Inmathematics, anadjunction space (orattaching space) is a common construction intopology where onetopological space is attached or "glued" onto another. Specifically, let${\displaystyle X}$ and${\displaystyle Y}$ be topological spaces, and let${\displaystyle A}$ be asubspace of${\displaystyle Y}$. Let${\displaystyle f:A\to X}$ be acontinuous map (called theattaching map). One forms the adjunction space${\displaystyle X{\cup }_{f}Y}$ (sometimes also written as${\displaystyle X{+}_{f}Y}$) by taking thedisjoint union of${\displaystyle X}$ and${\displaystyle Y}$ and identifying${\displaystyle a}$ with${\displaystyle f(a)}$ for all${\displaystyle a}$ in${\displaystyle A}$. Formally,

${\displaystyle X{\cup }_{f}Y=(X\bigsqcup Y)/\sim }$

where theequivalence relation${\displaystyle \sim }$ is generated by${\displaystyle a\sim f(a)}$ for all${\displaystyle a}$ in${\displaystyle A}$, and the quotient is given thequotient topology. As a set,${\displaystyle X{\cup }_{f}Y}$ consists of the disjoint union of${\displaystyle X}$ and (${\displaystyle Y-A}$). The topology, however, is specified by the quotient construction.

Intuitively, one may think of${\displaystyle Y}$ as being glued onto${\displaystyle X}$ via the map${\displaystyle f}$.

• A common example of an adjunction space is given whenY is a closedn-ball (orcell) andA 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 aCW complex.
• Adjunction spaces are also used to defineconnected sums ofmanifolds. Here, one first removes open balls fromX andY before attaching the boundaries of the removed balls along an attaching map.
• IfA is a space with one point then the adjunction is thewedge sum ofX andY.
• IfX is a space with one point then the adjunction is the quotientY/A.

The continuous mapsh :X ∪_fY →Z are in 1-1 correspondence with the pairs of continuous mapsh_X :X →Z andh_Y :Y →Z that satisfyh_X(f(a))=h_Y(a) for alla inA.

In the case whereA is aclosed subspace ofY one can show that the mapX →X ∪_fY is a closedembedding and (Y −A) →X ∪_fY is an open embedding.

The attaching construction is an example of apushout in thecategory of topological spaces. That is to say, the adjunction space isuniversal with respect to the followingcommutative diagram:

Herei is theinclusion map andΦ_X,Φ_Y are the maps obtained by composing the quotient map with the canonical injections into the disjoint union ofX andY. One can form a more general pushout by replacingi with an arbitrary continuous mapg—the construction is similar. Conversely, iff is also an inclusion the attaching construction is to simply glueX andY together along their common subspace.

• Quotient space
• Mapping cylinder

• 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".

• Topology
• Topological spaces