Cone (topology) explained

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) .

Definitions

Formally, the cone of X is defined as:

CX = (X x [ 0 , 1 ])\cup _{p} v = \varinjlim l ( (X x [ 0 , 1 ]) \hookleftarrow (X x \{ 0 \}) \xrightarrow{p} v r ),

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

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

R ^{2} , \{p\} x [ 0 , 1 ] .

• 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

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

is the curved surface of the solid cone:

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

This in turn is homeomorphic to the closed disc.More general examples:^[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 Top \to 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 x [ 0 , 1 ]) / (X x \left\{ 0 \right\} \cup\left\{x _{0\right\} x} [ 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. and

Notes and References

1. , Section 4.3