Fixed-Point Theorems  

In General > s.a. posets.
* Simple example: If I hold a map of an area including the point where I stand, there must be a point on the map which is exactly above the corresponding point of the real world (even if I deform or crumple the map).
* Motivation: If A is any differential operator, the existence of solutions of the equation A f = 0 is equivalent to the existence of fixed points for A + I; We are interested in equations like df = 0 for the study of critical points (> see morse theory, etc).
* Applications: The earliest one was to the existence of periodic orbits for 3 bodies.
@ General references: Granas & Dugundji 03; Farmakis & Moskowitz 13 [and applications].
@ Various types: Prykarpatsky a0902 [Leray-Schauder, Borsuk-Ulam type generalization].
> Online resources: see Wikipedia page.

Simple Case: Real Functions of One Variable
$ Def: Given any interval [a, b] ⊂ \(\mathbb R\), any f : [a, b] → [a, b] must have at least one fixed point (the graph must cross at least once the line x = y).

Brouwer Fixed-Point Theorem
$ Def: Any continuous f : Dn → Dn has (at least) one fixed point (Dn is the n-dimensional ball).
* Special cases: For n = 2, it can be proved using the fundamental group, through the fact that S1 is not a retract of D2 (in general, it is done using higher-order homotopy groups, or considering the index of the vector field v(x):= xf(x), at different points with respect to different loops).

Lefschetz Fixed-Point Theorem
* Idea: Let S be an n-dimensional manifold (or "almost"), and f a map f : SS; Then f induces a map f* of the homology groups,

f*: Hi → Hi,    for    i = 1, ..., n,

and from these we can define some numbers ni, which are something like the trace of f*, and

L = ∑i (–1)i ni ,    the Lefschetz number ;

The theorem states that this number can also be obtained as a sum of contributions of all the fixed points of f.
* Relationships: It is like a finite (as opposed to infinitesimal) generalization of the concept of Euler characteristic.
* Example: Consider the inversion map on S2, A: S2 → S2; We know that this does not have any fixed points; H0 = \(\mathbb R\), H1 = 0, H2 = \(\mathbb R\), and

L = ∑i (–1)i ni = 1 · 1 + (–1) · 0 + 1 · (–1) = 0 ;

If the map had been orientation-preserving, there would have been fixed points and L ≠ 0.
* Example: Every Lorentz transformation fixes at least one null direction.
@ References: van Lon MS-a1509 [quantum mechanical path integral methods, and other index theorems].

Borel Fixed-Point Theorem
$ Def: A connected solvable linear algebraic group over an algebraic closed field, when acting on a complete variety, has a fixed point.
@ References: Borel AM(56).

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