# Four-manifolds up to connected sum with complex projective planes

## TL;DR

This paper classifies closed, connected 4-manifolds up to connected sum with complex projective planes using algebraic invariants like the fundamental group and second homotopy group, simplifying the classification in certain cases.

## Contribution

It provides a classification framework for 4-manifolds up to connected sum with complex projective planes based on algebraic topological invariants, extending previous results.

## Key findings

- Classification in terms of fundamental group, orientation character, and extension class.
- Simplification to homotopy 2-type for certain fundamental groups.
- Reduction of classification complexity for torsion-free or one-ended groups.

## Abstract

We show that closed, connected 4-manifolds up to connected sum with copies of the complex projective plane are classified in terms of the fundamental group, the orientation character and an extension class involving the second homotopy group. For fundamental groups that are torsion free or have one end, we reduce this further to a classification in terms of the homotopy 2-type.