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!



consequence of Novikov conjecture
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. 


