Novikov conjeture is a famous open problem in Geometric topology.It predicts that higher signature is oriented-homotopy invariant.
http://en.wikipedia.org/wiki/Novikov_conjecture
I am a student interested in geometric topology.My question is:
What are the possible consequences of this famous conjecture? Thank you!
The Novikov conjecture is known for some groups, and open for other groups. Often Novikov conjecture is implied by a more general conjecture (e.g. Borel, Farrell-Jones, or Baum-Connes) which in my amateur view are more intersting. All these conjectures are true e.g. for $G=\mathbb Z^n$, $n>4$ and in many other cases.
Out of those the Borel conjecture is the easiest to state, it says: a homotopy equivalence of closed $K(G,1)$ manifolds is homotopic to a homeomorphism. There is also a relative Borel conjecture: the homotopy equivalence of compact aspherical manifolds that is a homeomorphism on the boundary is homotopic to a homeomorphism.
Here is a common way in which the conjectures get used: obstructions to various topological problems lie in groups associated with $G$ such as the Whitehead group (is an h-cobordism trivial?), projective class group (can one attach a boundary to an open manifold?) etc, which vanish if the appropriate Novikov-type conjecture hold for $G$.
There is a friendly textbook "The Novikov conjecture'' by Kreck and Lueck which starts as a slow introduction and ends with a survey.
