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

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kakarotyjn
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I just learned the Brouwer fixed-point theorem of dimension 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?
 

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