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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Global analysis on PL-manifolds

Author: Nicolae Teleman
Journal: Trans. Amer. Math. Soc. 256 (1979), 49-88
MSC: Primary 58G10; Secondary 57R10
MathSciNet review: 546907
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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.

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Keywords: Riemannian structure on PL-manifold, constant volume density, distance function, distributions, Sobolev spaces, Hodge theory, signature operator
Article copyright: © Copyright 1979 American Mathematical Society