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Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)$
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Ehud Hrushovski and François Loeser

Print publication date: 2016

Print ISBN-13: 9780691161686

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691161686.001.0001

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The smooth case

The smooth case

(p.177) Chapter Twelve The smooth case
Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)

Ehud Hrushovski

François Loeser

Princeton University Press

This chapter examines the simplifications occurring in the proof of the main theorem in the smooth case. It begins by stating the theorem about the existence of an F-definable homotopy h : I × unit vector X → unit vector X and the properties for h. It then presents the proof, which depends on two lemmas. The first recaps the proof of Theorem 11.1.1, but on a Zariski dense open set V₀ only. The second uses smoothness to enable a stronger form of inflation, serving to move into V₀. The chapter also considers the birational character of the definable homotopy type in Remark 12.2.4 concerning a birational invariant.

Keywords:   main theorem, smooth case, homotopy, Zariski dense open set, smoothness, inflation, definable homotopy type, birational invariant

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