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"Gluing" constructions are common in topology: by gluing two disks along their boundaries we get a sphere; by gluing a cylindical "handle" to a sphere we get a torus, and so forth.

If the original spaces have a metric in addition to topology, when and how do these metrics carry over to the result of gluing? What are the compatibility conditions required for the glued space to have a metric of its own?

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  • $\begingroup$ With the latest edit, this is a reasonable question. The prime example of the gluing construction that I know is Urysohn's universal metric space. Many other examples can be found in the book by Bridson and Haefliger. $\endgroup$ – Moishe Kohan Aug 6 '14 at 23:50
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To perform the gluing, we need:

  • two metric spaces $X$ and $Y$
  • a set $A\subset X$, this is the part of $X$ covered in glue
  • an isometric embedding $f:A\to Y$, which is a way to put the glue-covered part of $X$ over $Y$.

After we firmly press the spaces together and let them sit for a while, a point $x\in A$ becomes identical with the point $f(x)\in Y$. The resulting space can be described as the quotient $(X\sqcup Y)/(x\sim f(x))$. It is usually denoted $X\cup_A Y$. Its metric is the standard quotient metric; however, in this case the formula for quotient metric can be simplified to $$ \tilde d(p,q) = \begin{cases} d_X(p, q) & \mbox{if } p, q \in X \\ d_Y(p, q) & \mbox{if } p, q \in Y \\ \inf_{a\in A} [d_X(p, a)+d_Y(q, f(a))] & \mbox{if } p \in X \mbox{ and } q \in Y \\ \inf_{a\in A} [d_Y(p, f(a)) + d_X(q, a)] & \mbox{if } p \in Y \mbox{ and } q \in X \end{cases} $$

A simple example to start with: $X=(-\infty,0]$, $A=\{0\}$, Y=$[0,\infty)$, $f(0)=0$. This means we glue two half-lines together at point $0$. The result is $\mathbb R$ with the standard metric.

Another example: glue two closed disks of the same radius along their boundaries. This means $A$ is the boundary of one disk, and $f(A)$ is the boundary of the other. The resulting space is homeomorphic to a sphere, though the metric on each disk is still flat. It's like a sphere that someone sat on.

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  • $\begingroup$ Could you (or someone) explain what is meant by the quotient $(X\sqcup Y)/(x\sim f(x))$? I thought to define a quotient space, one needs an equivalence relation defined on a topological space. But here we are only defining $x$ to be equivalent to $f(x)$ (where $f:A\to Y$)? What happens to the other points in $X\sqcup Y$ (which I take to mean the disjoint union of $X$ and $Y$) in defining the quotient? $\endgroup$ – Fang Jing Nov 12 '14 at 2:16
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    $\begingroup$ It's the quotient by the smallest equivalence relation containing all pairs $\{(x,f(x)):x\in A\}$. So, if a point is not in $A$ or $f(A)$, then it is in the trivial, one-point equivalence class. $\endgroup$ – user147263 Nov 12 '14 at 3:21

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