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 d*f* = 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* :
D^{n} → D^{n}
has (at least) one fixed point (D^{n} is the
*n*-dimensional ball).

* __Special cases__: For *n* = 2, it
can be proved using the fundamental group, through the fact that S^{1} is
not a retract of D^{2} (in general, it is done using
higher-order homotopy groups, or considering the index of the vector field
*v*(*x*):= *x* − *f*(*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 *: *S* → *S*; Then *f* induces a
map *f*_{*} of the homology groups,

*f*_{*}:
H_{i}
→ H_{i},
for * i*
= 1, ..., *n*,

and from these we can define some numbers *n*_{i},
which are something like the trace of *f*_{*}, and

*L* = ∑_{i}
(−1)^{i}
*n*_{i} ,
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 S^{2}, *A*:
S^{2} → S^{2};
We know that this does not have any fixed points;
*H*_{0} = \(\mathbb R\),
*H*_{1} = 0,
*H*_{2} = \(\mathbb R\),
and

*L* = ∑_{i}
(−1)^{i}
*n*_{i}
= 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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