Novikov conjecture

Let ${\displaystyle G}$ be a discrete group and ${\displaystyle BG}$ its classifying space, which is an Eilenberg–MacLane space of type ${\displaystyle K(G,1)}$, and therefore unique up to homotopy equivalence as a CW complex. Let

${\displaystyle f\colon M\rightarrow BG}$

be a continuous map from a closed oriented ${\displaystyle n}$-dimensional manifold ${\displaystyle M}$ to ${\displaystyle BG}$, and

${\displaystyle x\in H^{n-4i}(BG;\mathbb {Q} ).}$

Novikov considered the numerical expression, found by evaluating the cohomology class in top dimension against the fundamental class ${\displaystyle [M]}$, and known as a higher signature:

${\displaystyle \left\langle f^{*}(x)\cup L_{i}(M),[M]\right\rangle \in \mathbb {Q} }$

where ${\displaystyle L_{i}}$ is the ${\displaystyle i^{\rm {th}}}$ Hirzebruch polynomial, or sometimes (less descriptively) as the ${\displaystyle i^{\rm {th}}}$ ${\displaystyle L}$-polynomial. For each ${\displaystyle i}$, this polynomial can be expressed in the Pontryagin classes of the manifold's tangent bundle. The Novikov conjecture states that the higher signature is an invariant of the oriented homotopy type of ${\displaystyle M}$ for every such map ${\displaystyle f}$ and every such class ${\displaystyle x}$, in other words, if ${\displaystyle h\colon M'\rightarrow M}$ is an orientation preserving homotopy equivalence, the higher signature associated to ${\displaystyle f\circ h}$ is equal to that associated to ${\displaystyle f}$.

The Novikov conjecture is equivalent to the rational injectivity of the assembly map in L-theory. The Borel conjecture on the rigidity of aspherical manifolds is equivalent to the assembly map being an isomorphism.

• Davis, James F. (2000), "Manifold aspects of the Novikov conjecture", in Cappell, Sylvain; Ranicki, Andrew; Rosenberg, Jonathan (eds.), Surveys on surgery theory. Vol. 1 (PDF), Annals of Mathematics Studies, Princeton University Press, pp. 195–224, ISBN 978-0-691-04937-3, MR 1747536
• John Milnor and James D. Stasheff, Characteristic Classes, Annals of Mathematics Studies 76, Princeton (1974).
• Sergei P. Novikov, Algebraic construction and properties of Hermitian analogs of k-theory over rings with involution from the point of view of Hamiltonian formalism. Some applications to differential topology and to the theory of characteristic classes. Izv.Akad.Nauk SSSR, v. 34, 1970 I N2, pp. 253–288; II: N3, pp. 475–500. English summary in Actes Congr. Intern. Math., v. 2, 1970, pp. 39–45.

• Biography of Sergei Novikov
• Novikov Conjecture Bibliography
• Novikov Conjecture 1993 Oberwolfach Conference Proceedings, Volume 1
• Novikov Conjecture 1993 Oberwolfach Conference Proceedings, Volume 2
• 2004 Oberwolfach Seminar notes on the Novikov Conjecture (pdf)
• Scholarpedia article by S.P. Novikov (2010)
• The Novikov Conjecture at the Manifold Atlas