# 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}$$; The aim of these exercises is to follow the proof of {{citeD|Lück2001|Theorem 3.45}}. Given the map BO\stackrel{BJ}\to{BG}, define G/O to be the homotopy fibre of BJ: G/O:=\text{hofib}(BJ). In {{citeD|Lück2001|p. 66}} there is defined a group of stable fibre homotopy trivialisations of smooth vector bundles over a space X which is denoted \mathcal{G}/\mathcal{O}(X). {{beginthm|Exercise}} Show that for any space X there is a bijection [X, G/O] \equiv \mathcal{G}/\mathcal{O}(X). {{endthm}} [[Category:Exercises]] [[Category:Exercises without solution]]BO\stackrel{BJ}\to{BG}$, define $G/O$$G/O$ to be the homotopy fibre of $BJ$$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$$X$ which is denoted $\mathcal{G}/\mathcal{O}(X)$$\mathcal{G}/\mathcal{O}(X)$.

Exercise 0.1. Show that for any space $X$$X$ there is a bijection $\displaystyle [X, G/O] \equiv \mathcal{G}/\mathcal{O}(X).$