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In some fields of studies, for example, Amenability of Banach algebras and $L^2$-Betti numbers, some chain complexes are studied, why is the study of these creatures important? When and why do these objects and their related structures enter mathematics?

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closed as too broad by Anton Fetisov, LSpice, Neil Hoffman, Stefan Waldmann, Yemon Choi May 13 at 23:11

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ You can blame Poincare for this. algtop.net/wp-content/uploads/2012/02/… . en.wikipedia.org/wiki/Homology_(mathematics) $\endgroup$ – Liviu Nicolaescu May 1 at 9:40
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    $\begingroup$ Homology is an invariant that is usually easy to compute; and that nonetheless has some distinguishing power : in that way it is a good invariant. $\endgroup$ – Max May 1 at 9:42
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    $\begingroup$ The “two broad” closer has a point: see e.g. hsm.stackexchange.com/a/7414 $\endgroup$ – Francois Ziegler May 1 at 10:49
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    $\begingroup$ Abstractly, exact sequences provide powerful tools in computations, because a simple check that something is $0$ will give you an existence result. Therefore measuring the defect of a chain complex from being exact is just as important. But I also agree that this question is too broad for this forum. $\endgroup$ – Ulrich Pennig May 1 at 11:01
  • $\begingroup$ This is a natural question but I don't think one can give a proper answer on MO. As others have said it is really too broad. (Incidentally, most people who have worked on versions of amenability of Banach algebras do not work very much with the Hochschild chain complex -- there is a lot more to Hochschild cohomology of Banach algebras than the "usual stuff" on amenability) $\endgroup$ – Yemon Choi May 13 at 23:06