1. CJM Online first
|Variations of Integrals in Diffeology|
We establish the formula for the variation of integrals of differential forms on cubic chains, in the context of diffeological spaces. Then, we establish the diffeological version of Stoke's theorem, and we apply that to get the diffeological variant of the Cartan-Lie formula. Still in the context of Cartan-De-Rham calculus in diffeology, we construct a Chain-Homotopy Operator $\mathbf K$ we apply it here to get the homotopic invariance of De Rham cohomology for diffeological spaces. This is the Chain-Homotopy Operator which used in symplectic diffeology to construct the Moment Map.
Keywords:diffeology, differential geometry, Cartan-De-Rham calculus
Categories:58A10, 58A12, 58A40
2. CJM 2012 (vol 65 pp. 544)
|Iterated Integrals and Higher Order Invariants|
We show that higher order invariants of smooth functions can be written as linear combinations of full invariants times iterated integrals. The non-uniqueness of such a presentation is captured in the kernel of the ensuing map from the tensor product. This kernel is computed explicitly. As a consequence, it turns out that higher order invariants are a free module of the algebra of full invariants.
Keywords:higher order forms, iterated integrals
Categories:14F35, 11F12, 55D35, 58A10