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A fixed-point theorem is a result saying that a function $F$ will have at least one fixed point (a point $x$ for which $F(x) = x$), under some conditions on $F$ that can be stated in general terms.

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A question on fixed point theory

I don't think $\mathbb{C}P^n/\mathbb{C}P^k$ (or $\mathbb{R}P^n/\mathbb{R}P^k$) ever has the fixed point property in the range you describe. I haven't thought about this very long so could be wrong, b …
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