Global analysis on PL-manifolds

Abstract:

The paper deals mainly with combinatorial structures; in some cases we need refinements of combinatorial structures. Riemannian metrics are defined on any combinatorial manifold M. The existence of distance functions and of Riemannian metrics with “constant volume density” implies smoothing. A geometric realization of ${\text {PL}}\left ( m \right ){\text {/O}}\left ( m \right )$ is given in terms of Riemannian metrics. A graded differential complex ${\Omega ^ {\ast } }( M )$ is constructed: it appears as a subcomplex of Sullivan’s complex of piecewise differentiable forms. In the complex ${\Omega ^{\ast }}( M )$ the operators $d$, $\ast$, $\delta$, $\Delta$ are defined. A Rellich chain of Sobolev spaces is presented. We obtain a Hodge-type decomposition theorem, and the Hodge homomorphism is defined and studied. We study also the combinatorial analogue of the signature operator.