A Parametrization of Integral Homology 3-Spheres by the Fourth Johnson subgroup

By results of Morita, Pitsch and, more recently, Faes, it is known that any integral homology 3-sphere can be constructed as a Heegaard splitting with a gluing map an element of the fourth Johnson subgroup. In this work we prove that the equivalence relation on the fourth Johnson subgroup induced by this construction admits an intrinsic description in terms of the fourth Johnson handlebody subgroups. In addition, we give an ``antisymmetic’’ Lagrangian trace map inspired in the Lagrangian trace map introduced by Faes and compute the image of the third Johnson handlebody subgroups by the third Johnson homomorphism.

💡 Research Summary

The paper investigates the relationship between integral homology 3‑spheres and the fourth Johnson subgroup of the mapping class group of a surface with one boundary component. Classical results of Morita, Pitsch and Faes show that every integral homology sphere can be obtained by a Heegaard splitting whose gluing map lies in the fourth Johnson subgroup ( \mathcal{M}{g,1}(4) ). However, the Heegaard map is far from injective: two different mapping classes may give the same 3‑manifold because each can be altered by elements that extend over the two handlebodies of the splitting. The paper formalises this redundancy by introducing the handlebody subgroups ( \mathcal{A}{g,1} ) (maps extending over the inner handlebody), ( \mathcal{B}{g,1} ) (maps extending over the outer handlebody) and their intersection ( \mathcal{AB}{g,1} ) (maps extending over the whole 3‑sphere).

The main theorem (Theorem 1.1) asserts that, after passing to the stable limit ( g\to\infty ), the double coset space \