Wedge Sum

Xee Y = (X\sqcup Y)\;/ \;\{x_0 \sim y_0\}

More generally, suppose (''X''_''i'')_{''i''∈''I''} is a Family of pointed spaces with basepoints {''p''_''i''}. The wedge sum of the family is given by:
:

\bigvee_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 {''p''_''i''} unless the spaces {''X''_''i''} are Homogeneous .

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 ''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 '', while a wedge product of arbitrary spheres is often called a "bouquet of spheres".

Van Kampen's Theorem gives certain conditions (which are usually fulfilled for Well-behaved spaces, such as CW Complex es) 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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Wedge Sum