How to judge whether a space has the fixed-point property?

[kakarotyjn]

I just learned the Brouwer fixed-point theorem of dimention 1 and 2.But the exercises make me sad,I can't solve them.

Suppose X and Y are of the same homotopy type and X has the fixed-point property.Does Y also have it?

 

[Office_Shredder]

Have you tried coming up with some counterexamples? Find a space that you know has the fixed point theorem, find a homotopy equivalent space, and see if you can come up with a function without a fixed point. Either you'll end up getting the answer for free or you might come up with some intuition as to how the proof should go

 

[kakarotyjn]

Yes,I've tried,but not very much.I will retry deep on it.Thank you