Novikov conjecture

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This page concerns mathematician Sergei Novikov's topology conjecture. For astrophysicist Igor Novikov's conjecture regarding time travel, see Novikov self-consistency principle.

The Novikov conjecture is one of the most important unsolved problems in topology. It is named for Sergei Petrovich Novikov who originally posed the conjecture in 1965.

The Novikov conjecture concerns the homotopy invariance of certain polynomials in the Pontryagin classes of a manifold, arising from the fundamental group. According to the Novikov conjecture, the higher signatures, which are certain numerical invariants of smooth manifolds, are homotopy invariants.

The conjecture has been proved for finitely generated abelian groups. It is not yet known whether the Novikov conjecture holds true for all groups. There are no known counterexamples to the conjecture.

[edit] Precise formulation of the conjecture

Let G be a discrete group and BG its classifying space, which is up to homotopy uniquely determined by the fact that it is an aspherical CW-complex, with G as fundamental group. Let

$f: M\rightarrow BG$

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

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

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

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

where L_i is the i^th Hirzebruch polynomial, or sometimes (less descriptively) as the i^th 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.

[edit] References

J. Milnor and J.D. Stasheff, Characteristic Classes, Ann. Math. Stud. 76, Princeton (1974).

S.P. Novikov, Analytic 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, P. 253-288; II: N3, P. 475-500. English transl. in Actes Congr. Intern. Math., V. 2, 1970, P. 39-45.