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  "body": "\n## Statement\n\nLet $M$ be a closed, oriented, [[smooth|F45VQ47GPV]] [[4-manifold|8H58Q6YJCH]] that admits a [[spin structure|RSPZ9TCFQ8]].\nLet $b_2(M)$ be the rank of $H_2(M; \\Z)$ and $\\sigma(M)$ the [[signature|07EZA80NEH]] of its [[intersection form|WK8GKRKAVD]].\nThen\n$$b_2(M) \\ge \\frac{11}{8} \\abs{\\sigma(M)}.$$\n\n## Remarks\n\nThe conjecture is open.\nThe K3 surface has $b_2 = 22$ and $\\sigma = -16$, so it attains equality.\nFor a simply connected $M$, the intersection form is even, and by [[Donaldson's theorem|34B7K77P31]] it is indefinite unless it is zero.\nSo it is a sum of copies of $\\pm E_8$ and of the hyperbolic plane.\nThe conjecture then says that each pair of $E_8$ summands needs at least three hyperbolic summands, as in the K3 surface.\nMikio Furuta proved the $10/8$ theorem: if $\\sigma(M) \\ne 0$, then $b_2(M) \\ge \\frac{10}{8} \\abs{\\sigma(M)} + 2$.\nLater work has improved the constant term of this bound.\nWithout smoothness the inequality fails: the [[$E_8$ manifold|BNDRN0G9BR]] is a closed topological manifold with an even form, $b_2 = 8$ and $\\sigma = 8$.\n\n## Explanations\n\n### L1 human\n\nFor a smooth four-dimensional space of spin type, the conjecture compares two numbers read off from how surfaces cross inside it.\nIt predicts that the total number of independent surfaces is at least $11/8$ times the size of the signature.\nThe K3 surface, a famous complex surface, shows that $11/8$ cannot be improved.\n",
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      "first_stated": "1982",
      "posed_by": "Yukio Matsumoto"
    },
    "kind": "conjecture",
    "msc": [
      "57K40",
      "57R55",
      "57R15"
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    "name": "11/8 conjecture",
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  "link": "https://math2.tamnd.com/node/90RA6TAT7H/",
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  "msc": [
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  "name": "11/8 conjecture",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/90R/90RA6TAT7H.md",
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