homotopic maps from the sphere to the sphere

Question

i have some problems with the exercise 2.2.6 in hatcher book "algebraic topology". Hope that someone could help me out. One has to show that every map $S^{n} \rightarrow S^{n}$ can be homotoped to a map with a fixed point. Hope this is not too trivial. Thanks in advance.

beno

Answer 1

One thing that is proved in that section is that any map without fixed points can be homotoped to the antipodal map. So it remains only to show that the antipodal map is homotopic to a map with a fixed point. A particular example of a homotopy is composition with a family of rotations. So the antipodal map is homotopic to its composition with a half rotation through some axis. Now does this map have any fixed points?