Bredon-Illman cohomology with local coefficients

A Mukherjee
1996 Quarterly Journal of Mathematics  
§ 1. Introduction LET A' be a G-space, where G is a topological group. In this paper we construct a cohomology H^{X\ M), where Af is a suitable equivariant local coefficients system on X. The cohomology satisfies all the equivariant Eilenberg-Steenrod axioms with reference to the system M (cf. [9], [6]), and can also be described in terms of equivariant cellular structure when A" is a G-CW-complex. Moreover, it reduces to the equivariant singular cohomology with contravariant coefficient system
more » ... of Illman [6] when M is simple in some sense, and to the Steenrod cohomology with the classical local coefficients system [9] when G is trivial. We present this theory in § § 2-7. An interesting feature of the Steenrod cohomology of a topological space X with local coefficients M is that it can be realized as certain cohomology of its universal covering X. If p: X-+X is the covering projection, then n = n x {X,x°) acts on X, and M o = M(x°) is a jr-module. Let C^(X;M 0 ) be the gToup of ^-invariant singular n-cochains, and H^(X;M 0 ) be the corresponding cohomology. Then a classical theorem of Eilenberg [3], [9] says that p induces an isomorphism H n (X;M)=*H n x (X;Mo). Motivated by this consideration, we next attempt to find a similar alternative description of our cohomology. In § 8, we introduce the notion of universal 0 c -covering space ^ of a G-space X when X satisfies certain G-connectivity conditions. The universal coverings X H of fixed point sets X H , where H is a closed subgroup of G, incorporate into % and there is a natural action of an O c -group n on °U which reflects the action of each ni(X H , x°) on X H where x° s X c . Moreover, an equivariant local coefficients system Af on A" induces an O c -group Af 0 with an action of the O c -group n on it. We then formulate a cohomology of °U with coefficients in Afo, and denote it by //£ |G (<&; Af 0 ). This cohomology is obtained by allowing the equivariant cochain complex of Eilenberg to dominate on universal covering of each fixed point set of X, and then piecing them together in a natural way by the group action. In § 9, we exploit this idea and prove The result reduces to the theorem of Eilenberg when G is trivial.
doi:10.1093/qjmath/47.185.199 fatcat:q3gndlh5tveezhzazpmktaoz4m