# Embedding lens spaces in definite 4-manifolds

## TL;DR

This paper investigates the embedding properties of lens spaces into complex projective planes, establishing limitations on their smooth embeddings and providing new insights into 4-manifold topology.

## Contribution

It proves that while every lens space can embed in a connected sum of 8 complex projective planes, there is no universal n for smooth embeddings into n copies of the plane.

## Key findings

- Every lens space embeds in 8 copies of CP^2
- No single n allows all lens spaces to smoothly embed in n copies
- Differentiates between locally flat and smooth embedding capabilities

## Abstract

Every lens space has a locally flat embedding in a connected sum of 8 copies of the complex projective plane and a smooth embedding in n copies of the complex projective plane for some positive integer n. We show that there is no n such that every lens space smoothly embeds in n copies of the complex projective plane.