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The Kirby-Siebenmann class can be used to prove the existence of topological manifolds that do not admit a PL-structure. Concrete examples of such manifolds are
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It is the only such obstruction, which can be phrased as the weak equivalence
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Foundational Essays on
Topological Manifolds, Smoothings, and Triangulations
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is an obstruction for topological manifolds to allow a
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121:{\displaystyle \kappa (M)\in H^{4}(M;\mathbb {Z} /2)}
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Piecewise linear structures on topological manifolds
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208:{\displaystyle TOP/PL\sim K(\mathbb {Z} /2,3)}
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271:{\displaystyle E_{8}\times T^{n},n\geq 1}
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298:{\displaystyle E_{8}}
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