This paper studies the homology and cohomology of the Temperley-Lieb algebra TL n (a), interpreted as appropriate Tor and Ext groups. Our main result applies under the common assumption that a = v + v −1 for some unit v in the ground ring, and states that the homology and cohomology vanish up to and including degree (n − 2). To achieve this we simultaneously prove homological stability and compute the stable homology. We show that our vanishing range is sharp when n is even.Our methods are inspired by the tools and techniques of homological stability for families of groups. We construct and exploit a chain complex of 'planar injective words' that is analogous to the complex of injective words used to prove stability for the symmetric groups. However, in this algebraic setting we encounter a novel difficulty: TL n (a) is not flat over TL m (a) for m < n, so that Shapiro's lemma is unavailable. We resolve this difficulty by constructing what we call 'inductive resolutions' of the relevant modules.We believe that these results, together with the second author's work on Iwahori-Hecke algebras, are the first time the techniques of homological stability have been applied to algebras that are not group algebras.

This paper investigates the homology of the Brauer algebras, interpreted as appropriate $${{\,\mathrm{Tor}\,}}$$ Tor -groups, and shows that it is closely related to the homology of the symmetric group. Our main results show that when the defining parameter $$\delta $$ δ of the Brauer algebra is invertible, then the homology of the Brauer algebra is isomorphic to the homology of the symmetric group, and that when $$\delta $$ δ is not invertible, this isomorphism still holds in a range of degrees that increases with n.

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We prove that certain sequences of Artin monoids containing the braid monoid as a submonoid satisfy homological stability. When the K(π, 1) conjecture holds for the associated family of Artin groups this establishes homological stability for these groups. In particular, this recovers and extends Arnol'd's proof of stability for the Artin groups of type A, B and D.

We give formulas for the second and third integral homology of an arbitrary finitely generated Coxeter group, solely in terms of the corresponding Coxeter diagram. The first of these calculations refines a theorem of Howlett, while the second is entirely new and is the first explicit formula for the third homology of an arbitrary Coxeter group.