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  "body": "\n## Definition\n\nLet $M$ be a closed, connected, [[oriented|913WEQERBE]] [[topological manifold|QQQ9B1DK4P]] of dimension $4$, with [[fundamental class|ZEA0PEM3B2]] $[M] \\in H_4(M; \\Z)$.\nLet $H^2$ be [[singular cohomology|6N7BC70CX0]] and $\\smile$ the [[cup product|JX47MS1R1H]].\nLet $\\langle \\cdot, \\cdot \\rangle$ be the Kronecker pairing of $H^4(M; \\Z)$ with $H_4(M; \\Z)$.\nThe *intersection form* of $M$ is the map\n$$Q_M \\colon H^2(M; \\Z) \\times H^2(M; \\Z) \\to \\Z, \\qquad Q_M(a, b) = \\langle a \\smile b, [M] \\rangle.$$\n\n## Remarks\n\n$Q_M$ is a [[symmetric bilinear form|0AXZNE5XNH]], since $a \\smile b = b \\smile a$ for classes of even degree.\nIt vanishes if $a$ or $b$ is a torsion class, so it induces a form on the free abelian group $H^2(M; \\Z)/\\text{torsion}$.\nBy [[Poincaré duality|9VY90HJVZX]] this induced form is unimodular: its matrix in any basis is a [[unimodular matrix|R6RBQHW191]], with determinant $\\pm 1$.\nIts rank is $b_2(M)$, and the [[signature|07EZA80NEH]] of its real extension is the signature of $M$.\n\nExample: for the complex projective plane $\\C P^2$ with its complex orientation, $H^2 \\cong \\Z$ is generated by $x$ with $\\langle x^2, [\\C P^2] \\rangle = 1$, so $Q = (1)$.\nFor $S^2 \\times S^2$ the form has matrix $\\begin{pmatrix} 0 \u0026 1 \\\\ 1 \u0026 0 \\end{pmatrix}$ in the basis given by the two factors.\n\nNon-example: on a closed oriented surface $\\Sigma$, the analogous form $H^1(\\Sigma) \\times H^1(\\Sigma) \\to \\Z$ is skew-symmetric, not symmetric.\nThe reason is that $a \\smile b = -\\,b \\smile a$ in degree $1$.\nA symmetric form needs dimension $4$, or more generally $4k$ with middle degree $2k$.\n",
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