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 coneof a topological space ${\displaystyle X}$ 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 ${\displaystyle CX}$ or by ${\displaystyle \operatorname {cone} (X)}$.

Definitions

Formally, the cone of X is defined as:

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

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

If ${\displaystyle X}$ is a non-empty compact subspace of Euclidean space, the cone on ${\displaystyle X}$ is homeomorphic to the union of segments from ${\displaystyle X}$ to any fixed point ${\displaystyle v\not \in X}$ such that these segments intersect only in ${\displaystyle v}$ 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: ${\displaystyle CX\simeq X\star \{v\}=}$ the join of ${\displaystyle X}$ with a single point ${\displaystyle v\not \in X}$. ^{[1] : 76}

Examples

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

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

is the curved surface of the solid cone:

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

This in turn is homeomorphic to the closed disc.

More general examples: ^{[1] : 77, Exercise.1}

• 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

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

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 ${\displaystyle CX}$ 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 ${\displaystyle CX}$ will be finer than the set of lines joining X to a point.

Cone functor

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

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

where square brackets denote equivalence classes.

Reduced cone

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

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

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

See also

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

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

• Allen Hatcher, Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN   0-521-79160-X and ISBN   0-521-79540-0
• "Cone". PlanetMath .