# Join

*of two topological spaces and *

The topological space, denoted by , and defined as the quotient space of the product by the decomposition whose elements are the sets (), (), and the individual points of the set .

Examples. If consists of a single point, then is the cone over . is homeomorphic to the -fold suspension over . In particular, . 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:

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

#### Comments

Let and be (abstract) simplicial complexes with vertices and , respectively. Then the join of and is the simplicial complex with vertices whose simplices are all subsets of the form for which is a simplex of and is a simplex of . If denotes a geometric realization of a simplicial complex , then is (homeomorphic to) .

#### 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. M.Sh. Farber (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Join&oldid=12786