View source for Join

''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" />''

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" />.

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:

<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>

<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>

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]].



====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" />.

====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>