Wedge sum

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 space of the disjoint union of X and Y by the identification x0 ∼ y0: 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 space of the disjoint union of X and Y by the identification x0 ∼ y0: where ∼ is the equivalence closure of the relation {(x0,y0)}.More generally, suppose (Xi )i ∈ I is a family of pointed spaces with basepoints {pi }. The wedge sum of the family is given by: where ∼ is the equivalence closure of the relation {(pi , pj ) | i,j ∈ I }.In other words, the wedge sum is the joining of several spaces at a single point. This definition is sensitive to the choice of the basepoints {pi}, unless the spaces {Xi } are homogeneous. The wedge sum is again a pointed space, and the binary operation is associative and commutative (up to homeomorphism). Sometimes the wedge sum is called the wedge product, but this is not the same concept as the exterior product, which is also often called the wedge product. The wedge sum of two circles is homeomorphic to a figure-eight space. The wedge sum of n circles is often called a bouquet of circles, while a wedge product of arbitrary spheres is often called a bouquet of spheres. A common construction in homotopy is to identify all of the points along the equator of an n-sphere S n {displaystyle S^{n}} . Doing so results in two copies of the sphere, joined at the point that was the equator: Let Ψ {displaystyle Psi } be the map Ψ : S n → S n ∨ S n {displaystyle Psi :S^{n} o S^{n}vee S^{n}} , that is, of identifying the equator down to a single point. Then addition of two elements f , g ∈ π n ( X , x 0 ) {displaystyle f,gin pi _{n}(X,x_{0})} of the n-dimensional homotopy group π n ( X , x 0 ) {displaystyle pi _{n}(X,x_{0})} of a space X at the distinguished point x 0 ∈ X {displaystyle x_{0}in X} can be understood as the composition of f {displaystyle f} and g {displaystyle g} with Ψ {displaystyle Psi } : Here, f , g : S n → X {displaystyle f,g:S^{n} o X} are maps which take a distinguished point s 0 ∈ S n {displaystyle s_{0}in S^{n}} to the point x 0 ∈ X . {displaystyle x_{0}in X.} Note that the above uses the wedge sum of two functions, which is possible precisely because they agree at s 0 , {displaystyle s_{0},} the point common to the wedge sum of the underlying spaces.