Five tori in $S^4$

Abstract

Ivansic proved that there is a link $L$ of five tori in $S^4$ with hyperbolic complement. We describe $L$ explicitly with pictures, study its properties, and discover that $L$ is in many aspects similar to the Borromean rings in $S^3$. In particular the following hold: (1) Any two tori in $L$ are unlinked, but three are not; (2) The complement $M=S^4\setminus L$ is integral arithmetic hyperbolic; (3) The symmetry group of $L$ acts $k$-transitively on its components for all $k$; (4) The double branched covering over $L$ has geometry ${\mathbb{H}}^2\times {\mathbb{H}}^2$; (5) The fundamental group of $M$ has a nice presentation via commutators; (6) The Alexander ideal has an explicit simple description; (7) Every class $x\in H^1(M,Z)=Z^5$ with non-zero xi is represented by a perfect circle-valued Morse function; (8) By longitudinal Dehn surgery along $L$ we get a closed 4-manifold with fundamental group $Z^5$; (9) The link $L$ can be put in perfect position. This leads also to the first descriptions of a cusped hyperbolic 4-manifold as a complement of tori in ${\mathbb{RP}}^4$ and as a complement of some explicit Lagrangian tori in the product of two surfaces of genus two.

35 pages, 24 figures