2004/12/23 by Mukul Patel, Patel, Mukul
Mathematics · #55Nxx #57Rxx #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology #Mathematics and Applications #math.AT #math.GT #msc:55Nxx #msc:57Rxx
paper · pdf · doi:10.48550/arxiv.math/0412481
8 pages
openalex publication_date 2004/12/23 · arxiv created 2005/03/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.