# Representations of $\pi_1M$ and Heegaard Splittings

I am reading Floer's Instanton-Invariant paper, and am stuck on a sentence. To set the stage:

Consider a closed connected oriented 3-manifold $M$ and the nonabelian group $SU_2$. Denote the equivalence classes of representations $\mathcal{R}(M)=Hom(\pi_1M,SU_2)/\text{ad}(SU_2)$.

Now given a Heegaard splitting $M=M_+\cup_SM_-$, one can consider $\mathcal{R}(M)$ as the intersection of $\mathcal{R}(M_+)$ and $\mathcal{R}(M_-)$ in $\mathcal{R}(S)$. Indeed, Seifert van-Kampen's theorem gives $\pi_1(M)\cong\pi_1(M_+)\ast_{\pi_1(S)}\pi_1(M_-)$ and then the statement follows by the universal property of amalgamated free products.

The resulting intersection number (ignoring the trivial representation) can be shown to be independent of the particular Heegaard splitting.
[The "result" refers to the integer-valued Casson invariant, which assigns a sign to each intersection $a\in\mathcal{R}$].

How is this done?

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I don't understand what you mean by "the resulting intersection." –  Qiaochu Yuan Oct 7 '12 at 7:21
Presumably what they're getting at is that $\mathcal R(M)$ is an idea that is independent of any particular presentation of $\pi_1 M$. –  Ryan Budney Oct 7 '12 at 7:47