Geometric structures in topology, geometry, global analysis and dynamics (Q6610518)

This article surveys a few results of the author on different aspects of geometric manifolds in dimension at least \(4\), mostly focused on dimension \(4\).\N\NThurston's geometrization on \(3\)-manifolds implies that any closed orientable prime \(3\)-manifold either supports one of eight geometric structures, or can be decomposed to such geometric manifolds by cutting along a disjoint union of incompressible tori. This picture of \(3\)-manifolds was confirmed by Perelman, and has many important applications on almost all aspects of \(3\)-manifolds.\N\NGeometric structures on manifolds of dimension at least \(4\) are less studied, since an analogy of geometrization fails in these dimensions. Geometric structures were classified in dimension \(4\) and \(5\), and were also partially classified in dimension \(6\) and \(7\). There are nineteen geometries in dimension \(4\) and fifty-eight geometries in dimension \(5\). This article mentions the following three results of the author on geometric manifolds.\N\begin{itemize}\N\item[1.] The existence and non-existence of non-zero degree maps between geometric \(4\)-manifolds.\N\item[2.] An axiomatic definition of the geometric Kodaira dimension, and its monotonicity under non-zero degree maps between geometric \(4\)-manifolds or \(5\)-manifolds.\N\item[3.] The non-existence of Anosov diffeomorphisms on most geometric \(4\)-manifolds.\N\end{itemize}\N\NFor \(3\)-manifolds, in [Math. Z. 208, No. 1, 147--160 (1991; Zbl 0737.57006)], \textit{S. Wang} studied the existence and non-existence of non-zero degree maps between aspherical \(3\)-manifolds, and gave a partial order on \(3\)-dimensional geometries. In [J. Topol. Anal. 10, No. 4, 853--872 (2018; Zbl 1421.57002)], the author gave a similar partial order on \(4\)-dimensional geometries other than the hyperbolic geometry. This result is summarized in Figure 9.1 of this article. The proof of this result highly depends on studying manifolds dominated or not dominated by products of manifolds, in both the topological sense and the group-theoretical sense.\N\NIn the author's joint work with \textit{W. Zhang} [Pac. J. Math. 315, No. 1, 209--233 (2021; Zbl 1487.57028)], they defined the geometric Kodaira dimension for a big class of manifolds, including all geometric manifolds of dimension at most \(5\). This definition is inspired by the Kodaira dimension of complex manifolds, but they can be different for the same manifold. The author and Zhang computed geometric Kodaira dimensions for geometric manifolds in dimensions \(4\) and \(5\), and \(4\)-dimensional fiber bundles. For geometric \(4\)- and \(5\)-manifolds, they proved that the geometric Kodaira dimension is non-increasing under a domination between manifolds, and a geometric manifold has the top geometric Kodaira dimension if and only if it has positive simplicial volume.\N\NA diffeomorphism \(f:M\to M\) is called Anosov if \(TM\) can be decomposed into the direct sum of two \(f\)-invariant sub-bundles, so that \(df\) is expanding on one sub-bundle and contracting on the other. In [Geom. Dedicata 213, 325--337 (2021; Zbl 1480.37044)], the author proved that a \(4\)-manifold carrying a geometry other than \(\mathbb{R}^4\) or \(\mathbb{H}^2\times \mathbb{H}^2\) does not admit transitive Anosov diffeomorphisms. One main ingredient of the proof is that the Lefschetz numbers of powers of \(f\) are unbounded.\N\NThe proofs of these three results all run the case-by-case argument, while they use a wide range of different tools in topology, geometry and dynamics.\N\NFor the entire collection see [Zbl 1537.57002].