A version of the classical Lefschetz fixed point formula is proved for the cohomology of the cone of a cochain mapping of elliptic complexes. As a particular case we show a Lefschetz formula for the relative de Rham cohomology.

On a compact closed manifold with edges live pseudodifferential operators which are block matrices of operators with additional edge conditions like boundary conditions in boundary value problems. They include Green, trace and potential operators along the edges, act in a kind of Sobolev spaces and form an algebra with a wealthy symbolic structure. We consider complexes of Fréchet spaces whose differentials are given by operators in this algebra. Since the algebra in question is a microlocalization of the Lie algebra of typical vector fields on a manifold with edges, such complexes are of great geometric interest. In particular, the de Rham and Dolbeault complexes on manifolds with edges fit into this framework. To each complex there correspond two sequences of symbols, one of the two controls the interior ellipticity while the other sequence controls the ellipticity at the edges. The elliptic complexes prove to be Fredholm, i.e., have a finite-dimensional cohomology. Using specific tools in the algebra of pseudodifferential operators we develop a Hodge theory for elliptic complexes and outline a few applications thereof.

Made available in DSpace on 2016-06-02T20:27:39Z (GMT). No. of bitstreams: 1 3703.pdf: 612745 bytes, checksum: 57763528b5a111b975b1122e35bbc887 (MD5) Previous issue date: 2011-06-16 / Universidade Federal de Minas Gerais / In this work, we extend some global L1 estimates proved by Bourgain-Brezis in the case of the de Rham complex on RN to the setup of local L1 estimates for elliptic complexes, namely, those associated to involutive elliptic structures spanned by a family of linearly independent smooth complex vector fields. In particular, we obtain a local version of Gagliardo-Nirenberg estimates for elliptic systems of vector fields. / Neste trabalho, estendemos algumas estimativas L1 provadas por Bourgain-Brezis no caso do complexo de de Rham em RN para o contexto local de estimativas L1 para complexos elíticos, a saber, aqueles associados a uma estrutura involutiva elítica gerada por uma família de campos vetoriais suaves e linearmente independentes. Em particular, obtemos uma versão local da desigualdade de Gagliardo-Nirenberg para um sistema de campos vetoriais elíticos.