On definite lattices bounded by a homology 3-sphere and Yang-Mills instanton Floer theory
type: Item
instance of: scholarly article
title: On definite lattices bounded by a homology 3-sphere and Yang-Mills instanton Floer theory
author: Christopher Scaduto
published in: Geometry \& Topology
publication date: 2024-08-30T00:00:00Z
review text: This paper is concerned with determining the definite lattices arising from smooth \(4\)-manifolds without \(2\)-torsion in their homology that are bounded by certain homology \(3\)-spheres. Given an integer homology \(3\)-sphere \(Y\), definite lattices arising as intersection forms of smooth \(4\)-manifolds bounded by \(Y\) are determined.\N\NThe methods used are an extension of the instanton homology methods of Frøyshov. The results extend Donaldson's \textit{Theorem A}, which answers the above question for the \(3\)-sphere.\N\NThe main result is given in:\N\N\textbf{Theorem 1.1}. Let \(Y\) be an integer homology \(3\)-sphere \(\mathbb{Z}/2\)-homology cobordant to \(+1\) surgery on a knot with smooth \(4\)-ball genus \(2\). If a smooth, compact, oriented and definite \(4\)-manifold with no \(2\)-torsion in its homology has boundary \(Y\), then its intersection form is equivalent to one of\N\[\N\langle\pm1\rangle^n,\quad E_8\oplus\langle+1\rangle^n,\quad\Gamma_{12}\oplus\langle+1\rangle^n,\N\]\Nwhere \(\Gamma_{12}\) is the unique indecomposable unimodular positive-definite lattice of rank \(12\).\N\NThe proof can be found in Section 4. Section 2 contains the instanton inequalities that serve as the main tools in this paper, for example:\N\N\textbf{Theorem 2.5}. Let \(X\) be a smooth, compact, oriented \(4\)-manifold with homology \(3\)-sphere boundary \(Y\) and \(b_2^+(X)=n\geq1\). For \(1\leq i\leq n\), let \(\Sigma_i\subset X\) be smooth, orientable, connected surfaces in \(X\) of genus \(g_i\) with \(\Sigma_i\cdot\Sigma_i=1\), which are pairwise disjoint. Denote by \(\mathcal{L}\subset H^2(X;\mathbb{Z})/\text{Tor}\) the unimodular lattice of vectors vanishing on the classes \([\Sigma_i]\). Then\N\[\Nh(Y)+\sum_{i=1}^n \lceil\frac{1}{2}g_i\rceil\geq e_0(\mathcal{L}).\N\]\NTheir proofs can be found in Section 7. Section 3 contains the proof of the following:\N\N\textbf{Theorem 1.3}. Let \(Y\) be an integer homology \(3\)-sphere \(\mathbb{Z}/2\)-homology cobordant …
reviewed by: Greg Weiler
Mathematics Subject Classification ID: 57R57
Mathematics Subject Classification ID: 57M99
zbMATH DE Number: 7904157
zbMATH Keywords: instanton Floer theory
zbMATH Keywords: lattice
zbMATH Keywords: homology 3-sphere
zbMATH Keywords: Frøyshov
zbMATH Keywords: instanton homology
zbMATH Keywords: instanton inequality
MaRDI profile type: Publication
cites work: Torsion Classes and a Universal Constraint on Donaldson Invariants for Odd Manifolds
cites work: The Yang-Mills equations over Riemann surfaces
cites work: Heegaard Floer correction terms, with a twist
cites work: Q4850925
cites work: Applications of Donaldson's theorems to classical knot concordance, homology 3-spheres and property P
cites work: On the enumeration of lattices of determinant one
cites work: Q4219187
cites work: Connections, cohomology and the intersection forms of 4-manifolds
cites work: The orientation of Yang-Mills moduli spaces and 4-manifold topology
cites work: Floer homology groups in Yang-Mills theory
cites work: Q3998602
cites work: Q4299405
cites work: A characterization of the \(\mathbb Z^ n\) lattice
cites work: Lattices and codes with long shadows
cites work: Definite 4-manifolds
cites work: Instanton Homology of Seifert Fibred Homology Three Spheres
cites work: The canonical class of a symplectic 4-manifold
cites work: An instanton-invariant for 3-manifolds
cites work: Q4850919
cites work: Equivariant aspects of Yang-Mills Floer theory
cites work: An inequality for the \(h\)-invariant in instanton Floer theory.
cites work: Monopole Floer homology for rational homology 3-spheres
cites work: Floer homology of connected sum of homology 3-spheres
cites work: Homology cobordism group of homology 3-spheres
cites work: On definite lattices bounded by integer surgeries along knots with slice genus at most 2
cites work: Embedded surfaces and the structure of Donaldson's polynomial invariants
cites work: Ring structure of the Floer cohomology of \(\Sigma\times S^1\)
cites work: Cosmetic surgeries on knots in \(S^3\)
cites work: Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary
cites work: Lens space surgeries and a conjecture of Goda and Teragaito
cites work: The \(E_{8}\)-boundings of homology spheres and negative sphere classes in \(E(1)\)
cites work: A perfect Morse function on the moduli space of flat connections
DOI: 10.2140/GT.2024.28.1587
description: scientific article; zbMATH DE number 7904157