Abstract
Given a compact and oriented 3-manifold and a representation of its fundamental group in the complex general linear group, we can define the characteristic class associated with the representation by taking the Chern classes of the complex vector bundle associated with the representation. Such a class is an element in the de Rham cohomology of the manifold. Cheeger and Simon gave a similar definition for the so-called secondary characteristic classes, but, this time, the classes are defined on the cohomology of the manifold with coefficients on the complexes modulo the integers. It is possible to identify this cohomology with the homomorphisms of the respective homology to the complexes modulo the integers. Characteristic numbers are defined by evaluating the characteristic class of the representation in the fundamental class of the manifold. In this talk we will give a way to calculate characteristic numbers by means of the determinant of representation and Atiyah Patodi Singer’s index theorem. We will give, as an example, the first characteristic numbers of representations of the fundamental group of spherical 3-manifolds. On the other hand, the representation induces a homomorphism between the homology of the manifold and the homology of the general linear group. Again, evaluating the fundamental class of the manifold defines an invariant of it, for the case of rational 3-spheres of homology, the invariant is defined as an element in the third algebraic K-group of the complexes. Furthermore, we will give a construction that recovers the spectrum of the spherical 3-manifolds already mentioned.
