# Proof for the existence of a second fixed point in Poincaré's last geometric theorem

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?

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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 $0$).