Donaldson invariants for connected sums along surfaces of genus 2

TL;DR

This paper explores how Donaldson invariants behave under connected sums along genus 2 surfaces, establishing relationships and constraints for four-manifolds with specific topological properties.

Contribution

It provides new formulas relating invariants of connected sum manifolds along genus 2 surfaces and demonstrates their implications for the structure of four-manifolds.

Findings

Connected sums along genus 2 surfaces preserve simple type under certain conditions.

Manifolds with genus 2 surfaces of self-intersection zero have constrained basic classes.

Any such four-manifold with an odd homology class is of finite type of second order.

Abstract

We relate the Donaldson invariants of two four-manifolds $X_i$ with embedded Riemann surfaces of genus 2 and self-intersection zero with the invariants of the manifold X which appears as a connected sum along the surfaces. When the original manifolds are of simple type with $b_1=0$ and $b^+>1$, X is of simple type with $b_1=0$ and $b^+>1$ as well, and the relationship between the invariants is expressed as constraints in the basic classes for X. Also we give some applications. For instance, if $X_i$ have both $b_1=0$ then X is of simple type with $b_1=0$, $b^+>1$, and has no basic classes evaluating zero on the Riemann surface. Finally, we prove that any four-manifold with $b^+>1$ and with an embedded surface of genus 2, self-intersection zero and representing an odd homology class, is of finite type of second order.