Adjunction space (original) (raw)

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

X ∪ f Y = ( X ⊔ Y ) / ∼ {\displaystyle X\cup _{f}Y=(X\sqcup Y)/\sim } {\displaystyle X\cup _{f}Y=(X\sqcup Y)/\sim }

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

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

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

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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.