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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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
DOI: https://doi.org/10.1090/S0002-9947-1993-1169081-5
MathSciNet review: 1169081
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Abstract: Consider a compact, connected, $ n$-dimensional, triangulable manifold $ M$ without boundary, embedded in $ {{\mathbf{R}}^{n + 1}}$ and a continuous vector field on $ M$, given as a map $ f$ from $ M$ to $ {S^n}$ of degree not equal to 0 or $ {( - 1)^{n + 1}}$. In this paper it is shown that there exists at least one pair of points $ x$, $ y \in M$ satisfying both $ f(x) = - f(y)$ and $ f(x) = \frac{{x - y}} {{\vert\vert x - y\vert\vert}}$. 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 $ f$ is not equal to 0 or $ 1$, then there exists at least one "attracting" pair of points $ x$, $ y \in M$ satisfying both $ f(x) = - f(y)$ and $ f(x) = \frac{{y - x}} {{\vert\vert y - x\vert\vert}}$. The total multiplicities are $ \frac{{k \bullet (k + {{( - 1)}^n})}} {2}$ for repelling pairs and $ \frac{{k \bullet (k - 1)}} {2}$ for attracting pairs.

In the proof, we work with close simplicial approximations of the map $ f$, using Simplicial, Singular and Čech Homology Theory, Künneth's Theorem, Hopf's Classification Theorem and the algebraic intersection number between two $ n$-dimensional homology cycles in a $ 2n$-dimensional space. In the case of repelling pairs, we intersect the graph of $ f$ in $ M \times {S^n}$ with the set of points $ (x,\frac{{x - y}} {{\vert\vert x - y\vert\vert}}) \in M \times {S^n}$, where $ x$ and $ y$ satisfy that $ f(x) = - f(y)$. In order to show that this set carries the homology $ (k,k) \in {H_n}(M \times {S^n},{\mathbf{Z}})$, we study the set $ {A_f} \equiv \{ (x,y) \in M \times M\vert f(x) = - f(y)\} $ in a simplicial setting. Let $ {f_j}$ be a close simplicial approximation of $ f$. It can be shown, that $ {A_{{f_j}}}$ is a homology cycle of dimension $ n$ with a natural triangulation and a natural orientation and that $ {A_f}$ and $ {A_f}_j$ carry the same homology.


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DOI: https://doi.org/10.1090/S0002-9947-1993-1169081-5
Article copyright: © Copyright 1993 American Mathematical Society