nLab join of topological spaces

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 John Milnor, which appears in the Milnor construction of universal principal 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.)

Related concepts

• suspension

• Milnor construction

• Hopf construction

• join type

• join of simplicial sets,

• join of categories, join of quasi-categories

References

• I.M. Hall, The generalized Whitney sum, The Quarterly Journal of Mathematics 16(4): 360–384, December 1965. 10.1093/qmath/16.4.360

• P.W.H. Lemmens, A note on the join of fibrations, Indagationes Mathematicae 73: 53–56 (1970) doi:10.1016/S1385-7258(70)80008-7

• John Milnor, Construction of Universal Bundles, II, Ann. of Math. 63 3 (1956) 430-436 [doi:10.2307/1970012]

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:

• Rudolf Fritsch, Marek Golasiński, Topological, simplicial and categorical joins, Archiv der Mathematik 82 5 (2004) 468-480

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

• Egbert Rijke, The join construction (arXiv:1701.07538)