Lefschetz-Pontrjagin duality for differential characters (Q2731759)

An analog of the Lefschetz-Pontryagin Duality Theorem for differential characters for compact manifolds with boundary \((X,\partial X)\) is presented. One asserts the existence of a pairing NEWLINE\[NEWLINE\widehat {\mathbb{H}}^k(X) \times \widehat {\mathbb{H}}^{n-k-1} (X,\partial X)\to S^1NEWLINE\]NEWLINE given by \((\alpha,\beta) \mapsto (\alpha* \beta)[X]\) and inducing injective maps with dense range. Coboundary maps \(\partial:\widehat {\mathbb{H}}^k(X) \to\widehat {\mathbb{H}}^{k+1} (X, \partial X)\) intertwine the duality mappings and reduce the standard picture under the natural transformation to integral cohomology.NEWLINENEWLINENEWLINESee also \textit{F. R. Harvey} and \textit{H. B. Lawson jun.}, `A theory of characteristic currents associated with a singular connection', Astérisque 213 (1993; Zbl 0804.53037) and Am. J. Math. 117, 829-873 (1995; Zbl 0851.58036).