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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Differential geometry on simplicial spaces
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by Michael A. Penna PDF
Trans. Amer. Math. Soc. 214 (1975), 303-323 Request permission


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 $\tau (M)$.
  • 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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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 214 (1975), 303-323
  • MSC: Primary 58A10; Secondary 53C20, 57C99
  • DOI:
  • MathSciNet review: 0391146