Adjunction space

In mathematics, an adjunction space is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let "X" and "Y" be a topological spaces with "A" a subspace of "Y". Let "f" : "A" → "X" be a continuous map (called the attaching map). One forms the adjunction space "X" ∪_"f" "Y" by taking the disjoint union of "X" and "Y" and identifying "x" with "f"("x") for all "x" in "A". Schematically,

:$Xcup\_f\; Y\; =\; (Xamalg\; Y)\; /\; \{f(A)\; sim\; A\}.$

Sometimes, the adjunction is written as $X+!\_f\; ,Y$. Intuitively, we think of "Y" as being glued onto "X" via the map "f".

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. 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.

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".

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 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.

References

* Stephen Willard, "General Topology", (1970) Addison-Wesley Publishing Company, Reading Massachusetts. "(Provides a very brief introduction.)"
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