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 $\operatorname {cone} (X)$ .

Cone of a circle. The original space X is in blue, and the collapsed end point v is in green.

Definitions

Formally, the cone of X is defined as:

CX=(X\times [0,1])\cup _{p}v\ =\ \varinjlim {\bigl (}(X\times [0,1])\hookleftarrow (X\times \{0\})\xrightarrow {p} 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 $X\times [0,1]$ by its face $X\times \{0\}$ to a point $v$ along the projection $p:{\bigl (}X\times \{0\}{\bigr )}\to v$ .

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 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 $v\not \in X$ .^[1]^: 76

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{\mbox{ 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}{\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

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 $C\colon \mathbf {Top} \to \mathbf {Top}$ 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)([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.

References

^ ^a ^b 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.

[:0-1] 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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