On certain exotic 4-manifolds of Akhmedov and Park

TL;DR

This paper provides an explicit homology splitting and computes canonical classes for certain exotic 4-manifolds constructed via fiber sums, offering a new proof of a known formula related to their symplectic structures.

Contribution

It introduces an explicit second homology splitting and calculates canonical classes, enhancing understanding of Akhmedov and Park's exotic 4-manifolds.

Findings

Explicit second homology splitting for the manifolds

Calculation of canonical classes of symplectic structures

New proof of a formula relating to the manifolds' topology

Abstract

In an article from 2008, A. Akhmedov and B. D. Park constructed irreducible symplectic 4-manifolds homeomorphic but not diffeomorphic to the manifolds CP^2#3CP^2bar and 3CP^2#5CP^2bar. These manifolds are constructed by using generalized fibre sums. In this note we describe an explicit splitting of the second (co-)homology of these manifolds adapted to their construction as fibre sums. We also calculate the canonical classes of the symplectic structures. This gives a new proof for a formula derived by A. Akhmedov, R. I. Baykur and B. D. Park.