Novikov Conjecture

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This page has not been refereed. The information given here might be incomplete or provisional.

1 Introduction

Let M be a closed oriented n-dimensional smooth manifold with a map f : M \to B\pi for some discrete group \pi and let \alpha \in H^{n-4*}(B\pi; \Qq) be a rational cohomology class. The higher signature of M defined by (f, \alpha) is the rational number

\displaystyle  \sigma_\alpha(M, f) = \langle L_M \cup f^*\alpha, [M] \rangle \in \Qq

where L_M \in H^{4*}(M; \Qq) is the Hirzebruch L-class of M. Let h: N \to M be a homotopy equivalence of closed oriented smooth manifolds. The Novikov conjecture states

\displaystyle  \sigma_\alpha(N, f \circ h) = \sigma_\alpha(M, f)

for all \pi, f, \alpha and for all homotopy equivalences h : N \to M. For the trivial group \pi the conjecture is true by Hirzebruch's signature theorem.

The original 1969 statement of the Novikov conjecture may be found in [Novikov1970] and [Novikov1970a]: a history and survey including the original statement in Russian with a translation into English may be found in [Ferry&Ranicki&Rosenberg1995b]. In the last 40 years the Novikov conjecture and the related conjectures of Borel and Farrell-Jones have been the subject of a great deal of research. In [Novikov2010] Novikov described how he came to formulate the conjecture.

More information may be found in [Ferry&Ranicki&Rosenberg1995], [Ferry&Ranicki&Rosenberg1995a], [Ranicki1995] and [Davis2000].

[edit] 2 References

[edit] 3 External links

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