Five tori in $S^4$

TL;DR

This work constructs and analyzes a five-tori link $L$ in $S^4$ (Ivanšić link) whose complement is hyperbolic, drawing deep parallels to the Borromean rings in $S^3$. Through an explicit Dehn-filling/polytope framework based on the right-angled 4-polytope $P^4$, the authors realize $M=S^4\setminus L$ as the orientable double cover of a Ratcliffe–Tschantz census manifold and establish a 640-element symmetry group acting k-transitively on the five components. They compute a minimal 5-generator presentation for $\pi_1(M)$, determine the Alexander ideal (with $\Delta=1$), and analyze infinite cyclic coverings, yielding conditions for perfect circle-valued Morse representatives and notable Dehn surgeries that yield $\mathbb{Z}^5$ as a fundamental group. The paper further describes the branched double cover $W$ with ${\mathbb{H}}^2\times{\mathbb{H}}^2$ geometry, relates $L$ to five simple quadrics, and exhibits realizations of the link in $\mathbb{RP}^4$ and in products of genus-2 surfaces as Lagrangian tori, highlighting new hyperbolic-link phenomena in dimension four.

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.

Figures (24)

• Figure 1: The Borromean rings $B$ in $S^3$.
• Figure 2: The equatorial 3-sphere $S^3 = \{x_5=0\}$ intersects the link $L$ in one black torus and 8 circles.
• Figure 3: The intersection of $L$ with the slice $x_5=t$ is obtained by doubling the corresponding picture in $D^3$ along the boundary $\partial D^3 = S^2$. At the times $t=\pm 0.9$ the picture consists of eight points, obtained by shrinking the eight circles shown at $t=\pm 0.8$.
• Figure 4: A genus two surface and the simple closed curves $\alpha, \beta, \gamma_1, \gamma_2, \delta_1, \delta_2, \epsilon$. Note that these curves cut the surface into 8 pentagons.
• Figure 5: The right-angled bipyramid $P^3\subset {\mathbb{H}}^3$ has two real vertices with valence 3 and three ideal vertices with valence 4 (left). We colour its faces with $1,2,3$ as shown in the dual graph, that is the 1-skeleton of a triangular prism (right).

Theorems & Definitions (23)

• Theorem 1
• Corollary 2
• proof
• Theorem 3
• Theorem 4
• Proposition 5
• proof
• Proposition 6
• proof
• Proposition 7