From Sheaf Cohomology to the Algebraic De Rham Theorem
ch2˙elzein˙tu˙11jan˙edited January 14, 2014 6x9 Chapter Two From Sheaf Cohomology to the Algebraic de Rham Theorem by Fouad El Zein and Loring W. Tu INTRODUCTION The concepts of homologyand cohomologytrace their origin to the work of Poincar´ein the late nineteenth century. They attach to a topological space algebraic structures such as groups or rings that are topological invariants of the space. There are actually many different theories, for example, simplicial, singular, and de Rham theories. In 1931, Georges de Rham proved a conjecture of Poincar´eon a relationship between cycles and smooth differential forms, which establishes for a smooth manifold an isomorphism between singular cohomology with real coefficients and de Rham cohomology. More precisely, by integrating smooth forms over singular chains on a smooth man- ifold M, one obtains a linear map Ak(M) → Sk(M, R) from the vector space Ak(M) of smooth k-forms on M to the vector space Sk(M, R) of real singular k-cochains on M. The theorem of de Rham asserts that this linear map induces an isomorphism ∗ ∼ ∗ R HdR(M) → H (M, ) ∗ ∗ arXiv:1302.5834v3 [math.AG] 11 Jan 2014 R between the de Rham cohomology HdR(M) and the singular cohomology H (M, ), under which the wedge product of classes of closed smooth differential forms corre- sponds to the cup product of classes of cocycles. Using complex coefficients, there is similarly an isomorphism ∼ h∗ A•(M, C) → H∗(M, C), where h∗ A•(M, C) denotes the cohomology of the complex A•(M, C) of smooth C-valued forms on M.
[Show full text]