Combinatorial Formulas for Products of Thom Classes

  • Victor Guillemin
  • Catalin Zara

Abstract

Let G be a torus of dimension n>1 and M be a compact Hamiltonian G-manifold with MG finite. A circle S1 in G is generic if MG=MS1. For such a circle the moment map associated with its action on M is a perfect Morse function. Let {Wp+;pMG} be the Morse-Whitney stratification of M associated with this function and let τp+ be the equivariant Thom class dual to Wp+. These classes form a basis of HG*(M) as a module over \( \mathbb{S}(\mathfrak{g}*) \) and, in particular,
$$ \tau _p^ + \tau _q^ + = \sum {c_{pq}^r \tau _r^ + }$$
with \( c_{pq}^r \in \mathbb{S}(\mathfrak{g}*) \). For a large class of manifolds of this type we obtain a combinatorial description of these τp+s and, from this description, a combinatorial formula for cpgr.

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Copyright information

© Springer-Verlag New York, Inc. 2002

Authors and Affiliations

  • Victor Guillemin
  • Catalin Zara

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