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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 main theorem

Chapter:
(p.154) Chapter Eleven The main theorem
Source:
Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691161686.003.0011

This chapter introduces the main theorem, which states: Let V be a quasi-projective variety over a valued field F and let X be a definable subset of V x Γ‎superscript Script Small l subscript infinity over some base set V ⊂ VF ∪ Γ‎, with F = VF(A). Then there exists an A-definable deformation retraction h : I × unit vector X → unit vector X with image an iso-definable subset definably homeomorphic to a definable subset of Γ‎superscript w subscript Infinity, for some finite A-definable set w. The chapter presents several preliminary reductions to essentially reduce to a curve fibration. It then constructs a relative curve homotopy and a liftable base homotopy, along with a purely combinatorial homotopy in the Γ‎-world. It also constructs the homotopy retraction by concatenating the previous three homotopies together with an inflation homotopy. Finally, it describes a uniform version of the main theorem with respect to parameters.

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