What is the bilinear intersection form of real homology of 4-manifold? I am currently reading Invariants of 3-manifolds via link polynomials and quantum groups by N. Reshetikhin and V.G. Turaev.
In section 3.2, given a framed link $L$, we can get a $4$-manifold $D_L$ by dehn surgery.
Its says that the bilinear intersection form
$$H_2(D_L;\mathbb{R})\times H_2(D_L;\mathbb{R})\rightarrow \mathbb{R}$$
may be diagonalized with respect to some basis in $H_2(D_L;\mathbb{R})=\mathbb{R}^m$ where $L$ is an $m$-component framed link in $S^3$. Denote by $\sigma_-(L)$ the number of non-positive squares on the diagonal.
May I know what exactly is the formula of the intersection form? I did some searching online and notice that intersection form of $4$-manifold is defined over homology over $\mathbb{Z}/2\mathbb{Z}$ on wikipedia. But I can't find definition for $\mathbb{R}$ case. I hope to know this since the quantity $\sigma_-(L)$ plays a role in the following context in the paper.
 A: If $M$ is a compact, oriented, $2n$-dimensional manifold with boundary $\partial M$ and $R$ is any ring, there is an intersection form $H_n(M;R)\times H_n(M;R)\rightarrow R$. This comes from identifying $H_n(M;R)\cong H^n(M,\partial M;R)$ via Poincaré-Lefschetz duality, taking the cup product $H^n(M,\partial M;R)\times H^n(M,\partial M;R)\rightarrow H^{2n}(M,\partial M;R)$ and identifying $H^{2n}(M,\partial M;R)\cong R$ by evaluating on the fundamental class. The commutativity of the cup product implies that this bilinear form is $(-1)^n$-symmetric. In particular, it is symmetric if (and only if) the dimension of $M$ is divisible by $4$.
Assume $n$ is even. If we choose $R=\mathbb{R}$, the intersection is a real symmetric bilinear form, so it can be diagonalized by linear algebra and the number of diagonal entries with a given sign ($-1$, $0$ or $+1$) does not depend on the diagonalization by Sylvester's law of inertia. Note that this form is non-degenerate if $\partial M=\emptyset$, by Poincaré duality, but can in general be degenerate (its radical is $\ker(H_n(M;R)\rightarrow H_n(M,\partial M;R)$). In the non-degenerate case, knowing the number of non-positive diagonal entries is equivalent to knowing what is called the signature of $M$. In general, it is less data.
