Agustin Romano, UNAM: Secondary characteristics classes for normal surfaces singularities II

Agustin Romano, Institute of Mathematics, UNAM: Secondary characteristics classes for normal surfaces singularities II
Cheeger-Chern-Simons classes are secondary characteristic classes of a vector bundle E over a smooth manifold M with a flat connection. They were defined by Cheeger and Simons using differential characters. In this talk, we show how to use these invariants to study new invariants of normal surface singularities.
i) In the case of a compact oriented 3-manifold L. For topologically trivial representations we will sketch the idea of how to compute the secondary characteristics classes using the Index Theorem for flat bundles by Atiyah, Patodi and Singer. Moreover, if L is a rational homology sphere we will provide a formula for these calculations.
ii) Our construction provides natural elements in algebraic K-theory. We will show this connection. We believe this may give a new perspective of the properties of normal surface singularities.
iii) Applications: As a consequence of our work, we have some applications to open problems and to a new interpretation of some invariants. More precisely, we will show the relationship between the secondary characteristic classes and the spectrum to Klenian singularities, and how to use these invariants to complete the classification of maximal Cohen-Macaulay modules over quotient surface singularities.