On the construction and topological invariance of the Pontryagin classes

Ranicki, Andrew; Weiss, Michael

doi:10.1007/s10711-010-9527-2

Cited by 8 publications

(3 citation statements)

References 26 publications

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“…From this point of view, the difficulties that we find in constructing the bordismtheory instead appear when one tries to obtain fundamental classes, whose construction beyond the manifold-case [35, §16] and pl-case [4] again becomes difficult. See also [36].…”

Section: Introductionmentioning

confidence: 99%

Topological Bordism of Singular Spaces and an Application to Stratified L-Classes

Rabel¹

2022

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A generalized-homology bordism-theory is constructed, such that for certain manifold homotopy stratified sets (MHSS; Quinn-spaces) homeomorphism-invariant geometric fundamental-classes exist. The construction combines three ideas: Firstly, instead of restricting geometric cycles by conditions on links only, a more flexible framework is built directly via geometric properties, secondly, controlled topology methods are used to give an accessible link-based criterion to detect suitable cycles and thirdly, a geometric argument is used to show, that these classes of cycles are suitable to study the transition to intrinsic stratifications. As an application, we give a construction of topologically (homeomorphism) invariant (homological) L-classes on MHSS Witt-spaces satisfying conditions on Whitehead-groups of links and the dimensional spacing of meeting strata. These L-classes agree, whenever those spaces are additionally pl-pseudomanifolds, with the Goresky-MacPherson L-classes.

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Section: Introductionmentioning

confidence: 99%

Topological Bordism of Singular Spaces and an Application to Stratified L-Classes

Rabel¹

2022

Preprint

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“…Recall that the rational Pontrjagin classes of a smooth manifold are defined using the smooth structure. A celebrated result of Novikov [Nov66] shows however that these classes in fact only depend on the underlying topological structure (other proofs were given in [Gro95], [Ran95], [RY06], and [RW10]). More precisely, if one has a pair of homeomorphic smooth manifolds, then the homeomorphism can be chosen to take the total rational Pontrjagin class to the total rational Pontrjagin class.…”

Section: Introductionmentioning

confidence: 99%

Some Kähler structures on products of 2-spheres

2019

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We consider a family of Kähler structures on products of 2-spheres, arising from complex Bott manifolds. These are obtained via iterated P 1 -bundle constructions, generalizing the classical Hirzebruch surfaces. We show that the resulting Kähler structures all have identical Chern classes. We construct Bott diagrams, which are rooted forests with an edge labelling by positive integers, and show that these classify these Kähler structures up to biholomorphism.2010 Mathematics Subject Classification. 32Q15 (primary), 32L05, 32Q10, 53C55 (secondary).

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“…9 See [17, Thm. 0] for the precise result used here and see [37,Appendix] for a history of the result. Igor Belegradek explains on MathOverflow (https://mathoverflow.net/q/442025) why compactness assumptions are not required.…”

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confidence: 99%