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“Neutrons are composed of fermions and so obey Fermi statistics”… cute. when the author learned how gluons can be the neutron’s constituents but allowing it to remain fermionic as a whole, he can become a great physicist
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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 (topology)|subspace]] of ''Y''. Let ''f'' : ''A'' → ''X'' be a [[continuous (topology)|continuous map]] (called the '''attaching map'''). One forms the adjunction space ''X'' ∪<sub>''f''</sub> ''Y'' by taking the [[disjoint union (topology)|disjoint union]] of ''X'' and ''Y'' and [[quotient space|identifying]] ''x'' with ''f''(''x'') for all ''x'' in ''A''. Schematically,


:<math>X\cup_f Y = (X\amalg Y) / \{f(A) \sim A\}.</math>


Sometimes, the adjunction is written as <math>X+\!_f \,Y</math>. Intuitively, we think of ''Y'' as being glued onto ''X'' via the map ''f''.
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As a set, ''X'' ∪<sub>''f''</sub> ''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 set|closed]] subspace of ''Y'' one can show that the map ''X'' → ''X'' ∪<sub>''f''</sub> ''Y'' is a closed [[embedding]] and (''Y'' − ''A'') → ''X'' ∪<sub>''f''</sub> ''Y'' is an open embedding.
 
==Examples==
*A common example of an adjunction space is given when ''Y'' is a closed ''n''-[[ball (mathematics)|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 sum]]s of [[manifold]]s. 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 (category theory)|pushout]] in the [[category of topological spaces]]. That is to say, the adjunction space is [[universal property|universal]] with respect to following [[commutative diagram]]:
 
<div style="text-align: center;">
[[Image:AdjunctionSpace-01.svg|175px]]
</div>
 
Here ''i'' is the [[inclusion map]] and φ<sub>''X''</sub>, φ<sub>''Y''</sub> 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'' &mdash; 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|id=3992|title=Adjunction space}}
 
* [[Ronald Brown (mathematician)|Ronald Brown]], [http://pages.bangor.ac.uk/~mas010/topgpds.html "Topology and Groupoids"], (2006) available from amazon sites. Discusses their homotopy type, and uses adjunction spaces as an introduction to (finite) cell complexes.
 
[[Category:Topology]]
[[Category:Topological spaces]]

Revision as of 15:27, 20 February 2014


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