# A Künneth formula in topological homology and its applications to the simplicial cohomology of $\ell \xb9\left(\mathbb{Z}{\u208a}^{k}\right)$

F. Gourdeau; Z. A. Lykova; M. C. White

Studia Mathematica (2005)

- Volume: 166, Issue: 1, page 29-54
- ISSN: 0039-3223

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topF. Gourdeau, Z. A. Lykova, and M. C. White. "A Künneth formula in topological homology and its applications to the simplicial cohomology of $ℓ¹(ℤ₊^{k})$." Studia Mathematica 166.1 (2005): 29-54. <http://eudml.org/doc/286692>.

@article{F2005,

abstract = {We establish a Künneth formula for some chain complexes in the categories of Fréchet and Banach spaces. We consider a complex of Banach spaces and continuous boundary maps dₙ with closed ranges and prove that Hⁿ(’) ≅ Hₙ()’, where Hₙ()’ is the dual space of the homology group of and Hⁿ(’) is the cohomology group of the dual complex ’. 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 $ℋⁿ(ℓ¹(ℤ₊^\{k\}),ℓ¹(ℤ₊^\{k\})^\{\prime \})$ and homology groups $ℋₙ(ℓ¹(ℤ₊^\{k\}),ℓ¹(ℤ₊^\{k\}))$ of the semigroup algebra $ℓ¹(ℤ₊^\{k\})$.},

author = {F. Gourdeau, Z. A. Lykova, M. C. White},

journal = {Studia Mathematica},

keywords = {exact sequence; Künneth formula},

language = {eng},

number = {1},

pages = {29-54},

title = {A Künneth formula in topological homology and its applications to the simplicial cohomology of $ℓ¹(ℤ₊^\{k\})$},

url = {http://eudml.org/doc/286692},

volume = {166},

year = {2005},

}

TY - JOUR

AU - F. Gourdeau

AU - Z. A. Lykova

AU - M. C. White

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

JO - Studia Mathematica

PY - 2005

VL - 166

IS - 1

SP - 29

EP - 54

AB - We establish a Künneth formula for some chain complexes in the categories of Fréchet and Banach spaces. We consider a complex of Banach spaces and continuous boundary maps dₙ with closed ranges and prove that Hⁿ(’) ≅ Hₙ()’, where Hₙ()’ is the dual space of the homology group of and Hⁿ(’) is the cohomology group of the dual complex ’. 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 $ℋⁿ(ℓ¹(ℤ₊^{k}),ℓ¹(ℤ₊^{k})^{\prime })$ and homology groups $ℋₙ(ℓ¹(ℤ₊^{k}),ℓ¹(ℤ₊^{k}))$ of the semigroup algebra $ℓ¹(ℤ₊^{k})$.

LA - eng

KW - exact sequence; Künneth formula

UR - http://eudml.org/doc/286692

ER -

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