Strongly minimal PD4-complexes

TL;DR

This paper classifies strongly minimal $PD_4$-complexes with certain fundamental groups, showing a finite bound on their homotopy types based on algebraic invariants, and applies results to Fox's 2-knot.

Contribution

It establishes a bound on the number of homotopy types of strongly minimal $PD_4$-complexes with specific fundamental groups, linking algebraic invariants to topological classification.

Findings

At most $2^eta$ orbits of $k$-invariants determine strongly minimal complexes.

Homotopy type of $PD_4$-complexes with $PD_2$-group fundamental group is determined by specific invariants.

Fox's 2-knot with metabelian group is classified up to TOP isotopy and reflection.

Abstract

We consider the homotopy types of $PD_4$-complexes $X$ with fundamental group $\pi$ such that $c.d.\pi=2$ and $\pi$ has one end. Let $\beta=\beta_2(\pi;F_2)$ and $w=w_1(X)$. Our main result is that (modulo two technical conditions on $(\pi,w)$) there are at most $2^\beta$ orbits of $k$-invariants determining "strongly minimal" complexes (i.e., those with homotopy intersection pairing $\lambda_X$ trivial). The homotopy type of a $PD_4$-complex $X$ with $\pi$ a $PD_2$-group is determined by $\pi$, $w$, $\lambda_X$ and the $v_2$-type of $X$. Our result also implies that Fox's 2-knot with metabelian group is determined up to TOP isotopy and reflection by its group.