On the involutive Heegaard Floer homology of negative semidefinite plumbed 3-manifolds with b1 = 1

Johnson, Peter K

doi:10.2140/agt.2025.25.827

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the Journal

Editorial Board

Subscriptions

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1472-2739 (online)

ISSN 1472-2747 (print)

Author Index

To Appear

Other MSP Journals

On the involutive Heegaard Floer homology of negative semidefinite plumbed $3$-manifolds with $b_{1}=1$

Peter K Johnson

Algebraic & Geometric Topology 25 (2025) 827–885

DOI: 10.2140/agt.2025.25.827

Abstract

Ozsváth and Szabó (2003) used Heegaard Floer homology to define numerical invariants $d_{1 ∕ 2}$ and $d_{- 1 ∕ 2}$ for $3$ -manifolds $Y$ with $H_{1} (Y; ℤ) ≅ ℤ$ . We define involutive Heegaard Floer theoretic versions of these invariants analogous to the involutive $d$ invariants $\bar{d}$ and $\underline{d}$ defined for rational homology spheres by Hendricks and Manolescu (2017). We prove their invariance under spin integer homology cobordism and use them to establish spin filling constraints and $0$ -surgery obstructions analogous to results by Ozsváth and Szabó for their Heegaard Floer counterparts $d_{1 ∕ 2}$ and $d_{- 1 ∕ 2}$ . We then apply calculation techniques of Dai and Manolescu (2019) and Rustamov (2004) to compute the involutive Heegaard Floer homology of some negative semidefinite plumbed $3$ -manifolds with $b_{1} = 1$ . By combining these calculations with the $0$ -surgery obstructions, we are able to produce an infinite family of small Seifert fibered spaces with weight 1 fundamental group and first homology $ℤ$ which cannot be obtained by $0$ -surgery on a knot in $S^{3}$ , extending a result of Hedden, Kim, Mark and Park (2019).

Keywords

involutive Heegaard Floer homology, plumbed manifolds, negative semidefinite, 0-surgery, Seifert fibered space

Mathematical Subject Classification

Primary: 57K18, 57K31, 57K41

References

Publication

Received: 16 July 2022

Revised: 18 September 2023

Accepted: 5 December 2023

Published: 16 May 2025

Authors

Peter K Johnson
Department of Mathematics University of Virginia Charlottesville, VA United States

Open Access made possible by participating institutions via Subscribe to Open.

Volume 25, issue 2 (2025)