Strongly minimal PD4-complexes

We consider the homotopy types of $PD_4$-complexes $X$ with fundamental group $π$ such that $c.d.π=2$ and $π$ has one end. Let $β=β_2(π;F_2)$ and $w=w_1(X)$. Our main result is that (modulo two technical conditions on $(π,w)$) there are at most $2^β$ orbits of $k$-invariants determining "strongly minimal" complexes (i.e., those with homotopy intersection pairing $λ_X$ trivial). The homotopy type of a $PD_4$-complex $X$ with $π$ a $PD_2$-group is determined by $π$, $w$, $λ_X$ 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.

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