The Fredholm index of a Cauchy–Riemann operator on the cylinder depends only on its asymptotic Conley–Zehnder indices

Let $1<p<\infty$ and $n\ge0$. Let $S,T:\mathbb{R}\times S^1\to\mathbb{R}^{2n\times2n}$ be continuous coefficients with $S(s,\cdot)\to S^\pm$ and $T(s,\cdot)\to T^\pm$ uniformly as $s\to\pm\infty$, where $S^\pm,T^\pm$ are continuous loops of symmetric matrices whose asymptotic operators $-J_0\partial_t-S^\pm$ and $-J_0\partial_t-T^\pm$ are nondegenerate. If

then the Cauchy–Riemann operators $D_S,D_T:W^{1,p}(\mathbb{R}\times S^1,\mathbb{R}^{2n})\to L^p(\mathbb{R}\times S^1,\mathbb{R}^{2n})$ have the same Fredholm index: $\operatorname{ind}D_S=\operatorname{ind}D_T$.

This says the index on the cylinder depends only on the Conley–Zehnder indices at the two ends. It is used to compute indices by comparing with any convenient operator that has the same asymptotic indices.

Formalization Note $D_S=\partial_s+J_0\partial_t+S$; the index is crIndex p S $=\dim\ker D_S-\dim\operatorname{coker}D_S$ from the setting module. Loops are continuous $1$-periodic maps $\mathbb{R}\to\mathbb{R}^{2n\times2n}$ (IsSymLoop), nondegeneracy is IsNondegenerate (no eigenvalue $1$ of the time-one map of $\Psi'=J_0S^\pm\Psi$), and the Conley–Zehnder index is asymptoticCZIndex. The asymptotic convergence is IsAsymptoticTo S Splus Sminus (positive end first).