Abstract
In Chapter 6, we studied the first de Rham cohomology H1(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 HP(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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© 2008 Birkhäuser Boston
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(2008). Integration of Forms and de Rham Cohomology. In: Differentiable Manifolds. Birkhäuser Advanced Texts / Basler Lehrbücher. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4767-4_8
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DOI: https://doi.org/10.1007/978-0-8176-4767-4_8
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4766-7
Online ISBN: 978-0-8176-4767-4
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