Exercise 9, § 4

Let $r\geqslant 1$ be an integer. If $f: M_1\rightarrow M_2$ is a morphism of class $C^r$ of manifolds of class $C^r$, one says that $x\in M_1$ is a critical point of $f$ if $f$ is not a submersion at $x$ (VAR, R1, 5.9.1). Let X be the set of critical points of $f$. The set $f(X)$ in $M_2$ is called the set of critical values of $f$, and $f(M_1)-f(X)$ is called the set of regular values of $f$.

If $M_1$ and $M_2$ are real manifolds of class $C^r$ (VAR, R1, 5.1.5, p. 35), a manifold morphism $f: M_1\rightarrow M_2$, of class $C^r$, is said to be Fredholm if, for every $x\in M_1$, the tangent linear mapping $T_x(f)$ is a Fredholm endomorphism of the Banach space $T_x(M_1)$ into the Banach space $T_{f(x)}(M_2)$. a) Let $f: M_1\rightarrow M_2$ be a Fredholm morphism. If $M_1$ is connected, the index ind(T$_x(f)$) is independent of $x\in M_1$. It is called the index of the Fredholm morphism $f$ and is denoted by ind($f$).

b) Let U be an open subset of a Banach space E such that $0\in U$ and let F be a Banach space. Let $f: U\rightarrow F$ be a Fredholm morphism of class $C^r$ and let $u= T_0(f)$. There exist a Banach space $E_1$, a chart $(U_1, \varphi ,E_1\times$Ker($u$)) of U such that $0\in U_1$, a chart $(U_2, \psi ,E_1\times$ Coker($u$)) of F such that $f(0)\in U_2$ and a Fredholm morphism $g:\varphi (U_1)\rightarrow E_1\times$ Coker($u$) of class $C^r$ such that

for every $(a, b)\in \varphi (U_1)\subset E_1\times$ Ker($u$).

c) Let $f: M_1\rightarrow M_2$ be a Fredholm morphism of class $C^r$ between manifolds of class $C^r$. For every $y\in M_2$, there exists an open neighbourhood V of $y$ such that $f$ defines by passing to the subspaces a proper mapping $\overset{-1}{f}(V)\rightarrow V$.

d) Let U be an open set of a Banach space E containing 0 and let F be a Banach space. Let $f: U\rightarrow F$ be a Fredholm morphism of class $C^{\infty}$. For every $x\in U$ and every neighbourhood V of $f(x)$ in F, there exists a regular value of $f$ in V. (Use the second Sard theorem, VAR, R2, 10.1.3, p. 37.) e) Let $f: M_1\rightarrow M_2$ be a Fredholm morphism of class $C^{\infty}$ of manifolds of class $C^{\infty}$. If $M_1$ and $M_2$ admit countable bases, then the set of critical values of $f$ is meagre ("Smale's theorem").

f) Let $f: M_1\rightarrow M_2$ be a Fredholm morphism of class $C^{\infty}$ of manifolds of class $C^{\infty}$ admitting countable bases. If ind($f$)$<0$, then the image of $f$ has empty interior.

g) Let $f: M_1\rightarrow M_2$ be a Fredholm morphism of class $C^{\infty}$ of manifolds of class $C^{\infty}$ admitting countable bases. For every $y\in M_2$ outside

$-1$

of a meagre set, the set $f(y)$ is a pure submanifold of dimension ind($f$).