Attracting and repelling point pairs for vector fields on manifolds. I
Author: Gabriele Meyer
Journal: Trans. Amer. Math. Soc. 336 (1993), 497-507
MSC: Primary 55M25; Secondary 55N05, 55N10, 57Q55, 57R25
MathSciNet review: 1169081
Abstract: Consider a compact, connected, -dimensional, triangulable manifold without boundary, embedded in and a continuous vector field on , given as a map from to of degree not equal to 0 or . In this paper it is shown that there exists at least one pair of points , satisfying both and . Geometrically, this means, that the points and the vectors lie on one straight line and the vector field is "repelling". Similarly, if the degree of is not equal to 0 or , then there exists at least one "attracting" pair of points , satisfying both and . The total multiplicities are for repelling pairs and for attracting pairs.
In the proof, we work with close simplicial approximations of the map , using Simplicial, Singular and Čech Homology Theory, Künneth's Theorem, Hopf's Classification Theorem and the algebraic intersection number between two -dimensional homology cycles in a -dimensional space. In the case of repelling pairs, we intersect the graph of in with the set of points , where and satisfy that . In order to show that this set carries the homology , we study the set in a simplicial setting. Let be a close simplicial approximation of . It can be shown, that is a homology cycle of dimension with a natural triangulation and a natural orientation and that and carry the same homology.
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