Witten-Hodge theory for manifolds with boundary and equivariant cohomology
- 作者：Qusay S.A. Al-Zamil Qusay.Abdul-Aziz@postgrad.manchester.ac.uk ; James Montaldi ; j.montaldi@manchester.ac.uk
- 关键词：58J32 ; 57R91 ; 55N91
- 刊名：Differential Geometry and its Applications
- 出版年：2012
- 期刊代码：18_09262245
- 类别：mt
- 出版时间：April, 2012
- 卷：30
- 期：2
- 页码：179-194
- 文件大小：295 K
摘要
We consider a compact, oriented, smooth Riemannian manifold M (with or without boundary) and we suppose G is a torus acting by isometries on M. Given X in the Lie algebra of G and corresponding vector field on M, one defines Witten始s inhomogeneous coboundary operator (even/odd invariant forms on M) and its adjoint . Witten (1982) showed that the resulting cohomology classes have -harmonic representatives (forms in the null space of ), and the cohomology groups are isomorphic to the ordinary de Rham cohomology groups of the set of zeros of . Our principal purpose is to extend these results to manifolds with boundary. In particular, we define relative (to the boundary) and absolute versions of the -cohomology and show the classes have representative -harmonic fields with appropriate boundary conditions. To do this we present the relevant version of the Hodge-Morrey-Friedrichs decomposition theorem for invariant forms in terms of the operators and . We also elucidate the connection between the -cohomology groups and the relative and absolute equivariant cohomology, following work of Atiyah and Bott. This connection is then exploited to show that every harmonic field with appropriate boundary conditions on has a unique -harmonic field on M, with corresponding boundary conditions. Finally, we define the -Poincar茅 duality angles between the interior subspaces of -harmonic fields on M with appropriate boundary conditions, following recent work of DeTurck and Gluck.