equivalent statements to statement that sphere is not contractible
Let be a normed space. Recall the definition of the sphere and the ball in :
Proposition. The following are equivalent:
is not contractible;
for each continous map there exists such that ;
there is no retraction from onto .
Proof. The proof of this proposition probably can be found in some books about topology. I present here the proof from my lecture due to Prof. .
Assume there exists a continous map such that for each we have . Define a map as follows:
Thanks to the condition this map is well defined and it is easy to check that this is a homotopy from the identity map to constant map. But is not contractible. Contradiction.
Assume there exists a retraction . Define a map by the formula . This map has no fixed point. Contradiction.
Assume that is contractible and take any homotopy from constant map to identity map, i.e. for all we have (for some ) and . Define a map as follows:
It is easy to see that this formula defines a retraction from onto . Contradiction.
Note that this proposition does not state that any of the conditions hold. It only states that they are equivalent. It is well known that all of them are true if is finite dimensional and all are false if is infinite dimensional.
|Title||equivalent statements to statement that sphere is not contractible|
|Date of creation||2013-03-22 18:07:53|
|Last modified on||2013-03-22 18:07:53|
|Last modified by||joking (16130)|