The Hauptvermutung according to Casson and Sullivan

Rourke, Colin

doi:10.1007/978-94-017-3343-4_6

Cited by 7 publications

(7 citation statements)

References 30 publications

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“…Here M n is a closed smooth manifold of dimension n ≥ 5 and the superscript x in our case is s for simple homotopy structures or h for homotopy structures (e.g., see Rourke [48] or Quinn [45] One useful feature of the preceding constructions is that they yield a reasonably well-behaved system of composition operations on the structure sets S DIF F ,x • (M n ). Proposition 7.4.…”

Section: Structure Sets and Self-equivalence Spacesmentioning

confidence: 99%

“…Here M n is a closed smooth manifold of dimension n ≥ 5 and the superscript x in our case is s for simple homotopy structures or h for homotopy structures (e.g., see Rourke [48] or Quinn [45]). A k -simplex in S DIF F ,x • (M n ) is represented by a suitable homotopy equivalence of manifold n-ads [59, Chapter 0] into the standard n-ad given by ∆ k ×M , the set F • (M n , F/O) is the simplicial function set as defined in May [38, Definition I.6.4, p. 17] or Goerss-Jardine [16, p. 20], and a k -simplex of L x • (−) is represented by a suitable surgery problem of manifold n-ads.…”

Section: Structure Sets and Self-equivalence Spacesmentioning

confidence: 99%

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Codimension two souls and cancellation phenomena

Belegradek

Kwasik

Schultz

2015

Advances in Mathematics

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For each m ≥ 0 we find an open (4m + 9) -dimensional simply connected manifold with complete nonnegatively curved metrics whose souls are nondiffeomorphic, homeomorphic, and have codimension 2 . We give a diffeomorphism classification of the pairs (N, soul) when N is the total space of a nontrivial complex line bundle over S 7 × CP 2 ; up to diffeomorphism there are precisely three such pairs, distinguished by their nondiffeomorphic souls.

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“…Here M n is a closed smooth manifold of dimension n ≥ 5 and the superscript x in our case is s for simple homotopy structures or h for homotopy structures (e.g., see Rourke [48] or Quinn [45] One useful feature of the preceding constructions is that they yield a reasonably well-behaved system of composition operations on the structure sets S DIF F ,x • (M n ). Proposition 7.4.…”

Section: Structure Sets and Self-equivalence Spacesmentioning

confidence: 99%

“…Here M n is a closed smooth manifold of dimension n ≥ 5 and the superscript x in our case is s for simple homotopy structures or h for homotopy structures (e.g., see Rourke [48] or Quinn [45]). A k -simplex in S DIF F ,x • (M n ) is represented by a suitable homotopy equivalence of manifold n-ads [59, Chapter 0] into the standard n-ad given by ∆ k ×M , the set F • (M n , F/O) is the simplicial function set as defined in May [38, Definition I.6.4, p. 17] or Goerss-Jardine [16, p. 20], and a k -simplex of L x • (−) is represented by a suitable surgery problem of manifold n-ads.…”

Section: Structure Sets and Self-equivalence Spacesmentioning

confidence: 99%

Codimension two souls and cancellation phenomena

Belegradek

Kwasik

Schultz

2015

Advances in Mathematics

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“…n D .X=@X / ‫ޒ.‬ n ; 0/ is a fiber homotopy equivalence of the absolute fiber ‫ޒ‬ n f 0 g ' S n 1 . The abelian group structure on OEX=@X; G=PL 0 is the 0 of the Whitney sum H -space structure on the -set Map 0 .X=@X; G=PL/ defined rigorously in Rourke [25,Proposition 2.3].…”

Section: Proofs In the Non-orientable Casementioning

confidence: 99%

On smoothable surgery for 4–manifolds

Khan

2007

Algebr. Geom. Topol.

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Under certain homological hypotheses on a compact 4-manifold, we prove exactness of the topological surgery sequence at the stably smoothable normal invariants. The main examples are the class of finite connected sums of 4-manifolds with certain product geometries. Most of these compact manifolds have non-vanishing second mod 2 homology and have fundamental groups of exponential growth, which are not known to be tractable by Freedman-Quinn topological surgery. Necessarily, the -construction of certain non-smoothable homotopy equivalences requires surgery on topologically embedded 2-spheres and is not attacked here by transversality and cobordism.57R67; 57N65, 57N75

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“…The morphisms in degree k are homeomorphisms respecting the reference maps to M ×∆ k . References: [Qun1], [Wa1, §17.A], [Ni], [Rou1]. There is a homotopy fiber sequence…”

Section: Stabilization and Descentmentioning

confidence: 99%

Automorphisms of manifolds

Weiss¹,

Williams²

2001

Princeton University Press eBook Package 2014

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This survey is about homotopy types of spaces of automorphisms of topological and smooth manifolds. Most of the results available are relative, i.e., they compare different types of automorphisms.In chapter 1, which motivates the later chapters, we introduce our favorite types of manifold automorphisms and make a comparison by (mostly elementary) geometric methods. Chapters 2, 3, and 4 describe algebraic models (involving L-theory and/or algebraic K-theory) for certain spaces of "structures" associated with a manifold M , that is, spaces of other manifolds sharing some geometric features with M . The algebraic models rely heavily onWall's work in surgery theory, e.g. [Wa1] , • Waldhausen's work in h-cobordism theory alias concordance theory, which includes a parametrized version of Wall's theory of the finiteness obstruction, [Wa2] .The structure spaces are of interest for the following reason. Suppose that two different notions of automorphism of M are being compared. Let X 1 (M ) and X 2 (M ) be the corresponding automorphism spaces; suppose that X 1 (M ) ⊂ X 2 (M ). As a rule, the space of cosets X 2 (M )/X 1 (M ) is a union of connected components of a suitable structure space.Chapter 5 contains the beginnings of a more radical approach in which one tries to calculate the classifying space BX 1 (M ) in terms of BX 2 (M ), rather than trying to calculate X 2 (M )/X 1 (M ). Chapter 6 contains some examples and calculations.

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The Hauptvermutung according to Casson and Sullivan

Cited by 7 publications

References 30 publications

Codimension two souls and cancellation phenomena

Codimension two souls and cancellation phenomena

On smoothable surgery for 4–manifolds

Automorphisms of manifolds

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