# zbMATH — the first resource for mathematics

Intersections of loops in two-dimensional manifolds. II: Free loops. (English. Russian original) Zbl 0543.57007
Math. USSR, Sb. 49, 357-366 (1984); translation from Mat. Sb., Nov. Ser. 121(163), No. 3, 359-369 (1983).
This paper is a sequel to Part I by the first author [ibid. 106(148), 566-588 (1978; Zbl 0384.57004)]. The following problems are approached (A being a connected 2-manifold): Which is the least number of intersection and self-intersection points (multiplicities taken into account) of loops belonging to given homotopy classes of maps $$S^ 1\to A?$$ Special cases of these are: Under which conditions are homotopy classes of maps $$S^ 1\to A$$ represented by nonintersecting loops? Under which conditions are such classes represented by simple loops?
The problems are formulated in three variants: (1) free homotopy classes; (2) classes $$\alpha \in \pi_ 1(A,a)$$ with $$a\in Int A$$; (3) classes $$\alpha \in \pi_ 1(A,a)$$ with $$a\in \partial A$$. Answers for (3) were given in Part I in terms of a bilinear map $${\mathbb{Z}}[\pi_ 1(A,a)]\times {\mathbb{Z}}[\pi_ 1(A,a)]\to {\mathbb{Z}}[\pi_ 1(A,a)];$$ an analogous pairing is used in variant (2) to derive necessary conditions for the special cases. Their sufficiency is proved here as a consequence of more general results for variant (1). The same pairing is used to provide, first, lower bounds for the numbers of intersections and self-intersections of loops and, then, upper bounds for the minimal values of these numbers; in a large number of cases, the exact minimal values are derived.
Reviewer: J.Weinstein

##### MSC:
 57N05 Topology of the Euclidean $$2$$-space, $$2$$-manifolds (MSC2010) 57M05 Fundamental group, presentations, free differential calculus
Full Text: