Class numbers, the novikov conjecture, and transformation groups

Abstract

EWING [7] showed that the G-signature of a smooth G-manifold for G = Z,, p an odd prime, is essentially unrestricted if and only if the class number of the cyclotomic field containing the p-th roots of unity is odd. Katz [9] has elaborated on this, using results from smooth equivariant cobordism theory, to prove precise integrality formulae; in particular, he connects the G-signature mod 4 to the signatures and local representations occurring at components of the fixed set. In a different direction the author [21] showed that the existence of certain group actions on nonsimply connected manifolds forces their higher structures to vanish. In fact, the Novikov conjecture is equivalent to a statement about group actions. In yet another direction, S. Cappell and the author constructed [S] certain characteristic classes for semifree PI!. (locally linear) G-actions; these were applied there to prove a splitting theorem for some classifying spaces. This splitting, crucial for many equivariant existence and classification problems, at the prime 2 depends on understanding peripheral invariants of free group actions on sphere bundles. For locally-nonlinear actions, the results are deduced by means of comparison to the locally linear case. In this paper, we study the cobordism of homologically trivial actions and use it to unify, extend, and improve our understanding of all the above phenomena. The method is to consider, say, QHT, (Z,, X) which is roughly speaking the cobordism group of n-manifolds with rationally homologically trivial Z,-action. mapping into X. This is not a representable functor, and one cannot reduce the problem to stable homotopy theory of some Thorn spectrum. (It is not always possible to make a transverse inverse image have a homologitally trivial action). The solution (Theorem 1) to this difficulty, similar to that for Poincart cobordism, measures the deviation through an exact sequence involving L.i(R[n,X]) (for an appropriate ring R) as the “third term”. At this point one easily recovers the result that the Novikov conjecture implies the vanishing of higher signatures-for homologically trivial actions. This then gives information about which classes in the bordism group R(B(n x G)) have such G-actions, while [21] only solves (in some cases) this for R(&r). (Of course, the exact sequence also computes how many actions there are corresponding to a given class.) Now, for X = point, the Atiyah-Singer invariant is essentially a cobordism invariant, and the next order of business is to compute its role in the theory. Using the connection