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(image from the book). The surfaces shown are images of immersions (i.e., locally smooth embeddings) in the complements of the singular points, shown in white. The vertical line in the right-side figure is a homologically non-trivial cycle.

(image from the book). The surfaces shown are images of immersions (i.e., locally smooth embeddings) in the complements of the singular points (white dots). The vertical line in the right-side figure is a homologically non-trivial cycle.

(image from the book). The 8-shaped level sets are immersions (i.e., not self-intersecting) except for where singular points are shown. The vertical line in the right-side figure is a homologically non-trivial cycle.

(image from the book). The surfaces shown are images of immersions (i.e., locally smooth embeddings) in the complements of the singular points, shown in white. The vertical line in the right-side figure is a homologically non-trivial cycle.

EDIT: The answer is trivially positive for any surface at all.

Consider the projective plane $\mathbb RP^2$ as the Boy surface (left) and the Klein bottle $K^2$ (right):

For even genera $g$ (except $g=2$, which is a different story), it is easy to do: e.g., connect the top and bottom of $K^2$ (right) by a tube (as if you drill a wormhole along the vertical axis), which will form a surface of genus $g=4$ immersed with two Bott-type extrema (circles) and two Morse-type saddles. (You can get any even genus $g\ge4$ by adding more handles.)

Unfortunately, I lack the skill to convert this into a formal proof, and even if I could do it for this particular type of immersion of $\mathbb RP^2$ (Boy surface), it would not prove the claim in the general case. Could you provide such a proof, or point to sources where a proof can be found? Detailed explanations would be greatly appreciated, since I am not an expert.

EDIT: The answer is trivially positive; the question arose from my misunderstanding of the figure below.

Consider the projective plane $\mathbb RP^2$ as the Boy surface (left) [EDIT: wrong, both figures are not immersions, and the left figure is not the Boy surface] and the Klein bottle $K^2$ (right):

For even genera $g$ (except $g=2$, which is a different story), it is easy to do: e.g., connect the top and bottom of $K^2$ (right) by a tube (as if you drill a wormhole along the vertical axis), which will form a surface of genus $g=4$ immersed [EDIT: wrong, this is not an immersion] with two Bott-type extrema (circles) and two Morse-type saddles. (You can get any even genus $g\ge4$ by adding more handles.)

Unfortunately, I lack the skill to convert this into a formal proof, and even if I could do it for this particular type of immersion [EDIT: wrong, this is not an immersion] of $\mathbb RP^2$ (Boy surface), it would not prove the claim in the general case. Could you provide such a proof, or point to sources where a proof can be found? Detailed explanations would be greatly appreciated, since I am not an expert.