On the Universal Coefficient Formula and Derived $\varprojlim ^{(i)} $ Functor

TL;DR

This paper investigates the non-commutativity of homology and inverse limit functors, exploring its implications for various homology theories and establishing their relations, properties, and applications in topology.

Contribution

It introduces new homology functors derived from non-free cochain complexes and analyzes their properties and relations to classical theories.

Findings

Homology and inverse limit functors do not commute.

Defined exact homology functors for topological spaces.

Established properties like tautness and continuity for these functors.

Abstract

It is known that homology and inverse limit functors do not commute. In the paper we consider this very problem and find its application for various homology theories. In particular, on the category of general topological spaces, there are defined exact homology functors induced by different non-free cochain complexes. Relation between them and other classical homology theories are given. In addition, for the defined homology functors the tautness and the continuous properties are obtained.