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The four sphere decomposes as a twisted double      N 2 ∪  f N 2 {N_2}{ \cup _f}{N_2}      , where      N 2 {N_2}       is the      2 2      -disk bundle over the real projective plane with Euler number 2. In this note the relationship of this splitting to the double branched cover of the complex projective plane over the four sphere as the quotient space under complex conjugation is made explicit.

Splitting 𝑆⁴ on 𝑅𝑃² via the branched cover of 𝐶𝑃² over 𝑆⁴