The Signature of a Manifold
  Let us consider a compact oriented riemannian manifold M without boundary and of dimension n=4k. The signature of M is defined as the signature of a given quadratic form Q. Two different products could be used to define Q and they render equivalent definitions: the exterior product of 2k-forms and the cup product of cohomology classes. The signature of a manifold is proved to yield a topological invariant. Additionally, using the metric, a suitable Dirac operator can be defined whose index coincides with the signature of the manifold. This second version includes corrections and many examples. 
Jose Rodriguez