# $\mathbb{CP}^2$-stable classification of $4$-manifolds with finite fundamental group

## TL;DR

This paper establishes a classification criterion for closed, connected 4-manifolds with finite fundamental groups based on their quadratic 2-types and Kirby-Siebenmann invariants, using $ ext{CP}^2$-stability.

## Contribution

It provides a complete classification of such 4-manifolds up to $ ext{CP}^2$-stability by linking their topological invariants.

## Key findings

- Two 4-manifolds are $ ext{CP}^2$-stably homeomorphic iff their quadratic 2-types are stably isomorphic and invariants match.
- The classification reduces to algebraic data involving quadratic 2-types and Kirby-Siebenmann invariants.
- The result extends understanding of 4-manifold topology with finite fundamental groups.

## Abstract

We show that two closed, connected $4$-manifolds with finite fundamental groups are $\mathbb{CP}^2$-stably homeomorphic if and only if their quadratic $2$-types are stably isomorphic and their Kirby-Siebenmann invariant agrees.