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''of two topological spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j0542501.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j0542502.png" />''
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The topological space, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j0542503.png" />, and defined as the quotient space of the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j0542504.png" /> by the decomposition whose elements are the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j0542505.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j0542506.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j0542507.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j0542508.png" />), and the individual points of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j0542509.png" />.
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Examples. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425010.png" /> consists of a single point, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425011.png" /> is the [[Cone|cone]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425012.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425013.png" /> is homeomorphic to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425014.png" />-fold [[Suspension|suspension]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425015.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425016.png" />. The operation of join is commutative and associative (at least in the category of locally compact Hausdorff spaces). For calculating the homology of a join (with coefficients in a principal ideal domain), an analogue of the [[Künneth formula|Künneth formula]] is used:
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''of two topological spaces  $  X $
 +
and $  Y $''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425017.png" /></td> </tr></table>
+
The topological space, denoted by  $  X \star Y $,
 +
and defined as the quotient space of the product  $  X \times Y \times [ 0, 1] $
 +
by the decomposition whose elements are the sets  $  \{ x \} \times Y \times \{ 0 \} $(
 +
$  x \in X $),
 +
$  X \times \{ y \} \times \{ 1 \} $(
 +
$  y \in Y $),
 +
and the individual points of the set  $  X \times Y \times [ 0, 1] \setminus  ( X \times Y \times \{ 0 \} \cup X \times Y \times \{ 1 \} ) $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425018.png" /></td> </tr></table>
+
Examples. If  $  X $
 +
consists of a single point, then  $  X \star Y $
 +
is the [[Cone|cone]] over  $  Y $.
 +
$  S  ^ {n} \star Y $
 +
is homeomorphic to the  $  ( n + 1) $-
 +
fold [[Suspension|suspension]] over  $  Y $.  
 +
In particular,  $  S  ^ {n} \star S  ^ {m} \approx S ^ {n + m + 1 } $.  
 +
The operation of join is commutative and associative (at least in the category of locally compact Hausdorff spaces). For calculating the homology of a join (with coefficients in a principal ideal domain), an analogue of the [[Künneth formula|Künneth formula]] is used:
  
The join of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425019.png" />-connected space and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425020.png" />-connected space is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425021.png" />-connected. The operation of join lies at the basis of Milnor's construction of a universal [[Principal fibre bundle|principal fibre bundle]].
+
$$
 +
\widetilde{H}  _ {r + 1 }  ( X \star Y)  \approx \
 +
\sum _ {i + j = r }
 +
\widetilde{H}  _ {i} ( X) \otimes
 +
\widetilde{H}  _ {j} ( Y) \oplus
 +
$$
  
 +
$$
 +
\oplus
 +
\sum _ {i + j = r - 1 }
 +
\mathop{\rm Tor} ( \widetilde{H}  _ {i} ( X), \widetilde{H}  _ {j} ( Y)).
 +
$$
  
 +
The join of an  $  r $-
 +
connected space and an  $  s $-
 +
connected space is  $  ( r + s + 2) $-
 +
connected. The operation of join lies at the basis of Milnor's construction of a universal [[Principal fibre bundle|principal fibre bundle]].
  
 
====Comments====
 
====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425023.png" /> be (abstract) simplicial complexes with vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425025.png" />, respectively. Then the join of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425027.png" /> is the simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425028.png" /> with vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425029.png" /> whose simplices are all subsets of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425030.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425031.png" /> is a simplex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425033.png" /> is a simplex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425034.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425035.png" /> denotes a geometric realization of a simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425036.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425037.png" /> is (homeomorphic to) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054250/j05425038.png" />.
+
Let $  K $
 +
and $  L $
 +
be (abstract) simplicial complexes with vertices $  \{ a  ^ {1} , a  ^ {2} , . . . \} $
 +
and $  \{ b  ^ {1} , b  ^ {2} , . . . \} $,  
 +
respectively. Then the join of $  K $
 +
and $  L $
 +
is the simplicial complex $  K \star L $
 +
with vertices  $  \{ a  ^ {1} , a  ^ {2} , . . . \} \cup \{ b  ^ {1} , b  ^ {2} , . . . \} $
 +
whose simplices are all subsets of the form $  \{ a ^ {i _ {1} } \dots a ^ {i _ {k} } \} \cup \{ b ^ {j _ {1} } \dots b ^ {j _ {l} } \} $
 +
for which $  \{ a ^ {i _ {1} } \dots a ^ {i _ {k} } \} $
 +
is a simplex of $  K $
 +
and  $  \{ b ^ {j _ {1} } \dots b ^ {j _ {l} } \} $
 +
is a simplex of $  L $.  
 +
If $  | K | $
 +
denotes a geometric realization of a simplicial complex $  K $,  
 +
then $  | K \star L | $
 +
is (homeomorphic to) $  | K | \star | L | $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Lefschetz,  "Topology" , Chelsea, reprint  (1965)  pp. Sect. 47 (Chapt. II §8)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)  pp. 25; 437–444</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C.R.F. Maunder,  "Algebraic topology" , v. Nostrand-Reinhold  (1970)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Lefschetz,  "Topology" , Chelsea, reprint  (1965)  pp. Sect. 47 (Chapt. II §8)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.H. Spanier,  "Algebraic topology" , McGraw-Hill  (1966)  pp. 25; 437–444</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C.R.F. Maunder,  "Algebraic topology" , v. Nostrand-Reinhold  (1970)</TD></TR></table>

Latest revision as of 22:14, 5 June 2020


of two topological spaces $ X $ and $ Y $

The topological space, denoted by $ X \star Y $, and defined as the quotient space of the product $ X \times Y \times [ 0, 1] $ by the decomposition whose elements are the sets $ \{ x \} \times Y \times \{ 0 \} $( $ x \in X $), $ X \times \{ y \} \times \{ 1 \} $( $ y \in Y $), and the individual points of the set $ X \times Y \times [ 0, 1] \setminus ( X \times Y \times \{ 0 \} \cup X \times Y \times \{ 1 \} ) $.

Examples. If $ X $ consists of a single point, then $ X \star Y $ is the cone over $ Y $. $ S ^ {n} \star Y $ is homeomorphic to the $ ( n + 1) $- fold suspension over $ Y $. In particular, $ S ^ {n} \star S ^ {m} \approx S ^ {n + m + 1 } $. The operation of join is commutative and associative (at least in the category of locally compact Hausdorff spaces). For calculating the homology of a join (with coefficients in a principal ideal domain), an analogue of the Künneth formula is used:

$$ \widetilde{H} _ {r + 1 } ( X \star Y) \approx \ \sum _ {i + j = r } \widetilde{H} _ {i} ( X) \otimes \widetilde{H} _ {j} ( Y) \oplus $$

$$ \oplus \sum _ {i + j = r - 1 } \mathop{\rm Tor} ( \widetilde{H} _ {i} ( X), \widetilde{H} _ {j} ( Y)). $$

The join of an $ r $- connected space and an $ s $- connected space is $ ( r + s + 2) $- connected. The operation of join lies at the basis of Milnor's construction of a universal principal fibre bundle.

Comments

Let $ K $ and $ L $ be (abstract) simplicial complexes with vertices $ \{ a ^ {1} , a ^ {2} , . . . \} $ and $ \{ b ^ {1} , b ^ {2} , . . . \} $, respectively. Then the join of $ K $ and $ L $ is the simplicial complex $ K \star L $ with vertices $ \{ a ^ {1} , a ^ {2} , . . . \} \cup \{ b ^ {1} , b ^ {2} , . . . \} $ whose simplices are all subsets of the form $ \{ a ^ {i _ {1} } \dots a ^ {i _ {k} } \} \cup \{ b ^ {j _ {1} } \dots b ^ {j _ {l} } \} $ for which $ \{ a ^ {i _ {1} } \dots a ^ {i _ {k} } \} $ is a simplex of $ K $ and $ \{ b ^ {j _ {1} } \dots b ^ {j _ {l} } \} $ is a simplex of $ L $. If $ | K | $ denotes a geometric realization of a simplicial complex $ K $, then $ | K \star L | $ is (homeomorphic to) $ | K | \star | L | $.

References

[a1] S. Lefschetz, "Topology" , Chelsea, reprint (1965) pp. Sect. 47 (Chapt. II §8)
[a2] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. 25; 437–444
[a3] C.R.F. Maunder, "Algebraic topology" , v. Nostrand-Reinhold (1970)
How to Cite This Entry:
Join. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Join&oldid=47466
This article was adapted from an original article by M.Sh. Farber (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article