Fredholm map between Banach manifolds

Let $M$ and $N$ be $C^1$ Banach manifolds (Countable base Banach manifold and smooth map) and let $f:M\to N$ be a $C^1$ map, so that $Df(p):T_{p}M\to T_{f(p)}N$ is a bounded linear operator for every $p\in M$ (Tangent space and differential on a Banach manifold, A bounded linear operator between normed spaces). Then:

• $f$ is Fredholm at $p$ when $Df(p)$ is a Fredholm operator, that is when $\ker Df(p)$ is finite dimensional, $\mathrm{ran}Df(p)$ is closed in $T_{f(p)}N$, and the cokernel $\mathrm{coker}Df(p)=T_{f(p)}N\mathrm{/}\mathrm{ran}Df(p)$ is finite dimensional (Fredholm operator cokernel and index);
• $f$ is a Fredholm map when it is Fredholm at every point of $M$;
• the index of a Fredholm map at $p$ is the integer $\mathrm{ind}Df(p)=\dim \ker Df(p)-\dim \mathrm{coker}Df(p)$, and $f$ has index $n$ when $\mathrm{ind}Df(p)=n$ for every $p\in M$.

The pointwise index is defined whenever $f$ is Fredholm at $p$; the phrase "has index $n$" is a separate, global condition, and the local constancy of $p\mapsto \mathrm{ind}Df(p)$ is a theorem below, not part of this definition.

• Well-definedness: chart changes conjugate the derivative. If $(\phi ,\psi )$ and $({\phi }^{\mathrm{\prime }},{\psi }^{\mathrm{\prime }})$ are two chart pairs around $p$ and $f(p)$, then the corresponding representatives of $f$ satisfy ${\psi }^{\mathrm{\prime }}\circ f\circ {\phi }^{\mathrm{\prime }-1}=({\psi }^{\mathrm{\prime }}\circ {\psi }^{-1})\circ (\psi \circ f\circ {\phi }^{-1})\circ (\phi \circ {\phi }^{\mathrm{\prime }-1})$, so their derivatives at the point in question are related by $D{\widehat{f}}^{\mathrm{\prime }}(x^{\mathrm{\prime }})=D({\psi }^{\mathrm{\prime }}\circ {\psi }^{-1})(\psi (f(p)))D\widehat{f}(x)D(\phi \circ {\phi }^{\mathrm{\prime }-1})(x^{\mathrm{\prime }})$, a conjugation by bounded linear isomorphisms (Banach manifold differentials are chart independent). Conjugation by isomorphisms preserves Fredholmness and the index: for bounded isomorphisms $U,V$ the composite $UTV$ is Fredholm exactly when $T$ is, and $\mathrm{ind}(UTV)=\mathrm{ind}T$, because $\mathrm{ind}(UT)=\mathrm{ind}U+\mathrm{ind}T$, isomorphisms have index $0$, and the same applies on the other side (Fredholm index is additive, Fredholm operator cokernel and index); that additivity theorem is proved under AC, which is therefore inherited by every use of the index made through charts.

• Index and the Fredholm condition are local in the base. Both are properties of the single operator $Df(p)$ in the appropriate tangent spaces; neither involves any choice of charts, by the previous remark. In particular the index at $p$ may be read in any chart pair around $p$ and $f(p)$.

• Nonconstant index is possible a priori. The definition allows the pointwise index to jump, and the local-constancy proposition below is what rules that out for $C^1$ Fredholm maps. It is not built into the definition, because the proof needs the openness of the set of Fredholm operators in operator norm, a theorem about operators rather than about manifolds.

• Finite-dimensional fibres. When the index is $\dim M-\dim N$ in the finite-dimensional model case one recovers the classical notion; the definition here is the infinite-dimensional one, in which neither tangent space need be finite dimensional and only the kernel and cokernel are required to be.