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4.3.4 Joins of Topological Spaces

The join operation on simplicial sets admits a topological interpretation.

Construction 4.3.4.1. Let $X$ and $Y$ be topological spaces, and let $[0,1] = | \Delta ^1 |$ denote the unit interval. We let $X \star Y$ denote the topological space given by the iterated pushout

\[ X \coprod _{ (X \times \{ 0\} \times Y)} (X \times [0,1] \times Y) \coprod _{ (X \times \{ 1\} \times Y) } Y. \]

We will refer to $X \star Y$ as the join of $X$ and $Y$.

Remark 4.3.4.2. Let $X$ and $Y$ be topological spaces. Then the join $X \star Y$ of Construction 4.3.4.1 is equipped with a pair of maps $\iota _{X}: X \hookrightarrow X \star Y$ and $\iota _{Y}: Y \hookrightarrow X \star Y$. It is not difficult to see that these maps are closed embeddings: that is, they induce homeomorphisms from $X$ and $Y$ onto closed subsets of $X \star Y$. We will generally abuse notation by identifying $X$ and $Y$ with their images under $\iota _{X}$ and $\iota _{Y}$, respectively.

Remark 4.3.4.3. Let $X$, $Y$, and $Z$ be topological spaces. Then the datum of a continuous function $X \star Y \rightarrow Z$ is equivalent to the datum of a triple $(f_ X, f_ Y, h)$, where $f_ X: X \rightarrow Z$ and $f_ Y: Y \rightarrow Z$ are continuous functions and $h: X \times [0,1] \times Y \rightarrow Z$ is a homotopy from $f_{X} \circ \pi _{X}$ to $f_{Y} \circ \pi _{Y}$; here $\pi _{X}: X \times Y \rightarrow X$ and $\pi _{Y}: X \times Y \rightarrow Y$ denote the projection maps.

Remark 4.3.4.4 (Symmetry). Let $X$ and $Y$ be topological spaces. Then there is a canonical homeomorphism $X \star Y \simeq Y \star X$, which is induced by the homeomorphism

\[ X \times [0,1] \times Y \rightarrow Y \times [0,1] \times X \quad \quad (x,t,y) \mapsto (y, 1-t, x). \]

Example 4.3.4.5 (Cones). Let $\ast $ denote the topological space consisting of a single point. For any topological space $X$, we write $X^{\triangleleft }$ for the join $\ast \star X$, and $X^{\triangleright }$ for the join $X \star \ast $, given more concretely by the formulae

\[ X^{\triangleleft } = \ast \coprod _{ (\{ 0\} \times X) }( [0,1] \times X) \quad \quad X^{\triangleright } = (X \times [0,1]) \coprod _{ (X \times \{ 1\} ) } \ast . \]

We will refer to both $X^{\triangleleft }$ and $X^{\triangleright }$ as the cone on $X$ (note that they are canonically homeomorphic, by virtue of Remark 4.3.4.4).

Remark 4.3.4.6. Let $X$ be a locally compact Hausdorff space. Then the functor

\[ \operatorname{Top}\rightarrow \operatorname{Top}_{X/} \quad \quad Y \mapsto X \star Y \]

preserves colimits. This follows from the fact that the functors $Y \mapsto X \times Y$ and $Y \mapsto X \times [0,1] \times Y$ preserve colimits.

Example 4.3.4.7. For each integer $n \geq 0$, let $| \Delta ^{n} | = \{ (u_0, \ldots , u_ n) \in \operatorname{\mathbf{R}}_{\geq 0}: u_0 + \cdots + u_ n = 1 \} $ denote the topological $n$-simplex. For $p,q \geq 0$, we have maps $| \Delta ^{p} | \xrightarrow {\iota } | \Delta ^{p+1+q} | \xleftarrow {\iota '} | \Delta ^{q} |$ given by the formulae

\[ \iota ( u_0, \ldots , u_ p) = (u_0, \ldots , u_ p, 0, \ldots , 0) \quad \quad \iota '( v_0, \ldots , v_ q) = (0, \ldots , 0, v_0, \ldots , v_ q). \]

There is a “straight-line” homotopy $h: | \Delta ^{p} | \times [0,1] \times | \Delta ^{q} | \rightarrow | \Delta ^{p+1+q} |$ from $\iota \circ \pi _{| \Delta ^{p} | }$ to $\iota ' \circ \pi _{ | \Delta ^{q} |}$, given concretely by the formula

\[ h( (u_0, \ldots , u_ p), t, (v_0, \ldots , v_ q) ) = ( (1-t) u_0, (1-t) u_1, \ldots , (1-t) u_ p, t v_0, \ldots , t v_ q ). \]

By virtue of Remark 4.3.4.3, the triple $(\iota , \iota ', h)$ can be identified with a continuous function $H_{p,q}: | \Delta ^{p} | \star | \Delta ^{q} | \rightarrow | \Delta ^{p+1+q} |$.

Proposition 4.3.4.8. Let $p$ and $q$ be nonnegative integers. Then the function $H_{p,q}: | \Delta ^{p} | \star | \Delta ^{q} | \rightarrow | \Delta ^{p+1+q} |$ of Example 4.3.4.7 is a homeomorphism of topological spaces.

Proof. Since $| \Delta ^ p | \star | \Delta ^ q |$ is compact and $| \Delta ^{p+1+q} |$ is Hausdorff, the continuous function $H_{p,q}$ is automatically closed. To complete the proof, it will suffice to show that $H_{p,q}$ is bijective. Fix a point $x$ of $| \Delta ^{p+1+q} |$, given by a sequence of nonnegative real numbers $( \overline{u}_0, \ldots , \overline{u}_ m, \overline{v}_0, \overline{v}_1, \ldots , \overline{v_ n} )$ satisfying

\[ \overline{u}_0 + \cdots + \overline{u}_ m + \overline{v}_0 + \cdots + \overline{v}_ n = 0. \]

Set $t = \overline{v}_0 + \cdots + \overline{v_ n}$. If $t = 0$, the set $H_{p,q}^{-1} \{ x\} $ consists of a single point of $| \Delta ^{p} |$ (regarded as a subset of $| \Delta ^{p} | \star | \Delta ^{q} |$), given by the sequence $( \overline{u}_0, \ldots , \overline{u}_ m)$. If $t = 1$, the set $H_{p,q}^{-1} \{ x \} $ consists of a single point of $| \Delta ^ q |$ (regarded as a subset of $| \Delta ^{p} | \star | \Delta ^{q} |$), given by the sequence $( \overline{v}_0, \ldots , \overline{v}_ m)$. In the case $0 < t < 1$, the set $H_{p,q}^{-1} \{ x\} $ consists of a single point of $| \Delta ^{p} | \star | \Delta ^{q} |$, given as the image of the triple

\[ ( \frac{\overline{u}_0}{1-t}, \ldots , \frac{ \overline{u}_ m}{1-t} ), t, ( \frac{ \overline{v}_0}{t}, \ldots , \frac{ \overline{v}_ n}{t} ) ) \in | \Delta ^{p} | \times [0,1] \times | \Delta ^{n} |. \]
$\square$

We now compare the join operation on topological spaces (given by Construction 4.3.4.1) to the join operation on simplicial sets (given by Construction 4.3.3.13).

Construction 4.3.4.9. Let $X$ and $Y$ be simplicial sets, and let $\sigma : \Delta ^ n \rightarrow X \star Y$ be a morphism. We define a continuous function $f(\sigma ): | \Delta ^ n | \rightarrow |X| \star |Y|$ as follows (see Remark 4.3.3.15):

  • If $\sigma $ factors through $X$, we let $f(\sigma )$ denote the composition

    \[ | \Delta ^ n | \xrightarrow { | \sigma | } |X| \xrightarrow {\iota _{|X|}} |X| \star |Y|, \]

    where the second map is the inclusion of Remark 4.3.4.2.

  • If $\sigma $ factors through $Y$, we let $f(\sigma )$ denote the composition

    \[ | \Delta ^ n | \xrightarrow { | \sigma | } |Y| \xrightarrow { \iota _{|Y|}} |X| \star |Y|, \]

    where the second map is the inclusion of Remark 4.3.4.2.

  • If $\sigma $ factors as a composition

    \[ \Delta ^ n = \Delta ^{p+1+q} \simeq \Delta ^{p} \star \Delta ^{q} \xrightarrow { \sigma _{-} \star \sigma _{+} } X \star Y, \]

    then we let $f(\sigma )$ denote the composite map

    \[ | \Delta ^ n | = | \Delta ^{p+1+q} | \xrightarrow { H_{p,q}^{-1} } | \Delta ^{p} | \star | \Delta ^{q} | \xrightarrow { | \sigma _{-} | \star | \sigma _{+} | } X \star Y, \]

    where $H_{p,q}$ denotes the homeomorphism of Proposition 4.3.4.8.

The construction $\sigma \mapsto f(\sigma )$ is compatible with face and degeneracy maps, and therefore determines a morphism of simplicial sets $f: X \star Y \rightarrow \operatorname{Sing}_{\bullet }( |X| \star |Y| )$. We will identify $f$ with a continuous function $T_{X,Y}: | X \star Y | \rightarrow |X| \star |Y|$, which we will refer to as the join comparison map.

Example 4.3.4.10. Let $X = \Delta ^{p}$ and $Y = \Delta ^{q}$ be standard simplices. Then the join comparison map $T_{X,Y}: | \Delta ^{p} \star \Delta ^{q} | \rightarrow |\Delta ^{p}| \star | \Delta ^{q} |$ fits into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ | \Delta ^{p} \star \Delta ^{q} | \ar [dr]^{ | \rho | } \ar [rr]^{ T_{X,Y} } & & | \Delta ^{p} | \star | \Delta ^{q} | \ar [dl]_{ H_{p,q} } \\ & | \Delta ^{p+1+q} |, & } \]

where $\rho : \Delta ^{p} \star \Delta ^{q} \simeq \Delta ^{p+1+q}$ denotes the isomorphism of simplicial sets appearing in Example 4.3.3.20 and $H_{p,q}$ is the homeomorphism of Proposition 4.3.4.8. In particular, $T_{X,Y}$ is a homeomorphism.

Proposition 4.3.4.11. Let $X$ and $Y$ be simplicial sets. If either $X$ or $Y$ is finite, then the join comparison map $T_{X,Y}: |X \star Y| \rightarrow |X| \star |Y|$ of Construction 4.3.4.9 is a homeomorphism.

Proof. Without loss of generality, we may assume that $X$ is finite. Then the geometric realization $|X|$ is a compact Hausdorff space (Corollary 3.5.1.10). Using Remarks 4.3.4.6 and 4.3.3.26, we see that the functors

\[ \operatorname{Set_{\Delta }}\rightarrow \operatorname{Top}_{|X|/} \quad \quad Y \mapsto |X \star Y|, Y \mapsto |X| \star |Y| \]

preserve colimits. Consequently, if we regard $X$ as fixed, then the collection of simplicial sets $Y$ for which $T_{X,Y}$ is a homeomorphism is closed under colimits. Since every simplicial set can be realized as a colimit of standard simplices (Corollary 1.1.8.17), it will suffice to prove Proposition 4.3.4.11 in the special case where $Y = \Delta ^{q}$ is a standard simplex. In this case, $Y$ is also finite. Repeating the preceding argument (with the roles of $X$ and $Y$ reversed), we are reduced to proving that $T_{X,Y}$ is a homeomorphism in the case where $X = \Delta ^{p}$ is also a standard simplex. In this case, the desired result follows from Example 4.3.4.10. $\square$

Corollary 4.3.4.12. Let $X$ be a simplicial set. Then the join comparison maps $T_{\Delta ^0,X}$ and $T_{X, \Delta ^0}$ supply homeomorphisms of topological spaces

\[ | X^{\triangleleft } | \simeq |X|^{\triangleleft } \quad \quad | X^{\triangleright } | \simeq | X |^{\triangleright }. \]

Here $X^{\triangleleft }$ and $X^{\triangleright }$ denote the left and right cones on $X$ in the category of simplicial sets (Construction 4.3.3.22), while $|X|^{\triangleleft }$ and $|X|^{\triangleright }$ denote the cone $|X|$ in the category of topological spaces (Example 4.3.4.5).

The join comparison map $T_{X,Y}: |X \star Y| \rightarrow |X| \star |Y|$ need not be a homeomorphism in general. However, we do have the following:

Corollary 4.3.4.13. Let $X$ and $Y$ be simplicial sets. Then the join comparison map $T_{X,Y}: |X \star Y| \rightarrow |X| \star |Y|$ is a bijection.

Proof. As a map of sets, we can realize $T_{X,Y}$ as a filtered colimit of join comparison maps $T_{X', Y}$, where $X'$ ranges over the finite simplicial subsets of $X$ (Remark 3.5.1.8). Each of these maps is bijective (even a homeomorphism), by virtue of Proposition 4.3.4.11. $\square$

Warning 4.3.4.14. Let $X$ and $Y$ be simplicial sets, and let $X \diamond Y$ denote the simplicial set given by the iterated coproduct

\[ X \coprod _{ (X \times \{ 0\} \times Y)} (X \times \Delta ^1 \times Y) \coprod _{ (X \times \{ 1\} \times Y) } Y. \]

Since the formation of geometric realization commutes with the formation of colimits, we have an evident comparison map of topological spaces

\[ |X \diamond Y | \rightarrow |X| \star |Y|. \]

This map is always bijective, and is a homeomorphism if either $X$ or $Y$ is finite (see Corollary 3.5.2.2). In this case, Corollary 4.3.4.13 supplies a homeomorphism of geometric realizations $| X \diamond Y | \simeq | X \star Y |$. Beware that this homeomorphism does not arise from a morphism of simplicial sets. In the case $X = \Delta ^ p$ and $Y = \Delta ^{q}$, it arises from the homotopy

\[ h: | \Delta ^{p} | \times | \Delta ^1 | \times | \Delta ^{q} | \rightarrow | \Delta ^{p+1+q} | \]
\[ h( (u_0, \ldots , u_ p), t, (v_0, \ldots , v_ q) ) = ( (1-t) u_0, (1-t) u_1, \ldots , (1-t) u_ p, t v_0, \ldots , t v_ q ). \]

appearing in Example 4.3.4.7, which is not piecewise-linear with respect to the natural triangulation of the polysimplex $| \Delta ^{p} | \times | \Delta ^1 | \times | \Delta ^ q |$.