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

F. 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 -