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EXISTENCE AND HOMOTOPY CLASSIFICATION 1 Nondegenerate singularities and their structure
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apply assume Atiyah and Dupont bijective bordant bordism class boundary canonical line bundle characteristic number classifying map closed connected n-manifold closed n-manifold closed smooth Coker commuting diagram compact connected manifold corollary corresponding CW-complex defined denote epimorphism exact sequence example extend fact finite singularities follows forgetful map Gysin sequence hence homomorphism homotopy classes homotopy group identify implies incl injective integer invariant isomorphism k-field with finite k-framefield k-morphism k+1)-morphism ker f kernel lemma manifold of dimension Math monomorphism Moreover morphism nondegenerate nonorientable normal bordism groups normal bundle obstruction obtain obvious odd torsion orientable manifold orientation bundle oriented n-manifold paracompact space projective space proof of theorem proposition pullback resp result singularity data smooth manifold span span(Sn stable span(M Stiefel-Whitney class suitable tangent tangent map theorem 9.3 topology trivial unoriented vanishes vector bundle vectorfield Wj(M wk(M zero bordism zero set