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Homology computations for complex braid groups. (English) Zbl 1302.20049
Complex braid groups are defined as follows: let \(W\) be a complex reflection group, i.e. a finite group generated by finite order endomorphisms of \(\mathrm{GL}_r(\mathbb C)\) that leave invariant some hyperplane in \(\mathbb C^r\). Let \(\mathcal A\) be the (central) hyperplane arrangement associated to the reflections of \(W\), and let \(X=\mathbb C^r\setminus\bigcup\mathcal A\) be the corresponding hyperplane complement. The generalized braid group \(B=\pi_1(X/W)\) is an extension of \(W\) by \(P=\pi_1(X)\). These braid groups are parametrized by three integers \(B(de,e,r)\) with 34 exceptional cases \(B_4,\ldots,B_{37}\). The full classification of these groups is not yet fully understood.
The authors give some partial classifications of these braid groups, some specific calculations of their homology in low degrees and some results about the stability behavior of their homology. Here are some examples of their results:
Theorem 6.4. Let \(B=B(e,e,r)\) and \(r\geq 3\). Then \[ H_2(B;\mathbb Z)\cong\begin{cases}\mathbb Z/e &\text{for }r=3,\\ \mathbb Z/e\times\mathbb Z/2 &\text{for }r=4\text{ and odd }e,\\ \mathbb Z/e\times(\mathbb Z/2)^2 &\text{for }r=4\text{ and even }e,\text{ and}\\ \mathbb Z/e\times\mathbb Z/2 &\text{for }r\geq 5.\end{cases} \] As for stability results, we have the following Theorem. Let \(p\) be an odd prime, the homology group \(H_*(B(2e,e,\infty);\mathbb F_p)\) is isomorphic to: \[ \lim_{r\to\infty}H_*(B(2e,e,r);\mathbb F_p)\cong\mathbb F_p[w_1,\overline y_1,\overline y_2,\ldots]\otimes\Lambda[\overline x_0,\overline x_1,\ldots], \] where \(\dim(w_1)=1\), \(\dim(\overline x_i)=2p^i-1\) and \(\dim(\overline y_i)=2p^i-2\). Moreover, the canonical morphism \[ H_i(B(2e,e,r);\mathbb F_p)\to H_i(B(2e,e,\infty);\mathbb F_p) \] is an isomorphism for \(r>(i-1)\frac{p}{p-1}+2\).

20J05 Homological methods in group theory
20J06 Cohomology of groups
20F36 Braid groups; Artin groups
20F55 Reflection and Coxeter groups (group-theoretic aspects)
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