TITLE: Cut-number of Klein bottle and other non-orientable surfaces
QUESTION [6 upvotes]: What is the maximum number $c$ (cut-number) of non-intersecting (edit: two-sided) circles on a Klein bottle $N_2$ and, in general, a surface $N_h$ with $h$ Möbius strips, such that cutting by these circles leaves $N_h$ connected? 
This should be (or is it?) the same as co-rank (maximum rank of a free homomorhic image) of its fundamental group $$G=\pi_1(N_h)=\langle a_1,a_2,\dots,a_h\mid a_1^2a_2^2\dots a_h^2=1\rangle.$$ Obviously $corank\ G\ge\left[\frac h2\right]$, but is it larger?
For an orientable surface of genus $M_g$, this is $g$.

REPLY [2 votes]: Corank $r$ cannot be larger than $c$. Otherwise, consider a PL map $f$ from $N_h$ to the bouquet $B$ of $r$ circles which induces the epimorphism $\pi_1(N_h)\to F_r$. Next, take preimages of generic points in the edges of $B$ under $f$. Due to genericity of the points, the preimages form a 2-sided 1-dimensional submanifold $L$ in $N_h$. Some of the circles in $L$ can be homologous to each other or null-homologous. Removing the homologous duplicates and null-homologous cicrles, we obtain a system $L'$ of $\ge r$ circles in $N_h$ cutting along which  results in a connected surface. To relate to $h$, use Lee Mosher's answer or the one I gave to this question.