Minimal surfaces in manifolds with \(S^ 1\) actions and the simple loop conjecture for Seifert fibered spaces. (English) Zbl 0627.57008

Hass, Joel

Proc. Am. Math. Soc. 99, 383-388 (1987).

The simple loop conjecture for 3-manifolds states that if a 2-sided map from a surface to a 3-manifold fails to inject on the fundamental group, then there is an essential simple loop in the kernel. The author solves the conjecture in the case where the 3-manifold is Seifert fibered. The techniques are geometric and involve studying least area surfaces and circle actions on Seifert fibered spaces.

Reviewer: J.Przytycki

MSC:

57M35	Dehn’s lemma, sphere theorem, loop theorem, asphericity (MSC2010)
53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
57N10	Topology of general \(3\)-manifolds (MSC2010)
57S15	Compact Lie groups of differentiable transformations

Keywords:

simple loop conjecture for 3-manifolds; 2-sided map from a surface to a 3-manifold; least area surfaces; circle actions on Seifert fibered spaces