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

### Tom 166 / 2005

Studia Mathematica 166 (2005), 29-54 MSC: Primary 46H20, 46J40, 22E41; Secondary 16E40, 43A20. DOI: 10.4064/sm166-1-3

#### 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 $\ell^1({\mathbb Z}_+^k)$.

#### Autorzy

• F. GourdeauDépartement de Mathématiques
Université Laval
Cité Universitaire (Québec)
e-mail
• Z. A. LykovaSchool of Mathematics and Statistics
University of Newcastle upon Tyne
Newcastle upon Tyne, NE1 7RU, UK
e-mail
• M. C. WhiteSchool of Mathematics and Statistics
University of Newcastle upon Tyne
Newcastle upon Tyne, NE1 7RU, UK
e-mail

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