On the universal coefficients formula for shape homology
Andrei V. Prasolov
TL;DR
This paper examines whether different shape homology theories satisfy the Universal Coefficients Formula, finding that some do under specific conditions and introducing new UCF-balanced theories with particular properties.
Contribution
It proves UCF for pro-homology and strong homology in certain classes and introduces new UCF-balanced shape homology theories with specific satisfaction conditions.
Findings
Pro-homology and strong homology satisfy UCF in the class FAB.
They do not satisfy UCF in the class AB.
New UCF-balanced theories are constructed with specific UCF satisfaction properties.
Abstract
In this paper it is investigated whether various shape homology theories satisfy the Universal Coefficients Formula (UCF). It is proved that pro-homology and strong homology satisfy UCF in the class FAB of finitely generated abelian groups, while they do not satisfy UCF in the class AB of all abelian groups. Two new shape homology theories (called UCF-balanced) are constructed. It is proved that balanced pro-homology satisfies UCF in the class AB, while balanced strong homology satisfies UCF only in the class FAB.
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