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Gromov and Lawson introduced the notion of enlargeability of a compact
manifold M.
They show that enlargeable spin manifolds do not admit a metric of
positive scalar curvature.
Gromov also introduced the notion of homological largeness for M,
meaning that the fundamental class
of M is non-zero in the cohomology or K-theory of the fundamental group.
In joint work with Hanke and partially Kotschick and Roe, we
systematically study these notions of largeness,
using methods from coarse geometry.
In particular, we prove that enlargeability implies homological largeness,
and we show even that the
fundamental class of M does not lie in the kernel of the Baum-Connes
assembly map. This proves a small
part of the strong Novikov conjecture.
The talk will discuss these results and the relavant homological and
geometrical notions and ideas.
