Generalized cohomology theory | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-02-18

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Cohomology


00:01:43 1 Singular cohomology
00:20:33 2 Examples
00:21:53 3 The diagonal
00:22:19 4 Poincaré duality
00:25:15 5 Characteristic classes
00:25:29 6 Eilenberg–MacLane spaces
00:25:57 7 Cap product
00:26:29 8 History, to the birth of singular cohomology
00:28:54 9 Sheaf cohomology
00:30:26 10 Cohomology of varieties
00:31:22 11 Axioms and generalized cohomology theories
00:33:57 12 Other cohomology theories
00:37:25 13 See also
00:41:34 14 Notes
00:44:57 15 References
00:51:14 Axioms and generalized cohomology theories
01:05:06 Other cohomology theories
01:05:27 See also



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SUMMARY
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In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.
From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do with functions and pullbacks in geometric situations: given spaces X and Y, and some kind of function F on Y, for any mapping f : X → Y, composition with f gives rise to a function F ∘ f on X. The most important cohomology theories have a product, the cup product, which gives them a ring structure.  Because of this feature, cohomology is usually a stronger invariant than homology.