Adjunction space
An adjunction space is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let X and Y be a topological spaces with A a subspace of Y. Let f : A → X be continuous map (called the attaching map). One forms the adjunction space X ∪f Y by taking the disjoint union of X and Y and identifying x ∼ f(x) for all x in A. Schematically,
Intuitively, we think of Y as being glued onto X via the map f.
As a set, X ∪f Y consists of the disjoint union of X and (Y − A). The topology, however, is specified by the quotient construction. 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.
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 following commutative diagram:
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AdjunctionSpace-01.png
Image:AdjunctionSpace-01.png
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.
Examples
- 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.