Hints for an exercise on Morse theory

Question

Exercise: Let $M$ be a $3$-dimensional smooth manifold with boundary $\partial M$ which is a surface of genus $g$. Moreover let $f:M\longrightarrow [0,1]$ be a Morse function with the following properties:

1. $f(\partial M)$ is a regular value.
2. Lets denote with $\mu_i(f)$ the number of critical points of $f$ of index $i$, then $\mu_0(f)=1$ and $\mu_2(f)=\mu_3(f)=0$

Prove that $M$ is connected and determine $\mu_1(f)$.

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I tried different approaches to solve it; here you can see my wrong reasonings:

$1)$ Lets reconstruct partially the Morse-Smale complex (over $\mathbb Z/2\mathbb Z$). We have one critical point of index $0$, therefore $C_0(M)\cong\mathbb Z/2\mathbb Z$ and in the same way we can conclude that $C_2(M)=C_3(M)=0$. For compact manifolds we have the following theorem:

Let $M$ be a compact manifold of dimension $n$, then the dimension as vector space of the Morse-Smile homology group $H_n(C^\bullet)$ (or $H_0(C^\bullet)$) is the number of connected components of $M$.

In my case $H_3(C^\bullet)=0$ since $C_3(M)=C_4(M)$ so maybe I'm on the wrong way. Probably I can't use the theorem for two reasons: $M$ has boundary and it is not compact.

$2)$ I tried to use the Morse inequalities but I have two main obstacles, $M$ ha boundary and I don't know the number of critical points of index $1$ (I have to find them).

$3)$ I don't know if I can use the fact that $\chi(M)=0$ if $\text{dim}(M)$ is odd because I've seen this theorem only for compact manifolds without boundary.

$4)$ If $M$ where without boundary I could conclude that it has the homotopy type of a CW-complex with only one $0$-cell. At this point I should prove that this type of $CW$-complexes are connected (it is easy).

I don't know how to use the fact that the homology of $\partial M$ is known, because I don't understand how to relate it with the homology of $M$.

Could you help me please?

Answer 1

On each connected component, $f$ admits a minimum and a maximum. Let $M_1$ the connected component of $M$ containing the boundary (it is connected, by assumption). If $f(\partial M)$ is a minimum, the maximum of $f$ restricted to this component is achieved at a critical point and $\mu_3(f)\geq 1$. So $f(\partial M)$ is not a minimum and $f$ has a minimum on $M_1$, achieved at a critical point. As $\mu_0(f)=1$, $f$ cannot have other connected components, and $M=M_1$ is connected. Now let $N$ be the double of $M$ glued along the boundary $\chi (N)=2\chi (M)- \chi(\partial M)$. Therefore $0=2\chi (M)- (2-2g)$, $\chi(M)=-g+1= 1-\mu_1+0-0$, $\mu_1=g$