A stability-like theorem for cohomology of pure braid groups of the series A, B and D

Details A stability-like theorem for cohomology of pure braid groups of the series A, B and D A stability-like theorem for cohomology of pure braid groups of the series A, B and D

2003-07-10

arxiv.org/abs/math/0307149v2

Mathematics

Group Theory

17 pages; added reference for section 1

Scientific paper

Consider the ring $R:=\Q[\tau,\tau^{-1}]$ of Laurent polynomials in the variable $\tau$. The Artin's Pure Braid Groups (or Generalized Pure Braid Groups) act over $R,$ where the action of every standard generator is the multiplication by $\tau$. In this paper we consider the cohomology of such groups with coefficients in the module $R$ (it is well known that such cohomology is strictly related to the untwisted integral cohomology of the Milnor fibration naturally associated to the reflection arrangement). We give a sort of \textit{stability} theorem for the cohomologies of the infinite series $A$, $B$ and $D,$ finding that these cohomologies stabilize, with respect to the natural inclusion, at some number of copies of the trivial $R$-module $\Q$. We also give a formula which compute this number of copies.

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