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 and homology groups of the semigroup algebra .
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm166-1-3,
author = {F. Gourdeau and Z. A. Lykova and M. C. White},
title = {A Kunneth formula in topological homology and its applications to the simplicial cohomology of $l1(Z+^{k})$
},
journal = {Studia Mathematica},
volume = {166},
year = {2005},
pages = {29-54},
zbl = {1065.46030},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm166-1-3}
}
F. Gourdeau; Z. A. Lykova; M. C. White. A Künneth formula in topological homology and its applications to the simplicial cohomology of $ℓ¹(ℤ₊^{k})$
. Studia Mathematica, Tome 166 (2005) pp. 29-54. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm166-1-3/