Struggling to answer this question from Lickorish's introduction to Knot Theory.
If M has a 3-manifold with a genus g Heegaard splitting, then the fundamental group of M has a presentation with g generators and g relators
I know that since the Heegaard splitting exists, then $M=X\cup Y$ for a pair of handlebodies X and Y, both of genus g, and that by Van-Kampen's Theorem, $$\pi_{1}(M)=\pi_{1}(X)_{*\pi_{1}(X\cap Y)}\pi_{1}(Y)$$ but I'm struggling to show that this shows that the fundamental group of M has a presentation with g generators and g relators.
Any help?
First of all notice that the fundamental group of a CW complex with one 0-cell, $b$ 1-cells, and $r$ 2-cells admits a presentation with $b$ generators and $r$ relations.
Suppose that $M = U_0 \cup_\partial U_1$ with $U_0$ and $U_1$ genus $g$ handlebodies. Pick a collection of properly embedded two-disks $D_1, \dots, D_g \subset U_1$ such that $B=U_1- (D_1 \cup \dots \cup D_g)$ is a three ball (you can see that such a collection exists just by looking at the standard picture of a genus $g$ handlebody). Since the fundamental group depends only on the 2-skeleton we have that
$$ \pi_1(M)= \pi_1(M - B) = \pi_1(U_1 \cup_\partial(D_1\cup \dots \cup D_g)) . $$
Now observe that $U_1$ has the homotopy type of a bouquet of $g$ circles and consequently that $X=U_1 \cup_\partial(D_1\cup \dots \cup D_g)$ has the homotopy type of a CW complex with one 0-cell, $g$ 1-cells, and g 2-cells.