Normal invariants and G/O (Ex)

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The aim of these exercises is to follow the proof of [Lück2001, Theorem 3.45].

Given the map BO\stackrel{BJ}\to{BG}, define G/O to be the homotopy fibre of BJ:

\displaystyle G/O:=\text{hofib}(BJ).

In [Lück2001, p. 66] there is defined a group of stable fibre homotopy trivialisations of smooth vector bundles over a space X which is denoted \mathcal{G}/\mathcal{O}(X).

Exercise 0.1. Show that for any space X there is a bijection

\displaystyle  [X, G/O] \equiv \mathcal{G}/\mathcal{O}(X).
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