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Assume that $0<k<n-1$, Note that $\mathbb{C}P^{k}$ can be considered as a closed subset of $\mathbb{C}P^{n}$, in a natural way. We collapse $\mathbb{C}P^{k}$ to a point. The resulting space is denoted by $\mathbb{C}P^{n}/\mathbb{C}P^{k}$

My fixed point question:

Does $\mathbb{C}P^{n}/\mathbb{C}P^{k}$ satisfies fixed point property?

This question is motivated by the following post:

Show $\mathbb{CP^2/CP^1}$ is not a retract of $\mathbb{CP^4/CP^1}$.

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