Localization via cobordisms and polytope decompositions

Connected Cerf Theory

We review the theory of Cerf describing the decomposition of cobordisms into elementary cobordisms, and the Cerf moves between different decompositions. We put special emphasis on connectedness and define Cerf decompositions as decompositions of morphisms in the category of connected manifolds and connected cobordisms. In addition, we discuss the cyclic Cerf theory of Morse functions to S. Prel...

New Polytope Decompositions and Euler-maclaurin Formulas for Simple Integral Polytopes

We use a version of localization in equivariant cohomology for the norm-square of the moment map, described by Paradan, to give several weighted decompositions for simple polytopes. As an application, we study Euler-Maclaurin formulas.

Symplectic Cobordisms and Open Book Decompositions

Dual Decompositions of 4-manifolds Iii: S-cobordisms

The main result is that an s-cobordism (topological or smooth) of 4-manifolds has a product structure outside a “core” sub-s-cobordism. These cores are arranged to have quite a bit of structure, for example they are smooth and abstractly (forgetting boundary structure) diffeomorphic to a standard neighborhood of a 1-complex. The decomposition is highly nonunique so cannot be used to define an i...

A COMPACTLY SUPPORTED FORMULA FOR EQUIVARIANT LOCALIZATION and SIMPLICIAL COMPLEXES OF BIAŁYNICKI-BIRULA DECOMPOSITIONS

The Duistermaat-Heckman formula for their induced measure on a moment polytope is nowadays seen as the Fourier transformof the Atiyah-Bott localization formula, applied to the T -equivariant Liouville class. From this formula one does not see directly that the measure is positive, nor that it vanishes outside the moment polytope. In [Knutson99] we gave a formula for the Duistermaat-Heckman meas...