Topological 4-manifolds with geometrically 2-dimensional fundamental groups

TL;DR

This paper classifies certain 4-manifolds with specific 2-dimensional fundamental groups up to s-cobordism, using invariants like $w_2$-type and intersection forms, and applies this to manifolds with solvable Baumslag-Solitar groups.

Contribution

It provides a complete classification of 4-manifolds with geometrically 2-dimensional fundamental groups based on their invariants, including a realization result for Baumslag-Solitar groups.

Findings

Classified 4-manifolds up to s-cobordism using invariants.

Achieved a homeomorphism classification for manifolds with solvable Baumslag-Solitar groups.

Provided a realization result for these manifolds.

Abstract

Closed oriented 4-manifolds with the same geometrically 2-dimensional fundamental group (satisfying certain properties) are classified up to $s$-cobordism by their $w_2$-type, equivariant intersection form and the Kirby-Siebenmann invariant. As an application, we obtain a complete homeomorphism classification of closed oriented 4-manifolds with solvable Baumslag-Solitar fundamental groups, including a precise realization result.