In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if X and Y are pointed spaces) the wedge sum of X and Y is the quotient space of the disjoint union of X and Y

is the equivalence closure is an indexed family are homogeneous.

The wedge sum is again a pointed space, and the binary operation is associative and commutative (up to homeomorphism).

Sometimes the wedge sum is called the wedge product, but this is not the same concept as the exterior product, which is also often called the wedge product.

Examples

The wedge sum of two circles is homeomorphic to a figure-eight space circles is often called a bouquet of circles, while a wedge product of arbitrary spheres is often called a bouquet of spheres.

A common construction in homotopy

-dimensional homotopy group

the point common to the wedge sum of the underlying spaces.

Categorical description

The wedge sum can be understood as the coproduct in the category of pointed spaces. Alternatively, the wedge sum can be seen as the pushout in the category of topological spaces is any one-point space).

Properties

Van Kampen's theorem gives certain conditions (which are usually fulfilled for well-behaved spaces, such as CW complexes) under which the fundamental group is the free product

See also

• Connected sum

• Smash product

• Hawaiian earring, a topological space resembling, but not the same as, a wedge sum of countably many circles

References

• Joseph J. Rotman. An Introduction to Algebraic Topology, Springer-Verlag, 2004, p.153.