Towards a nonabelian cohomology of forms

Mukul Patel

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Towards a nonabelian cohomology of forms

Mukul Patel

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Abstract

We consider a simple and natural coboundary operator, on the Lie algebra valued differential forms on a manifold, which in the abelian case reduces to usual exterior derivative of such forms. Using the corresponding de Rham cohomology Lie superalgebra H*(M,G) we obtain numerical smooth invariants--as opposed to homotopy invariants--for manifolds. The corresponding Hodge theory yields finiteness of nonabelian Betti numbers. A genralized Poincaré lemma, along with a Poincaré duality, a Mayer-Vietoris, and a particularly empowered Bockstein makes our cohomology computable. Bockstein also allows us to relate (nonabelian) diffeomorphism invariants to (abelian) homotopy invariants.

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Mukul Patel. 2005-03-07. Towards a nonabelian cohomology of forms. https://arxiv.org/abs/math/0412481

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