The Asymptotic Method in the Novikov Conjecture

The famous Hirzebruch signature theorem asserts that the signature of a closed oriented manifold is equal to the integral of the so called L-genus. An immediate corollary of this is the homotopy invariance of < L(M); [M ] >. The L-genus is a characteristic class of tangent bundles, so the above remark is a non-trivial fact. The problem of higher signatures is a generalization of the above consideration. Namely we investigate whether the higher signatures are homotopy invariants or not. The problem is called the Novikov conjecture. The characteristic numbers are closely related to the fundamental groups of manifolds. There are at least two proofs of the signature theorem. One is to use the cobordism ring. The other is to use the Atiyah-Singer index theorem. Recall that the signature is equal to the index of the signature operators. The higher signatures are formulated as homotopy invariants of bordism groups of B . The problem was solved using the Atiyah-Singer index theorem in many partial solutions. Here we have the index-theoretic approach in mind when considering the higher signatures. Roughly speaking, a higher signature is an index for a signature operator with some coe cients. To interpret the number as a generalized signature, one considers homology groups with rational group ring coe cients. By doing surgery on the homology groups, we obtain non degenerate symmetric form 2 L( ) over the group ring. It is called the Mishchenko-Ranicki symmetric signature. This element is a homotopy invariant of manifolds. Mishchenko introduced Fredholm representations, obtaining a number (F) from a Fredholm representation F and . On the other hand, one can construct a virtual bundle over K( ; 1) from a Fredholm representation. By pulling back the bundle through maps from the base manifolds to K( ; 1), we can make a signature operator with coe cients. Mishchenko discovered the generalized signature theorem which asserts the coincidence of the index of the operators and (F). Thus a higher signature coming from a Fredholm representation is an oriented homotopy invariant. In [CGM] the authors showed that all higher signatures come from Fredholm representations for large class of discrete groups, including word hyperbolic groups. They formulated the notion of a proper Lipschitz cohomology class in group cohomology. It corresponds to a Fredholm representation in K-theory. In fact for many discrete groups, any class of group cohomology can be represented by a proper Lipschitz cohomology class. Their method depends on the existence of nite dimensional spaces of Q type K( ; 1). On the other hand for larger classes of discrete groups, we cannot expect existence of such good spaces. In [G], Gromov introduced a very large class of discrete groups,