Now I want to calculate the intersection form of a logarithmic transformation which is defined as follows.
Let $X$ be an oriented, closed, simply-connected 4-manifold and $T^2\subset X$ be an embedded 2-torus which has zero self intersection number. Let $N(T^2)=T^2\times D^2$ denotes a closed tubular neighborhood in $X$. Suppose that the complement of $T^2$ is simply-connected. We consider an orientation preserving deffeomorphism $h$ of $T^2\times\partial D^2$ which satisfies the relation in integral 1st homology that $h_*[\partial D^2] = p[\alpha] + q[\beta] + r[\partial D^2]$, where $\alpha$ and $\beta$ are standard basis loop of $T^2$. Set $X(h):=(X-\text{int} N(T^2))\cup_h T^2\times D^2$. My question is:
How can we calculate the intersection form of $X(h)$?
Just easy application of Mayer-Vietoris exact sequence? I couldn't get it by this approach. So I would like to get more detailed explanation. Thank you.
EDIT1 : Yes, I already know that Euler characteristics and signatures of the two are same. So I really want to know the parity of intersection forms or spin structures. There is a reference (Six Lectures on Four 4-manifolds) which says that:
- If $X$ is odd, then $X(h)$ is odd.
- If $r$ is odd and $X$ is even, then $X(h)$ is even.
- If $r$ and $X$ are even, then $X(h)$ is odd.
How can we prove that? Any comments and answers are welcome!
EDIT2 : Here, $p,q$ and $r$ are an integer. Moreover, note that $\text{gcd}(p,q,r)=1$ holds because $[\partial D^2]$ is primitive in $H_1(T^2\times\partial D^2;\mathbb{Z})$.
