Contents

# Contents

## Definition

###### Definition

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

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

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

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

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

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

$\array{ & & X \times Y & & & & X \times Y & & \\ & \mathllap{\pi_1} \swarrow & & \searrow \mathrlap{i_0} & & \mathllap{i_1} \swarrow & & \searrow \mathrlap{\pi_2} & \\ X & & & & X \times I \times Y & & & & Y }$

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

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

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

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

## Examples

###### Example

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

$X \star \ast = C X \,.$
###### Example

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

$X \star S^0 \simeq \Sigma X \,.$
###### Example

The join of n-spheres with each other is

$S^m \star S^n \cong S^{m+n+1}$
###### Example

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

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

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

###### Example

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

$G \to G \star G \to G \star G \star G \to \ldots$

where the identity element of $G$ is used to embed each $G^{\ast n}$ into its successor $G^{\ast (n+1)}$. 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 $X$; an $H$-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.

Discussion of relation to homotopy limits etc.

• Jean-Paul Doeraene, Homotopy pull backs, homotopy push outs and joins, Bull. Belg. Math. Soc. Simon Stevin, Volume 5, Number 1 (1998), 15-37. pdf on Project Euclid

Comparison with other forms of join:

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