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Ordering Thurston’s geometries by maps of nonzero degree. (English) Zbl 1421.57002

A closed, oriented manifold \(M\) dominates a closed, oriented manifold \(N\) of the same dimension if there is a map of nonzero degree from \(M\) to \(N\). This defines a relation on the homotopy types of closed oriented manifolds of a given dimension. In dimension two, a surface of genus \(g\) dominates a surface of genus \(h\) if and only if \(g\geq h\).
In the present paper, the author shows the following result about the domination relation for non-hyperbolic closed \(4\)-manifolds that carry a Thurston aspherical geometry. Let \(\mathbb X\) and \(\mathbb Y\) be two such geometries. Depending on \(\mathbb X\) and \(\mathbb Y\), either any closed \(\mathbb Y\)-manifold is dominated by a closed \(\mathbb X\)-manifold or no closed \(\mathbb Y\)-manifold is dominated by a closed \(\mathbb X\)-manifold. In Figure 1, the author gives a complete picture on the behavior for any two geometries. This result is similar to the result obtained by S. Wang [Math. Z. 208, No. 1, 147–160 (1991; Zbl 0737.57006)] about closed aspherical \(3\)-manifolds.
Let \(M,N\) be two closed oriented geometric \(4\)-manifolds. As an application, the author shows that if \(M\) dominates \(N\) then the Kodaira dimension of \(M\) is greater than or equal to the Kodaira dimension of \(N\).

MSC:

57M05 Fundamental group, presentations, free differential calculus
57M10 Covering spaces and low-dimensional topology
57M50 General geometric structures on low-dimensional manifolds
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
55M25 Degree, winding number
22E25 Nilpotent and solvable Lie groups
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)

Citations:

Zbl 0737.57006
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References:

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