Gluing formulae for Donaldson invariants for connected sums along surfaces

TL;DR

This paper proves a conjecture relating the basic classes of four-manifolds connected along surfaces, extending understanding of Donaldson invariants and their behavior under such connected sums, consistent with Seiberg-Witten theory.

Contribution

It establishes a gluing formula for Donaldson invariants for connected sums along surfaces, confirming conjectured relationships of basic classes in this context.

Findings

Derived constraints on basic classes of connected sum manifolds

Confirmed compatibility with Seiberg-Witten invariants

Extended the understanding of Donaldson invariants in complex surgeries

Abstract

We solve a conjecture of Morgan and Szabo (Embedded genus 2 surfaces in four-manifolds, Preprint) about the relationship of the basic classes of two four-manifolds $X_i$ of simple type with $b_1=0$, $b^+>1$, such that there are embedded Riemann surfaces of genus $g \geq 2$ and self-intersection zero (and representing odd homology classes) with the basic classes of the manifold X which appears as a connected sum along the surfaces (supposing this latter one is of simple type). This is also expressed as constraints in the basic classes of X. The result is in accordance with the results on Seiberg-Witten invariants (Morgan, Szabo and Taubes, A product formula for the Seiberg-Witten invariants and the generalized Thom conjecture).