MathDB

Let

F

be a smooth (i.e.

C^{\infty}

) closed surface. Call a continuous map

f\colon F\rightarrow \mathbb{R}^2

an almost-immersion if there exists a smooth closed embedded curve

\gamma

(possibly disconnected) in

F

such that

f

is smooth and of maximal rank (i.e., rank 2) on

F\backslash \gamma

and each point

p\in\gamma

admits local coordinate charts

(x,y)

and

(u,v)

about

p

and

f(p)

, respectively, such taht the coordinates of

p

and

f(p)

are zero and the map

f

is given by

(x,y)\rightarrow (u,v), u=|x|, v=y

. Determine the genera of those smooth, closed, connected, orientable surfaces

F

that admit an almost-immersion in the plane with the curve

\gamma

having a given positive number

n

of connected components.

Problem Statement