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Mod 2 instanton homology and 4-manifolds with boundary

Kim A. Frøyshov

Abstract

Using instanton homology with coefficients in $Z/2$ we construct a homomorphism $q_2$ from the homology cobordism group in dimension 3 to the integers which is not a rational linear combination of the instanton $h$--invariant and the Heegaard Floer correction term $d$. If an oriented homology $3$--sphere $Y$ bounds a smooth, compact, negative definite $4$--manifold without $2$--torsion in its homology then $q_2(Y)\ge0$, with strict inequality if the intersection form is non-standard.

Mod 2 instanton homology and 4-manifolds with boundary

Abstract

Using instanton homology with coefficients in we construct a homomorphism from the homology cobordism group in dimension 3 to the integers which is not a rational linear combination of the instanton --invariant and the Heegaard Floer correction term . If an oriented homology --sphere bounds a smooth, compact, negative definite --manifold without --torsion in its homology then , with strict inequality if the intersection form is non-standard.
Paper Structure (36 sections, 86 theorems, 428 equations, 2 figures)

This paper contains 36 sections, 86 theorems, 428 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Outline
  4. The base-point fibration
  5. Moduli spaces
  6. Spaces of linearly dependent vectors
  7. "Generic" sections
  8. Instanton cohomology and cup products
  9. Instanton cohomology
  10. Cup products
  11. Commutators of cup products
  12. Definition of the invariant $q_2$
  13. Definite $4$-manifolds
  14. Reducibles
  15. $2$--torsion invariants of $4$--manifolds
  16. ...and 21 more sections

Key Result

Theorem 1.1

For any oriented homology $3$--spheres $Y_0$ and $Y_1$ one has

Figures (2)

  • Figure 1: The plumbing graph for $\Sigma(2,2k-1,4k-3)$
  • Figure 2: A portion of the surface $\Sigma$

Theorems & Definitions (99)

  • Theorem 1.1: Additivity
  • Theorem 1.2: Monotonicity
  • Theorem 1.3: Lower bounds
  • Corollary 1.1
  • Theorem 1.4
  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • ...and 89 more