From: lrudolph@panix.com (Lee Rudolph) Subject: Re: fixed point theorems & "hairy ball" type results for compact surfaces Date: 4 Apr 1999 14:57:13 -0400 Newsgroups: sci.math Keywords: When do smooth vector fields have zeroes? David Bernier writes: >Thanks for the detailed explanation. As I understand it, compact >orientable n-manifolds for n=>3 are not completely classified, and >in particular the Poincare conjecture (n=3) remains unsettled. > >What is known about the existence of zero points of smooth vector >fields , and the other problem of the existence of fixed points on >self-map homeomorphisms, for this vast (?) class of manifolds? Everything is known. Robin Chapman's reply gave the relevant information (but may have been phrased so that you didn't realize that): in *all* dimensions (not just for surfaces), a compact smooth manifold without boundary admits a nowhere- zero vectorfield if and only if its Euler characteristic is 0 if and only if it admits a fixed-point-free autohomeomorphism. A connected manifold with non-empty boundary, by contrast, always admits a nowhere-zero vectorfield. Lee Rudolph ============================================================================== From: lrudolph@panix.com (Lee Rudolph) Subject: Re: fixed point theorems & "hairy ball" type results for compact surfaces Date: 4 Apr 1999 19:09:41 -0400 Newsgroups: sci.math Inspired by Robin Chapman's "doh", I want to issue a "doh" of my own. I wrote: >a compact smooth manifold without boundary admits a nowhere- >zero vectorfield if and only if its Euler characteristic is 0 >if and only if it admits a fixed-point-free autohomeomorphism and of course I should have appended "homotopic to the identity" after "autohomeomorphism". Lee Rudohlph