Editing Lefschetz fixed-point theorem (section)

==Historical context==
Lefschetz presented his fixed-point theorem in {{r|Lef1926}}. Lefschetz's focus was not on fixed points of maps, but rather on what are now called [[coincidence point]]s of maps.

Given two maps <math>f</math> and <math>g</math> from an orientable [[manifold]] <math>X</math> to an orientable manifold <math>Y</math> of the same dimension, the ''Lefschetz coincidence number'' of <math>f</math> and <math>g</math> is defined as

:<math>\Lambda_{f,g} = \sum (-1)^k \mathrm{tr}( D_X \circ g^* \circ D_Y^{-1} \circ f_*),</math>

where <math>f_*</math> is as above, <math>g^*</math> is the homomorphism induced by <math>g</math> on the [[cohomology]] groups with rational coefficients, and <math>D_X</math> and <math>D_Y</math> are the [[Poincaré duality]] isomorphisms for <math>X</math> and <math>Y</math>, respectively.

Lefschetz proved that if the coincidence number is nonzero, then <math>f</math> and <math>g</math> have a coincidence point. He noted in his paper that letting <math>X= Y</math> and letting <math>g</math> be the identity map gives a simpler result, which is now known as the fixed-point theorem.