# Stabilization of spherical space forms

## Description

This dissertation is devoted to studying the following question: Let M1 and M2 be two nonhomeomorphic lens spaces, either linear or fake. When are M 1 and M2 stably homeomorphic, i.e., when is M1 x Rn homeomorphic to M2 x Rn for some n ≥ 0? We show that for fake lens spaces of dimension ≥ 5, given a tangential homotopy equivalence f : M1 → M2 the existence of an h-cobordism (W; M1, M2) is a necessary and sufficient condition for the existence of a homeomorphism between M1 x R1 and M2 x R1 . We also show that a proper h-cobordism (W; M1 x R1 , M2 x R1 ) is a necessary and sufficient condition for the existence of a homeomorphism between M1 x R2 and M2 x R2 . We also obtain an estimate of the cardinality of the set of fake lens spaces M2 which can appear in the h-cobordism (W; M1 x R1 , M2 x R1 ) For linear lens spaces of dimension ≥ 5 having fundamental group of order 2k, the existence of a tangential homotopy equivalence f : M1 → M2 implies M1 x R3 ≈ M2 x R3 . There is a long Sullivan-Wall type exact sequence &ldots;→Lhn +4&parl0;M2xR3 &parr0;→ShTOP &parl0;M2x R3&parr0;→&sqbl0;M 2xR3; G/TOP&sqbr0;→Lh n+3&parl0;M2x R3&parr0;→&cdots; relating surgery groups to the existence of a homeomorphism between M1 x R3 and M2 x R3 It turns out that in this case the surgery groups are trivial. As a consequence, the triviality of the normal invariant eta(f) ∈ [ M2; G/TOP] will show that f x id : M1 x R3 → M2 x R3 represents a trivial element in ShTOP&parl0; M2xR3 &parr0; . This when combined with some additional work leads to the conclusion that g = f x id : M1 x R3 → M2 x R3 is indeed properly homotopic to a homeomorphism Finally we apply these results to the stabilization of fake projective spaces, defined as the particular type of fake lens spaces which are quotients of 2n - 1-spheres by the action of Z2 . We prove that there are countably many nonhomeomorphic fake projective spaces, and that stabilization by multiplication with R3 is always possible