Adjunction space
In mathematics, an adjunction space is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let and be topological spaces, and let be a subspace of. Let be a continuous map. One forms the adjunction space by taking the disjoint union of and and identifying with for all in. Formally,
where the equivalence relation is generated by for all in, and the quotient is given the quotient topology. As a set, consists of the disjoint union of and. The topology, however, is specified by the quotient construction.
Intuitively, one may think of as being glued onto via the map.
Examples
- A common example of an adjunction space is given when Y is a closed n-ball and A is the boundary of the ball, the -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=hY 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 → 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.