A Survey of Bounded Surgery Theory and Applications

Ferry, Steven C.; Hambleton, Ian; Pedersen, Erik Helmer

doi:10.1007/978-1-4613-9526-3_3

Cited by 5 publications

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“…The extension of surgery theory for finite groups to dimension four by Freedman [61], [22, §11.10, p. 231] allows the statements in [117] to be extended to actions on (R 4 , 0). One-point compactification produces interesting orientation-preserving examples of semifree topological actions on S 4 fixing two points, by finite groups which are not subgroups of SO (5). Another general result on problem (A) is that every π with such a semi-free action on (S n+k , S k ) also acts semi-freely on (S n , S 0 ).…”

Section: Topological Euclidean Space Formsmentioning

confidence: 99%

Topological spherical space forms

Hambleton

2014

Preprint

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Section: Topological Euclidean Space Formsmentioning

confidence: 99%

Topological spherical space forms

Hambleton

2014

Preprint

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“…Let X be a metric space on which a finite group G acts by (quasi)isometries. Let Y ⊂ X be a closed G-invariant subspace, and let M m → Y , V m+q → X be finite free bounded G-CW Poincaré complexes [5,Def 2.7]. Then a finite bounded Poincaré embedding of M in V consists of (i) a (q − 1)-spherical G-fibration ξ, with projection p : E → M , (ii) a finite free bounded G-CW Poincaré pair (C, E) → (X, Y ), and (iii) a bounded G-homotopy equivalence h : C ∪ M (p) → V , bounded over X, where M (p) is the mapping cylinder of p and C ∩ M (p) = E. Such a Poincaré embedding is "induced" by a locally flat topological embedding if the normal block bundle and complement to the embedding give data which are G-h-cobordant to those of the given Poincaré embedding.…”

Section: Bounded Embedding Theoremsmentioning

confidence: 99%

Non-Linear Similarity Revisited

Hambleton¹,

Pedersen²

1996

Prospects in Topology (AM-138)

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Let G be a finite group and V, V finite dimensional real orthogonal representations of G. Then V is said to be topologically similar to V (V ∼ t V ) if there exists a homeomorphism h : V → V which is G-equivariant. If V, V are topologically similar, but not linearly isomorphic, then such a homeomorphism is called a nonlinear similarity.The topological classification of G-representations was first studied by de Rham [18]. He proved that if a topological similarity h : V → V of orthogonal representations preserves the unit spheres and restricts to a diffeomorphism between S(V ) and S(V ), then V and V are linearly isomorphic. The purpose of this paper is to give new restrictions on the existence of non-linear similarities using techniques from bounded topology.Theorem A. Let V, V be real orthogonal G-representations, where G is a finite cyclic group. Suppose that Res H V ∼ = Res H V for each proper subgroup H G, and thatThis result gives information about topological similarities for non-cyclic groups as well, since linear equivalence of representations is detected by character values. The formulation is also well-adapted to inductive arguments, and we get a new proof for the results of [10], [12].Corollary B. Let G be a finite group and V, V be real orthogonal G-representations. Suppose that for each cyclic subgroup C ⊆ G of 2-power order, the elements of C act All previously contructed topological similarities of cyclic groups G contain the non-trivial 1-dimensional representation (i.e. the representation with isotropy group H ⊂ G of index 2), or are induced from such examples.For example, suppose that G = Z/4q, q = 2 r−2 , r ≥ 4. Now let V 1 = t i + t j , with i ≡ 1 mod 4 and j ≡ ±i mod 8, and V 2 = t i+2q + t j+2q where t denotes a faithful 2-dimensional representation of G. Let (resp. δ) denote the 1-dimensional trivial (resp. non-trivial) representation of G. ThenG, so our result says that V 1 ⊕ W is not topologically similar to V 2 ⊕ W unless δ is a summand of W . More generally,is also a non-linear similarity and the non-trivial 1-dimensional H-representation is a summand of Res H W .Our techniques also give information about the existence and classification of nonlinear similarities. We recently noticed that [4, Thm 1(i)] is incorrect as stated. For example for G = Z/12, there are no 6-dimensional non-linear similarities. The problem is that the natural epimorphism π : Z/4q → Z/2 r where q = 2 r−2 s, s odd, does not induce an isomorphism π * :as claimed in [4, p. 732 l. -8]. These topics will be discussed elsewhere [8].

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Quadratic K-Theory and Geometric Topology

Williams

2005

Handbook of K-Theory

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A Survey of Bounded Surgery Theory and Applications

Cited by 5 publications

References 46 publications

Topological spherical space forms

Topological spherical space forms

Non-Linear Similarity Revisited

Quadratic K-Theory and Geometric Topology

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