(also nonabelian homological algebra)
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Basic definitions
Stable homotopy theory notions
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(braid lemma)
Given a commuting diagram of abelian groups of the following form
Consider the following four sequences inside the diagram
;
;
;
.
Then: if the first three of these are long exact sequences and the fourth is a chain complex, then the fourth is also long exact.
Given a generalized homology theory , then by definition for every inclusion of topological spaces, there is an long exact sequence
For an unreduced generalized homology theory and for two consecutive inclusions (a “triple” ) there is a long exact sequence of the form
Dually for generalized (Eilenberg-Steenrod) cohomology.
Consider the following braid diagram
(graphics from this Maths.SE comment)
The blue, purple and the black sequence are the exact sequences of the pairs , and , respectively. The orange sequence is a chain complex by these exact sequences and using the commutativity of the diagram, and because the diagonals factor through . Hence the braid lemma, prop. , implies the claim.
Created on April 12, 2016 at 08:23:02. See the history of this page for a list of all contributions to it.