The wedge sum of two circles has fixed point property?
I'm trying to find a continuous map from the wedge sum to itself, that this property fails, I couldn't find it, I need help.
Thanks
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
5
If by circle you mean $S^1$, and the fixed point property is the claim that every continuous map into itself has a fixed point, for two circles like so: 
consider the map that rotates $A$ by 90 degrees, and sends all of $B$ to the image of $x$.
-
$\begingroup$ can you give more details please? thank you for your answer :) $\endgroup$ Nov 25, 2012 at 12:07
-
$\begingroup$ if you rotate $A$ by 90 degrees, and send all of $B$ to the image of $x$, this is (i) a continuous map from the wedge of two circles to itself and (ii) sends no point to itself, i.e. has no fixed points $\endgroup$ Nov 25, 2012 at 22:38
-
$\begingroup$ can I make a map, that rotate $90^o$ both circles as you do in the circle A? I think this map is well-defined and doesn't have fixed points. $\endgroup$ Nov 29, 2012 at 3:46
-
$\begingroup$ Er it's not well defined: the point $x$ would be sent two different places. $\endgroup$ Nov 29, 2012 at 18:55
-