Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Yes, this was my logic: you can't get rid of the Whitney umbrella. The evolution of the levels of the Boy surface is shown in [dfgm.math.msu.su/files/bf-engl.pdf, page 75, fig. 2.21(a)]. Not that I understanded it much either :) It confused me that the umbrella was shown in the same figure as (b), so I thought it was the same.
Yes, now I see: it was a stupid question. I mistakenly thought the figure showed an immersion. So your answer is that on any surface there can be defined a simple Morse function, any such function is the height function of a suitable immersion (dfgm.math.msu.su/files/bf-engl.pdf, page 76), and any Morse center can be turned into a Bott-type circle by adding a handle. Obvious. Sorry to bother you with a stupid question :(
Than you! Yes, centers are what you say (I've edited the question). Could you please clarify how to make the cylinder connecting the two centers an immersion? The figure (left) is precisely what you say: a Morse height function on an immersion of $\mathbb RP^2$ with two centers, but I can't see how to connect them by an immersed cylinder without a second saddle on it.
