# construction of Riemann surface using paths

Note: All arcs and curves are assumed to be smooth in this entry.

Let $f$ be a complex function defined in a disk $D$ about a point ${z}_{0}\in \u2102$. In this entry, we shall show how to construct a Riemann surface such that $f$ may be analytically continued to a function on this surface by considering paths in the complex plane.

Let $\mathcal{P}$ denote the class of paths on the complex plane having ${z}_{0}$ as an endpoint^{} along which $f$ may be analytically continued. We may define an equivalence relation^{} $\sim $ on this set — ${C}_{1}\sim {C}_{2}$ if ${C}_{1}$ and ${C}_{2}$ have the same endpoint and there exists a one-parameter family of paths along which $f$ can be analytically continued which includes ${C}_{1}$ and ${C}_{2}$.

Define $\mathcal{S}$ as the quotient of $\mathcal{P}$ modulo $\sim $. It is possible to extend $f$ to a function on $\mathcal{S}$. If $C\in \mathcal{P}$, let $f(C)$ be the value of the analytic continuation of $f$ at the endpoint of $C$ (not ${z}_{0}$, of course, but the other endpoint). By the monodromy theorem^{}, if ${C}_{1}\sim {C}_{2}$, then $f({C}_{1})=f({C}_{2})$. Hence, $f$ is well defined on the quotient $\mathcal{S}$.

Also, note that there is a natural projection^{} map $\pi :\mathcal{S}\to \u2102$. If $C$ is an equivalence class^{} of paths in $\mathcal{S}$, define $\pi (C)$ to be the common endpoint of those paths (not ${z}_{0}$, of course, but the other endpoint).

Next, we shall define a class of subsets of $\mathcal{S}$. If $f$ can be analytically continued from along a path $C$ from ${z}_{0}$ to ${z}_{1}$ then there must exist an open disk ${D}^{\prime}$ centered about ${z}_{1}$ in which the continuation of $f$ is analytic^{}. Given any $z\in {D}^{\prime}$, let $C(z)$ be the concatenation of the path $C$ from ${z}_{0}$ to ${z}_{1}$ and the straight line segment from ${z}_{1}$ to $z$ (which lies inside ${D}^{\prime}$). Let $N(C,{D}^{\prime})\subset \mathcal{S}$ be the set of all such paths.

We will define a topology of $\mathcal{S}$ by taking all these sets $N(C,{D}^{\prime})$ as a basis. For this to be legitimate, it must be the case that, if ${C}_{3}$ lies in the intersection^{} of two such sets, $N({C}_{1},{D}_{1})$ and $N({C}_{2},{D}_{2})$ there exists a basis element $N({C}_{3},{D}_{3})$ contained in the intersection of $N({C}_{1},{D}_{1})$ and $N({C}_{2},{D}_{2})$. Since the endpoint of ${C}_{3}$ lies in the intersection of ${D}_{1}$ and ${D}_{2}$, there must exist a disk ${D}_{3}$ centered about this point which lies in the intersection of ${D}_{1}$ and ${D}_{2}$. It is easy to see that $N({C}_{3},{D}_{3})\subset N({C}_{1},{D}_{1})\cap N({C}_{2},{D}_{2})$.

Note that this topology has the Hausdorff property. Suppose that ${C}_{1}$ and ${C}_{2}$ are distinct elements of $\mathcal{S}$. On the one hand, if $\pi ({C}_{1})\ne \pi ({C}_{2})$, then one can find disjoint open disks ${D}_{1}$ and ${D}_{2}$ centered about ${C}_{1}$ and ${C}_{2}$. Then $N({C}_{1},{D}_{1})\cap N({C}_{2},{D}_{2})=\mathrm{\varnothing}$ because $\pi (N({C}_{1},{D}_{1}))\cap \pi (N({C}_{2},{D}_{2}))={D}_{1}\cap {D}_{2}=\mathrm{\varnothing}$. On the other hand, if $\pi ({C}_{1})=\pi ({C}_{2})$, then let ${D}_{3}$ be the smaller of the disks ${D}_{1}$ and ${D}_{2}$. Then $N({C}_{1},{D}_{3})\cap N({C}_{2},{D}_{3})=\mathrm{\varnothing}$.

To complete^{} the proof that $\mathcal{S}$ is a Riemann surface, we must exhibit coordinate neighborhoods^{} and homomorphisms^{}. As coordinate neighborhoods, we shall take the neighborhoods $N(C,D)$ introduced above and as homomorphisms we shall take the restrictions^{} of $\pi $ to these neighborhoods. By the way that these neighborhoods have been defined, every element of $\mathcal{S}$ lies in at least one such neighborhood. When the domains of two of these homomorphisms overlap, the composition^{} of one homomorphism with the inverse^{} of the restriction of the other homomorphism to the overlap region is simply the identity map in the overlap region, which is analytic. Hence, $\mathcal{S}$ is a Riemann surface.

Title | construction of Riemann surface using paths |
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Canonical name | ConstructionOfRiemannSurfaceUsingPaths |

Date of creation | 2013-03-22 14:44:23 |

Last modified on | 2013-03-22 14:44:23 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 21 |

Author | rspuzio (6075) |

Entry type | Proof |

Classification | msc 30F99 |