Editing Möbius transformation (section)

=== Topological proof ===
Topologically, the fact that (non-identity) Möbius transformations fix 2 points (with multiplicity) corresponds to the [[Euler characteristic]] of the sphere being 2:
<math display="block"> \chi(\hat{\Complex}) = 2.</math>

Firstly, the [[projective linear group]] {{nowrap|PGL(2, ''K'')}} is [[n-transitive|sharply 3-transitive]]&nbsp;– for any two ordered triples of distinct points, there is a unique map that takes one triple to the other, just as for Möbius transforms, and by the same algebraic proof (essentially [[dimension counting]], as the group is 3-dimensional). Thus any map that fixes at least 3 points is the identity.

Next, one can see by identifying the  Möbius group with <math>\mathrm{PGL}(2,\Complex)</math> that any Möbius function is homotopic to the identity. Indeed, any member of the [[general linear group]] can be reduced to the identity map by [[Gaussian elimination|Gauss-Jordan elimination]], this shows that the projective linear group is path-connected as well, providing a homotopy to the identity map. The [[Lefschetz–Hopf theorem]] states that the sum of the indices (in this context, multiplicity) of the fixed points of a map with finitely many fixed points equals the [[Lefschetz number]] of the map, which in this case is the trace of the identity map on homology groups, which is simply the Euler characteristic.

By contrast, the projective linear group of the real projective line, {{nowrap|PGL(2, '''R''')}} need not fix any points&nbsp;– for example <math>(1+x) / (1-x)</math> has no (real) fixed points: as a complex transformation it fixes ±''i''<ref group="note">Geometrically this map is the [[stereographic projection]] of a rotation by 90° around ±''i'' with period 4, which takes <math>0 \mapsto 1 \mapsto \infty \mapsto -1 \mapsto 0.</math></ref>&nbsp;– while the map 2''x'' fixes the two points of 0 and ∞. This corresponds to the fact that the Euler characteristic of the circle (real projective line) is 0, and thus the Lefschetz fixed-point theorem says only that it must fix at least 0 points, but possibly more.