Cone (topology)

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 <math>X</math> 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 <math>CX</math> or by <math>\operatorname{cone}(X)</math>.

DefinitionsEdit

Formally, the cone of X is defined as:

<math>CX = (X \times [0,1])\cup_p v\ =\ \varinjlim \bigl( (X \times [0,1]) \hookleftarrow (X\times \{0\}) \xrightarrow{p} v\bigr),</math>

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

If <math>X</math> is a non-empty compact subspace of Euclidean space, the cone on <math>X</math> is homeomorphic to the union of segments from <math>X</math> to any fixed point <math>v \not\in X</math> such that these segments intersect only in <math>v</math> 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: <math>CX \simeq X\star \{v\} = </math> the join of <math>X</math> with a single point <math>v\not\in X</math>.<ref name=":0" />Template:Rp

ExamplesEdit

Here we often use a geometric cone (<math>C X</math> where <math>X</math> 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 <math>\mathbb{R}^2</math>, <math>\{p\} \times [0,1]</math>.
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

is the curved surface of the solid cone:

This in turn is homeomorphic to the closed disc.

More general examples:<ref name=":0">Template:Cite Matousek 2007, Section 4.3</ref>Template:Rp

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.

PropertiesEdit

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

<math>h_t(x,s) = (x, (1-t)s)</math>.

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 <math>CX</math> 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 <math>CX</math> will be finer than the set of lines joining X to a point.

Cone functorEdit

The map <math>X\mapsto CX</math> induces a functor <math>C\colon \mathbf{Top}\to\mathbf{Top}</math> on the category of topological spaces Top. If <math>f \colon X \to Y</math> is a continuous map, then <math>Cf \colon CX \to CY</math> is defined by

<math>(Cf)([x,t])=[f(x),t]</math>,

where square brackets denote equivalence classes.

Reduced coneEdit

If <math>(X,x_0)</math> is a pointed space, there is a related construction, the reduced cone, given by

<math>(X\times [0,1]) / (X\times \left\{0\right\}

\cup\left\{x_0\right\}\times [0,1])</math>

where we take the basepoint of the reduced cone to be the equivalence class of <math>(x_0,0)</math>. With this definition, the natural inclusion <math>x\mapsto (x,1)</math> becomes a based map. This construction also gives a functor, from the category of pointed spaces to itself.

ReferencesEdit

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Allen Hatcher, Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp. Template:ISBN and Template:ISBN
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