PreprintCC BY 4.0arXiv
Barbell twists are natural
Abstract
For any oriented smooth $4$--manifold X diffeomorphic to (S2× D2)natural n (n≥0), the author establishes a natural isomorphism of abelian groups: Mod(X,∂ X)≅ Mod(D4,∂ D4)×∧2H2(X;ℤ), concerning the (smooth) boundary-fixing mapping class group of X. For n=2, the Budney--Gabai barbell twist φ∈Mod(𝒩,∂𝒩) is identified with a generator of the factor subgroup ∧2H2(𝒩;ℤ)≅ℤ. Up to boundary-fixing diffeotopy, the barbell spines of 𝒩 are completely classified by the bases of H2(𝒩;ℤ)≅ℤ2, forming a homogeneous set modeled on the group GL(H2(𝒩;ℤ))≅GL(2,ℤ). Any barbell spine of 𝒩 gives rise to an implanted barbell twist equal to φ or φ−1 in Mod(𝒩,∂ 𝒩), according to the sign of the homological basis orientation.
Related
BibTeX
@misc{liu2026barbell,
title = {{Barbell twists are natural}},
author = {Yi Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.02801},
eprint = {2608.02801},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.02801},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Barbell twists are natural
AU - Yi Liu
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2608.02801
UR - https://arxiv.org/abs/2608.02801
AB - For any oriented smooth $4$--manifold X diffeomorphic to (S^2× D^2)^natural n (n≥0), the author establishes a natural isomorphism of abelian groups: Mod(X,∂ X)≅ Mod(D^4,∂ D^4)×∧^2H_2(X;ℤ), concerning the (smooth) boundary-fixing mapping class group of X. For n=2, the Budney--Gabai barbell twist φ∈Mod(𝒩,∂𝒩) is identified with a generator of the factor subgroup ∧^2H_2(𝒩;ℤ)≅ℤ. Up to boundary-fixing diffeotopy, the barbell spines of 𝒩 are completely classified by the bases of H_2(𝒩;ℤ)≅ℤ^2, forming a homogeneous set modeled on the group GL(H_2(𝒩;ℤ))≅GL(2,ℤ). Any barbell spine of 𝒩 gives rise to an implanted barbell twist equal to φ or φ^−1 in Mod(𝒩,∂ 𝒩), according to the sign of the homological basis orientation.
ER -