I am sorry to ask it, but I am not an algebraic topologist. I just guessed that F is the space of sections of a sphere bundle over $RP^n$. I do not understand the terminology. What do "1", "+" and above all "-" stand for in the formula $W=n+2-L$ ? I just need a reference to be able to understand what you said. Thank you again, Neil Strickland, for the time you have dedicated to this question.

Thanks a lot ! It is more elementary than I expected. This method works thanks to the vectorial product in $R^3$, but I failed to find a canonical way to do a quarter turn of $X$ homotopically for all $x$ in higher dimensions. Can we generalize this result to higher dimensions ?