The cohomology of the braid group \(B_3\) and of \(SL_2(\mathbb Z)\) with coefficients in a geometric representation. (Q2851204)

Let \(B_3\) be the full braid group on 3 strings of the disk. There is a well known action of \(B_3\) into \(\mathbb Z+\mathbb Z\) which corresponds to the natural symplectic representation \(B_3\to SL_2(\mathbb Z)\). It is known that this representation is surjective and the kernel is \(\mathbb Z\). The action above extends to the symmetric power \(M=\mathbb Z[x,y]\). It is known the free part of \(H^*(B_3,M)\), \(H^*(SL_2(\mathbb Z),M)\), and \(H^*(B_3,\mathbb Z_p)\), \(H^*(SL_2(\mathbb Z),\mathbb Z_p)\), for a prime \(p>3\). The purpose of this work is to extend the calculation above. In few words the authors give a complete description of \(H^*(B_3,M)\), \(H^*(SL_2(\mathbb Z),M)\), which is contained in Theorems 3.7, 3.8, 3.10 and 3.11. Notoriously, for the primes \(p=2,3\) the \(2^i\)-torsion as well the \(3^i\)-torsion are determined. The main two tools used are a spectral sequence which arises from the short exact sequence of groups \(1\to\mathbb Z\to B_3\to SL_2(\mathbb Z)\to 1\) and the cohomology of the group \(\mathbb Z_4*_{\mathbb Z_2}\mathbb Z_6\). Towards the end they observe that the groups obtained have a strong relation with the groups which are the cohomology of certain loops spaces, which in turn are closely connected with the size of the \(p\)-torsion of the homotopy groups of the spheres.NEWLINENEWLINE The paper is quite well organized where many details are provided as well as many of the basic material needed in the proofs and statements. Finally the paper suggests the possibility to study similar questions for other Artin braid groups.NEWLINENEWLINE To give a sample of the results, perhaps not the most striking, but easy to state, they show:NEWLINENEWLINE Theorem 3.10. The following equalities hold: NEWLINE\[NEWLINEH^2(B_3;M)_{(p)}=H^1(B_3;M)_{(p)}=\Gamma_p^+[\mathcal P_p,\mathcal Q_p]\text{ for }p\neq 2,3,\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINEFH^2(B_3,M_n)=FH^1(B_3;M_n)=\mathbb Z^{f_n},\tag{2}NEWLINE\]NEWLINE where the ranks \(f_n\) are defined using the Poincaré series of the quotient \(t^4(1+t^4-t^{12}+t^{16})/(1-t^8)(1-t^{12})\), and \(\Gamma_p\) is defined in terms of certain divided polynomial algebra over \(\mathbb Z\).