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 <bernier@my-dejanews.com> 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