# Wedge product (topology)

Wedge product of two circles

With the wedge product (after wedge English wedge; also called one-point union or bouquet ) of two dotted topological spaces and one designates their disjoint union, which is glued at one point (the base point). Formally, the definition is as follows: ${\ displaystyle X \ vee Y}$ ${\ displaystyle X}$${\ displaystyle Y}$

${\ displaystyle X \ vee Y = (X \ coprod Y) / (pt \ coprod pt)}$

Here denotes the respective base point. ${\ displaystyle pt}$

The construction can also be generalized to any set of spaces:

${\ displaystyle \ bigvee _ {i \ in I} X_ {i} = (\ coprod _ {i \ in I} X_ {i}) / (\ coprod _ {i \ in I} pt_ {i})}$

In a more abstract way, the wedge product can be understood as the co -product in the category of dotted topological spaces.

## Role in algebraic topology

The wedge product behaves well with respect to some functors in algebraic topology . For example, for the fundamental group for locally contractible spaces${\ displaystyle X_ {i}}$

${\ displaystyle \ pi _ {1} (\ bigvee _ {i \ in I} X_ {i}) = * _ {i \ in I} \ pi _ {1} (X_ {i}),}$

where denotes the free product of the groups. ${\ displaystyle *}$

In the singular homology :

${\ displaystyle H_ {n} (\ bigvee _ {i \ in I} X_ {i}, pt) = \ bigoplus _ {i \ in I} H_ {n} (X_ {i}, pt)}$

One can wedge sum in an obvious way into the product embed the quotient${\ displaystyle X \ vee Y}$ ${\ displaystyle X \ times Y}$

${\ displaystyle X \ wedge Y: = X \ times Y / X \ vee Y}$

is the Smash product .

In particular, the reduced suspension is of importance in the stable homotopy theory. ${\ displaystyle \ Sigma X: = S ^ {1} \ wedge X}$

The wedge product is also used in the definition of the link in the homotopy groups .