Adjunction space

In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let <math>X</math> and <math>Y</math> be topological spaces, and let <math>A</math> be a subspace of <math>Y</math>. Let <math>f : A \rightarrow X</math> be a continuous map (called the attaching map). One forms the adjunction space <math>X \cup_f Y</math> (sometimes also written as <math> X +_f Y</math>) by taking the disjoint union of <math>X</math> and <math>Y</math> and identifying <math>a</math> with <math>f(a)</math> for all <math>a</math> in <math>A</math>. Formally,

<math>X\cup_f Y = (X\sqcup Y) / \sim</math>

where the equivalence relation <math> \sim</math> is generated by <math> a\sim f(a)</math> for all <math>a</math> in <math>A</math>, and the quotient is given the quotient topology. As a set, <math>X \cup_f Y</math> consists of the disjoint union of <math>X</math> and (<math> Y-A</math>). The topology, however, is specified by the quotient construction.

Intuitively, one may think of <math>Y</math> as being glued onto <math>X</math> via the map <math>f</math>.

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.

Properties

The continuous maps h : X ∪_f Y → Z are in 1-1 correspondence with the pairs of continuous maps h_X : X → Z and h_Y : Y → Z that satisfy h_X(f(a))=h_Y(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 Φ_X, Φ_Y 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

• Quotient space
• Mapping cylinder

References

• Stephen Willard, General Topology, (1970) Addison-Wesley Publishing Company, Reading Massachusetts. (Provides a very brief introduction.)
• Template:Planetmath reference
• Ronald Brown, "Topology and Groupoids" pdf available , (2006) available from amazon sites. Discusses the homotopy type of adjunction spaces, and uses adjunction spaces as an introduction to (finite) cell complexes.
• J.H.C. Whitehead "Note on a theorem due to Borsuk" Bull AMS 54 (1948), 1125-1132 is the earliest outside reference I know of using the term "adjuction space".