Homotopy types of 4-manifolds with 3-manifold fundamental groups

TL;DR

This paper demonstrates that the homotopy type of certain 4-manifolds is fully determined by their fundamental group, second homotopy group, first k-invariant, and intersection pairing, linking algebraic invariants to topological classification.

Contribution

It establishes that for 4-manifolds with fundamental groups as finitely presentable PD_3-groups, the homotopy type is uniquely determined by specific algebraic and intersection data.

Findings

Homotopy type determined by fundamental group and invariants

Extension of classification results to 4-manifolds with PD_3-group fundamental groups

Explicit relation between algebraic invariants and topological structure

Abstract

We show that the homotopy type of a 4-manifold $M$ whose fundamental group is a finitely presentable $PD_3$-group $\pi$ and with $w_1(M)=w_1(\pi)$ is determined by $\pi$, $\pi_2(M)$, $k_1(M)$ and the equivariant intersection pairing $\lambda_M$.