On Dold-Whitney's parallelizability of 4-manifolds

IF 0.6 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2024-11-19 DOI:10.1016/j.topol.2024.109144

Valentina Bais

引用次数: 0

Abstract

We present a proof of a theorem by Dold and Whitney, according to which a closed orientable 4-manifold is parallelizable if and only if its second Stiefel-Whitney class, first Pontryagin class and Euler characteristics vanish. This follows from a stronger result due to Dold and Whitney on the classification of oriented sphere bundles over a 4-complex. Our proof is based on an argument by R. Kirby on the classification of

S O (4)

-principal bundles over the 4-sphere by means of their Euler and first Pontryagin classes.

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论多尔德-惠特尼的 4 维平行性

我们提出了多尔德和惠特尼定理的证明，根据该定理，当且仅当封闭的可定向 4-manifold 的第二 Stiefel-Whitney 类、第一 Pontryagin 类和欧拉特征消失时，它是可平行的。这源于多尔德和惠特尼关于 4 复合体上定向球束分类的更强结果。我们的证明基于柯比（R. Kirby）通过欧拉级和第一庞特里亚金级对 4 球上 SO(4)- 主束分类的论证。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.