Strongly minimal PD4-complexes

Abstract

We consider the homotopy types of $P{D}_4$-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 $P{D}_4$-complex $X$ with $\pi$ a $P{D}_2$-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.

17 pages

References in corpus (2)

• Poincare duality complexes in dimension four
• Topological 4-manifolds with geometrically 2-dimensional fundamental groups

Cited by in corpus (7)

• Poincare duality complexes in dimension four
• On minimal Poincaré $4$-complexes
• Topological 4-manifolds with geometrically 2-dimensional fundamental groups
• Homotopy classification of $P{D}_4$-complexes relative an order relation
• $P{D}_4$-complexes and 2-dimensional duality groups
• 2-knots with solvable group
• Rank 1 abelian normal subgroups of 2-knot groups