I will describe an invariant for a smooth manifold with some additional topological structure. This will be a meromorphic function on the algebraic set of complex representations of its fundamental group defined using either spectral geometry (of a Riemannian metric) or a triangulation or the dynamics of a nice vector field.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/3946