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On a philosophical level, homological algebra teaches us that certain chain complexes associated with algebraic or geometric objects (topological spaces, simplicial complexes, R-modules) contain a lot of valuable algebraic information about them, with the homology being only the most readily available part. On a technical level, homological ...
A "homology-like" theory satisfying all of the Eilenberg–Steenrod axioms except the dimension axiom is called an extraordinary homology theory (dually, extraordinary cohomology theory). Important examples of these were found in the 1950s, such as topological K-theory and cobordism theory , which are extraordinary co homology theories, and ...
In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages.The most direct usage of the term is to take the homology of a chain complex, resulting in a sequence of abelian groups called homology groups.
MO is a rather weak cobordism theory, as the spectrum MO is isomorphic to H(π * (MO)) ("homology with coefficients in π * (MO)") – MO is a product of Eilenberg–MacLane spectra. In other words, the corresponding homology and cohomology theories are no more powerful than homology and cohomology with coefficients in Z/2Z. This was the first ...
Homology Theory — An Introduction to Algebraic Topology, James W. Vick (1994, 2nd ed., ISBN 978-0-387-94126-4) Computability — A Mathematical Sketchbook, Douglas S. Bridges (1994, ISBN 978-0-387-94174-5) Algebraic K-Theory and Its Applications, Jonathan Rosenberg (1994, ISBN 978-0-387-94248-3)
Differential Topology. New York: Springer. ISBN 978-0-387-90148-0. (see p. 72–73 for Hirsch's proof utilizing non-existence of a differentiable retraction) Hilton, Peter J.; Wylie, Shaun (1960). Homology theory: An introduction to algebraic topology. New York: Cambridge University Press. ISBN 0521094224. MR 0115161.
Peter J. Hilton, An introduction to homotopy theory, Cambridge Tracts in Mathematics and Mathematical Physics, no. 43, Cambridge University Press, 1953. [36] ISBN 0-521-05265-3 MR 0056289; Peter J. Hilton, Shaun Wylie, Homology theory: An introduction to algebraic topology, Cambridge University Press, New York, 1960. [37] ISBN 0-521-09422-4 MR ...
Homological algebra is a collection of algebraic techniques that originated in the study of algebraic topology but has also found applications to group theory and algebraic geometry The main article for this category is Homological algebra .
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