An irreducible real projective plane in the 4-sphere

TL;DR

The paper constructs an irreducible embedded real projective plane in the 4-sphere, providing counterexamples to existing conjectures and answering open problems in 4-manifold topology.

Contribution

It introduces a novel construction of an irreducible projective plane in S^4, addressing longstanding questions and conjectures in the field.

Findings

Counterexample to the Kinoshita conjecture.

Connected sum R#R is a Klein bottle with extremal normal Euler number.

The projective plane R is irreducible with a kernel of order 2 in the peripheral map.

Abstract

We construct an irreducible embedded projective plane in $S^4$. This gives a counterexample to the Kinoshita conjecture and answers Problem 4.37 of the K3 problem list. Moreover, we answer both Questions (i) and (ii) of Problem 4.37: (i) the connected sum $R\# R$ is a Klein bottle in $S^4$ with extremal normal Euler number that does not admit an unknotted projective plane summand, and (ii) we show that our projective plane $R$ is irreducible by showing that the peripheral map $\pi_1 (\partial (S^4\setminus\mathring{N}(R)))\to \pi_1 (S^4 \setminus \mathring{N}(R))$ has kernel of order $2$.