Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

In "Geometry and Billiards" by S. Tabachnikov the author proves Poincaré's last geometric theorem:

"An area-preserving transformation of an annulus that moves the boundary circles in opposite directions has at least two distinct fixed points."

He states that the hardest part of the argument is to prove the existence of one fixed point and that the existence of the second point follows from a standard topological argument involving Euler characteristic.

Can anybody tell me what this "standard topological argument" is?

share|improve this question

1 Answer 1

Given an automorphism $f:A\to A$ of the annulus, we can define a vector field on $A$ by $v(x)=f(x)-x$, with fixed points of $f$ corresponding to zeros of $v$. By the Poincare-Hopf theorem, the sum of the indices of the zeros of $v$ must be $\chi(A)=0$, so $v$ cannot have only one zero (since the index of a zero is never $0$).

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.