Contractible space

Contractible spaces - also referred to as contractible or contractible spaces - are considered in the mathematical sub-area of topology . From the point of view of homotopy theory , contractible spaces are considered trivial . Many of the invariants defined in algebraic topology disappear for contractible spaces.

definition

A topological space is called contractible or contractible or contractible if it is homotopy equivalent to a one-point subspace , that is, if it is a continuous mapping ${\displaystyle X\neq \emptyset }$

${\displaystyle H\colon X\times [0,1]\to X}$

and there is a fixed point so ${\displaystyle p\in X}$

• ${\displaystyle H(x,0)=x}$for everyone and ${\displaystyle x\in X}$
• ${\displaystyle H(x,1)=p}$ for all ${\displaystyle x\in X}$

applies.

example

• Euclidean space is contractible: set ${\displaystyle \mathbb {R} ^{n}}$

  ${\displaystyle H(x,t)=(1-t)x}$for and . ${\displaystyle x\in \mathbb {R} ^{n}}$ ${\displaystyle 0\leq t\leq 1}$

  Note that the space is not "constantly deformed to a point" in the graphic sense: the image of the illustration

  ${\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{n},\quad x\mapsto H(x,t)}$

  is always the entire space, only for the picture is only the origin. ${\displaystyle t<1}$ ${\displaystyle t=1}$

• More generally, star-shaped sets are contractible.

Slightly contractible spaces

A topological space is said to be weakly contractible or weakly contractible if the homotopy groups are trivial for all , i.e. H. ${\displaystyle X}$ ${\displaystyle x\in X}$ ${\displaystyle \pi _{n}(X,x)}$

${\displaystyle \pi _{0}(X,x)=\mathbb {Z} }$and for everyone . ${\displaystyle \pi _{n}(X,x)=0}$ ${\displaystyle n\geq 1}$

If a room is contractible, then it is also weakly contractible. ${\displaystyle X}$

The reverse also applies to CW-complexes : From and for all it follows that the CW-complex is contractible. The converse i applies to any topological space. A. not. ${\displaystyle \pi _{0}(X,x)=\mathbb {Z} }$ ${\displaystyle \pi _{n}(X,x)=0}$ ${\displaystyle n\geq 0}$ ${\displaystyle X}$

More results

The following results are available:

• A non-empty convex subset of Euclidean space is always contractible.
• Every contractible space is path-connected .
• Each retract of a contractible space is contractible.
• A non-empty topological product of non-empty contractible spaces is always contractible. ( Inheritance set )

Counterexamples

• The unit sphere (or more generally: a corresponding sphere with a fixed radius) is not contractible, although it is simply connected . ${\displaystyle \mathbb {S} ^{n}}$ ${\displaystyle n\geqslant 2}$
• The space that can be seen as a union of

${\displaystyle \left\{\left(x,\sin {\frac {1}{x}}\right):x\in (0,1]\right\}\cup \{(0,0)\}}$

with an arc connecting (0, -1) and (1, sin (1)) is not contractible, although all of its homotopy groups are trivial.

This shows that Whitehead's Theorem does not generally have to hold for topological spaces that are not a CW complex .

literature

• Thorsten Camps, Stefan Kühling, Gerhard Rosenberger: Introduction to set-theoretical and algebraic topology (=  Berlin study series on mathematics . Volume 15 ). Heldermann Verlag, Lemgo 2006, ISBN 3-88538-115-X , p. 110 ff . ( MR2172813 ).
• Horst Schubert : Topology . 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 , pp. 156 ff . ( MR0423277 ).
• Stephen Willard: General Topology (=  Addison-Wesley Series in Mathematics ). Addison-Wesley, Reading MA (et al.) 1970, pp. 224 ff . ( MR0264581 ).

Individual evidence

1. ^ Edwin H. Spanier : Algebraic Topology. 1st corrected Springer edition, reprint. Springer, New York NY a. a. 1995, ISBN 3-540-90646-0 , p. 25.
2. ↑ Stephen Willard: General Topology. 1970, p. 224
3. ↑ Thorsten Camps et al .: Introduction to set theoretical and algebraic topology. 2006, p. 112
4. ↑ ^{a b} Stephen Willard: General Topology. 1970, p. 226
5. ↑ ^{a b} Thorsten Camps et al., Op.cit., P. 111
6. ↑ Horst Schubert: Topology. 1975, p. 162