Cone (topology)

In topology, especially algebraic topology, the cone of a topological space $X$ 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 $CX$ or by $\operatorname(X)$.

Definitions

Formally, the cone of ''X'' is defined as:

:CX = (X \times ,1\cup_p v\ =\ \varinjlim \bigl( (X \times ,1 \hookleftarrow (X\times \) \xrightarrow v\bigr),

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

If $X$ is a non-empty compact subspace of Euclidean space, the cone on $X$ is homeomorphic to the union of segments from $X$ to any fixed point $v \not\in X$ such that these segments intersect only by $v$ 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: $CX \simeq X\star \ =$ the join of $X$ with a single point $v\not\in X$.''''

Examples

Here we often use a geometric cone ($C X$ where $X$ 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 $\mathbb^2$, \ \times ,1/math>.
* The cone over two points is a "V" shape with endpoints at and .
* 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:'', Section 4.3''
* 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

:$h_t(x,s) = (x, (1-t)s)$.

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 $CX$ 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 $CX$ will be finer than the set of lines joining ''X'' to a point.

Cone functor

The map $X\mapsto CX$ induces a functor $C\colon \mathbf\to\mathbf$ on the category of topological spaces Top. If $f \colon X \to Y$ is a continuous map, then $Cf \colon CX \to CY$ is defined by
:(Cf)( ,t= (x),t/math>,
where square brackets denote equivalence classes.

Reduced cone

If $(X,x_0)$ is a pointed space, there is a related construction, the reduced cone, given by
:(X\times ,1 / (X\times \left\
\cup\left\\times ,1

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

See also

* Cone (disambiguation)
*Suspension (topology)
* Desuspension
*Mapping cone (topology)
*Join (topology)

References

*Allen Hatcher
''Algebraic topology.''
Cambridge University Press, Cambridge, 2002. xii+544 pp. and
*{{planetmath reference, urlname=Cone, title=Cone

Topology
Algebraic topology