Stabilization of spherical space forms

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Title 
Stabilization of spherical space forms 
Author 
Webb, Joel 
School 
Tulane University 
Academic Field 
Mathematics 
Abstract 
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 hcobordism (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 hcobordism (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 hcobordism (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 SullivanWall 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  1spheres 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 
Language 
eng 
Advisor(s) 
Kwasik, Slawomir 
Degree Date 
2006 
Degree 
Ph.D 
Publisher 
Tulane University 
Publication Date 
2006 
Source 
Source: 66 p., Dissertation Abstracts International, Volume: 6705, Section: B, page: 2598 
Identifier 
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Rights 
Copyright is in accordance with U.S. Copyright law 
Contact Information 
acase@tulane.edu 
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