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 $E^1\_\omega$?**




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**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.



Intuition behind the definition of the Siegel-Eichler transformation
This is explained very clearly in Wall's paper, Diffeomorphisms of 4-manifolds, J. London Math. Soc. (1964) 131-140. The idea is that if you do surgery on a circle $C$ in a simply-connected $4$-manifold $X$, you get $X \# T\_r$ where $T$ is a 2-sphere bundle over $S^2$ with fiber $S^2$, and intersection form generated by $x,y$ as above, except that maybe $y\cdot y = r$ may be non-zero. (Up to diffeomorphism, only the parity of $r$ matters.) Now if you perform an isotopy of that $C$, ending up at $C$, you get a self-diffeomorphism of $X \# T\_r$. (You can get $r\neq 0$ by twisting the framing of the circle in this isotopy.) If the $2$-dimensional homology class swept out by the circle (ie a map of a torus) in this isotopy is $\omega$, then the induced map on homology is of the form $E^1\_\omega$.


I don't know what was the original intuition behind the algebraic formula (due to Siegel), but Wall's geometric picture makes it a lot clearer to me.


