Splitting homomorphisms and the Geometrization Conjecture

Kernels of splitting homomorphisms

Lei and Wu have given a description of the second homotopy group of a closed orientable 3-manifold in terms of the kernels of the epimorphisms from the fundamental group of a Heegaard splitting surface onto the fundamental groups of the two handlebody sides. In this note, we give a geometric derivation of this result and collect some observations about the relation between the various groups and the topology of the 3-manifold and the Heegaard splitting.

Quasi-homomorphisms on mapping class groups vanishing on a handlebody group

We construct infinitely many linearly independent quasi-homomorphisms on the mapping class group of a Riemann surface with genus at least two which vanish on a handlebody subgroup. As a corollary, we disprove a conjecture of Reznikov on bounded width in Heegaard splittings. Another corollary is that there are infinitely many linearly independent quasi-invariants on the Heegaard splittings of three-manifolds.

自由群乘积与3-流形群中的方程，I

Equations in Products of Free Groups and 3-Manifold Groups, I

本文构造了所有分裂坐标满射同态，给出了对应的平衡表示（包含所有闭可定向3-流形群），并分析了包括Olshanskii同态在内的一大类同态，证明分裂满射非常罕见且对应的平凡群平衡表示可通过Andrews-Curtis变换化为标准形式。