# The main theorem

# The main theorem

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.

*Keywords:*
main theorem, valued field, definable subset, deformation retraction, iso-definable subset, curve fibration, homotopy, inflation homotopy

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