On the mapping class group of 4-dimensional 1-handlebodies via Budney-Gabai invariants

2025/12/17 by Niu, Weizhe
#FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2512.15099

Abstract

We define an invariant (W₃)_m for π₀Diff(\natural_m S¹× D³,∂) for m≥ 1 that generalizes Budney--Gabai's W₃ invariant. We give a computational framework inspired by Budney--Gabai and use it to calculate the invariant for all unknotted barbell difeomorphisms of \natural_m S¹× D³ for m=1,2. This allows us to detect more linearly independent elements in π₀Diff(S¹× D³,∂), and to prove that π₀Diff( \natural₂ S¹× D³,∂)/ ( π₀ Diff(S¹× D³,∂))² admits infinitely generated subgroups generated by unknotted barbell diffeomorphisms, leading to infinitely many properly embedded separating 3-balls that are non-isotopic relative to the boundary.

Citations

• Splitting spheres for unlinked S²'s in S⁴
• Knotted 3-balls in S⁴
• Lecture notes on CAT(0) cube complexes
• Manifold theoretic compactifications of configuration spaces
• The topology of spaces of knots
• On the automorphism groups of hyperbolic manifolds

Related

• On the mapping class groups of 4-manifolds with 1-handles
• Grasper families of spheres in S² × D² and barbell diffeomorphisms of S¹× S² × I
• Realization of the mapping class group of handlebody by diffeomorphisms
• Knotted handlebodies in the 4-sphere and 5-ball
• Isotopy Uniqueness of Self-diffeomorphism of Handlebodies