Linear continuous division for exterior and interior products

Abstract:

We consider the complex \[ \begin {CD} 0@>>> {\Lambda _ 0(M;E)} @>{\partial _ \omega }>> {\Lambda _ 1(M;E)} @>{\partial _\omega }>>{\dots } @>{\partial _\omega }>> {\Lambda _ m(M;E)}, \end {CD} \] where $E$ is a finite-dimensional vector bundle over a suitable differential manifold $M$, $\Lambda _ q(M;E)$ denotes the space of all smooth or real analytic or holomorphic sections of the $q$-exterior product of $E$ and $\partial _\omega (\eta ):=\omega \wedge \eta$ for $\omega \in \Lambda _ 1(M;E)$. We give sufficient and necessary conditions for the above complex to be exact and, in smooth and holomorphic cases, we give sufficient conditions for its splitting, i.e., for existence of linear continuous right inverse operators for $\partial _\omega :\Lambda _ q(M;E)\to \operatorname {Im} \partial _\omega \subseteq \Lambda _{q+1}(M;E)$. Analogous results are obtained whenever $M$ is replaced by a suitable closed subset $X$ or $\partial _\omega$ are replaced by the interior product operators $\partial _ Z$, $\partial _ Z(\eta ):=Z\rfloor \eta$ for a given section $Z$ of the dual bundle $E^ *$.