Recently I am reading Wall's paper "On the Orthogonal Groups of Unimodular Quadratic Forms II". In this paper, I encountered with the map $E^1_\omega$, which now I am interested in.
Let $X$ be an unimodular integral lattice, and $H$ denotes the unimodular lattice which has a basis $x$, $y$ satisfying $x\cdot x=y\cdot y=0$, and $x\cdot y=1$. For $\omega \in X$ with $\omega\cdot\omega\in 2\mathbb{Z}$, we define the isometry $E^1_\omega$ of $X\oplus H$ as follows.
- For $\xi\in X$, $E^1_\omega(\xi)=\xi-(\xi\cdot\omega)x$.
- $E^1_\omega(x)=x$.
- $E^1_\omega(y)=y+\omega-2^{-1}(\omega\cdot\omega)x$.
My question is: What is algebraic intuition behind the definition of $E^1_\omega$?
EDIT 1 : As D. Ruberman answered below, there exists intuition behind the map $E^1_\omega$ from the view point of differential topology. In fact, Wall proved that, for any closed simply-connected 4-manifold $M$ which has indefinite unimodular form $X$ as intersection form, any automorphism of $X\oplus H$ is realized by some diffeomorphisms of $M\#S^2\times S^2$.
He proved it as follows.
Step 1 : He had showed that the automorphism group of $X\oplus H$ is generated by $E^1_\omega$ and $E^2_\omega$, where $E^2_\omega$ is defined by interchanging the roles of $x$ and $y$ in $E^1_\omega$.
Step 2 : He also proved that each of $E^i_\omega$ is induced from a diffeomorphism of $M\#S^2\times S^2$. This completes the proof.
I already knew the proof of Step 2 which intuitively explains the map $E^i_\omega$ from the topological point of view. But I want to know some intuition behind Step 1, which is algebraic part of the proof.
EDIT 2 : The map $E^i_\omega$ is called the Siegel-Eichler transformation in a lot of literature.
