Contents

# Contents

## Idea

The join $X \star Y$ of topological spaces $X$ and $Y$ consists of $X$, $Y$, and convex combinations of a point of $X$ and a point of $Y$ (with some suitable topology).

## 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 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.

## Milnor’s join construction

There is another topology on the join of spaces, due to Milnor, which appears in his construction of universal bundles. (Milnor 1956, section 2)

###### Definition

Milnor’s join $X \overline{\star} Y$ has the same underlying set as $X \star Y$ and the coarsest topology which makes $t \colon X \overline{\star} Y \to [0,1]$ (given on $X \times I \times Y$ by $(x,t,y)\mapsto t$, constantly $0$ on $X$, and constantly $1$ on $Y$) and the projections $\pi_1 \colon t^{-1}([0,1)) \to X$ and $\pi_2 \colon t^{-1}((0,1]) \to Y$ continuous.

If $p\colon X \to S$ and $p\colon Y \to S$ are spaces over $S$, then their fibrewise join or Whitney sum $X \overline{\star}_S Y$ is the space $\{ x \in X \overline{\star} Y : t(x) \in (0,1) \implies p(\pi_1(x))=q(\pi_2(y)) \}$.

The inclusions $X \hookrightarrow X \overline{\star} Y$ and $Y \hookrightarrow X \overline{\star} Y$ are closed Hurewicz cofibrations: the open set $t^{-1}[[0,1)]$ deforms onto $X$ and $t^{-1}[(0,1]]$ deforms onto $Y$.

###### Proposition

The identity map $X \star Y \to X \overline{\star} Y$ is a homotopy equivalence. If $X$ and $Y$ are compact, then it is a homeomorphism.

(See Fritsch–Golasiński 2004, section 1)

###### Proposition

If $p \colon X \to S$ and $q \colon Y \to S$ are Hurewicz fibrations, then so is $X \overline{\star}_S Y \to S$

(See Hall 1965, Lemmens 1970, theorem I)

## Examples

###### Example

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

$X \star 1 = C X \,.$
###### Example

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

$X \star S^0 \cong S X \,.$
###### 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

$G \to G \overline{\star} G \to G \overline{\star} G \overline{\star} G \to \ldots$

where the identity element of $G$ is used to embed each $G^{\bar \star n}$ into its successor $G^{\bar \star (n+1)}$. 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.

###### Example

If $p \colon X \to Y$ is a fibration, then $p$ factors as a Hurewicz cofibration $X \hookrightarrow X \overline{\star}_Y Y$ followed by an acyclic Hurewicz fibration $X \overline{\star}_Y Y \to Y$ by proposition : the embedding $Y \hookrightarrow X \overline{\star}_Y Y$ and retraction $X \overline{\star}_Y Y \to Y$ exhibit $Y$ as a strong deformation retract of $X \overline{\star}_Y Y$.

(This fact is used in Strøm’s construction of his model structure on topological spaces.)

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