A 4-dimensional pseudo-Anosov homeomorphism

Abstract

We know from previous work with Italiano and Migliorini that there exists some hyperbolic 5-manifold that fibers over the circle. Here we build one example where the monodromy is a "pseudo-Anosov homeomorphism" of the 4-dimensional fiber, in a way that is surprisingly similar to the familiar and beautiful two-dimensional picture of Nielsen and Thurston for surfaces. This fact has various consequences: (1) There is a compact smooth 4-manifold $M$ such that no non-trivial class in $H_2(M)$ is represented by immersed tori, and infinitely many classes are represented by smoothly embedded genus two surfaces. (2) There is a compact locally CAT(0) space $Y$ such that $π_1(Y)$ is not hyperbolic and does not contain $\mathbb Z \times \mathbb Z$. The latter answers a question of Gromov, known as the Closing Flat Problem.

Figures (27)

• Figure 1: Singular points with cone angles $\pi$ and $3\pi$ in a foliated flat cone surface.
• Figure 2: The flat orbifold $\mathsf{O} = \mathsf{T}/D_{10}$ is $S^4$ with singular set a torus $T\subset S^4$ that is locally flatly embedded except at five points $P_1,\ldots,P_5$ whose link (drawn in red) is the figure-eight knot $K\subset S^3$ shown in Figure \ref{['figure-8:fig']}.
• Figure 3: The figure eight knot $K$ in $S^3$. The double branched covering over $K$ is the elliptic manifold $L(5,2)$, while the triple branched covering is the flat 3-manifold ${{\rm HW}}$. Both these facts are important here. By quotienting $S^3$ via the standard dihedral group $D_{10}$ of isometries we get an elliptic orbifold, that is $S^3$ with cone angle $\pi$ along $K$, doubly covered by $L(5,2)$. The triple branched covering ${{\rm HW}}$ inherits a spherical cone structure from $S^3/D_{10}$, whose singular locus has cone angle $3\pi$. The 3-manifold ${{\rm HW}}$ has both a flat and a cone spherical structure.
• Figure 4: The monodromy $f$ of the fibered hyperbolic Gieseking 3-manifold sends one ideal triangulation $\Delta$ of the punctured torus to another ideal triangulation $\Delta'$ that differs from $\Delta$ only by a flip. By juxtaposing the two triangulations we get an ideal triangulation of the Gieseking manifold with one ideal tetrahedron $T$.
• Figure 5: The middle squares $Q_1,\ldots, Q_5$ form the singular torus $T$ of $\mathsf{O}$, with vertices $P_1,\ldots, P_5$. Edges with similar arrows should be identified. This is indeed a square torus: it suffices to take the red square as a fundamental domain.

Theorems & Definitions (95)

• Theorem 1
• Theorem 2
• Corollary 3
• Theorem 4
• Theorem 6
• Theorem 7
• Theorem 8
• proof
• Theorem 9
• Theorem 10