Wedge sum


In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if X and Y are pointed spaces (i.e. topological spaces with distinguished basepoints and ) the wedge sum of X and Y is the quotient space of the disjoint union of X and Y by the identification

A wedge sum of two circles

where is the equivalence closure of the relation More generally, suppose is a indexed family of pointed spaces with basepoints The wedge sum of the family is given by:

where is the equivalence closure of the relation In other words, the wedge sum is the joining of several spaces at a single point. This definition is sensitive to the choice of the basepoints unless the spaces 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 Edit

The wedge sum of two circles is homeomorphic to a figure-eight space. The wedge sum of   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 is to identify all of the points along the equator of an  -sphere  . Doing so results in two copies of the sphere, joined at the point that was the equator:


Let   be the map   that is, of identifying the equator down to a single point. Then addition of two elements   of the  -dimensional homotopy group   of a space   at the distinguished point   can be understood as the composition of   and   with  :


Here,   are maps which take a distinguished point   to the point   Note that the above uses the wedge sum of two functions, which is possible precisely because they agree at   the point common to the wedge sum of the underlying spaces.

Categorical description Edit

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 of the diagram   in the category of topological spaces (where   is any one-point space).

Properties Edit

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

See also Edit

References Edit

  • Rotman, Joseph. An Introduction to Algebraic Topology, Springer, 2004, p. 153. ISBN 0-387-96678-1