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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Linear continuous division for exterior and interior products


Authors: P. Domanski and B. Jakubczyk
Journal: Proc. Amer. Math. Soc. 131 (2003), 3163-3175
MSC (2000): Primary 46E10, 58A10
Published electronically: May 9, 2003
MathSciNet review: 1992857
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Abstract: We consider the complex

\begin{displaymath}\begin{CD}0@>>> {\Lambda\sb 0(M;E)} @>{\partial\sb \omega}>> ... ...>>{\dots} @>{\partial\sb\omega}>> {\Lambda\sb m(M;E)}, \end{CD}\end{displaymath}

where $E$ is a finite-dimensional vector bundle over a suitable differential manifold $M$, $\Lambda\sb q(M;E)$ denotes the space of all smooth or real analytic or holomorphic sections of the $q$-exterior product of $E$ and $\partial\sb\omega(\eta):=\omega\wedge\eta$ for $\omega\in \Lambda\sb 1(M;E)$. 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 $\partial\sb\omega:\Lambda\sb q(M;E)\to \operatorname{Im} \partial\sb\omega\subseteq \Lambda\sb{q+1}(M;E)$.

Analogous results are obtained whenever $M$ is replaced by a suitable closed subset $X$ or $\partial\sb\omega$ are replaced by the interior product operators $\partial\sb Z$, $\partial\sb Z(\eta):=Z\rfloor \eta$ for a given section $Z$ of the dual bundle $E\sp *$.


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Additional Information

P. Domanski
Affiliation: Faculty of Mathematics and Computer Science, A. Mickiewicz University and Institute of Mathematics (Poznań branch), Polish Academy of Sciences, Umultowska 87, 61-614 Poznań, Poland
Email: domanski@math.amu.edu.pl

B. Jakubczyk
Affiliation: Institute of Applied Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warszawa, Poland On leave from: Institute of Mathematics, Polish Academy of Sciences, Warszawa, Poland
Email: B.Jakubczyk@impan.gov.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07107-7
PII: S 0002-9939(03)07107-7
Keywords: Exact complexes, splitting, exterior and interior multiplication, division properties, spaces of smooth functions, spaces of holomorphic functions, spaces of real analytic functions
Received by editor(s): May 7, 2002
Published electronically: May 9, 2003
Additional Notes: The research of the second named author was partially supported by the Committee for Scientific Research, Poland, grant KBN 2P03A 03516
Communicated by: Jozef Dodziuk
Article copyright: © Copyright 2003 American Mathematical Society