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Consider the English alphabet in this font with serifs

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Consider any letter in this font as a topological space (assume that letters don't have weight and are genersted by lines) and consider a continuous mapping from any letter to itself. For which letters any such mapping have a fixed point and for which it has not? For O the answer is negative, for C, Z, S it is positive.

The main difficulty for me is serifs. For the sans-serif font like Arial the problem is much easier. Here we can't even easily divide letters into topologically equal groups and these groups aren't obvious: G ~ J, T ~ I ~ U ~ W, E ~ F, C ~ Z ~ S etc.

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Each of the letters deformation retracts to either a point, a circle, or a figure 8.

  • If a letter deformation retracts to a point, Lefschetz fixed point theorem tells us that every map to itself has a fixed point.

  • If a letter retracts to a circle, then the composition of the retraction with a rotation gives us a map which does not have a fixed point.

  • If a letter retracts to a figure 8, the composition of the retraction with the map that collapses one of the loops to a point and rotates the other some angle does not have any fixed points.

In particular, the serifs play no role here.

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    @Nimza: Since all the higher homology groups are zero, the Lefschetz number is simply the trace of the map induced on $H_0$. You can easily compute that!2012-02-27