Alladi Ramakrishnan Hall

Analytic surgery exact sequences and $\ell^2$ spectral invariants

Indrava Roy

ISI New Delhi

The Atiyah-Singer index theorem describes a deep relation between the
analytic and geometric-topological properties of a compact smooth manifold.
Further development of the theory by Atiyah, Patodi and Singer established
the existence of certain spectral invariants which are again intricately
related with the underlying geometry and topology. The Baum-Connes
conjecture is a far reaching generalization of the index theorem, and deep
results due to Keswani and Piazza-Schick assert that the spectral
invariants defined by Atiyah-Patodi-Singer have various stability
properties when the Baum-Connes conjecture holds. Similar stability results
hold for $\ell^2$ rho-invariants, a generalization of classical
rho-invariants that was first introduced by Cheeger and Gromov. Building on
earlier work by Higson and Roe on the analytic surgery exact sequence, we
give a more conceptual framework for the relation of $\ell^2$-rho
invariants with the Baum-Connes conjecture and give new proofs of the above
classical results, introducing new analytic and geometric invariants in the
process. (joint work with M.-T. Benameur)