Math 6510 • Differentiable Manifolds

For \(t\in (0,1)\) define a map \(f_t:S^2\to S^2\) by \[ f_t(x,y,z)=\left ( \frac {x}{\sqrt {x^2+y^2+(z+t)^2}}, \frac {y}{\sqrt {x^2+y^2+(z+t)^2}}, \frac {z+t}{\sqrt {x^2+y^2+(z+t)^2}} \right ). \] Show that \(f_t\) is a Lefschetz map with isolated fixed points at the two antipodal points with \(z=\pm 1\) and the Lefschetz number is \(L(f)=2\). Conclude that every diffeomorphism of \(S^2\) that is homotopic to the identity has a fixed point. (Hint: One can simply compute the derivative of \(f_t\) to see that on the tangent spaces of the two fixed points it is a multiple of the identity. However, if you use that \(f_t\) is a composition of the map \((x,y,z)\mapsto (x,y,z+t)\) and the 3-dimensional version of the map \(r\) from (2) then the computation is easier.)