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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.


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up vote 4 down vote accepted

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?

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it is not so clear for me. Could you write which function is homotopy to antipodal map ? – rmznyzgyr Jun 6 at 17:38

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