2010/04/15 by Qusay S. A. Al-Zamil, Al-Zamil, Qusay S. A., James Montaldi +1 · 1 citation
Mathematics · Physics and Astronomy · #57R95 #58J32 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #math-ph #math.AT #math.DG #math.MP #msc:57R95 #msc:58J32
paper · pdf · doi:10.48550/arxiv.1004.2687
17 pages
openalex publication_date 2010/04/15 · arxiv created 2011/05/06 · arxiv updated 2011/05/09 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
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 and corresponding vector field XM on M, one defines Witten's inhomogeneous coboundary operator dXM = d+ιXM: ΩG^± →ΩG^∓ (even/odd invariant forms on M) and its adjoint δXM. In the 1980s Witten showed that the resulting cohomology classes have XM-harmonic representatives (forms in the null space of ΔXM = (dXM+δXM)2), and the cohomology groups are isomorphic to the ordinary de Rham cohomology groups of the set N(XM) of zeros of XM. 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 XM-cohomology and show the classes have representative XM-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 dXM and δXM. We also elucidate the connection between the XM-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 N(XM) has a unique XM-harmonic field on M, with corresponding boundary conditions. Finally, we define the XM-Poincaré duality angles between the interior subspaces of XM-harmonic fields on M with appropriate boundary conditions, following recent work of DeTurck and Gluck.