Talk:Bundle structures and lifting problems (Ex)
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(Created page with "<wikitex>; '''Exercise 2.1''' Assume that there exists a lift $\overline{f}$: $X\to\mathrm{hofib}(p)$ of $f$. Then there exist maps $f_1$: $X\to Y$ and $f_2$: $X\times[0,1]\...") |
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Furthermore we have for all $x\in X$: $f_2(x,1)=g(f(x))$ and $f_2(x,0)=z_0$. | Furthermore we have for all $x\in X$: $f_2(x,1)=g(f(x))$ and $f_2(x,0)=z_0$. | ||
Thus $f_2$ defines a homotopy from $g\circ f$ to a constant map. | Thus $f_2$ defines a homotopy from $g\circ f$ to a constant map. | ||
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+ | Assume that there exists $h$: $X\times[0,1]\to Z$ such that for all $x\in X$ we have $h(x,0)=z_0$ and $h(x,1)=g(f(x))$. | ||
+ | Define $\overline{f}$: $X\to\mathrm{hofib}(p)$ by $\overline{f}(x)=(f(x),h(x,.))$. | ||
+ | By the definitions of $h$ and of $\mathrm{hofib}(p)$ we find that $\overline{f}$ is well defined and a lift of $f$. | ||
</wikitex> | </wikitex> |
Revision as of 19:12, 2 April 2012
Exercise 2.1
Assume that there exists a lift : of . Then there exist maps : and : such that for all we have . Since we find that . Furthermore we have for all : and . Thus defines a homotopy from to a constant map.
Assume that there exists : such that for all we have and . Define : by . By the definitions of and of we find that is well defined and a lift of .