# 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 *X* and *Y* be topological spaces with *A* a subspace of *Y*. Let *f* : *A* → *X* be a 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* with *f*(*x*) for all *x* in *A*. Schematically,

Sometimes, the adjunction is written as . 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.

## 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*.

## 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 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.)* - Adjunction space at PlanetMath.org.

- Ronald Brown, "Topology and Groupoids", (2006) available from amazon sites. Discusses their homotopy type, and uses adjunction spaces as an introduction to (finite) cell complexes.