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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Attracting and repelling point pairs for vector fields on manifolds. I
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by Gabriele Meyer PDF
Trans. Amer. Math. Soc. 336 (1993), 497-507 Request permission

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 $f(x) = \frac {{x - y}} {{||x - y||}}$. 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 $f(x) = \frac {{y - x}} {{||y - x||}}$. 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 $(x,\frac {{x - y}} {{||x - y||}}) \in M \times {S^n}$, 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 ${A_f} \equiv \{ (x,y) \in M \times M|f(x) = - f(y)\}$ 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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Additional Information
  • © Copyright 1993 American Mathematical Society
  • 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