By Milgram R.J.
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Extra resources for Algebraic and Geometric Surgery
D1 × S 1 and the inverse image of ω(S 1 ) Lemma 12. Let V ⊂ M m be any subcomplex of the manifold M m having dimension ≤ m − 3 and suppose m ≥ 5.
It follows that for k > m each map [X, BOk ] −−→ [X, BOk+1 ] −−→ . . −−→ [X, BO] is a bijection. 6. THOM SPACES AND TRANSVERSALITY 29 The relation between stable and unstable bundles. An n-plane bundle, Ω, on an n-dimensional complex, Xn , which is stably trivial can, by 8, be assumed to be trivial and trivialized (this means a particular homotopy of the classifying map is given) on the (n − 1)-skeleton Xn−1 . Hence the classifying map BΩ : X → BOn can be assumed to factor in the form W µi Sin −−→ BOn , X −−→ X/Xn−1 = i∈I Sn where the µi : → BOn represent bundles over S n which, together, induce stably trivial bundles over X.
Algebraic and Geometric Surgery by Milgram R.J.