= Fredholm Kuranishi reduction
{c}
{title2=$\kappa:\ker L\to\operatorname{coker}L$}
For a smooth Hilbert-space map $F$ with $F(0)=0$ and Fredholm derivative $L$, split the source into $\ker L$ and a complement and the target into $\operatorname{im}L$ and a finite-dimensional cokernel. The implicit function theorem solves the range component uniquely as a graph over $\ker L$. The remaining equation is a smooth map $\kappa:\ker L\to\operatorname{coker}L$ with $\kappa(0)=d\kappa(0)=0$. Thus the local zero set is $\kappa^{-1}(0)$. Compact symmetry groups permit equivariant splittings.
// Destination: geometry-and-topology.bigb
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