Kernels of splitting homomorphisms

TL;DR

This paper provides a geometric derivation of a description of the second homotopy group of closed orientable 3-manifolds, relating it to kernels of splitting homomorphisms and their topological implications.

Contribution

It offers a geometric proof of Lei and Wu's result and explores the relationship between homotopy groups, splitting kernels, and 3-manifold topology.

Findings

Geometric derivation of second homotopy group description

Insights into the relation between kernels and 3-manifold topology

Observations on the interaction between various groups and topology

Abstract

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.