Fredholm Kuranishi reduction ($\kappa :\ker L\to \mathrm{coker}L$)

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 $\mathrm{im}L$ 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 $\kappa :\ker L\to \mathrm{coker}L$ 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