TITLE: Knotted projective planes and fake complex projective space
QUESTION [6 upvotes]: Paul Melvin gave a talk at Knots in Washington last year in which he asked whether the connected sum of an odd twist-spin of a classical knot and a standard cross-cap embedding of ${\mathbb R}P^2$ is a standard cross-cap. I believe this question to be a standard one, and I have two questions about it.
First, I think that Paul stated that if the result is knotted, then there is a non-standard ${\mathbb C}P^2$. Please may I have an outline of the argument that gives the construction of the non-standard ${\mathbb C}P^2?$
Second, for aesthetic reasons, I imagine that there is a difference in using a cross-cap with normal Euler class $2$ and using one with normal Euler class $-2$. Won't the construction give a $\pm {\mathbb C}P^2$ (or better ${\mathbb C}P^2$ or $\overline{{\mathbb C}P}^2$) depending on the normal Euler class of the cross-cap?
Any other folk-lore would be appreciated.

REPLY [2 votes]: I can't think of another way to get $CP^2$ from an embedded $RP^2$ in the 4-sphere, so I would guess you are right.  But why would the knottedness of the $RP^2$ imply that the $CP^2$ is exotic?  After all, there are plenty of knotted 2-spheres in $S^4$ whose double branched covers are $S^4$.  In other words, a knotted $RP^2$ might produce an exotic involution on the genuine (I don't want to say real, for obvious reasons!) $CP^2$ rather than an exotic $CP^2$.