Homotopy invariance of 4-manifold decompositions: connected sums

TL;DR

This paper proves that the existence and uniqueness of connected sum decompositions of oriented 4-manifolds are homotopy invariants under certain conditions, linking topological invariance to fundamental group properties.

Contribution

It establishes homotopy invariance of 4-manifold decompositions up to h-cobordism, assuming the fundamental groups are 'good' and relates the Borel Conjecture to s-cobordism in dimension 4.

Findings

Connected sum decompositions are homotopy invariants up to h-cobordism.

The Borel Conjecture holds in dimension 4 up to s-cobordism under Farrell--Jones conjecture.

Decomposition uniqueness is preserved under homotopy equivalence for certain fundamental groups.

Abstract

We show, up to h-cobordism, that the existence and uniqueness of connected sum decompositions of oriented 4-dimensional manifolds is an invariant of homotopy equivalence, assuming that the fundamental group of each summand is "good" in the sense of Freedman and Quinn. On a separate note, we observe that the Borel Conjecture is true in dimension 4 up to s-cobordism, assuming that the fundamental group satisfies the Farrell--Jones Conjecture.