Cone (topology)

Circle cone. The source space is highlighted in blue, the constricted endpoint is highlighted in green.

A cone in topology is a topological space obtained from the original space $X$ ${\ displaystyle X}$ $X$ contraction of subspace $X\times \{0\}$ ${\ displaystyle X \ times \ {0 \}}$ $X \ times \ {0 \}$ its cylinder ( $X\times [0,1]$ ${\ displaystyle X \ times [0,1]}$ $X \ times [0, 1]$ ) at one point, i.e., factor space $(X\times [0,1])/(X\times \{0\})$ ${\ displaystyle (X \ times [0,1]) / (X \ times \ {0 \})}$ $(X \ times [0, 1]) / (X \ times \ {0 \})$ . Cone over space $X$ ${\ displaystyle X}$ $X$ is indicated $\mathrm {C} X$ ${\ displaystyle \ mathrm {C} X}$ $\ mathrm CX$ .

If $X$ ${\ displaystyle X}$ $X$ Is a compact subset of Euclidean space , then the cone over $X$ ${\ displaystyle X}$ $X$ homeomorphic to the union of segments from $X$ ${\ displaystyle X}$ $X$ to a distinguished point in space, that is, the definition of a topological cone is consistent with the definition of a geometric cone . However, a topological cone is a more general construction.

Content

1 Examples
2 Properties
3 Conical functor
4 reduced cone
5 See also
6 notes
7 Literature

Examples

Cone over point $p$ ${\ displaystyle p}$ real line is the interval $\{p\}\times [0,1]$ ${\ displaystyle \ {p \} \ times [0,1]}$ , the cone over the interval of the real line is a filled triangle (2-simplex), the cone over the polygon $P$ ${\ displaystyle P}$ Is a base pyramid $P$ ${\ displaystyle P}$ . A cone above a circle is a classic cone (with an inside); cone over a circle - lateral surface of a classic cone:

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

{\ displaystyle \ {(x, y, z) \ in \ mathbb {R} ^ {3} \ mid x ^ {2} + y ^ {2} = z ^ {2} \ wedge 0 \ leqslant z \ leqslant one\}}

,

homeomorphic to a circle .

In the general case, a cone over a hypersphere is homeomorphic to a closed $(n+1)$ ${\ displaystyle (n + 1)}$ -dimensional ball . Cone over $n$ ${\ displaystyle n}$ - simplex - $(n+1)$ ${\ displaystyle (n + 1)}$ simplex.

Properties

Cone $\mathrm {C} X$ ${\ displaystyle \ mathrm {C} X}$ can be designed as a constant display cylinder $X\to \{0\}$ ${\ displaystyle X \ to \ {0 \}}$ ^[1] .

All cones are linearly connected , since any point can be connected to the vertex. Moreover, any cone is contractible to the vertex using the homotopy defined by the formula $h_{t}(x,s)=(x,(1-t)s)$ ${\ displaystyle h_ {t} (x, s) = (x, (1-t) s)}$ .

If $X$ ${\ displaystyle X}$ is compact and Hausdorff , then the cone $\mathrm {C} X$ ${\ displaystyle \ mathrm {C} X}$ can be represented as the space of segments connecting each point $X$ ${\ displaystyle X}$ with a single point; if $X$ ${\ displaystyle X}$ is not compact or Hausdorff, this is not so, since in the general case the topology on the quotient space $\mathrm {C} X$ ${\ displaystyle \ mathrm {C} X}$ will be thinner than many pieces connecting $X$ ${\ displaystyle X}$ with a dot.

In algebraic topology, cones are widely used due to the fact that they represent spaces as embeddings in a contractible space; In this regard, the following result is also important: space $X$ ${\ displaystyle X}$ is contractible if and only if it is a retract of its cone.

Conic functor

Display $X\mapsto \mathrm {C} X$ ${\ displaystyle X \ mapsto \ mathrm {C} X}$ generates a conic functor - endofunctor $\mathrm {C} :\mathbf {Top} \to \mathbf {Top}$ ${\ displaystyle \ mathrm {C}: \ mathbf {Top} \ to \ mathbf {Top}}$ over the category of topological spaces $\mathbf {Top}$ ${\ displaystyle \ mathbf {Top}}$ .

Reduced Cone

Reduced cone - a construction over a ^[2] $(X,x_{0})$ ${\ displaystyle (X, x_ {0})}$ :

\mathrm {C} (X,x_{0})={\big (}X\times [0,1]{\big )}/{\big (}(X\times \left\{0\right\})\cup (\left\{x_{0}\right\}\times [0,1]){\big )}

{\ displaystyle \ mathrm {C} (X, x_ {0}) = {\ big (} X \ times [0,1] {\ big)} / {\ big (} (X \ times \ left \ {0 \ right \}) ​​\ cup (\ left \ {x_ {0} \ right \} \ times [0,1]) {\ big)}}

.

Natural investment $x\mapsto (x,1)$ ${\ displaystyle x \ mapsto (x, 1)}$ allows us to consider any punctured space as a closed subset of its reduced cone ^[3] .

Notes

↑ Speyer, 1971 , p. 77.
↑ Sweitzer, 1985 , p. 13.
↑ Speyer, 1971 , p. 469.

Literature

Alain Hatcher. Algebraic topology. - Moscow: Publishing House MTsNMO, 2011. - ISBN 978-5-940-57-748-5 .
R. M. Switzer. Algebraic topology - homotopy and homology. - Moscow: “Science”, Main Edition of the Physics and Mathematics Literature, 1985.
E. Spenier. Algebraic topology. - Moscow: Mir, 1971.

[_872cfd646194b15c-1] Speyer, 1971 , p. 77.

[_526978a900710671-2] Sweitzer, 1985 , p. 13.

[_46276c91cfac6cc1-3] Speyer, 1971 , p. 469.