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A symplectic homology basis of a genus-two surface

Statement

Assume the Axiom of Choice (The Axiom of Choice) through the polygonal normal form, surface classification, integral cup-pairing, and Poincaré-duality interfaces. Let $Σ_{2}$ be the quotient of an oriented octagon with boundary word $a_{1} b_{1} a_{1}^{- 1} b_{1}^{- 1} a_{2} b_{2} a_{2}^{- 1} b_{2}^{- 1}$ , the standard genus-two model (Polygonal normal forms for compact connected surfaces, Classification of compact connected surfaces, Polygonal schemas and paired boundary edges). Let $e_{1} = [a_{1}], e_{2} = [b_{1}], e_{3} = [a_{2}], e_{4} = [b_{2}]$ . Then:

$H_{1} (Σ_{2}; Z) = Z^{4}$ with basis $e_{1}, e_{2}, e_{3}, e_{4}$ (Cellular homology of the one-polygon surface model).
In this ordered basis, the intersection matrix of The intersection form on the homology of a closed oriented surface is $(\begin{array}{cccc} 0 & 1 & 0 & 0 \\ - 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & - 1 & 0 \end{array}) .$ Equivalently, $a_{i} \cdot b_{j} = δ_{i j}$ , $b_{i} \cdot a_{j} = - δ_{i j}$ , and all same-type products vanish. Its determinant is $1$ , so this is a symplectic basis and the form is unimodular.
The endpoint cases are consistent: at genus $0$ the paired digon has $H_{1} = 0$ and the empty intersection matrix; at genus $1$ the commutator square has $H_{1} ≅ Z^{2}$ and matrix $J_{2} = (\begin{array}{cc} 0 & 1 \\ - 1 & 0 \end{array})$ (Polygonal normal forms for compact connected surfaces, Classification of compact connected surfaces, Cellular homology of the one-polygon surface model, Integral surface cup pairing from the oriented polygon).

Facts & Assumptions

Given: The oriented octagon and side-pairings of the Statement.

[F1]

The one-polygon schema has its corner classes as vertices, paired sides as edges, and disk interior as a face; the commutator word with two handle blocks is the genus-two normal form, and opposite-exponent side pairs are orientation-compatible (Polygonal schemas and paired boundary edges, Polygonal normal forms for compact connected surfaces, Classification of compact connected surfaces, The Axiom of Choice).

[F2]

Under AC, for a genus- $g$ commutator surface the cellular calculation gives the ordered side-loop classes as a $Z$ -basis of $H_{1}$ of rank $2 g$ , with $H_{1} = 0$ at genus zero (The Axiom of Choice, Cellular homology of the one-polygon surface model).

[F3]

Under AC, in the evaluation-dual cohomology basis $x_{a_{i}}, x_{b_{i}}$ and positive generator $ω$ of $H^{2}$ , the polygon cup computation is $x_{a_{i}} ⌣ x_{b_{j}} = δ_{i j} ω$ , $x_{b_{i}} ⌣ x_{a_{j}} = - δ_{i j} ω$ , and same-type products are zero; it also covers the empty genus-zero basis (The Axiom of Choice, Integral surface cup pairing from the oriented polygon, Kronecker evaluation pairing).

[F4]

Under AC, cap with $[Σ_{g}]$ gives $D (a) = a \cap [Σ_{g}]$ ; the intersection form is $⟨ γ, δ ⟩ = ⟨ D^{- 1} (γ) ⌣ D^{- 1} (δ), [Σ_{g}] ⟩$ , and cap-cup adjunction is $⟨ a ⌣ b, [Σ_{g}] ⟩ = ⟨ b, D (a) ⟩$ (The Axiom of Choice, The intersection form on the homology of a closed oriented surface, Kronecker evaluation pairing).

[F5]

The sphere digon and commutator square are the standard genus-zero and genus-one schemas; the two commutator blocks specify the genus-two model (The Axiom of Choice, Polygonal schemas and paired boundary edges, Polygonal normal forms for compact connected surfaces, Classification of compact connected surfaces).

Proof

technique · direct

1.1F1construct

The eight corners of the octagon lie in one vertex class: side pairings give $v_{0} \sim v_{3} \sim v_{2} \sim v_{1} \sim v_{4} \sim v_{7} \sim v_{6} \sim v_{5} \sim v_{0}$ , and the corresponding corner sectors form one link cycle. There are four paired edges and one face. Every side pair has opposite exponents, so the face orientation descends to an orientation of the closed connected surface. The word has two commutator blocks, hence is the genus-two normal form by [F1].

1.2F2

Applying [F2] to this model gives $H_{1} (Σ_{2}; Z) = Z^{4}$ with ordered basis $e_{1}, e_{2}, e_{3}, e_{4}$ .

2.1F3step 1.2

Let $J = diag (J_{2}, J_{2})$ and let $x_{1}, x_{2}, x_{3}, x_{4}$ be the evaluation-dual cohomology basis. By [F3], $⟨ x_{p} ⌣ x_{q}, [Σ_{2}] ⟩ = J_{p q}$ .

3.1F3F4step 2.1

By the adjunction in [F4], $⟨ x_{q}, D (x_{p}) ⟩ = ⟨ x_{p} ⌣ x_{q}, [Σ_{2}] ⟩ = J_{p q}$ , so $D (x_{p}) = \sum_{q} J_{p q} e_{q}$ . Put $z_{q} = \sum_{p} J_{p q} x_{p}$ . Since $⟨ x_{r}, D (z_{q}) ⟩ = \sum_{p} J_{p q} J_{p r} = (J^{T} J)_{q r} = δ_{q r}$ , we have $D (z_{q}) = e_{q}$ and $D^{- 1} (e_{q}) = z_{q}$ .

4.1F4step 3.1

Substituting these coordinates into [F4] gives $⟨ e_{p}, e_{q} ⟩ = \sum_{r, s} J_{r p} J_{s q} J_{r s} = (J^{T} J J)_{p q} = J_{p q}$ . Thus the displayed matrix is $diag (J_{2}, J_{2})$ ; its determinant is $\det (J_{2})^{2} = 1$ , proving the symplectic and unimodular claims.

5.1F2F3F4F5step 4.1∎

The same A-page suppliers [F2–F5] give the endpoint cases: for genus zero, $H_{1} = 0$ and the unique form on the zero group has empty matrix and determinant $1$ by convention; for genus one, the standard square has $H_{1} ≅ Z^{2}$ with matrix $J_{2}$ and determinant $1$ . These computations use the cellular, cup-pairing, and intersection-form suppliers rather than importing an examples-page result.

Remarks

The octagon is a concrete two-handle instance of the commutator normal form. The finite cell and matrix computations are choice-free after the normal form and integral cup/duality interfaces are fixed.

Depends on

Used by

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Dependency tree · two levels

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Sources

Allen Hatcher, Algebraic Topology (author-hosted PDF) (standard reference, not scraped)
Jean Gallier and Dianna Xu, A Guide to the Classification Theorem for Compact Surfaces (standard reference, not scraped)