Novikov Conjecture - Precise Formulation of The Conjecture

Precise Formulation of The Conjecture

Let G be a discrete group and BG its classifying space, which is a K(G,1) and therefore unique up to homotopy equivalence as a CW complex. Let

be a continuous map from a closed oriented n-dimensional manifold M to BG, and

Novikov considered the numerical expression, found by evaluating the cohomology class in top dimension against the fundamental class, and known as the higher signature:

where L_i is the ith Hirzebruch polynomial, or sometimes (less descriptively) as the ith L-polynomial. For each i, this polynomial can be expressed in the Pontryagin classes of the manifold's tangent bundle. The Novikov conjecture states that the higher signature is a homotopy invariant for every such map f and every such class x.