On Homology Groups of Infinite Complexes and Compact

By Eilenberg, Samuel

Book Id: WPLBN0000674362
Format Type: PDF eBook:
File Size: 4.51 MB
Reproduction Date: 2005

Title:	On Homology Groups of Infinite Complexes and Compact
Author:	Eilenberg, Samuel
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Language:	English
Subject:	Science., Mathematics, Logic
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Citation APA MLA Chicago Eilenberg, B. S. (n.d.). On Homology Groups of Infinite Complexes and Compact. Retrieved from http://gutenberg.us/

Description
Mathematics document containing theorems and formulas.

Excerpt
Excerpt: The results of (111, 18) on universal coefficient groups for finite complexes suggest the consideration of similar problems for the homology theory of infinite complexes. To what extent are the homology groups constructed from a general group G of coefficients determined by the homology (or cohomology) groups with specially chosen universal coefficient groups? Results of this nature have already been found by Cech [dl and Steenrod [a]. The present summary describes results on this problem recently obtained by the authors. They are based on a complete analysis of the homology groups under consideration, utilizing an important concept not previously occurring in topology: the group of group extensions of one given group by another group. This analysis yields a formula (5.1) expressing the homology group of the infinite cycles over G in terms of the finite integral cohomology groups. This formula also applies to the &ch homology groups of a compactum. Modified formulas can also be found for other varieties of homology theories. For further details the reader is referred to a forthcoming article by the authors [a].