Wedge sum

From Academic Kids

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 x0 and y0) the wedge sum of X and Y is the quotient of the disjoint union of X and Y by the identification x0y0:

<math>X\vee Y = (X\cup Y)\;/ \;\{x_0 \sim y_0\}<math>

More generally, suppose (Xi)iI is a family of pointed spaces with basepoints {pi}. The wedge sum of the family is given by:

<math>\bigvee_i X_i := \coprod_i X_i\;/ \;\{p_i\sim p_j \mid i,j \in I\}<math>

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 spaces. Alternatively, the wedge sum can be see as the pushout of the diagram X ← {•} → Y in the category of topological spaces (where {•} is any one point space).

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.

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