Historical Context
Lefschetz presented his fixed point theorem in . Lefschetz's focus was not on fixed points of mappings, but rather on what are now called coincidence points of mappings.
Given two maps f and g from an orientable manifold X to an orientable manifold Y of the same dimension, the Lefschetz coincidence number of f and g is defined as
where f∗ is as above, g∗ is the mapping induced by g on the cohomology groups with rational coefficients, and DX and DY are the Poincaré duality isomorphisms for X and Y, respectively.
Lefschetz proves that if the coincidence number is nonzero, then f and g have a coincidence point. He notes in his paper that letting X = Y and letting g be the identity map gives a simpler result, which we now know as the fixed point theorem.
Read more about this topic: Lefschetz Fixed-point Theorem
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