# Join

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.

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) $| K | \star | L |$.