# 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, and let *A* be a subspace of *Y*. Let *f* : *A* → *X* be a continuous map (called the **attaching map**). One forms the adjunction space *X* ∪_{f} *Y* (sometimes also written as *X* +_{f} *Y*) by taking the disjoint union of *X* and *Y* and identifying *a* with *f*(*a*) for all *a* in *A*. Formally,

where the equivalence relation ~ is generated by *a* ~ *f*(*a*) for all *a* in *A*, and the quotient is given the quotient topology. 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.

Intuitively, one may think of *Y* as being glued onto *X* via the map *f*.

## ExamplesEdit

- 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*.

## PropertiesEdit

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 descriptionEdit

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 alsoEdit

## ReferencesEdit

- Stephen Willard,
*General Topology*, (1970) Addison-Wesley Publishing Company, Reading Massachusetts.*(Provides a very brief introduction.)* - "Adjunction space".
*PlanetMath*. - 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".