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 \rightarrow X be a continuous map (called the attaching map). One forms the adjunction space X \cup_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 \sim is generated by a\sim f(a) for all a in A , and the quotient is given the quotient topology. As a set, X \cup_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 .

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 \rightarrow X be a continuous map (called the attaching map). One forms the adjunction space X \cup_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 \sim is generated by a\sim f(a) for all a in A , and the quotient is given the quotient topology. As a set, X \cup_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.) 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".

Text derived from https://en.wikipedia.org/w/index.php?oldid=1376192001 under CC BY-SA 4.0; modifications have been made.