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.

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)