Let $X$ be the quotient topological space obtained by identifying the matrices $A$ and $\overline{A}$ in the topological group $\mathrm{SU}(n)$ (here $\overline{A}$ denotes entry-wise complex conjugation). Are there any techniques for finding the second homotopy group of $X$? For example, is the quotient map $p:\mathrm{SU}(n) \rightarrow X$ a fibration?