I would like to know some applications of Anick's resolution in noncommutative algebras.
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The first paragraph of David Anick's paper, "On the Homology of Associative Algebras" (http://www.jstor.org/stable/2000383):



In view of Yemon's reference to group cohomology, I would like to mention Graham Ellis' work on "Homological Algebra Programming". The key point is that he constructs free resolutions inductively together with a contracting homotopy: it is the latter that gives the computational aspect. There is an explanation of some of this in Section 9.3 of the book Nonabelian algebraic topology, in terms of constructing a "home for a contracting homotopy", as against the more traditional "killing kernels", a method which is notably non algorithmic. The spirit of this derives from Homological Perturbation Theory, in which also the homotopies are crucial. 

