Preliminary exam prep
Let $X,Y$ be manifolds embedded in $\R^n$, and $f:X\to Y$ a map, with $df_x:T_xX\to T_{f(x)}Y$ the induced map on tangent spaces.
Definition: The map $f$ is a
Definition: The manifolds $X$ and $Y$ are transverse if $T_pX\oplus T_pY \cong \R^n$ for every $p\in X\cap Y$. The map $f$ and $Y$ are transverse if $\text{im}(f)$ and $Y$ are transverse.
Note that being transverse (or transversal) is a symmetric, but not a reflexive, nor a transitive relation. Recall that a regular value of $f$ is $y\in Y$ such that $df_x:T_xX \to T_{f(x)}Y$ is surjective for all $x\in f^{-1}(y)$. If $y$ is not in the image of $f$, then $f^{-1}(y)$ is empty, so $y$ is trivially a regular value. Every value that is not a regular value is a critical value.
Theorem: (Preimage theorem) For every regular value $y$ of $f$, the subset $f^{-1}(y)\subset X$ is a submanifold of $X$ of dimension $\dim(X)-\dim(Y)$.
Now let $M$ be a submanifold of $Y$.
Corollary: If $f$ is transverse to $M$, then $f^{-1}(M)$ is a manifold, with $\codim_Y(M)=\codim_X(f^{-1}(M))$.
Theorem: (Transversality theorem) Let $\{g_s:X\to Y\ |\ s\in S\}$ be a smooth family of maps. If $g:X\times S\to Y$ is transverse to $M$, then for almost every $s\in S$ the map $g_s$ is transverse to $M$.
If we replace $f$ with $df$, and ask that it be transverse to $M$, then $df|_s$ is also transverse to $M$.
Example: Consider the map $g_s:X\to \R^n$ given by $g_s(X)=i(X)+s=X+s$, where $i$ is the embedding of $X$ into $\R^n$. Since $g(X\times \R^n)=\R^n$ and $g$ varies smoothly in both variables, we have that $g$ is transverse to $X$. Hence by the transversality theorem, $X$ is transverse to its translates $X+s$ for almost all $s\in \R^n$.
Theorem: (Sard) For $f$ smooth and $\dy Y=\emptyset$, almost every $y\in Y$ is a regular value of $f$ and $f|_{\dy X}$. Equivalently, the set of critical values of $f$ has measure zero.
Resources: Guillemin and Pollack (Differential topology, Chapters 1, 2), Lee (Introduction to smooth manifolds, Chapter 6)
Definition: The map $f$ is a
- homeomorphism if it is continuous and has a continuous inverse,
- diffeomorphism if it is smooth and has a smooth inverse,
- injection if $f(a)=f(b)$ implies $a=b$,
- immersion if $df_x$ is injective for all $x\in X$,
- embedding if it is an immersion and $df_x$ is a homeomorphism onto its image,
- submersion if $df_x$ is surjective for all $x\in X$.
Definition: The manifolds $X$ and $Y$ are transverse if $T_pX\oplus T_pY \cong \R^n$ for every $p\in X\cap Y$. The map $f$ and $Y$ are transverse if $\text{im}(f)$ and $Y$ are transverse.
Note that being transverse (or transversal) is a symmetric, but not a reflexive, nor a transitive relation. Recall that a regular value of $f$ is $y\in Y$ such that $df_x:T_xX \to T_{f(x)}Y$ is surjective for all $x\in f^{-1}(y)$. If $y$ is not in the image of $f$, then $f^{-1}(y)$ is empty, so $y$ is trivially a regular value. Every value that is not a regular value is a critical value.
Theorem: (Preimage theorem) For every regular value $y$ of $f$, the subset $f^{-1}(y)\subset X$ is a submanifold of $X$ of dimension $\dim(X)-\dim(Y)$.
Now let $M$ be a submanifold of $Y$.
Corollary: If $f$ is transverse to $M$, then $f^{-1}(M)$ is a manifold, with $\codim_Y(M)=\codim_X(f^{-1}(M))$.
Theorem: (Transversality theorem) Let $\{g_s:X\to Y\ |\ s\in S\}$ be a smooth family of maps. If $g:X\times S\to Y$ is transverse to $M$, then for almost every $s\in S$ the map $g_s$ is transverse to $M$.
If we replace $f$ with $df$, and ask that it be transverse to $M$, then $df|_s$ is also transverse to $M$.
Example: Consider the map $g_s:X\to \R^n$ given by $g_s(X)=i(X)+s=X+s$, where $i$ is the embedding of $X$ into $\R^n$. Since $g(X\times \R^n)=\R^n$ and $g$ varies smoothly in both variables, we have that $g$ is transverse to $X$. Hence by the transversality theorem, $X$ is transverse to its translates $X+s$ for almost all $s\in \R^n$.
Theorem: (Sard) For $f$ smooth and $\dy Y=\emptyset$, almost every $y\in Y$ is a regular value of $f$ and $f|_{\dy X}$. Equivalently, the set of critical values of $f$ has measure zero.
Resources: Guillemin and Pollack (Differential topology, Chapters 1, 2), Lee (Introduction to smooth manifolds, Chapter 6)