A Künneth formula in topological homology and its applications to the simplicial cohomology of $\ell^1({\Bbb Z}_+^k)$

F. Gourdeau¹ ; Z. A. Lykova² ; M. C. White²

¹Département de Mathématiques Université Laval Cité Universitaire (Québec) G1K 7P4, Canada
²School of Mathematics and Statistics University of Newcastle upon Tyne Newcastle upon Tyne, NE1 7RU, UK

Studia Mathematica, Tome 166 (2005) no. 1, pp. 29-54

Résumé

We establish a Künneth formula for some chain complexes in the categories of Fréchet and Banach spaces. We consider a complex ${\cal X}$ of Banach spaces and continuous boundary maps $d_n$ with closed ranges and prove that $H^n({\cal X}') \cong H_n({\cal X})'$, where $H_n({\cal X})'$ is the dual space of the homology group of ${\cal X}$ and $H^n({\cal X}')$ is the cohomology group of the dual complex ${\cal X}'$. A Künneth formula for chain complexes of nuclear Fréchet spaces and continuous boundary maps with closed ranges is also obtained. This enables us to describe explicitly the simplicial cohomology groups ${\cal H}^n(\ell^1({\mathbb Z}_+^k), \ell^1({\mathbb Z}_+^k)')$ and homology groups ${\cal H}_n(\ell^1({\mathbb Z}_+^k), \ell^1({\mathbb Z}_+^k))$ of the semigroup algebra $\ell^1({\mathbb Z}_+^k)$.

Détail

DOI : 10.4064/sm166-1-3

Keywords: establish nneth formula chain complexes categories chet banach spaces consider complex nbsp cal banach spaces continuous boundary maps closed ranges prove cal cong cal where cal dual space homology group cal cal cohomology group dual complex cal nneth formula chain complexes nuclear chet spaces continuous boundary maps closed ranges obtained enables describe explicitly simplicial cohomology groups cal ell mathbb ell mathbb homology groups cal ell mathbb ell mathbb semigroup algebra ell mathbb