A Künneth formula in topological homology and its applications to the simplicial cohomology of
Tom 166 / 2005
Streszczenie
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).
