Adjunction space

• A common example of an adjunction space is given when Y is a closed n-ball (or cell) and A is the boundary of the ball, the (n−1)-sphere. Inductively attaching cells along their spherical boundaries to this space results in an example of a CW complex. • Adjunction spaces are also used to define connected sums of manifolds. Here, one first removes open balls from X and Y before attaching the boundaries of the removed balls along an attaching map. • If A is a space with one point then the adjunction is the wedge sum of X and Y. • If X is a space with one point then the adjunction is the quotient Y/A. ==Properties==

The continuous maps h : X ∪f Y → Z are in 1-1 correspondence with the pairs of continuous maps hX : X → Z and hY : Y → Z that satisfy hX(f(a))=hY(a) for all a in A. In the case where A is a closed subspace of Y one can show that the map X → X ∪f Y is a closed embedding and (Y − A) → X ∪f Y is an open embedding. ==Categorical description==

The attaching construction is an example of a pushout in the category of topological spaces. That is to say, the adjunction space is universal with respect to the following commutative diagram: Here i is the inclusion map and ΦX, ΦY are the maps obtained by composing the quotient map with the canonical injections into the disjoint union of X and Y. One can form a more general pushout by replacing i with an arbitrary continuous map g—the construction is similar. Conversely, if f is also an inclusion the attaching construction is to simply glue X and Y together along their common subspace. ==See also==