G space

In geometry, a G-space is a topological space provided with a continuous group effect . Continuous group effects and the general terms defined in this context occur naturally in many mathematical problems.

definition

Let be a topological space, a ( topological or discrete ) group and ${\displaystyle M}$ ${\displaystyle G}$

${\displaystyle G\times M\rightarrow M}$

${\displaystyle (g,x)\mapsto gx}$

a continuous effect of on , that is, a continuous mapping with ${\displaystyle G}$ ${\displaystyle M}$

${\displaystyle g(hx)=(gh)x}$

for everyone as well ${\displaystyle g,h\in G,x\in M}$

${\displaystyle ex=x}$

for the neutral element and all , then it is called G-space. ${\displaystyle e\in G}$ ${\displaystyle x\in M}$ ${\displaystyle M}$

More terms

In the following, let it be a G-space, the product topology and the path space the quotient topology . ${\displaystyle M}$ ${\displaystyle G\times M}$ ${\displaystyle G\backslash M}$

Transitive effect

One effect is transitive , if for every pair one with there. ${\displaystyle G\times M\rightarrow M}$ ${\displaystyle (x,y)\in M\times M}$ ${\displaystyle g\in G}$ ${\displaystyle gx=y}$

If transitive acts on, then is homeomorphic to with the quotient topology, where the stabilizer is an (arbitrary) element . ${\displaystyle G}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle G/G_{x}}$ ${\displaystyle G_{x}=\left\{g\in G:gx=x\right\}}$ ${\displaystyle x\in M}$

Free effect

An effect is called free if it always follows from (with and ) . ${\displaystyle G\times M\rightarrow M}$ ${\displaystyle gx=x}$ ${\displaystyle g\in G}$ ${\displaystyle x\in M}$ ${\displaystyle g=e}$

An effect is free if, for all of them, the stabilizer consists only of the neutral element. ${\displaystyle x\in M}$ ${\displaystyle G_{x}\subset G}$

Effective effect

One effect is effective (or true ), if for every one with there. ${\displaystyle g\not =e}$ ${\displaystyle x\in M}$ ${\displaystyle gx\not =x}$

An effect is then only effective if the corresponding homomorphism of in the group of homeomorphisms of is a monomorphism . ${\displaystyle G}$ ${\displaystyle X}$

Fixed points

The fixed points of an element are the elements with . ${\displaystyle g\in G}$ ${\displaystyle x\in M}$ ${\displaystyle gx=x}$

A point is called the global fixed point of the group effect if it applies to all . ${\displaystyle x\in M}$ ${\displaystyle gx=x}$ ${\displaystyle g\in G}$

Actual effect

An effect actually means when it is through ${\displaystyle G\times M\rightarrow M}$

${\displaystyle (g,x)\rightarrow (x,gx)}$

given figure is an actual figure . ${\displaystyle \rho :G\times X\rightarrow X\times X}$

If the effect of on is actually Hausdorffsch and all orbits are closed. The stabilizer of each point is compact and the map is a homeomorphism . ${\displaystyle G}$ ${\displaystyle M}$ ${\displaystyle G\backslash M}$ ${\displaystyle Gx}$ ${\displaystyle G/G_{x}\rightarrow Gx}$

Actually discontinuous action, discontinuity area

An effect is actually called discontinuous if there is an environment for everyone for which ${\displaystyle G\times M\rightarrow M}$ ${\displaystyle x\in M}$ ${\displaystyle U}$

${\displaystyle \sharp \left\{g\in G:gU\cap U\not =\emptyset \right\}<\infty }$.

A free effect is actually discontinuous if and only if the projection is an overlay . ${\displaystyle M\rightarrow G\backslash M}$

An -invariant, open subset is called a discontinuity domain if the effect of on is actually discontinuous. In general, a maximum discontinuity area need not be uniquely determined. ${\displaystyle G}$ ${\displaystyle \Omega \subset M}$ ${\displaystyle G}$ ${\displaystyle U}$

In the case of a Klein group and its effect on the sphere at infinity there is a definite maximum discontinuity area, this is the complement of the Limes set and is often referred to as the discontinuity area of ​​the Klein group. (This also applies more generally to discrete groups of isometries of Hadamard manifolds and their effect on the sphere at infinity.)

Co-compact effect

An effect is called co-compact when the orbit space is compact. ${\displaystyle G\times M\rightarrow M}$ ${\displaystyle G\backslash M}$

An effect is co-compact when there is a compact fundamental domain .

Geometric effect

An effect is called geometric (English: geometric action ) if it is actually discontinuous and co-compact.

swell

1. ^ Tammo tom Dieck : Algebraic Topology . European Mathematical Society Publishing House, Zurich 2008, ISBN 978-3-03719-048-7 , pp. 17 .
2. ^ Properly discontinuous actions