join of topological spaces




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

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


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topological homotopy theory




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.



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

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

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 \,.

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}

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.


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.


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

Last revised on May 18, 2020 at 16:45:58. See the history of this page for a list of all contributions to it.