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10 local classification theorems
11o Stability of local maps
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admissible extension admissible map basic path called coefficient group cohomology compact-open topology completely regular completes the proof complex numbers conic neighborhood conic point connected around xQ continuous real function Corollary countable basis define a continuous define a homotopy Define a path Define an admissible deformation retract deg g denote the natural element exists a path exists an open f and g f induces following property following Theorem given local map given triplet global theory grotp Hausdorff space homology theory homotopy classes homotopy groups Hopf theorem identity map immediate consequence inclusion map ind f induced homomorphisms induces an isomorphism induces isomorphisms infinite cyclic isomorphisms Lemma let f let us consider Let xQ map f maps f,g natural projection neighborhood of xQ non-negative real number obvious open neighborhood pathwise accessible pathwise connected point xQ positive real singular homology sub space tangent space topological space X,xQ