Novikov conjecture

The Novikov conjecture is one of the most important unsolved problems in topology. It is named for Sergei 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.
Precise formulation of the conjecture.
Let formula_1 be a discrete group and formula_2 its classifying space, which is an Eilenberg–MacLane space of type formula_3, and therefore unique up to homotopy equivalence as a CW complex. Let
be a continuous map from a closed oriented formula_5-dimensional manifold formula_6 to formula_2, and
Novikov considered the numerical expression, found by evaluating the cohomology class in top dimension against the fundamental class formula_9, and known as a higher signature:
where formula_11 is the formula_12 Hirzebruch polynomial, or sometimes (less descriptively) as the formula_12 formula_14-polynomial. For each formula_15, 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 formula_6 for every such map formula_17 and every such class formula_18, in other words, if formula_19 is an orientation preserving homotopy equivalence, the higher signature associated to formula_20 is equal to that associated to formula_17.
Connection with the Borel conjecture.
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.