Normal invariants and G/O (Ex)

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 $<wikitex>; 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 $BJ$: $$G/O:=\text{hofib}(J)$$. 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 $\mathcal{G}/\mathcal{O}(X)$. {{beginthm|Exercise}} Show that for any space $X$ there is a bijection $$ [X, G/O] \equiv \mathcal{G}/\mathcal{O}(X).$$ {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Exercises]]BO\stackrel{BJ}\to{BG}$, define $G/O$ to be the homotopy fibre of $BJ$:

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

In [Lück2001, p 61] there is defined a group of stable fibre homotopy trivialisations of smooth vector bundles over a space $X$ which is denoted $\mathcal{G}/\mathcal{O}(X)$.

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

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

References