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 $X$ 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 $\mathrm{cone}(X)$.

Definitions

Formally, the cone of X is defined as:

$CX=(X\times [0,1]){\cup }_{p}v=\underrightarrow{\lim }((X\times [0,1])\hookleftarrow (X\times \{0\})\stackrel{p}{\to }v),$

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 $X\times [0,1]$ by its face $X\times \{0\}$ to a point $v$ along the projection $p:(X\times \{0\})\to v$.

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 $v\in ̸X$ such that these segments intersect only in $v$ 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: $CX\simeq X\star \{v\}=$ the join of $X$ with a single point $v\in ̸X$.

Examples

Here we often use a geometric cone ($CX$ 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 ${\mathbb{ℝ}}^2$, $\{p\}\times [0,1]$.
• 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

$\{(x,y,z)\in {\mathbb{ℝ}}^3\mid x^2+y^2=1\text{ and }z=0\}$

is the curved surface of the solid cone:

$\{(x,y,z)\in {\mathbb{ℝ}}^3\mid x^2+y^2=(z-1{)}^2\text{ and }0\le z\le 1\}.$

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

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

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

$(Cf)([x,t])=[f(x),t]$,

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

$(X\times [0,1])/(X\times \left\{0\right\}\cup \left\{x_0\right\}\times [0,1])$

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. ISBN:0-521-79160-X and ISBN:0-521-79540-0
• "Cone". http://planetmath.org/?op=getobj&from=objects&id={{{id}}}.