On the homeomorphism problem for 4-manifolds

TL;DR

This paper proves that there is no algorithm to determine whether a 4-manifold is homeomorphic to a connected sum of 12 S^2 imes S^2, highlighting fundamental limits in 4-manifold topology.

Contribution

It establishes the undecidability of the homeomorphism problem for certain 4-manifolds, advancing understanding of computational complexity in topology.

Findings

No algorithm can decide the homeomorphism to the connected sum of 12 S^2 imes S^2.

The result demonstrates fundamental undecidability in 4-manifold topology.

Highlights limits of algorithmic approaches in high-dimensional topology.

Abstract

We show that there is no algorithm to decide whether or not a given 4-manifold is homeomorphic to the connected sum of 12 copies of S^2 \times S^2.