Wedge sum

In topology, the wedge sum is a "one-point union" of a family of topological spaces. More precisely, let (pi) be respective points of a family {Xi | i ∈ I} of spaces. The wedge sum is the quotient of the disjoint union of the family (Xi) by identification of the (pi):

\vee_i X_i := \coprod_i X_i\;/ \;\{p_i\sim p_j \mid i,j \in I\}

In other words, the wedge sum is the joining of several spaces at a single point. This definition of course depends on the choice of (pi) unless the spaces (Xi) are homogeneous. The wedge sum can be understood as the coproduct in the category of pointed topological spaces.

For example, the wedge product of two circles is homeomorphic to a "figure eight space". The wedge product of n-circles is often called a bouquet of circles.

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 X and Y is the free product of the fundamental groups of X and Y.

See also: Smash product


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