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A Künneth formula in topological homology and its applications to the simplicial cohomology of ℓ¹(ℤ+^k)

Lookup NU author(s): Dr Zinaida LykovaORCiD, Dr Michael White

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Abstract

We establish a Künneth formula for some chain complexes in the categories of Fréchet and Banach spaces. We consider a complex Χ of Banach spaces and continuous boundary maps dn with closed ranges and prove that Hn(Χ′) ≅ Hn(Χ)′, where Hn(Χ)′ is the dual space of the homology group of Χ and Hn(Χ′) is the cohomology group of the dual complex Χ′. 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 script H signn(ℓ1(ℤ+k), ℓ1 (ℤ+k)′) and homology groups script H signn(ℓ1(ℤ +k), ℓ1(ℤ+k)) of the semigroup algebra ℓ1(ℤ+k).

Publication metadata

Author(s): Gourdeau F, Lykova ZA, White MC

Publication type: Article

Publication status: Published

Journal: Studia Mathematica

Year: 2005

Volume: 166

Issue: 1

Pages: 29-54

Print publication date: 01/01/2005

ISSN (print): 0039-3223

ISSN (electronic): 1730-6337

Publisher: Instytut Matematyczny Polska Akademia Nauk,

URL: http://dx.doi.org/10.4064/sm166-1-3

DOI: 10.4064/sm166-1-3

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