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## 2010.31: Witten-Hodge theory for manifolds with boundary and equivariant cohomology

2010.31: Qusay Al-Zamil and James Montaldi (2012) Witten-Hodge theory for manifolds with boundary and equivariant cohomology. Differential Geometry and its Applications, 30 (2). pp. 179-194. ISSN 0926-2245

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## Abstract

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 $X_M$ on $M$, one defines Witten's inhomogeneous coboundary operator $\d_{X_M} = \d+\iota_{X_M}: \Omega_G^\pm \to\Omega_G^\mp$ (even/odd invariant forms on $M$) and its adjoint $\delta_{X_M}$. Witten \cite{Witten} showed that the resulting cohomology classes have $X_M$-harmonic representatives (forms in the null space of $\Delta_{X_M} = (\d_{X_M}+\delta_{X_M})^2$), and the cohomology groups are isomorphic to the ordinary de Rham cohomology groups of the set $N(X_M)$ of zeros of $X_M$. 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 $X_M$-cohomology and show the classes have representative $X_M$-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 $\d_{X_M}$ and $\delta_{X_M}$. We also elucidate the connection between the $X_M$-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 $N(X_M)$ has a unique $X_M$-harmonic field on $M$, with corresponding boundary conditions. Finally, we define the $X_M$-Poincar\'{e} duality angles between the interior subspaces of $X_M$-harmonic fields on $M$ with appropriate boundary conditions, following recent work of DeTurck and Gluck.

Item Type: Article Hodge theory, manifolds with boundary, equivariant cohomology, Killing vector fields MSC 2000 > 35 Partial differential equationsMSC 2000 > 53 Differential geometryMSC 2000 > 55 Algebraic topologyMSC 2000 > 57 Manifolds and cell complexes 2010.31 Dr James Montaldi 29 April 2011