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Cone (topology)

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(Redirected from Reduced cone)
Cone of a circle. The original space X is in blue, and the collapsed end point v is in green.

In topology, especially algebraic topology, the cone of a topological space X {\displaystyle X} is intuitively obtained by stretching X into a cylinder and then collapsing one of its end faces to a point. The cone of X is denoted by C X {\displaystyle CX} or by cone ( X ) {\displaystyle \operatorname {cone} (X)} .

Definitions

Formally, the cone of X is defined as:

C X = ( X × [ 0 , 1 ] ) p v   =   lim ( ( X × [ 0 , 1 ] ) ( X × { 0 } ) p v ) , {\displaystyle CX=(X\times )\cup _{p}v\ =\ \varinjlim {\bigl (}(X\times )\hookleftarrow (X\times \{0\})\xrightarrow {p} v{\bigr )},}

where v {\displaystyle v} is a point (called the vertex of the cone) and p {\displaystyle p} is the projection to that point. In other words, it is the result of attaching the cylinder X × [ 0 , 1 ] {\displaystyle X\times } by its face X × { 0 } {\displaystyle X\times \{0\}} to a point v {\displaystyle v} along the projection p : ( X × { 0 } ) v {\displaystyle p:{\bigl (}X\times \{0\}{\bigr )}\to v} .

If X {\displaystyle X} is a non-empty compact subspace of Euclidean space, the cone on X {\displaystyle X} is homeomorphic to the union of segments from X {\displaystyle X} to any fixed point v X {\displaystyle v\not \in X} such that these segments intersect only in v {\displaystyle v} itself. That is, the topological cone agrees with the geometric cone for compact spaces when the latter is defined. However, the topological cone construction is more general.

The cone is a special case of a join: C X X { v } = {\displaystyle CX\simeq X\star \{v\}=} the join of X {\displaystyle X} with a single point v X {\displaystyle v\not \in X} .

Examples

Here we often use a geometric cone ( C X {\displaystyle CX} where X {\displaystyle X} is a non-empty compact subspace of Euclidean space). The considered spaces are compact, so we get the same result up to homeomorphism.

  • The cone over a point p of the real line is a line-segment in R 2 {\displaystyle \mathbb {R} ^{2}} , { p } × [ 0 , 1 ] {\displaystyle \{p\}\times } .
  • The cone over two points {0, 1} is a "V" shape with endpoints at {0} and {1}.
  • The cone over a closed interval I of the real line is a filled-in triangle (with one of the edges being I), otherwise known as a 2-simplex (see the final example).
  • The cone over a polygon P is a pyramid with base P.
  • The cone over a disk is the solid cone of classical geometry (hence the concept's name).
  • The cone over a circle given by
{ ( x , y , z ) R 3 x 2 + y 2 = 1  and  z = 0 } {\displaystyle \{(x,y,z)\in \mathbb {R} ^{3}\mid x^{2}+y^{2}=1{\mbox{ and }}z=0\}}
is the curved surface of the solid cone:
{ ( x , y , z ) R 3 x 2 + y 2 = ( z 1 ) 2  and  0 z 1 } . {\displaystyle \{(x,y,z)\in \mathbb {R} ^{3}\mid x^{2}+y^{2}=(z-1)^{2}{\mbox{ and }}0\leq z\leq 1\}.}
This in turn is homeomorphic to the closed disc.

More general examples:

  • The cone over an n-sphere is homeomorphic to the closed (n + 1)-ball.
  • The cone over an n-ball is also homeomorphic to the closed (n + 1)-ball.
  • The cone over an n-simplex is an (n + 1)-simplex.

Properties

All cones are path-connected since every point can be connected to the vertex point. Furthermore, every cone is contractible to the vertex point by the homotopy

h t ( x , s ) = ( x , ( 1 t ) s ) {\displaystyle h_{t}(x,s)=(x,(1-t)s)} .

The cone is used in algebraic topology precisely because it embeds a space as a subspace of a contractible space.

When X is compact and Hausdorff (essentially, when X can be embedded in Euclidean space), then the cone C X {\displaystyle CX} can be visualized as the collection of lines joining every point of X to a single point. However, this picture fails when X is not compact or not Hausdorff, as generally the quotient topology on C X {\displaystyle CX} will be finer than the set of lines joining X to a point.

Cone functor

The map X C X {\displaystyle X\mapsto CX} induces a functor C : T o p T o p {\displaystyle C\colon \mathbf {Top} \to \mathbf {Top} } on the category of topological spaces Top. If f : X Y {\displaystyle f\colon X\to Y} is a continuous map, then C f : C X C Y {\displaystyle Cf\colon CX\to CY} is defined by

( C f ) ( [ x , t ] ) = [ f ( x ) , t ] {\displaystyle (Cf)()=} ,

where square brackets denote equivalence classes.

Reduced cone

If ( X , x 0 ) {\displaystyle (X,x_{0})} is a pointed space, there is a related construction, the reduced cone, given by

( X × [ 0 , 1 ] ) / ( X × { 0 } { x 0 } × [ 0 , 1 ] ) {\displaystyle (X\times )/(X\times \left\{0\right\}\cup \left\{x_{0}\right\}\times )}

where we take the basepoint of the reduced cone to be the equivalence class of ( x 0 , 0 ) {\displaystyle (x_{0},0)} . With this definition, the natural inclusion x ( x , 1 ) {\displaystyle x\mapsto (x,1)} becomes a based map. This construction also gives a functor, from the category of pointed spaces to itself.

See also

References

  1. ^ Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler , Section 4.3
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