Wedge product (topology)

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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:

Here denotes the respective base point.

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

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

where denotes the free product of the groups.

In the singular homology :

One can wedge sum in an obvious way into the product embed the quotient

is the Smash product .

In particular, the reduced suspension is of importance in the stable homotopy theory.

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