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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Degree one maps between geometric $ 3$-manifolds


Author: Yong Wu Rong
Journal: Trans. Amer. Math. Soc. 332 (1992), 411-436
MSC: Primary 57M99; Secondary 57M25, 57Q35
MathSciNet review: 1052909
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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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DOI: http://dx.doi.org/10.1090/S0002-9947-1992-1052909-6
PII: S 0002-9947(1992)1052909-6
Keywords: Degree, Gromov's norm, hyperbolic Dehn surgery, Haken-number
Article copyright: © Copyright 1992 American Mathematical Society