Riemann surfaces · The Lumenary

MotivationH

Making multivalued functions single-valued

Every non-zero complex number has two square roots, and no continuous choice of one of them works on all of $\C \setminus \{0\}$ : follow $\sqrt z$ once round the origin and it comes back as $-\sqrt z$ . The logarithm is worse, with infinitely many values. For a century these "multivalued functions" were handled by cutting the plane and keeping track of branches. In his 1851 dissertation Bernhard Riemann proposed a radical alternative: do not change the function, change its domain. Build a surface lying over the plane, with one sheet for each value, glued so that walking round a branch point carries you from one sheet to the next. On that surface the square root is an honest, single-valued holomorphic function.

These Riemann surfaces turned out to be the natural home of an astonishing amount of mathematics. Compact Riemann surfaces are the same thing as complex algebraic curves (solution sets of polynomial equations $F(z, w) = 0$ ); their topology is a single number, the genus; the integrals of algebraic functions — elliptic integrals, which compute the arc length of an ellipse or the period of a pendulum — are integrals of $1$ -forms over loops on them; and the same equations studied over the rational numbers are the elliptic curves of number theory.

This lesson builds on complex analysis, the fundamental group and covering spaces, and the classification of surfaces by their Euler characteristic. It uses the language of manifolds, with holomorphic charts in place of smooth ones.

IntuitionH

Two sheets glued crosswise

Take two copies of the plane, cut each along the negative real axis, and glue the upper edge of the cut in sheet 1 to the lower edge of the cut in sheet 2, and vice versa. A path going once round $0$ starts on sheet 1, crosses the cut and continues on sheet 2, and only after a second turn returns to its starting point. On this two-sheeted surface (completed by one point over $0$ and one over $\infty$ ) the function " $\sqrt z$ " takes the value $\sqrt z$ on sheet 1 and $-\sqrt z$ on sheet 2, and is continuous. In fact the surface is just another copy of the plane, with coordinate $w$ , lying over the $z$ -plane by $z = w^2$ : the "two sheets" are the two preimages of each $z \ne 0$ .

The Riemann surface of $\sqrt z$: two sheets over the $z$-plane, cut along the negative real axis and cross-glued. A loop round $0$ on the surface (gold) has to go round twice below before it closes. The point over $0$ is a branch point of order $2$.

The same recipe for $w = \sqrt{z^3 - z}$ needs two cuts — say from $-1$ to $0$ and from $1$ to $\infty$ — because a loop round one of the branch points $-1, 0, 1, \infty$ swaps the sheets while a loop round two of them does not. Two cut spheres glued crosswise along two slits make a torus: the Riemann surface of $\sqrt{z^3 - z}$ has genus $1$ . Counting branch points to find the genus is the Riemann–Hurwitz formula.

DefinitionsH

Riemann surfaces and holomorphic maps

Definition— Riemann surface

A Riemann surface is a connected, Hausdorff, second countable space $X$ with an atlas of charts $\varphi_\alpha : U_\alpha \to \varphi_\alpha(U_\alpha) \subseteq \C$ whose transition maps $\varphi_\beta \circ \varphi_\alpha^{-1}$ are holomorphic (a maximal such atlas is a complex structure). Equivalently, a connected complex manifold of complex dimension $1$ . (Second countability is in fact automatic, by Radó's theorem.)

Definition— holomorphic and meromorphic maps

A continuous map $f : X \to Y$ of Riemann surfaces is holomorphic if it is holomorphic in local coordinates. A meromorphic function on $X$ is a holomorphic map $X \to \mathbb P^1$ to the Riemann sphere not identically $\infty$ ; equivalently, a function holomorphic outside a discrete set of poles. The meromorphic functions form a field $\mathcal M(X)$ .

Definition— ramification, degree, genus

By the local normal form (Theorem 1), a non-constant holomorphic $f$ is near each $p$ given in suitable coordinates by $z \mapsto z^k$ with $k = e_p \ge 1$ , the ramification index (or multiplicity) of $f$ at $p$ . Points with $e_p \ge 2$ are ramification points, and their images branch points. For compact $X$ the degree of $f$ is $\deg f = \sum_{p \in f^{-1}(q)} e_p$ (independent of $q$ , by Theorem 2). The genus of a compact Riemann surface is the genus of the underlying closed orientable surface, so $\chi(X) = 2 - 2g$ .

Holomorphic maps preserve orientation (the Jacobian determinant of $z \mapsto f(z)$ is $|f'(z)|^2 \ge 0$ ), so every Riemann surface is orientable: the Klein bottle and the projective plane carry no complex structure. Conversely, every orientable surface carries one (for example from a Riemannian metric and isothermal coordinates).

Worked examplesH

Examples

Example— The plane, the sphere and their quotients

$\C$ , the unit disc $\mathbb D$ , and every connected open subset of $\C$ .
The Riemann sphere $\mathbb P^1 = \C \cup \{\infty\}$ with the two charts $z$ on $\C$ and $w = 1/z$ on $\mathbb P^1 \setminus \{0\}$ ; the transition $z \mapsto 1/z$ is holomorphic on $\C \setminus \{0\}$ . Rational functions are exactly the meromorphic functions on $\mathbb P^1$ , and a rational function $p/q$ (in lowest terms) has degree $\max(\deg p, \deg q)$ .
Complex tori. Let $\Lambda = \Z\omega_1 + \Z\omega_2$ with $\omega_1, \omega_2$ linearly independent over $\R$ . The quotient $\C/\Lambda$ is a compact Riemann surface of genus $1$ : charts are local inverses of the projection $\pi : \C \to \C/\Lambda$ on small discs, and transition maps are translations $z \mapsto z + \omega$ . Holomorphic functions on $\C/\Lambda$ are $\Lambda$ -periodic entire functions — hence constant (Liouville; formalised below) — while meromorphic functions on $\C/\Lambda$ are the elliptic functions, such as the Weierstrass function $\wp(z) = \frac{1}{z^2} + \sum_{\omega \in \Lambda \setminus 0}\big(\frac{1}{(z - \omega)^2} - \frac{1}{\omega^2}\big)$ .
Quotients of the disc. If $\Gamma$ is a group of Möbius transformations acting freely and properly discontinuously on $\mathbb D$ or on the upper half-plane $\mathbb H$ , then $\mathbb H/\Gamma$ is a Riemann surface. Modular curves $\mathbb H/\Gamma(N)$ are of this kind (after removing elliptic fixed points or treating them as orbifold points).

Example— Algebraic curves

Let $F(z, w)$ be a polynomial and $C = \{(z, w) \in \C^2 : F(z, w) = 0\}$ . At points where $\partial F/\partial w \ne 0$ the implicit function theorem (holomorphic version) makes $z$ a local coordinate; where $\partial F/\partial z \ne 0$ , $w$ is one. So if $C$ is smooth (no common zeros of $F$ , $F_z$ , $F_w$ ), it is a Riemann surface (if connected), and $(z, w) \mapsto z$ is a holomorphic map whose ramification points are where $F_w = 0$ . Adding finitely many points at infinity (in $\mathbb P^2$ , or by gluing in charts) gives a compact Riemann surface.

For $w^2 = f(z)$ with $f$ a squarefree polynomial of degree $d$ , the projection $(z, w) \mapsto z$ has degree $2$ , ramified exactly over the roots of $f$ — and over $\infty$ when $d$ is odd. With $d = 3$ , $f = z^3 - z$ , there are four branch points $-1, 0, 1, \infty$ , and the compact curve is a torus (Riemann–Hurwitz, below).

Example— z² as a branched double cover

$f(w) = w^2$ from $\mathbb P^1$ to $\mathbb P^1$ has degree $2$ . Every $z \ne 0, \infty$ has two preimages $\pm\sqrt z$ ; over $0$ and $\infty$ there is one preimage, with $e = 2$ . Near $w = 0$ angles at the origin are doubled — the conformal map $z^2$ fails to be conformal exactly at its critical point — and the visualization below shows this. The inverse of this map is the Riemann surface of $\sqrt z$ from the intuition section.

ProofH

Local structure of holomorphic maps

Theorem— Theorem 1 (local normal form)

Let $f : X \to Y$ be a non-constant holomorphic map and $p \in X$ . There is a unique integer $k \ge 1$ and charts $z$ centred at $p$ and $w$ centred at $f(p)$ in which $f$ is $w = z^k$ .

Proof

Choose any charts centred at $p$ and $f(p)$ , so that $f$ becomes a holomorphic function $F$ near $0$ with $F(0) = 0$ . Since $X$ is connected and $f$ is not constant, $F$ is not identically $0$ (identity theorem), so $F(z) = z^k g(z)$ with $k \ge 1$ and $g$ holomorphic, $g(0) \ne 0$ . On a small disc $g$ takes values in a disc not containing $0$ , on which a holomorphic logarithm exists; so $g = h^k$ with $h = \exp\big(\tfrac1k \log g\big)$ holomorphic and $h(0) \ne 0$ . Then $\zeta = z\,h(z)$ has $\zeta'(0) = h(0) \ne 0$ , so by the inverse function theorem it is a new coordinate near $p$ , and in it $F = \zeta^k$ . The integer $k$ is determined by $f$ : it is the number of preimages near $p$ of each point near (but not equal to) $f(p)$ . ∎

∎

Consequences: a non-constant holomorphic map is open (since $z \mapsto z^k$ is), its fibres are discrete, and it is a local biholomorphism exactly where $e_p = 1$ , i.e. where its derivative in charts is non-zero. Ramification points are isolated, so on a compact surface there are finitely many.

Theorem— Theorem 2 (maps of compact surfaces are branched coverings)

Let $f : X \to Y$ be a non-constant holomorphic map with $X$ compact. Then $f$ is surjective, $Y$ is compact, every fibre is finite, and $d = \sum_{p \in f^{-1}(q)} e_p$ is the same for all $q \in Y$ . Away from the finitely many branch points, $f$ is a $d$ -sheeted covering map.

Proof

$f(X)$ is open (open mapping) and compact, hence closed; $Y$ is connected, so $f(X) = Y$ , which is compact. Fibres are discrete and closed in the compact $X$ , hence finite. Fix $q$ with $f^{-1}(q) = \{p_1, \dots, p_r\}$ and normal-form charts $z \mapsto z^{e_i}$ round each $p_i$ , on disjoint neighbourhoods $U_i$ . Shrink a neighbourhood $V$ of $q$ so that $f^{-1}(V) \subseteq \bigcup U_i$ (possible: $f(X \setminus \bigcup U_i)$ is compact and misses $q$ ). Then each $q' \in V \setminus \{q\}$ has exactly $e_i$ preimages in $U_i$ , all with multiplicity $1$ , so $\sum_{p \in f^{-1}(q')} e_p = \sum_i e_i$ . Thus the function $q \mapsto \sum_{f^{-1}(q)} e_p$ is locally constant, hence constant on the connected $Y$ . ∎

∎

Theorem— Corollary 3

On a compact Riemann surface: (a) every holomorphic function is constant; (b) a non-constant meromorphic function has as many zeros as poles, counted with multiplicity; (c) if there is a meromorphic function of degree $1$ , then $X \cong \mathbb P^1$ .

Proof

(a) A holomorphic $f : X \to \C$ composed with $\C \subset \mathbb P^1$ is a holomorphic map to $\mathbb P^1$ that misses $\infty$ ; by Theorem 2 it must be constant. (b) A meromorphic $f : X \to \mathbb P^1$ has $\deg f = \sum_{f^{-1}(0)} e_p = \sum_{f^{-1}(\infty)} e_p$ . (c) A degree- $1$ map is a bijective holomorphic map, hence a biholomorphism (its inverse is holomorphic by the local normal form with $k = 1$ ). ∎

∎

ProofH

The Riemann–Hurwitz formula

Theorem— Theorem 4 (Riemann–Hurwitz)

Let $f : X \to Y$ be a non-constant holomorphic map of degree $n$ between compact Riemann surfaces of genera $g_X$ and $g_Y$ . Then

2g_X - 2 = n\,(2g_Y - 2) + \sum_{p \in X}(e_p - 1).

Proof

The sum is finite (ramification points are isolated). Choose a triangulation of $Y$ — with geodesic or smooth edges — whose vertices include all branch points, fine enough that each closed triangle lies in a disc over which $f$ is a trivial $n$ -sheeted covering away from its vertices. Its preimage is a triangulation of $X$ : each open face and open edge, avoiding branch points, has exactly $n$ preimages, so $F_X = nF_Y$ and $E_X = nE_Y$ . A vertex $q$ has $\sum_{f^{-1}(q)} 1 = n - \sum_{p \in f^{-1}(q)}(e_p - 1)$ preimages, by Theorem 2. Hence $V_X = nV_Y - \sum_p (e_p - 1)$ , and

\chi(X) = V_X - E_X + F_X = n\,\chi(Y) - \sum_p (e_p - 1).

Substituting $\chi = 2 - 2g$ gives the formula. ∎

∎

Example— Genera from branched covers

$w = z^n$ on $\mathbb P^1$ : $e = n$ at $0$ and $\infty$ , so $-2 = n(-2) + 2(n - 1)$ . ✓
Hyperelliptic curves $w^2 = f(z)$ , $f$ squarefree of degree $d$ : a double cover of $\mathbb P^1$ with $r$ branch points, each with $e = 2$ , where $r = d$ for $d$ even and $r = d + 1$ for $d$ odd. Then $2g - 2 = 2(-2) + r$ , so $g = (r - 2)/2$ : genus $\lfloor (d - 1)/2\rfloor$ . For $w^2 = z^3 - z$ : $r = 4$ , $g = 1$ . In particular a double cover of the sphere has an even number of branch points.
Fermat curves $x^d + y^d = 1$ (compactified in $\mathbb P^2$ ): $(x, y) \mapsto x$ has degree $d$ , total ramification over the $d$ points $x^d = 1$ (where $y = 0$ , $e = d$ ), and is unramified at the $d$ points at infinity. So $2g - 2 = -2d + d(d - 1)$ and $g = (d-1)(d-2)/2$ , in agreement with the genus formula for smooth plane curves of degree $d$ .
Unramified covers of tori are tori: if $g_Y = 1$ and $f$ is unramified, then $2g_X - 2 = 0$ . And there is no non-constant holomorphic map from a torus to a surface of genus $\ge 2$ , or from $\mathbb P^1$ to a torus: in both cases the right-hand side exceeds the left.

ProofH

Complex tori: a one-parameter family

All complex tori $\C/\Lambda$ are homeomorphic (to $S^1 \times S^1$ ) and even diffeomorphic. They are not all biholomorphic.

Theorem— Theorem 5 (classification of tori)

$\C/\Lambda \cong \C/\Lambda'$ (biholomorphically) if and only if $\Lambda' = a\Lambda$ for some $a \in \C^\times$ . Every lattice is homothetic to $\Lambda_\tau = \Z + \Z\tau$ with $\operatorname{Im}\tau > 0$ , and $\Lambda_\tau$ , $\Lambda_{\tau'}$ are homothetic exactly when $\tau' = \frac{a\tau + b}{c\tau + d}$ for some $\begin{pmatrix} a & b \\ c & d\end{pmatrix} \in SL(2, \Z)$ .

Proof

If $\Lambda' = a\Lambda$ , then $z \mapsto az$ descends to a biholomorphism. Conversely, let $\phi : \C/\Lambda \to \C/\Lambda'$ be a biholomorphism; composing with a translation we may assume $\phi(0) = 0$ . Since $\C$ is simply connected, $\phi \circ \pi$ lifts through the covering $\pi' : \C \to \C/\Lambda'$ to a holomorphic $F : \C \to \C$ with $F(0) = 0$ . For each $\omega \in \Lambda$ , $F(z + \omega) - F(z) \in \Lambda'$ depends continuously on $z$ and $\Lambda'$ is discrete, so it is a constant $c_\omega$ . Hence $F'$ is $\Lambda$ -periodic and entire, so constant (Liouville), and $F(z) = az$ . Applying the same to $\phi^{-1}$ shows $a \ne 0$ and $a\Lambda = \Lambda'$ .

For the second part: rotating and scaling, any lattice $\Z\omega_1 + \Z\omega_2$ (ordered so that $\operatorname{Im}(\omega_2/\omega_1) > 0$ ) is homothetic to $\Z + \Z\tau$ with $\tau = \omega_2/\omega_1$ . Two oriented bases of the same lattice differ by a matrix in $SL(2, \Z)$ , $(\omega_2', \omega_1') = (a\omega_2 + b\omega_1, c\omega_2 + d\omega_1)$ , which changes $\tau$ to $(a\tau + b)/(c\tau + d)$ . ∎

∎

So the isomorphism classes of tori correspond to points of $\mathbb H/SL(2, \Z)$ , whose standard fundamental domain is $\{|\operatorname{Re}\tau| \le \frac12, |\tau| \ge 1\}$ ; the $j$ -invariant $j : \mathbb H/SL(2, \Z) \to \C$ is a bijection. Tori form a family with one complex parameter: the square torus ( $\tau = i$ , $j = 1728$ ) and the hexagonal torus ( $\tau = e^{2\pi i/3}$ , $j = 0$ ) are different Riemann surfaces, and they have more automorphisms fixing $0$ than other tori ( $\Z/4$ and $\Z/6$ instead of $\Z/2$ ). This is the first moduli space; for genus $g \ge 2$ , the moduli space $\mathcal M_g$ has complex dimension $3g - 3$ .

Theorem— Theorem 6 (tori are cubic curves; stated)

The Weierstrass function satisfies $\wp'(z)^2 = 4\wp(z)^3 - g_2\wp(z) - g_3$ , with $g_2 = 60\sum_{\omega \ne 0}\omega^{-4}$ , $g_3 = 140\sum_{\omega \ne 0}\omega^{-6}$ , and the map $z \mapsto [\wp(z) : \wp'(z) : 1]$ (with $0 \mapsto [0 : 1 : 0]$ ) is a biholomorphism from $\C/\Lambda$ onto the smooth projective cubic $y^2 = 4x^3 - g_2x - g_3$ . Conversely, every smooth cubic arises this way.

The differential equation is proved by noting that $\wp'^2 - 4\wp^3 + g_2\wp + g_3$ is an elliptic function whose Laurent expansion at $0$ has no pole and vanishing constant term, so by Corollary 3(a) it is identically $0$ . Mathlib proves exactly this, and we cite it below. One more consequence of Corollary 3: no elliptic function has a single simple pole (per fundamental parallelogram). Indeed, the integral of $f$ round the boundary of a fundamental parallelogram $P$ (shifted to avoid poles) vanishes, because opposite sides cancel by periodicity; by the residue theorem it equals $2\pi i$ times the sum of residues in $P$ . So the residues of an elliptic function sum to $0$ , and a single simple pole is impossible — which also follows from Corollary 3(c), since a function of degree $1$ would make the torus a sphere.

ProofH

Uniformisation and Riemann–Roch

Theorem— Theorem 7 (uniformisation; Koebe and Poincaré, 1907; stated)

Every simply connected Riemann surface is biholomorphic to exactly one of $\mathbb P^1$ , $\C$ and the unit disc $\mathbb D$ .

Since the universal cover of a Riemann surface is a simply connected Riemann surface on which $\pi_1$ acts by biholomorphisms, every Riemann surface is $\mathbb P^1$ , or $\C$ , $\C^\times$ or a torus $\C/\Lambda$ , or a quotient $\mathbb D/\Gamma$ by a discrete group of Möbius transformations. For compact surfaces this is decided by the genus: $g = 0$ gives $\mathbb P^1$ , $g = 1$ gives a torus, $g \ge 2$ gives $\mathbb D/\Gamma$ with $\Gamma$ a cocompact Fuchsian group. Geometrically: every compact Riemann surface carries a conformal metric of constant curvature $+1$ , $0$ or $-1$ , matching the sign of $\chi$ — and by Gauss–Bonnet, a hyperbolic surface of genus $g$ has area $-2\pi\chi = 4\pi(g - 1)$ . See Riemannian geometry.

A divisor on a compact Riemann surface is a finite formal sum $D = \sum n_p\, p$ of points, of degree $\sum n_p$ . Let $L(D)$ be the space of meromorphic functions $f$ with $\operatorname{div}(f) + D \ge 0$ (poles allowed only where $D$ permits) together with $0$ , and $\ell(D) = \dim L(D)$ . Let $K$ be the divisor of any non-zero meromorphic $1$ -form (a canonical divisor).

Theorem— Theorem 8 (Riemann–Roch; Riemann 1857, Roch 1865; stated)

For every divisor $D$ on a compact Riemann surface of genus $g$ ,

\ell(D) - \ell(K - D) = \deg D + 1 - g .

Example— Riemann–Roch at work

Consequences for $K$ . $D = 0$ gives $\ell(K) = g$ : there are exactly $g$ independent holomorphic $1$ -forms. $D = K$ then gives $\deg K = 2g - 2$ , matching Riemann–Hurwitz.
Genus $0$ . On $\mathbb P^1$ , $L(n\cdot\infty)$ is the space of polynomials of degree $\le n$ , of dimension $n + 1 = n + 1 - 0$ (and $\ell(K - n\infty) = 0$ since $\deg K = -2$ ). In general, on a surface of genus $0$ , $\ell(p) = 2$ for any point $p$ , so there is a meromorphic function with a single simple pole: a degree- $1$ map to $\mathbb P^1$ , and $X \cong \mathbb P^1$ by Corollary 3(c).
Genus $1$ . For $n \ge 1$ , $\ell(n\cdot 0) = n$ : on $\C/\Lambda$ , $L(0) = \C$ , $L(1\cdot 0) = \C$ (no function with one simple pole), $L(2\cdot 0) = \langle 1, \wp\rangle$ , $L(3\cdot 0) = \langle 1, \wp, \wp'\rangle$ . The seven functions $1, \wp, \wp', \wp^2, \wp\wp', \wp^3, \wp'^2$ lie in the $6$ -dimensional $L(6\cdot 0)$ , so they satisfy a linear relation — the Weierstrass cubic. Every compact surface of genus $1$ is a cubic curve, hence a torus.
Genus $2$ . $\ell(K) = 2$ and $\deg K = 2$ , so the quotient of two holomorphic $1$ -forms is a meromorphic function of degree $2$ : every surface of genus $2$ is hyperelliptic, $w^2 = f(z)$ with $\deg f = 5$ or $6$ .

Riemann–Roch also shows that every compact Riemann surface has enough meromorphic functions to embed it in some $\mathbb P^N$ (for $\deg D \ge 2g + 1$ , the functions in $L(D)$ give an embedding), and Chow's theorem then makes the image algebraic: every compact Riemann surface is a smooth projective algebraic curve. The hard analytic input is the existence of non-constant meromorphic functions, proved via harmonic functions (the Dirichlet principle) or Hodge theory; the proofs are in Forster, Chapters 2–3, and Miranda, Chapter VI.

Formal mathematicsF

Riemann surfaces in Lean and Mathlib

Mathlib's complex manifolds are manifolds modelled on $\C^n$ with holomorphic transition maps; a Riemann surface is the case $n = 1$ . The theory of branched coverings, Riemann–Hurwitz, uniformisation and Riemann–Roch is not yet formalised. What we can do is prove the analytic facts on which the lesson's examples rest.

First, the starting point of the whole subject: there is no continuous square root on $\C \setminus \{0\}$ . The proof uses connectedness of $[0, 2\pi]$ in the form "two continuous functions with the same square and no zeros agree up to a global sign":

no_continuous_sqrtLean 4Proved here from basic factsFormally verified

/-- **There is no continuous square root on `ℂ ∖ {0}`.** Suppose `g` were one. Along the circle,
`f(t) = g(e^{it})` and `h(t) = e^{it/2}` are continuous on `[0, 2π]` with `f² = h²` and `h ≠ 0`;
since `[0, 2π]` is connected, `f = h` or `f = -h` throughout. But `f(0) = f(2π) = g(1)`, while
`h(0) = 1` and `h(2π) = e^{iπ} = -1`: contradiction in both cases. The square root becomes a
single-valued continuous (indeed holomorphic) function only on its two-sheeted Riemann surface. -/
theorem no_continuous_sqrt :
    ¬ ∃ g : ℂ → ℂ, ContinuousOn g {z | z ≠ 0} ∧ ∀ z, z ≠ 0 → g z ^ 2 = z := by
  rintro ⟨g, hg, hsq⟩
  set S : Set ℝ := Set.Icc 0 (2 * Real.pi)
  let f : ℝ → ℂ := fun t => g (exp (t * I))
  let h : ℝ → ℂ := fun t => exp (t * I / 2)
  have hexp_ne : ∀ t : ℝ, exp (t * I) ≠ 0 := fun t => exp_ne_zero _
  have hf : ContinuousOn f S := by
    refine hg.comp (Continuous.continuousOn (by fun_prop)) ?_
    intro t _
    exact hexp_ne t
  have hh : ContinuousOn h S := Continuous.continuousOn (by fun_prop)
  have hfh : Set.EqOn (f ^ 2) (h ^ 2) S := by
    intro t _
    simp only [Pi.pow_apply, f, h]
    rw [hsq _ (hexp_ne t), ← exp_nat_mul]
    congr 1
    push_cast
    ring
  have hne : ∀ {t}, t ∈ S → h t ≠ 0 := fun {t} _ => exp_ne_zero _
  have h0 : (0 : ℝ) ∈ S := ⟨le_rfl, by positivity⟩
  have h2 : 2 * Real.pi ∈ S := ⟨by positivity, le_rfl⟩
  have hf0 : f 0 = g 1 := by simp [f]
  have hf2 : f (2 * Real.pi) = g 1 := by
    simp only [f]
    congr 1
    push_cast
    exact exp_two_pi_mul_I
  have hh0 : h 0 = 1 := by simp [h]
  have hh2 : h (2 * Real.pi) = -1 := by
    simp only [h]
    rw [show ((2 * Real.pi : ℝ) : ℂ) * I / 2 = Real.pi * I by push_cast; ring]
    exact exp_pi_mul_I
  rcases (isPreconnected_Icc).eq_or_eq_neg_of_sq_eq hf hh hfh hne with heq | heq
  · have a := heq h0
    have b := heq h2
    rw [hf0, hh0] at a
    rw [hf2, hh2, a] at b
    norm_num at b
  · have a := heq h0
    have b := heq h2
    simp only [Pi.neg_apply] at a b
    rw [hf0, hh0] at a
    rw [hf2, hh2, a] at b
    norm_num at b