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## Ehud Hrushovski and François Loeser

Print publication date: 2016

Print ISBN-13: 9780691161686

Published to University Press Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691161686.001.0001

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date: 13 December 2017

# The smooth case

Chapter:
(p.177) Chapter Twelve The smooth case
Source:
Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691161686.003.0012

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.

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