Attracting and Repelling Point Pairs for Vector Fields on Manifolds. I
Gabriele Meyer · Transactions of the American Mathematical Society · 1993
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.