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Transactions of the American Mathematical Society

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



Differential geometry on simplicial spaces

Author: Michael A. Penna
Journal: Trans. Amer. Math. Soc. 214 (1975), 303-323
MSC: Primary 58A10; Secondary 53C20, 57C99
MathSciNet review: 0391146
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Abstract: A simplicial space M is a separable Hausdorff topological space equipped with an atlas of linearly related charts of varying dimension; for example every polyhedron is a simplicial space in a natural way. Every simplicial space possesses a natural structure complex of sheaves of piecewise smooth differential forms, and the homology of the corresponding de Rham complex of global sections is isomorphic to the real cohomology of M. A cosimplicial bundle is a continuous surjection $\xi :E \to M$ from a topological space E to a simplicial space M which satisfies certain criteria. There is a category of cosimplicial bundles which contains a subcategory of vector bundles. To every simplicial space M a cosimplicial bundle $\tau (M)$ over M is associated; $\tau (M)$ is the cotangent object of M since there is an isomorphism between the module of global piecewise smooth one-forms on M and sections of $\tau (M)$.

References [Enhancements On Off] (What's this?)

  • Harley Flanders, Differential forms with applications to the physical sciences, Academic Press, New York-London, 1963. MR 0162198
  • F. Hirzebruch, Neue topologische Methoden in der algebraischen Geometrie, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 9, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1956 (German). MR 0082174
  • M. Penna, Differential geometry on simplicial manifolds, Dissertation, University of Illinois, Urbana-Champaign, Ill., 1974.

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Keywords: Polyhedron, manifold, differential forms, de Rham theorem, cotangent bundle
Article copyright: © Copyright 1975 American Mathematical Society