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 $BJ$: | ||
− | $$G/O:=\text{hofib}(J)$$ | + | $$G/O:=\text{hofib}(J).$$ |
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 $\mathcal{G}/\mathcal{O}(X)$. | 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 $\mathcal{G}/\mathcal{O}(X)$. | ||
{{beginthm|Exercise}} | {{beginthm|Exercise}} |
Revision as of 10:42, 27 March 2012
The aim of these exercises is to follow the proof of [Lück2001, Theorem 3.45].
Given the map , define to be the homotopy fibre of :
In [Lück2001, p 61] there is defined a group of stable fibre homotopy trivialisations of smooth vector bundles over a space which is denoted .
Exercise 0.1. Show that for any space there is a bijection