Bundle theories for topological manifolds

Hughes, C. Bruce; Taylor, Laurence R.; Williams, E. Bruce

doi:10.1090/s0002-9947-1990-1010410-8

Cited by 21 publications

(19 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However to Hutt's arguments one must add foundational results of Hughes, Taylor and Williams [10] and a folk theorem proved by Hughes [9] about mapping cylinder neighbourhoods, MCNs, and manifold approximate fibrations, MAFs. We summarise these results in Theorem 3.1 and Corollary 3.3 and use them to show that Hutt's map is defined.…”

Section: Periodicity In Topological Surgerymentioning

confidence: 99%

The additivity of theρ–invariant and periodicity in topological surgery

Crowley¹,

Macko

2011

Algebr. Geom. Topol.

View full text Add to dashboard Cite

For a closed topological manifold M with dim(M ) ≥ 5 the topological structure set S(M ) admits an abelian group structure which may be identified with the algebraic structure group of M as defined by Ranicki. If dim(M ) = 2d − 1, M is oriented and M is equipped with a map to the classifying space of a finite group G, then the reduced ρ-invariant defines a function,to a certain sub-quotient of the complex representation ring of G. We show that the function ρ is a homomorphism when 2d − 1 ≥ 5.Along the way we give a detailed proof that a geometrically defined map due to Cappell and Weinberger realises the 8-fold Siebenmann periodicity map in topological surgery.

show abstract

Section: Periodicity In Topological Surgerymentioning

confidence: 99%

The additivity of theρ–invariant and periodicity in topological surgery

Crowley¹,

Macko

2011

Algebr. Geom. Topol.

View full text Add to dashboard Cite

show abstract

“…More specifically, the methods used combine the explicit construction of the pseudoisotopy relaxation given in [31] and the controlled methods used in [10,19,21,25,26]. Using the construction in [31], we give an explicit calculation of the relaxation map: r : P(X ×S 1 ) → RP(X ×S 1 ) that splits the "forget-control" map.…”

Section: Theorem (Bass-heller-swan Splitting For Pseudoisotopy Groupsmentioning

confidence: 99%

“…A controlled map from p 0 to p 1 is a map There is an alternate definition in [20]. The two definitions are equivalent when the base space B is a compact metric space [25]. This will be the definition that we will use in the rest of this paper.…”

Section: Pseudoisotopies In Topmentioning

confidence: 99%

A Bass–Heller–Swan formula for pseudoisotopies

Kinsey

Prassidis

2009

Geom Dedicata

View full text Add to dashboard Cite

show abstract

“…First, π −1 ((0, +∞)) is the infinite cyclic cover of a closed manifoldÂ with a manifold approximate fibrationÂ → S 1 [11]. One can pull back handles inÂ to get handles with a periodic structure in A. Alternatively, use the Approximate Isotopy Covering Property of manifold approximate fibrations [5], [11], [13] as follows. For i = 1, 2, 3, .…”

Section: Completion Of the Proof Of The Main Theoremmentioning

confidence: 99%

NEIGHBORHOODS OF STRATA IN MANIFOLD STRATIFIED SPACES Supported in part by NSF Grant DMS-9971367.

Hughes

2004

Glasgow Math. J.

View full text Add to dashboard Cite

Strata in manifold stratified spaces are shown to have neighborhoods that are teardrops of manifold stratified approximate fibrations (under dimension and compactness assumptions). This is the best possible version of the tubular neighborhood theorem for strata in the topological setting. Applications are given to replacement of singularities, to the structure of neighborhoods of points in manifold stratified spaces, and to spaces of manifold stratified approximate fibrations.2000 Mathematics Subject Classification. Primary 57N80, 57N40; Secondary 55R65, 58A35.1. Introduction. The foundations of differential topology include the tubular neighborhood theorem: a smooth submanifold of a smooth manifold has a neighborhood that is the total space of a disc bundle over the submanifold. For locally flat topological submanifolds, the best result about neighborhoods (in high dimensions) is due to Edwards [3]: the submanifold has a mapping cylinder neighborhood given by a manifold approximate fibration (see also [14]).For stratified spaces, the stratifications of Whitney are considered to be the correct theory in the smooth category. For Whitney stratified spaces, the tubular neighborhood theorem of Thom [23] and Mather [16], [17] says that each stratum has a neighborhood that is the total space of a bundle over the stratum, and the fiber of the bundle is the cone on the stratified link (see Goresky and MacPherson [4] for an exposition). As is the case for submanifolds, the structure on the neighborhoods is not part of the definition, and the proof of their existence is non-trivial.In the topological category, Quinn [19] has introduced a natural stratification theory. The purpose of this paper is to establish the existence of a type of tubular neighborhood for strata in Quinn's stratified spaces, or manifold stratified spaces.

show abstract

Bundle theories for topological manifolds

Cited by 21 publications

References 28 publications

The additivity of theρ–invariant and periodicity in topological surgery

The additivity of theρ–invariant and periodicity in topological surgery

A Bass–Heller–Swan formula for pseudoisotopies

NEIGHBORHOODS OF STRATA IN MANIFOLD STRATIFIED SPACES Supported in part by NSF Grant DMS-9971367.

Contact Info

Product

Resources

About