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  "body": "\n## Definition\n\nLet $M$ be a [[topological manifold|QQQ9B1DK4P]].\nLet $\\mathrm{TOP}$ and $\\mathrm{PL}$ be the stable groups of homeomorphisms and of piecewise linear homeomorphisms of Euclidean spaces, with classifying spaces $B\\mathrm{TOP}$ and $B\\mathrm{PL}$.\nThe stable tangent microbundle of $M$ has a classifying map $\\tau_M \\colon M \\to B\\mathrm{TOP}$.\nThe forgetful map $B\\mathrm{PL} \\to B\\mathrm{TOP}$ is a fibration up to homotopy, with fibre $\\mathrm{TOP}/\\mathrm{PL}$.\nBy a theorem of Kirby and Siebenmann, $\\mathrm{TOP}/\\mathrm{PL}$ is an Eilenberg–MacLane space $K(\\Z/2, 3)$: its only nonzero [[homotopy group|P976WYEFMB]] is $\\pi_3 \\cong \\Z/2$.\nThe *Kirby–Siebenmann class* $\\operatorname{ks}(M) \\in H^4(M; \\Z/2)$ is the obstruction class, in [[singular cohomology|6N7BC70CX0]] with coefficients $\\pi_3(\\mathrm{TOP}/\\mathrm{PL}) \\cong \\Z/2$, to lifting $\\tau_M$ to a map $M \\to B\\mathrm{PL}$.\n\n## Remarks\n\nThe $4$-dimensional $E_8$ manifold $M_{E_8}$ has $\\operatorname{ks}(M_{E_8}) \\ne 0$ in $H^4(M_{E_8}; \\Z/2) \\cong \\Z/2$.\nFor a closed simply connected topological $4$-manifold with even intersection form, $\\operatorname{ks}$ is the signature divided by $8$, taken modulo $2$, and the $E_8$ form has signature $8$.\n\nA space that is not a manifold, such as a finite graph with a vertex of degree three, has no tangent microbundle, so the definition does not apply to it.\n\nIf $M$ has a [[piecewise linear structure|15MKGRNM9K]], or a smooth structure, then $\\operatorname{ks}(M) = 0$.\nConversely, if $\\dim M \\ge 5$ and $\\operatorname{ks}(M) = 0$, then $M$ has a piecewise linear structure; for any $M$, $\\operatorname{ks}(M) = 0$ if and only if $M \\times \\R$ has one.\nThe class is natural: for an open subset $U \\subseteq M$, $\\operatorname{ks}(U)$ is the restriction of $\\operatorname{ks}(M)$.\n",
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