Analytic Continuation, Monodromy, and Riemann Surfaces

If $[f{]}_a=[f_1{]}_a$ and $[g{]}_a=[g_1{]}_a$, then [L1] gives a neighbourhood of $a$ on which $f=f_1$ and a neighbourhood on which $g=g_1$; on their intersection one has $f+g=f_1+g_1$ and $fg=f_1{g}_1$. Thus the displayed sum and product are well defined on germs.

If $f(a)\mathrm{\ne }0$, [L2] gives a neighbourhood $U$ of $a$ on which $f$ never vanishes. For each $z\in U$, the reciprocal rule in [L2] applies to $f$ at $z$, so $1\mathrm{/}f$ is holomorphic on $U$ and $[f{]}_a[1\mathrm{/}f{]}_a=[1{]}_a$. Hence $[f{]}_a$ is a unit.

If $f(a)=0$ and $[f{]}_a[g{]}_a=[1{]}_a$, then [L1] lets us evaluate at $a$ and obtain $0=f(a)g(a)=1$, impossible. So a germ vanishing at $a$ is not a unit.

Pointwise addition and multiplication on representatives give associative and commutative operations on ${\mathcal{O}}_a$, the constant germs $[0{]}_a$ and $[1{]}_a$ are additive and multiplicative identities, and $[-f{]}_a$ is an additive inverse of $[f{]}_a$. So step 1.1 makes ${\mathcal{O}}_a$ a commutative ring with identity $[1{]}_a$.

Steps 1.2 and 1.3 show that the nonunits are exactly the germs in ${\mathfrak{m}}_a$. This set is an ideal because sums of germs vanishing at $a$ still vanish at $a$, additive inverses still vanish at $a$, and $h(a)f(a)=0$ for every $[h{]}_a\in {\mathcal{O}}_a$ and $[f{]}_a\in {\mathfrak{m}}_a$. Also $[1{]}_a\mathrm{\notin }{\mathfrak{m}}_a$, so ${\mathfrak{m}}_a$ is proper.

Every proper ideal consists entirely of nonunits, hence step 2.2 places it inside ${\mathfrak{m}}_a$. Therefore ${\mathfrak{m}}_a$ is the unique maximal ideal, and [L3] makes ${\mathcal{O}}_a$ a local ring.

By [L1], each set ${\gamma }^{-1}(U_i)$ is open in $[0,1]$, and the containment $\gamma ([t_i,t_{i+1}])\subseteq U_i$ gives $[t_i,t_{i+1}]\subseteq {\gamma }^{-1}(U_i)$. Hence the finite family $\mathcal{U}:=\{{\gamma }^{-1}(U_i):0\le i<m\}$ is an open cover of $[0,1]$. Likewise $\mathcal{V}:=\{{\gamma }^{-1}(V_j):0\le j<n\}$ is an open cover of $[0,1]$.

Apply [L2] to the compact interval $[0,1]$ and the two open covers $\mathcal{U}$ and $\mathcal{V}$. Let ${\delta }_{\mathcal{U}},{\delta }_{\mathcal{V}}>0$ be corresponding Lebesgue numbers, put $\delta :=\min \{{\delta }_{\mathcal{U}},{\delta }_{\mathcal{V}}\}$, and choose a subdivision $0=u_0<u_1<\cdots <u_r=1$ whose mesh is less than $\delta$. Then every interval $[u_k,u_{k+1}]$ has diameter less than both ${\delta }_{\mathcal{U}}$ and ${\delta }_{\mathcal{V}}$.

For each $k<r$, the interval $[u_k,u_{k+1}]$ has diameter less than ${\delta }_{\mathcal{U}}$ and less than ${\delta }_{\mathcal{V}}$, so the Lebesgue-number property gives indices $i,j$ with $[u_k,u_{k+1}]\subseteq {\gamma }^{-1}(U_i)$ and $[u_k,u_{k+1}]\subseteq {\gamma }^{-1}(V_j)$. Equivalently, $\gamma ([u_k,u_{k+1}])\subseteq U_i\cap V_j$. Restricting $f_i$ and $g_j$ to these smaller intervals produces the required common refinement.

By [L1], refine both chains so that they use the same subdivision $0=u_0<\cdots <u_r=1$, and on each interval $[u_k,u_{k+1}]$ the path image lies in both a function element $(f_k,U_k)$ from the first chain and a function element $(g_k,V_k)$ from the second.

At the initial point $\gamma (u_0)=\gamma (0)$ the two first representatives have germ ${\xi }_0$, so [L2] gives an open neighbourhood of $\gamma (0)$ on which $f_0=g_0$.

Assume inductively that $f_k$ and $g_k$ have the same germ at the left endpoint $\gamma (u_k)$. The path segment $\gamma ([u_k,u_{k+1}])$ is connected, so it lies in one connected component $W_k$ of $U_k\cap V_k$. By [L2] the functions $f_k$ and $g_k$ agree on a neighbourhood of $\gamma (u_k)$ contained in $W_k$, and [L3] therefore gives $f_k=g_k$ on all of $W_k$. In particular they have the same germ at the right endpoint $\gamma (u_{k+1})$.

Applying step 2.1 successively for $k=0,\dots ,r-1$ shows that the two refined chains have the same germ at every subdivision point, hence especially at $\gamma (1)$.

The terminal germs of the original chains equal those of the refinements, so the terminal germ depends only on ${\xi }_0$ and $\gamma$, not on the chosen admissible chain.

By [L1], any analytic continuation of ${\xi }_0$ along $\gamma$ is represented by an admissible continuation chain whose successive representatives agree at the joining points of the subdivision.

If two such continuations existed with different terminal germs, their underlying admissible chains would contradict [L2]. Therefore the terminal germ is unique whenever continuation along $\gamma$ exists.

For $t\in [0,1]$, write ${\alpha }_t(s):=H(s,t)$. By [L2], each ${\alpha }_t$ is a path from $a_0$ to the common endpoint, so continuation of ${\xi }_0$ along ${\alpha }_t$ exists by hypothesis.

Fix $t_0\in [0,1]$ and choose an admissible continuation chain $(f_j,U_j)$ over a subdivision $0=s_0<\cdots <s_m=1$ for ${\alpha }_{t_0}$. For each $j<m$, the compact set ${\alpha }_{t_0}([s_j,s_{j+1}])$ lies in the open set $U_j$. Continuity of $H$ therefore gives ${\epsilon }_j>0$ such that

For every $t$ with $\mathrm{\mid }t-t_0\mathrm{\mid }<\epsilon$, step 1.2 keeps the subpath ${\alpha }_t([s_j,s_{j+1}])$ inside $U_j$ for every $j<m$. It also keeps each joining point ${\alpha }_t(s_{j+1})$ inside $W_j$, where $f_j=f_{j+1}$. So the same function elements $(f_j,U_j)$ and the same subdivision form an admissible continuation chain for ${\alpha }_t$. By [L1], the terminal germ of continuation along ${\alpha }_t$ is therefore the terminal germ of this fixed chain, so it is independent of $t$ on that neighbourhood of $t_0$.

Step 2.1 shows that the terminal germ depends locally constantly on $t\in [0,1]$. Since $[0,1]$ is connected, this terminal germ is constant on the whole interval. In particular the terminal germs at $t=0$ and $t=1$, namely the continuations along $\alpha$ and $\beta$, are equal.

Fix $z\in \mathrm{\Omega }$. Because $\mathrm{\Omega }$ is path-connected by [L2], there is at least one path ${\gamma }_z$ from $a_0$ to $z$. Let $T_{\gamma }$ denote the terminal germ obtained by continuing ${\xi }_0$ along a path $\gamma$ from $a_0$.

If $\gamma$ and $\beta$ are two paths from $a_0$ to $z$, then $\lambda :=\bar{\gamma }\ast \beta$ is a based loop at $z$. By [L2] and [L3], its loop class is the identity, so there is an endpoint-fixed path homotopy $K:[0,1]\times [0,1]\to \mathrm{\Omega }$ from $\lambda$ to the constant loop $c_z$.

Define $e:[0,1]\to [0,1{]}^2$ by $e(t)=(0,3t)$ for $0\le t\le \frac{1}{3}$, $e(t)=(3t-1,1)$ for $\frac{1}{3}\le t\le \frac{2}{3}$, and $e(t)=(1,3-3t)$ for $\frac{2}{3}\le t\le 1$. For $(s,t)\in [0,1{]}^2$, put $q_t(s):=(1-s)(\frac{1}{2},0)+se(t)$ and $H(s,t):=K(q_t(s))$. Then $H$ is continuous, $H(0,t)=K(\frac{1}{2},0)=\lambda (\frac{1}{2})=a_0$, and $H(1,t)=K(e(t))=z$ because $e(t)$ lies on the three edges where $K$ is constantly $z$. Also $q_0(s)=((1-s)\mathrm{/}2,0)$ and $q_1(s)=((1+s)\mathrm{/}2,0)$, so $H(s,0)=\lambda ((1-s)\mathrm{/}2)=\gamma (s)$ and $H(s,1)=\lambda ((1+s)\mathrm{/}2)=\beta (s)$. Thus $H$ is a path homotopy from $\gamma$ to $\beta$ relative to the endpoints.

Fact [L1] applied to the path homotopy of step 2.1 gives $T_{\gamma }=T_{\beta }$. Therefore the value of the terminal germ over $z$ is independent of the chosen path from $a_0$ to $z$.

Define $F(z)$ to be the value at $z$ of this common germ. This is well defined by step 3.1.

Let $z\in \mathrm{\Omega }$ and choose a path $\gamma$ from $a_0$ to $z$. Let $(f,U)$ represent the terminal germ $T_{\gamma }$ at $z$. For every $w\in U$, the same function element $(f,U)$ continues that germ from $z$ to $w$, so step 3.1 forces the terminal germ over $w$ to be $[f{]}_w$. Hence $F{\mathrm{\mid }}_U=f$, and $F$ is holomorphic on $U$. Since $z$ was arbitrary, $F$ is holomorphic on all of $\mathrm{\Omega }$, and its germ at each point is the continued germ.

Every point of $\mathcal{R}({\xi }_0,\mathrm{\Omega })$ is, by [L1], a germ $[f{]}_z$ coming from some function element $(f,U)$, and then $[f{]}_z\in N(f,U)$. So the family $\{N(f,U)\}$ covers the germ space.

Suppose $\xi =[f{]}_z$ lies in $N(f,U)\cap N(g,V)$. Then $\xi =[g{]}_z$ as well, so [L1] gives equality of the germs of $f$ and $g$ at $z$. Hence there is a disc $D$ centered at $z$ with $D\subseteq U\cap V$ and $f=g$ on $D$. For each $w\in D$ this implies $[f{]}_w=[g{]}_w$, so $N(f,D)=N(g,D)\subseteq N(f,U)\cap N(g,V)$. Thus [L2] makes the family $\{N(f,U)\}$ a basis for a topology on the germ space.

The space is Hausdorff. If $[f{]}_z$ and $[g{]}_w$ have $z\mathrm{\ne }w$, choose disjoint discs $D_z\ni z$ and $D_w\ni w$; then $N(f,D_z)$ and $N(g,D_w)$ are disjoint basis neighbourhoods. If $z=w$ but $[f{]}_z\mathrm{\ne }[g{]}_z$, choose discs $D_f\subseteq U$ and $D_g\subseteq V$ centered at $z$ so small that $D_f\cap D_g$ is connected. If $N(f,D_f)$ and $N(g,D_g)$ met, then $f$ and $g$ would agree as germs at some point of $D_f\cap D_g$, and [L3] would force $f=g$ on that connected overlap, hence near $z$, contradiction. So distinct germs have disjoint neighbourhoods, exactly as [L4] requires.

To prove second countability, let $\mathcal{D}$ be the countable family of rational open discs contained in $\mathrm{\Omega }$. For every finite chain $D_0,\dots ,D_m$ of discs in $\mathcal{D}$ with $a_0\in D_0$ and $D_{j-1}\cap D_j\mathrm{\ne }\mathrm{\varnothing }$, at most one branch of the complete analytic function is determined on $D_m$ by continuing the initial germ successively across that chain. So the basis sets arising from such rational-disc chains form a countable family.

On each basis element, ${\varphi }_{f,U}$ is bijective with inverse $z\mapsto [f{]}_z$. If $N(f,U)\cap N(g,V)\mathrm{\ne }\mathrm{\varnothing }$, then step 1.2 gives a disc $D\subseteq U\cap V$ on which $f=g$, so on ${\varphi }_{f,U}(N(f,D))=D$ the transition map ${\varphi }_{g,V}\circ {\varphi }_{f,U}^{-1}:D\to D$ is the identity. Therefore the charts are holomorphically compatible.

Let $[f{]}_z\in N(f,U)$. By [L5], there is a polygonal path in $\mathrm{\Omega }$ from $a_0$ to $z$. Cover its compact image by finitely many rational discs from $\mathcal{D}$ that lie inside the function-element neighborhoods of one continuation chain to $[f{]}_z$, and choose them in the order encountered along the path, with the last disc contained in $U$ and containing $z$. The resulting rational-disc chain determines the same terminal branch on that last disc, so it produces a countable-basis neighbourhood of $[f{]}_z$ contained in $N(f,U)$. Thus the topology has a countable basis, and [L4] makes the germ space second countable.

Let $\xi =[f{]}_z$ be a point of the germ surface. By [L1], the basis neighbourhood $N(f,U)$ of $\xi$ is mapped by the projection $p$ exactly as the chart ${\varphi }_{f,U}$, namely $p{\mathrm{\mid }}_{N(f,U)}={\varphi }_{f,U}:N(f,U)\to U$, and its inverse is $u\mapsto [f{]}_u$.

Fact [L1] makes both $p{\mathrm{\mid }}_{N(f,U)}$ and its inverse holomorphic in the chosen charts. Therefore $p{\mathrm{\mid }}_{N(f,U)}$ is a biholomorphism onto the open set $U$, and [L2] shows that $p$ is a local biholomorphism.

For each $w\in \mathbb{C}$, let $V_w:=e^{w}B(1,1\mathrm{/}2)$ and define ${\mathrm{\ell }}_w(u):=w+\mathrm{Log}(e^{-w}u)$ for $u\in V_w$. Because $B(1,1\mathrm{/}2)\subseteq S$, fact [L1] makes ${\mathrm{\ell }}_w$ holomorphic on $V_w$, with $\exp ({\mathrm{\ell }}_w(u))=e^w{e}^{\mathrm{Log}(e^{-w}u)}=u$ and ${\mathrm{\ell }}_w(e^w)=w$. Thus $\mathrm{\Phi }(w):=[{\mathrm{\ell }}_w{]}_{e^w}$ is a well-defined logarithm germ over $e^w$.

The germ $\mathrm{\Phi }(w)$ lies on ${\mathcal{R}}_{\log }$. Indeed, along the path ${\gamma }_w(t)=e^{tw}$ from $1$ to $e^w$, choose a subdivision fine enough that for every adjacent pair $t_j,t_{j+1}$ and every $t\in [t_j,t_{j+1}]$ one has $\mathrm{\mid }{e}^{(t-t_j)w}-1\mathrm{\mid }<\frac{1}{2},$ and also $\mathrm{\mid }\mathrm{Im}((t_{j+1}-t_j)w)\mathrm{\mid }<\pi$. The first condition puts the entire subpath ${\gamma }_w([t_j,t_{j+1}])$ inside $V_{t_{j}w}$. At the joining point, the second condition gives ${\mathrm{\ell }}_{t_{j}w}(e^{t_{j+1}w})=t_{j}w+\mathrm{Log}(e^{(t_{j+1}-t_j)w})=t_{j+1}w,$ so consecutive branches agree there as germs. Starting at $t_0=0$ gives the principal logarithm germ at $1$, and the final germ is $\mathrm{\Phi }(w)$.

If $\mathrm{\Phi }(w)=\mathrm{\Phi }(v)$, then the two germs have the same base point and the same value at that base point. Therefore $e^w=e^v$ and $w=v$. So $\mathrm{\Phi }$ is injective.

Let $\xi =[\mathrm{\ell }{]}_z$ be any point of ${\mathcal{R}}_{\log }$ and put $w:=\mathrm{\ell }(z)$. Shrink the representative domain of $\mathrm{\ell }$ to a disc $U\subseteq V_w$ on which $\mathrm{\ell }(U)\subseteq B(w,\pi )$. On $U$ one has $\exp (\mathrm{\ell }(u))=u=\exp ({\mathrm{\ell }}_w(u))$, so [L2] gives $\mathrm{\ell }(u)-{\mathrm{\ell }}_w(u)\in 2\pi i\mathbb{Z}$ for every $u\in U$. The difference is continuous, takes the value $0$ at $z$, and its image lies in the discrete set $2\pi i\mathbb{Z}$; because $U$ is connected, the difference is identically $0$. Hence $\xi =\mathrm{\Phi }(w)$, and $\mathrm{\Phi }$ is surjective.

The map $\mathrm{\Lambda }$ is the inverse of $\mathrm{\Phi }$ by steps 2.1 and 2.2. In a chart $N(\mathrm{\ell },U)$ on the germ surface one has $\mathrm{\Lambda }\circ {\varphi }_{\mathrm{\ell },U}^{-1}(u)=\mathrm{\ell }(u)$, so [L3] makes $\mathrm{\Lambda }$ holomorphic. Near any $w_0\in \mathbb{C}$, if $\mathrm{\mid }w-w_0\mathrm{\mid }<1\mathrm{/}4$ then $\mathrm{\Phi }(w)\in N({\mathrm{\ell }}_{w_0},V_{w_0})$ and ${\varphi }_{{\mathrm{\ell }}_{w_0},V_{w_0}}(\mathrm{\Phi }(w))=e^w$, which is holomorphic in $w$; so $\mathrm{\Phi }$ is holomorphic as well. Therefore $\mathrm{\Lambda }$ is a biholomorphism.

For every $[\mathrm{\ell }{]}_z\in {\mathcal{R}}_{\log }$ one has $p([\mathrm{\ell }{]}_z)=z=\exp (\mathrm{\ell }(z))=\exp (\mathrm{\Lambda }([\mathrm{\ell }{]}_z))$, so $p=\exp \circ \mathrm{\Lambda }$.