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A symplectic homology basis of a compact Riemann surface

Statement

Assume the Axiom of Choice (The Axiom of Choice), used through the polygonal normal form, the classification, the integral universal coefficient theorem, Poincaré duality, and the geometric-intersection theorem. Let $X$ be a compact connected Riemann surface of genus $g$ with its canonical orientation (Topological classification of compact Riemann surfaces, Genus and Euler characteristic of a compact Riemann surface). Choose an orientation-compatible one-polygon normal form; for $g = 0$ it is the sphere digon, and for $g \geq 1$ its boundary word is $\prod_{i = 1}^{g} a_{i} b_{i} a_{i}^{- 1} b_{i}^{- 1}$ (Polygonal normal forms for compact connected surfaces, Classification of compact connected surfaces). Let $e_{1}, e_{2}, \dots, e_{2 g}$ be the homology classes of its ordered side loops $a_{1}, b_{1}, \dots, a_{g}, b_{g}$ . Then:

Rank and basis. $H_{1} (X; Z)$ is free of rank $2 g$ , with basis $e_{1}, \dots, e_{2 g}$ ; for $g = 0$ it is zero (Cellular homology of the one-polygon surface model).
Standard intersection matrix. In the ordered basis $(a_{1}, b_{1}, \dots, a_{g}, b_{g})$ the intersection matrix is $(⟨ e_{p}, e_{q} ⟩_{X})_{p, q} = diag (J_{2}, \dots, J_{2}), J_{2} = (\begin{array}{cc} 0 & 1 \\ - 1 & 0 \end{array}),$ equivalently $⟨ a_{i}, a_{j} ⟩_{X} = ⟨ b_{i}, b_{j} ⟩_{X} = 0$ and $⟨ a_{i}, b_{j} ⟩_{X} = δ_{i j}$ . Its determinant is $1$ (including the empty matrix convention at $g = 0$ ), so the intersection form is unimodular. In the evaluation-dual basis of $H^{1} (X; Z)$ , the cup pairing has the same matrix and is unimodular.
Symplectic basis. The side-loop classes form a symplectic basis: a $Z$ -basis with the pairings in (2).
Geometric meaning. For any two smooth closed oriented embedded-curve representatives of classes $e_{p}, e_{q}$ that are transverse, their signed geometric intersection number is $⟨ e_{p}, e_{q} ⟩_{X}$ (The geometric intersection pairing on a closed oriented manifold, The geometric intersection number is the Poincare-dual cup pairing).

Reversing the surface orientation negates all intersection entries; replacing each $b_{i}$ by $- b_{i}$ restores the displayed symplectic basis convention.

Facts & Assumptions

Given: $X$ is a compact connected Riemann surface of genus $g$ with its canonical orientation.

[F1]

Under AC, the topological classification and polygonal normal form give an orientation-compatible one-polygon model: the sphere digon for $g = 0$ and the commutator $4 g$ -gon for $g \geq 1$ . The genus is the unique handle number (The Axiom of Choice, Genus and Euler characteristic of a compact Riemann surface, Topological classification of compact Riemann surfaces, Classification of compact connected surfaces, Polygonal normal forms for compact connected surfaces).

[F2]

The cellular calculation gives $H_{1} (X; Z) = 0$ when $g = 0$ , and gives the ordered side-loop classes as a $Z$ -basis of rank $2 g$ when $g \geq 1$ ; connectedness gives $H_{0} (X; Z) ≅ Z$ (Cellular homology of the one-polygon surface model).

[F3]

The AC-stated universal coefficient sequence maps $H^{1} (X; Z)$ to $Hom (H_{1} (X; Z), Z)$ by Kronecker evaluation; since $H_{0} (X; Z) ≅ Z$ is free, its ${Ext}^{1}$ term vanishes (The Axiom of Choice, Topological universal coefficient short exact sequence for cohomology, Kronecker evaluation pairing).

[F4]

For the ordered side-loop basis, its evaluation-dual cohomology basis $x_{1}, \dots, x_{2 g}$ and the positive generator $ω \in H^{2} (X; Z)$ satisfy $⟨ x_{p} ⌣ x_{q}, [X] ⟩ = J_{p q}$ , where $J = diag (J_{2}, \dots, J_{2})$ (Integral surface cup pairing from the oriented polygon).

[F5]

Under AC, Poincaré duality makes $D_{X} (a) = a \cap [X]$ an isomorphism; the intersection-form definition states both $⟨ γ, δ ⟩_{X} = ⟨ D_{X}^{- 1} (γ) ⌣ D_{X}^{- 1} (δ), [X] ⟩_{X}$ and the cap-cup adjunction $⟨ a ⌣ b, [X] ⟩ = ⟨ b, D_{X} (a) ⟩$ (The Axiom of Choice, The intersection form on the homology of a closed oriented surface, Poincaré duality for oriented topological manifolds).

[F6]

The geometric-intersection theorem, invoked under AC, identifies for closed oriented smooth embedded curves $A, B$ in a closed oriented smooth surface that are transverse the signed count $I (A, B)$ with $⟨ PD [A] ⌣ PD [B], [X] ⟩$ in the stated first-factor convention; the finite count itself is choice-free (The Axiom of Choice, The geometric intersection pairing on a closed oriented manifold, The geometric intersection number is the Poincare-dual cup pairing).

[F7]

The complex atlas makes $X$ smooth because its holomorphic transitions are smooth in real coordinates; the compact Riemann surface is closed and its complex structure gives its canonical orientation (Riemann surfaces and holomorphic atlases, Holomorphic functions are real analytic and smooth in their two real coordinates, Topological classification of compact Riemann surfaces).

Proof

technique · direct

1.1F1F2

Choose the one-polygon normal form in [F1] with its orientation matching the canonical orientation of $X$ . If this orientation gives the inverse commutator word, use the finite relabeling $a_{i} : = b_{g + 1 - i}$ and $b_{i} : = a_{g + 1 - i}$ ; since $[a, b]^{- 1} = [b, a]$ , this changes the reversed word back to $\prod_{i} a_{i} b_{i} a_{i}^{- 1} b_{i}^{- 1}$ without changing the quotient orientation. By [F2], for $g \geq 1$ the ordered side loops are a basis of $H_{1} (X; Z)$ , and for $g = 0$ the group is zero.

1.2F2F3

Suppose $g \geq 1$ . The universal coefficient sequence [F3] has kernel ${Ext}_{Z}^{1} (H_{0} (X; Z), Z) = {Ext}_{Z}^{1} (Z, Z) = 0$ , since $Z$ is free. Thus evaluation $β : H^{1} (X; Z) \to Hom (H_{1} (X; Z), Z)$ is an isomorphism; let $e_{p}^{*}$ be the coordinate functional with $e_{p}^{*} (e_{q}) = δ_{p q}$ and set $x_{p} = β^{- 1} (e_{p}^{*})$ . These classes form the cohomology basis dual to the side-loop basis.

2.1F4step 1.2

Suppose $g \geq 1$ and let $J = diag (J_{2}, \dots, J_{2})$ . The polygon cup-pairing computation [F4] gives $⟨ x_{p} ⌣ x_{q}, [X] ⟩ = J_{p q}$ , with the positive generator $ω$ normalized by $⟨ ω, [X] ⟩ = 1$ . Thus the cup-pairing matrix on this evaluation-dual basis is $J$ , so it is unimodular.

3.1F3F5step 1.2step 2.1

Suppose $g \geq 1$ . By the adjunction identity in [F5], $⟨ x_{q}, D_{X} (x_{p}) ⟩ = ⟨ x_{p} ⌣ x_{q}, [X] ⟩ = J_{p q}$ . Since $x_{q}$ is evaluation-dual to $e_{q}$ by step 1.2, these are the coordinates of $D_{X} (x_{p})$ , so $D_{X} (x_{p}) = \sum_{q} J_{p q} e_{q}$ . For each $q$ put $z_{q} = \sum_{p} J_{p q} x_{p}$ . Then $⟨ x_{r}, D_{X} (z_{q}) ⟩ = \sum_{p} J_{p q} J_{p r} = (J^{T} J)_{q r} = δ_{q r}$ , hence $D_{X} (z_{q}) = e_{q}$ and $D_{X}^{- 1} (e_{q}) = z_{q}$ .

4.1F5step 3.1

Suppose $g \geq 1$ . Substituting the inverse-duality coordinates from step 3.1 into the definition in [F5] gives $⟨ e_{p}, e_{q} ⟩_{X} = \sum_{r, s} J_{r p} J_{s q} J_{r s} = (J^{T} J J)_{p q} = J_{p q},$ because $J^{T} J = I$ . Thus the matrix is $J$ , each $2 \times 2$ block has determinant $1$ , and the form is unimodular for $g \geq 1$ .

5.1F2F3F6F7step 4.1∎

If $g = 0$ , [F2] gives $H_{1} (X; Z) = 0$ , and [F3] then gives $H^{1} (X; Z) = 0$ because $H_{0} (X; Z) ≅ Z$ is free. Both pairings are on the zero group, whose empty matrix has determinant $1$ by convention, so both are unimodular. For $g \geq 1$ , whenever smooth transverse embedded-curve representatives of the side-loop classes are given, [F6,F7] identifies their signed counts with the same intersection form. Reversing orientation changes $[X]$ to $- [X]$ and negates the intersection form; replacing each $b_{i}$ by $- b_{i}$ restores the displayed symplectic basis convention.

Depends on

Used by

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Sources

Allen Hatcher, Algebraic Topology (author-hosted PDF) (standard reference, not scraped)
Curtis T. McMullen, Riemann Surfaces, Harvard Math 213b course notes (2026) (standard reference, not scraped)
Eduard Looijenga, Riemann Surfaces (2007 author lecture notes) (standard reference, not scraped)