Homotopical Height
Mahan Mj (RKM Vivekananda University)
Abstract:
We introduce the notion of homotopical height ht_{C(G)} of a
finitely presented group G within a class C of smooth manifolds
with an extra structure (e.g. symplectic, contact, Kahler etc).
Homotopical height provides an obstruction to finding a K(G,1) space
within the given class C. This leads to a hierarchy of these
classes in terms of "softness" or "hardness" a la Gromov. We show that
the classes of closed contact, CR, and almost complex manifolds as
well as the class of (open) Stein manifolds are soft.
The classes SP and CA of closed symplectic and complex manifolds
exhibit intermediate "softness" in the sense that every finitely
presented group G can be realized as the fundamental group of a
manifold in SP and a manifold in CA. For these classes,
ht_{C(G)} provides a numerical invariant for finitely presented
groups. We give explicit computations of these invariants for some
standard finitely presented groups.
We use the notion of homotopical height within the "hard" category of
Kahler groups to obtain partial answers to questions of Toledo
regarding second cohomology and second group cohomology of Kahler
groups. We also modify and generalize a construction due to Dimca,
Papadima and Suciu to give a large class of projective groups
(fundamental group of complex projective manifolds) violating property
FP. These provide counterexamples to a question of Kollar.