The [<b>Lefschetz Fixed Point Theorem</b>][1] is wonderful. It generalizes the Fixed Point Theorem of Brouwer, and is an indispensable tool in topological analysis of dynamical systems.

The weakest form goes like this. For any continuous function $f:X \to X$ from a triangulable space $X$ to itself, let $H_\ast f:H_\ast X\to H_\ast X$ denote the induced endomorphism of the Rational homology groups. If the alternating sum (over dimension) of the traces

$$\Lambda(f) := \sum_{d \in \mathbb{N}}(-1)^d\text{ Tr}(H_df)$$

is non-zero, then $f$ has a fixed point! Since everything is defined in terms of homology, which is a homotopy invariant, one gets to add "for free" the conclusion that any other self-map of $X$ homotopic to $f$ also has a fixed point.

When $f$ is the identity map, $\Lambda(f)$ equals the Euler characteristic of $X$.

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**Update:** [Here][2] is a lively document written by James Heitsch as a tribute to Raoul Bott. Along with an outline of the standard proof of the LFPT, you can find a large list of interesting applications.


  [1]: https://en.wikipedia.org/wiki/Lefschetz_fixed-point_theorem
  [2]: http://www.math.uic.edu/~heitsch/HeitschBC.pdf "James L. Heitsch: The Lefschetz Principle, Fixed Point Theory, and Index Theory"