nLab
join of topological spaces

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

Definition

Write I[0,1]I \coloneqq [0,1] for the unit interval, regarded as a topological space.

For X,YX,Y two topological spaces, the join XYX \star Y is the quotient space

XY(X×I×Y) / X \star Y \coloneqq (X \times I \times Y)_{/\sim}

of the product space X×I×YX \times I \times Y by the equivalence relation

(x,0,y 1)(x,0,y 2),(x 1,1,y)(x 2,1,y). (x, 0, y_1) \simeq (x,0,y_2) \;\;,\;\; (x_1,1,y) \sim (x_2, 1, y) \,.

In other words, the join is the colimit in Top of the diagram

X×Y X×Y π 1 i 0 i 1 π 2 X X×I×Y Y \array{ & & X \times Y & & & & X \times Y & & \\ & \mathllap{\pi_1} \swarrow & & \searrow \mathrlap{i_0} & & \mathllap{i_1} \swarrow & & \searrow \mathrlap{\pi_2} & \\ X & & & & X \times I \times Y & & & & Y }

where i 0i_0 is the inclusion (x,y)(x,0,y)(x, y) \mapsto (x, 0, y), and i 1i_1 is the inclusion (x,y)(x,1,y)(x, y) \mapsto (x, 1, y).

(This in turn, by the discussion at mapping cone, is a model for the homotopy type of the homotopy pushout of the two projections XX×YYX \leftarrow X \times Y \to Y.)

Intuitively, XYX \star Y is the union of all line segments connecting XX to YY when these are placed in general position in an ambient Euclidean space.

The join is associative; intuitively, a join of three spaces X,Y,ZX, Y, Z is the union of 2-simplices whose vertices lie in X,Y,ZX, Y, Z respectively when these are placed in general position. Thus the join endows TopTop with a monoidal category structure whose unit is the empty space.

Examples

Example

The join of any topological space XX with the point is the cone on XX:

X*=CX. X \star \ast = C X \,.
Example

The join of any topological space XX with the 0-sphere S 0S^0 is the suspension of XX:

XS 0ΣX. X \star S^0 \simeq \Sigma X \,.
Example

The join of n-spheres with each other is

S mS nS m+n+1 S^m \star S^n \cong S^{m+n+1}
Example

For X=S 1,Y=S 1X = S^1, Y = S_1, we may consider XX to consist of quaternions a+bia + b i and YY to consist of quaternions cj+dkc j + d k such that a 2+b 2=1=c 2+d 2a^2 + b^2 = 1 = c^2 + d^2. Then XX and YY are in general position in 4\mathbb{H} \cong \mathbb{R}^4 with respect to each other, and the quotient map

X×I×YS 3X \times I \times Y \to S^3
\,
(a+bi,t,c+di)t 1/2(a+bi)+(1t) 1/2(c+di)(a + b i, t, c + d i) \mapsto t^{1/2}(a + b i) + (1 - t)^{1/2}(c + d i)

is an explicit realization of the unit sphere S 3S^3 as XYX \star Y.

Example

For a topological group GG, the Milnor construction of the total space EGE G of the classifying bundle is an iterated join, i.e., the colimit of a diagram of inclusions

GGGGGGG \to G \star G \to G \star G \star G \to \ldots

where the identity element of GG is used to embed each G *nG^{\ast n} into its successor G *(n+1)G^{\ast (n+1)}. The idea is that passing to higher joins kills off more and more lower-dimensional homotopy groups, until one reaches the colimit which is then weakly contractible.

The same idea applies to a general space XX; an HH-space structure and higher homotopy associativities (collectively embodied in a structure of algebra over the Stasheff operad) may be used to build a classifying bundle in iterative fashion, with the Hopf construction giving the first stage of the iteration.

References

Discussion in homotopy type theory (applied to n-image factorization) is in

Revised on May 12, 2017 05:49:00 by Urs Schreiber (92.218.150.85)