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On definite lattices bounded by a homology 3-sphere and Yang-Mills instanton Floer theory

Christopher Scaduto

Abstract

Using instanton Floer theory, extending methods due to Froyshov, we determine the definite lattices that arise from smooth 4-manifolds bounded by certain homology 3-spheres. For example, we show that for +1 surgery on the (2,5) torus knot, the only non-diagonal lattices that can occur are E8 and the indecomposable unimodular definite lattice of rank 12, up to diagonal summands. We require that our 4-manifolds have no 2-torsion in their homology.

On definite lattices bounded by a homology 3-sphere and Yang-Mills instanton Floer theory

Abstract

Using instanton Floer theory, extending methods due to Froyshov, we determine the definite lattices that arise from smooth 4-manifolds bounded by certain homology 3-spheres. For example, we show that for +1 surgery on the (2,5) torus knot, the only non-diagonal lattices that can occur are E8 and the indecomposable unimodular definite lattice of rank 12, up to diagonal summands. We require that our 4-manifolds have no 2-torsion in their homology.

Paper Structure

This paper contains 12 sections, 27 theorems, 77 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The inequalities
  3. Genus 1 applications
  4. Genus 2 applications
  5. More examples
  6. Relations for a circle times a surface
  7. Adapting Frø yshov's argument
  8. Proofs of Theorem \ref{['thm:char4']} and Proposition \ref{['prop:mod8']}
  9. $\mu$-classes of loops
  10. Other adaptations
  11. Alternative proofs
  12. The lattice $E_7^2$

Key Result

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 where $\Gamma_{12}$ is the unique indecomposable unimodular positive definite lattice of rank 12.

Figures (3)

  • Figure 1:
  • Figure 2: The (2,3) torus knot, depicted on the left, is transformed into the (2,5) torus knot, on the right, by changing the encircled negative crossing to a positive one.
  • Figure 3: The (3,4) torus knot, also known as $8_{19}$ in Rolfsen notation, is transformed into the (2,5) torus knot by changing the encircled positive crossing to a negative crossing.

Theorems & Definitions (48)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4: cf. froyshov-thesis
  • Theorem 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Theorem 2.5: cf. froyshov-inequality Thm.2
  • ...and 38 more