Today in my topology class (undergraduate) we introduced the basics for Fixed Point theory for topological spaces (i.e.Fixed point property, retracts, contractability, etc.) and we were discussing the different ways to obtain other spaces which have the fixed point property. The question came up of on what conditions can we say the quotient topology under some equivalence relation has the fixed point property, and we were all stumped teacher included.
I tried google, but I didn't seem to find any general theory on the fixed point property on these types of spaces. Do you have any recommendations on papers regarding the subject? It would be preferable if the results were contained in a point set topology context, but Algebraic topology would be OK to a certain extent.
Maybe someone could point me in a direction to think about this question? I know that if I generate any closed convex set in $\mathbb{R}^n$ it has the fixed point property given the fact that $[0,1]^n$ has the $fpp$ property. Possibly we may pose the problem in the context of when can we generate convex closed sets in $\mathbb{R}^n$ via quotient maps and generalize from there?