Abstract
In Chapter 6, we studied the first de Rham cohomology H 1 (M) of a manifold This measures the difference between exactness and local exactness of 1-forms on M and was shown to have interesting topological applications. Here we generalize these ideas, using the full Grassmann algebra A* (M) to produce a graded algebra H* (M), the de Rham cohomology algebra. The proper generalization of “locally exact 1-form” is “closed p-form”, defined as a p-form that is annihilated by “exterior differentiation”. Exact forms are closed and H P(M) measures the extent to which closed p-forms may fail to be exact. By Stokes’ theorem, the geometric boundary operator and exterior differentiation of forms are mutually adjoint operations in a certain precise sense. This is a generalization of the fundamental theorem of calculus and a powerful tool for computing cohomology. The reader who would like to pursue this theory further could hardly do better than to consult [5].
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© 1993 Springer Science+Business Media New York
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Conlon, L. (1993). Integration of Forms and de Rham Cohomology. In: Differentiable Manifolds. Birkhäuser Advanced Texts. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4757-2284-0_8
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DOI: https://doi.org/10.1007/978-1-4757-2284-0_8
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4757-2286-4
Online ISBN: 978-1-4757-2284-0
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