# Novikov Conjecture

## 1 Introduction


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

where $L_M \in H^{4*}(M; \Qq)$$L_M \in H^{4*}(M; \Qq)$ is the Hirzebruch L-class of $M$$M$. Let $h: N \to M$$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$$\pi$, $f$$f$, $\alpha$$\alpha$ and for all homotopy equivalences $h : N \to M$$h : N \to M$. For the trivial group $\pi$$\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.