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  "body": "\n## Statement\n\nLet $M$ be a closed oriented [[smooth manifold|F45VQ47GPV]] of dimension $4k$ with fundamental class $[M]$.\nLet $\\sigma(M)$ be the [[signature|07EZA80NEH]] of its [[intersection form|WK8GKRKAVD]] on $H^{2k}(M; \\R)$, the symmetric bilinear form $(a, b) \\mapsto \\langle a \\smile b, [M] \\rangle$.\nLet $p_i = p_i(TM) \\in H^{4i}(M; \\Q)$ be the [[Pontryagin classes|DN11FKWN63]] of the tangent bundle.\nLet $L_k(p_1, \\ldots, p_k)$ be the degree $4k$ part of the multiplicative sequence of Hirzebruch with characteristic power series $\\sqrt{z} / \\tanh \\sqrt{z}$.\nThen\n$$\\sigma(M) = \\langle L_k(p_1, \\ldots, p_k), [M] \\rangle.$$\n\n## Remarks\n\nThe first two polynomials are $L_1 = \\tfrac{1}{3} p_1$ and $L_2 = \\tfrac{1}{45}(7 p_2 - p_1^2)$.\nIdea of the proof: both sides are invariants of oriented [[cobordism|T8BVGPPSZP]], additive under disjoint union and multiplicative under products.\nSo both sides are ring homomorphisms from the rational oriented cobordism ring to $\\Q$.\nThom showed that this ring is a polynomial ring on the classes of the complex projective spaces $\\C P^{2i}$.\nThe power series is chosen so that the right side equals $1 = \\sigma(\\C P^{2i})$ on each generator, and so the two homomorphisms agree.\n\nExample: for $M = \\C P^2$ with generator $h \\in H^2$, the total Pontryagin class is $(1 + h^2)^3$, so $p_1 = 3h^2$ and $\\langle L_1, [M] \\rangle = 1$.\nThe intersection form is $(1)$, so $\\sigma(\\C P^2) = 1$, as the theorem says.\nFor a closed oriented smooth $4$-manifold the theorem gives $p_1[M] = 3 \\sigma(M)$.\n\n## Explanations\n\n### L1 human\n\nThe signature is a whole number read off from how middle-dimensional cycles in a manifold intersect.\nPontryagin classes measure how the tangent bundle is twisted.\nThe theorem says that one fixed combination of the Pontryagin numbers always equals the signature.\nSo a number defined by pure topology is computed by curvature-type data.\n\n### L2 human\n\nThe theorem shows that a certain rational combination of Pontryagin numbers is always an integer, which constrains which manifolds can exist.\nMilnor used the case $k = 2$ to show that some $7$-manifolds homeomorphic to $S^7$ are not diffeomorphic to it, the first [[exotic spheres|6WYEMXSHD9]].\nThe theorem is a special case of the Atiyah–Singer index theorem applied to the [[signature operator|4NCYA7GME1]].\nIt is also the model for the [[Hirzebruch–Riemann–Roch theorem|G3X02FKQCR]].\nIn surgery theory, the signature formula is what makes the simply connected surgery obstruction in dimensions $4k$ computable.\n",
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