Algebraic and geometric intersection numbers for free groups (Q1019136)

The geometric intersection number of a pair of homotopy classes of simple closed curves on a surface is the minimal number of intersection points of curves representing the classes. Based on the associated partition of the ends of a group, \textit{P. Scott} and \textit{G. A. Swarup} introduced an algebraic intersection number for pairs of splittings of a group; they showed that, in the case of surfaces (where splittings of the fundamental group correspond to homotopy classes of simple closed curves), the algebraic and geometric intersection numbers coincide [Geom. Topol. 4, 179--218 (2000; Zbl 0983.20024)]. In the present paper, an analogous result is proved for free groups, considered as fundamental groups of connected sums of copies of \(S^2 \times S^1\): the geometric intersection number of a pair of isotopy classes of embedded 2-spheres (the minimal number of components in an intersection) coincides with the algebraic intersection number for the corresponding splittings of the free group. The principal method of the proof is the normal form for embedded 2-spheres developed by \textit{A. E. Hatcher} [Comment. Math. Helv. 70, No.~1, 39--62 (1995; Zbl 0836.57003)].