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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.\nIt is also the model for the Hirzebruch–Riemann–Roch theorem.\nIn surgery theory, the signature formula is what makes the simply connected surgery obstruction in dimensions $4k$ computable.\n",
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