A Künneth formula in topological homology and its applications to the simplicial cohomology of $\ell^1({\Bbb Z}_+^k)$
Volume 166 / 2005
Abstract
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 $\ell^1({\mathbb Z}_+^k)$.
