates on X without fixed points and that the ordinary homology groups of X vanish up to a certain dimension. It is then shown that the equivariant and residual groups of the space X are isomorphic with the appropriate groups of the group W. In Chapter IV these results are applied to prove some theorems about continuous mappings of spaces with operators. In Chapter V we take up the homology theory with local coefficient introduced recently by Steenrod [17]. Roughly speaking a homology (or cohomology) group with local coefficients arises whenever the fundamental group irl(X) operates on the coefficient group G. If Xdenotes the universal covering space of X, then W=7r1(X) operates both on Xand on G thus giving rise to the equivariant groups of X-. We show that the equivariant