Algebraic data of the intersection form of a closed oriented $4$-manifold: second Betti number $b_2 = b_2^+ + b_2^-$, signature $\sigma = b_2^+ - b_2^-$, parity, and a symmetric bilinear form on $H^2(M; \mathbb{Z}) \cong \mathbb{Z}^{b_2}$.
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An almost-complex structure $J$ on a smooth $4$-manifold (placeholder, witnessing existence).
- mk' :: (
- )
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Topological data attached to a closed smooth $4$-manifold $M$: its intersection form Q
and Euler characteristic $\chi(M) = 2 + b_2(M)$ (for simply connected $M$).
- euler : ℤ
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Chern and Pontrjagin number data for a closed $4$-manifold equipped with an almost complex structure: $c_1^2 \cdot [M]$, $c_2 \cdot [M]$, with $c_2 = \chi(M)$.
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First Pontrjagin number $p_1 \cdot [M] = c_1^2 - 2 c_2$, the Whitney sum relation for an almost complex $4$-manifold.
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The rearranged Chern–Pontrjagin relation: $c_1^2 = 2 c_2 + p_1$.
Data of a closed oriented $4$-manifold: Pontrjagin number $p_1$, Betti numbers $b_2^\pm$, signature $\sigma$, and a symmetric intersection form on $H^2 \cong \mathbb{Z}^{b_2}$.
- p₁ : ℤ
- b₂_plus : ℕ
- b₂_minus : ℕ
- signature : ℤ
- intersectionForm_symm (x y : Fin (self.b₂_plus + self.b₂_minus) → ℤ) : self.intersectionForm x y = self.intersectionForm y x
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Hirzebruch signature theorem (axiomatized): $p_1 \cdot [M] = 3 \sigma(M)$ for closed oriented $4$-manifolds.
Hirzebruch signature theorem (Theorem 1 of the chapter): $p_1(TM) \cdot [M] = 3 \sigma(M)$.
Build closed-oriented manifold data from Chern–Pontrjagin data plus topological data.
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Bundled data for a closed oriented almost-complex $4$-manifold: the underlying topology, Euler number $\chi$, $c_1^2$, $c_2 = \chi$, and the Whitney relation $p_1 = c_1^2 - 2c_2$.
- mfd : ClosedOriented4ManifoldData M
- euler : ℤ
- c₁_sq : ℤ
- c₂ : ℤ
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The Chern–Pontrjagin relation $c_1^2 = 2 c_2 + p_1$ for closed almost complex $4$-manifolds.
Construct the bundled closed Chern–Pontrjagin data from HasChernPontrjaginData together
with Mfd4Topology, assuming the simply-connected case.
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Corollary 1 (closed form): $c_1^2 \cdot [M] = 2 \chi(M) + 3 \sigma(M)$, obtained by combining the Chern–Pontrjagin relation with the Hirzebruch signature theorem.
Corollary 1: for a closed almost complex $4$-manifold, $c_1^2 \cdot [M] = 2 \chi(M) + 3 \sigma(M)$.
Generic algebraic data needed to state the almost-complex existence criterion: rank of $H^2$, Euler number $\chi$, signature $\sigma$, and a symmetric bilinear pairing modeling the intersection form $Q$.
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The square $\alpha^2 \cdot [M] = Q(\alpha, \alpha)$ evaluated via the bilinear pairing.
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Self-pairing $Q([A], [A])$ of a homology class with itself under the intersection form.
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Witness that an almost complex structure J has first Chern class with prescribed coordinates
in the basis of $H^2(M; \mathbb{Z})$.
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Theorem 2 (almost-complex existence): there exists an almost complex structure $J$ on $M^4$ with $\alpha = c_1(TM, J)$ iff $\alpha^2 \cdot [M] = 2\chi + 3\sigma$ and $\alpha \cdot [A] \equiv Q([A], [A]) \pmod 2$ for every $[A] \in H_2(M; \mathbb{Z})$.
Obstruction data for an almost-complex structure, tied to the ambient Mfd4Topology: the
rank of $H^2$ matches $b_2(M)$, and a symmetric pairing models the intersection form.
- rank_H2 : ℕ
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Forget the link to Mfd4Topology and produce general almost-complex obstruction data.
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Witness that an almost complex structure J has a first Chern class with the given
integer coordinates in the obstruction basis.
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Obstruction example: $\mathbb{CP}^2 \# \mathbb{CP}^2$ (with $b_2 = 2$, $\sigma = 2$, $\chi = 4$) admits no almost-complex structure, since the Chern number formula forces $c_1^2 = 14$, which cannot be written as a sum of two squares.