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Instanton Floer homology and Milnor Fibers

Kyoung-Seog Lee, Anatoly Libgober, Nikolai Saveliev

Abstract

In the early days of the Floer theory, Atiyah asked if there is a Milnor fiber description of the Floer homology of the links of singularities. We answer this question for the Brieskorn-Hamm complete intersection singularities. The resulting combinatorial formulas lead to an independent proof of the equality of the Casson invariants in the Donaldson and Seiberg-Witten theories. We use similar techniques to express the Heegaard Floer d-invariant of torus knots in terms of their Milnor fibers.

Instanton Floer homology and Milnor Fibers

Abstract

In the early days of the Floer theory, Atiyah asked if there is a Milnor fiber description of the Floer homology of the links of singularities. We answer this question for the Brieskorn-Hamm complete intersection singularities. The resulting combinatorial formulas lead to an independent proof of the equality of the Casson invariants in the Donaldson and Seiberg-Witten theories. We use similar techniques to express the Heegaard Floer d-invariant of torus knots in terms of their Milnor fibers.

Paper Structure

This paper contains 10 sections, 13 theorems, 100 equations.

Table of Contents

  1. Classical results
  2. Compactification of the Milnor fiber
  3. Instanton homology and integer lattice points
  4. The statement
  5. Brieskorn singularities
  6. Proof of Theorem \ref{['T:spectrum']}
  7. The Seiberg--Witten equations
  8. Comments and generalizations
  9. The $d$--invariant of torus knots
  10. A formula for the $\bar{\mu}$--invariant

Key Result

Theorem 2.1

Let $K_Y$ be the canonical divisor of $Y$. Then

Theorems & Definitions (30)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • ...and 20 more