A homotopic intersection theory on surfaces: applications to mapping class group and braids.

*(English)*Zbl 1161.57009Let \(S_{g,b,n}\) be a connected orientable surface of genus \(g\) with \(b\geq 1\) boundary components from which \(n\geq 0\) interior points are removed. Let \(H\) denote the first homology group of \(S_{g,b,n}\), let \(\langle\, ,\rangle\) denote the intersection number on \(H\), and \(\Gamma\) the fundamental group of \(S_{g,b,n}\) with a base point on the boundary, so that \(\Gamma\) is a free group. Let \({\mathbb Z}[\Gamma]\) denote the integral group ring of \(\Gamma\) and \(e:{\mathbb Z}[\Gamma]\to {\mathbb Z}\) the augmentation map, let \(\pi:\Gamma\to H\) denote the natural map and let \(\overline{( \,\, )}:{\mathbb Z}[\Gamma] \to {\mathbb Z}[\Gamma]\) denote the anti-isomorphism defined by \(g_i\mapsto g_i^{-1}\) on elements of \(\Gamma\) and extended to \({\mathbb Z}[\Gamma]\) linearly.

The main result of this paper states that there exists a map \(\omega :\Gamma\times \Gamma \to {\mathbb Z}[\Gamma]\) (the homotopic intersection form) such that

1) \( e(w(x,y))=\langle \pi(x), \pi (y) \rangle \)

2) \(\omega (y,x)= -\overline {\omega (x,y)} +(y-1)(x^{-1}-1)\),

3) \(\omega (xy,z)=\omega (x,z)+x\omega (y,z)\),

4) \(\omega (x,yz)=\omega (x,y)+\omega (x,z)y^{-1}\).

A fundamental formula of C. D. Papakyriakopoulos [Knots, Groups, 3-Manif.; Pap. dedic. Mem. R. H. Fox, 261–292 (1975; Zbl 0325.55002)] which is used to get a necessary and sufficient condition for a regular covering to be planar is concluded from the main result. As a second application, the author obtains a simple proof of a theorem of S. Morita [Duke Math. J. 70, No. 3, 699–726 (1993; Zbl 0801.57011)] on the symplectic nature of the Magnus representation of the mapping class group of a compact surface with one boundary component, and produces many elements in the kernel of representation of the Torelli group induced by the Magnus representation, which was shown to be not injective by M. Suzuki [Proc. Am. Math. Soc. 130, No. 3, 909–914 (2002; Zbl 0993.57006)]. As a final application, the author gets some restrictions on the image of the Burau and Gassner matrices of a braid.

The author states at the end that after the paper had been written he was informed that the main result of this paper was obtained earlier by V. G. Turaev [Math. USSR, Sb. 35, 229–250 (1979; Zbl 0422.57005)].

The main result of this paper states that there exists a map \(\omega :\Gamma\times \Gamma \to {\mathbb Z}[\Gamma]\) (the homotopic intersection form) such that

1) \( e(w(x,y))=\langle \pi(x), \pi (y) \rangle \)

2) \(\omega (y,x)= -\overline {\omega (x,y)} +(y-1)(x^{-1}-1)\),

3) \(\omega (xy,z)=\omega (x,z)+x\omega (y,z)\),

4) \(\omega (x,yz)=\omega (x,y)+\omega (x,z)y^{-1}\).

A fundamental formula of C. D. Papakyriakopoulos [Knots, Groups, 3-Manif.; Pap. dedic. Mem. R. H. Fox, 261–292 (1975; Zbl 0325.55002)] which is used to get a necessary and sufficient condition for a regular covering to be planar is concluded from the main result. As a second application, the author obtains a simple proof of a theorem of S. Morita [Duke Math. J. 70, No. 3, 699–726 (1993; Zbl 0801.57011)] on the symplectic nature of the Magnus representation of the mapping class group of a compact surface with one boundary component, and produces many elements in the kernel of representation of the Torelli group induced by the Magnus representation, which was shown to be not injective by M. Suzuki [Proc. Am. Math. Soc. 130, No. 3, 909–914 (2002; Zbl 0993.57006)]. As a final application, the author gets some restrictions on the image of the Burau and Gassner matrices of a braid.

The author states at the end that after the paper had been written he was informed that the main result of this paper was obtained earlier by V. G. Turaev [Math. USSR, Sb. 35, 229–250 (1979; Zbl 0422.57005)].

Reviewer: Mustafa Korkmaz (Ankara)

##### MSC:

57M50 | General geometric structures on low-dimensional manifolds |

57M05 | Fundamental group, presentations, free differential calculus |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

20F65 | Geometric group theory |