NDLI: Stiefel—Whitney currents

Content Provider	Springer Nature Link
Author	Harvey, Reese Zweck, John
Copyright Year	1998
Abstract	A canonically defined mod 2 linear dependency current is associated to each collection v of sections, v$_{1}$,…,v$_{m}$, of a real rank n vector bundle. This current is supported on the linear dependency set of v. It is defined whenever the collection v satisfies a weak measure theoretic condition called “atomicity.” Essentially any reasonable collection of sections satisfies this condition, vastly extending the usual general position hypothesis. This current is a mod 2 d-closed locally integrally flat current of degree q = n −m + 1 and hence determines a ℤ$_{2}$-cohomology class. This class is shown to be well defined independent of the collection of sections. Moreover, it is the qth Stiefel-Whitney class of the vector bundle.More is true if q is odd or q = n. In this case a linear dependency current which is twisted by the orientation of the bundle can be associated to the collection v. The mod 2 reduction of this current is the mod 2 linear dependency current. The cohomology class of the linear dependency current is 2-torsion and is the qth twisted integral Stiefel-Whitney class of the bundle.In addition, higher dependency and general degeneracy currents of bundle maps are studied, together with applications to singularities of projections and maps.These results rely on a theorem of Federer which states that the complex of integrally flat currents mod p computes cohomology mod p. An alternate approach to Federer’s theorem is offered in an appendix. This approach is simpler and is via sheaf theory.
Starting Page	809
Ending Page	844
Page Count	36
File Format	PDF
ISSN	10506926
Journal	Journal of Geometric Analysis
Volume Number	8
Issue Number	5
e-ISSN	1559002X
Language	English
Publisher	Springer-Verlag
Publisher Date	1998-01-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Differential Geometry Convex and Discrete Geometry Fourier Analysis Abstract Harmonic Analysis Dynamical Systems and Ergodic Theory Global Analysis and Analysis on Manifolds
Content Type	Text
Resource Type	Article
Subject	Geometry and Topology

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