nLab join of topological spaces

Contents

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

Idea

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

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 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.

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¯YX \overline{\star} Y has the same underlying set as XYX \star Y and the coarsest topology which makes t:X¯Y[0,1]t \colon X \overline{\star} Y \to [0,1] (given on X×I×YX \times I \times Y by (x,t,y)t(x,t,y)\mapsto t, constantly 00 on XX, and constantly 11 on YY) and the projections π 1:t 1([0,1))X\pi_1 \colon t^{-1}([0,1)) \to X and π 2:t 1((0,1])Y\pi_2 \colon t^{-1}((0,1]) \to Y continuous.

If p:XSp\colon X \to S and p:YSp\colon Y \to S are spaces over SS, then their fibrewise join or Whitney sum X¯ SYX \overline{\star}_S Y is the space {xX¯Y:t(x)(0,1)p(π 1(x))=q(π 2(y))}\{ x \in X \overline{\star} Y : t(x) \in (0,1) \implies p(\pi_1(x))=q(\pi_2(y)) \}.

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

Proposition

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

(See Fritsch–Golasiński 2004, section 1)

Proposition

If p:XSp \colon X \to S and q:YSq \colon Y \to S are Hurewicz fibrations, then so is X¯ SYSX \overline{\star}_S Y \to S

(See Hall 1965, Lemmens 1970, theorem I)

Examples

Example

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

X1=CX. X \star 1 = C X \,.
Example

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

XS 0SX. X \star S^0 \cong S 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

GG¯GG¯G¯GG \to G \overline{\star} G \to G \overline{\star} G \overline{\star} G \to \ldots

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

Example

If p:XYp \colon X \to Y is a fibration, then pp factors as a Hurewicz cofibration XX¯ YYX \hookrightarrow X \overline{\star}_Y Y followed by an acyclic Hurewicz fibration X¯ YYYX \overline{\star}_Y Y \to Y by proposition : the embedding YX¯ YYY \hookrightarrow X \overline{\star}_Y Y and retraction X¯ YYYX \overline{\star}_Y Y \to Y exhibit YY as a strong deformation retract of X¯ YYX \overline{\star}_Y Y.

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

References

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 June 9, 2022 at 06:41:07. See the history of this page for a list of all contributions to it.