Towards a nonabelian cohomology of forms

Details Towards a nonabelian cohomology of forms Towards a nonabelian cohomology of forms

2004-12-23

arxiv.org/abs/math/0412481v2

Mathematics

Geometric Topology

8 pages

Scientific paper

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 Poincar\'{e} lemma, along with a Poincar\'{e} 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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