Degree one maps between geometric 3-manifolds

Rong, Yong Wu

doi:10.1090/S0002-9947-1992-1052909-6

Degree one maps between geometric $3$-manifolds
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Trans. Amer. Math. Soc. 332 (1992), 411-436 Request permission

Abstract:

Let $M$ and $N$ be two compact orientable $3$-manifolds, we say that $M \geq N$, if there is a degree one map from $M$ to $N$. This gives a way to measure the complexity of $3$-manifolds. The main purpose of this paper is to give a positive answer to the following conjecture: if there is an infinite sequence of degree one maps between Haken manifolds, then eventually all the manifolds are homeomorphic to each other. More generally, we prove a theorem which says that any infinite sequence of degree one maps between the so-called "geometric $3$-manifolds" must eventually become homotopy equivalences.

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