Bordism, and Free Group Actions

In this paper we give characteristic class formulae for all semicharacteristic classes of all compact, closed manifolds with finite fundamental groups. These invariants are identified with elements in certain odd L-groups, and exactly which elements occur is specified. An appendix calculates the cohomology of the model groups needed. A second appendix determines the structure of the L-groups needed. The Euler characteristic of an odd-dimensional manifold is zero; a natural substitute is a semicharacteristic—an alternating sum of the homology up to the middle dimension. Study of semicharacteristics was initiated by Kervaire [K] who examined their role in differential topology and geometry. The first semicharacteristic bordism invariant was introduced by DeRham. It turns out k Xx/2(M2k+X;K) = ¿(-l)'d¡ffl(ff¿(*f;A)j i=i is dependent on the characteristic of the field K and is not a bordism invariant. However, if k is even then the difference DR(M) = xxll(M ; Q) Xl/2(M ; F2) mod 2 is an invariant of the oriented bordism group ß2*+i • Characteristic class formulae for this invariant were given in [L-M-P]. Equivariant bordism invariants were introduced by Lee [L] and studied by Stong [S]. If a group G acts on an orientable manifold M, define the orientation character w:G^{±l}, by w(g) = +1 if and only if g is orientation-preserving. An action of G with orientation character w is called a (G, w)-acXion. Let Cln(G,w) denote the bordism group of closed oriented manifolds with a free (G,ifj)-action; an element is zero if it is the boundary of a compact manifold with a free (G ,w)action. If G is a finite group acting on a manifold M of dimension 2k + 1 Received by the editors Received by the editors November 23, 1987. 1980 Mathematics Subject Classification. Primary 57R67, 55N22, 55R40, 57Q20; Secondary 55M35, 55N25, 57N65, 57R20. Research of the first author supported by an NSF Posdoctoral Fellowship. Research of the second author supported by an NSF grant. © 1989 American Mathematical Society 0002-9947/89 $1.00 + $.25 per page