Normal invariants and G/O (Ex)

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Given the map $BO\stackrel{BJ}\to{BG}$, define $G/O$ to be the homotopy fibre of $BJ$:
Given the map $BO\stackrel{BJ}\to{BG}$, define $G/O$ to be the homotopy fibre of $BJ$:
$$G/O:=\text{hofib}(J)$$.
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$$G/O:=\text{hofib}(BJ).$$
In {{citeD|Lück2001|p 61}} there is defined a group of stable fibre homotopy trivialisations of smooth vector bundles over a space $X$ which is denoted $\mathcal{G}/\mathcal{O}(X)$.
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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}}
{{beginthm|Exercise}}
Show that for any space $X$ there is a bijection
Show that for any space $X$ there is a bijection
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{{endthm}}
{{endthm}}
</wikitex>
</wikitex>
== References ==
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<!-- == References ==
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{{#RefList:}} -->
[[Category:Exercises]]
[[Category:Exercises]]
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[[Category:Exercises without solution]]

Latest revision as of 17:29, 2 April 2012

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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