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,

:$X\cup_f Y = (X\sqcup Y) / \sim$

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''.

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.)''
* {{planetmath reference, urlname=AdjunctionSpace, title=Adjunction space
* 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