Genericity exercises

For most of this page, we refer to Golubitsky and Guillemin for definitions etc. We always implictly endow $C^k(X,Y)$ with the Whitney $C^k$ topology.

Definition transversality of vector spaces

Assume We have two vectorsubspaces $A,B\subset X$ of some vector space $X$. We write $A⋔B$ if $A+B=X$.

Definition transversality of manifolds

Assume we have two submanifolds $A,B\subset M$ of some manifold $M$. We write $A⋔B$ if for each $p\in A\cap B,T_{p}A⋔T_{p}B$ (as vector spaces).

Definition 4.1. (p.50) Transversality

Let X, Y be smooth manifolds and $f:X\to Y$ be a smooth mapping. Let $W$ be a submanifold of $Y$ and $x\in X$. Then $f$ intersects $W$ transversally at $x$ (denoted by $f⋔W$ at $x$) if either:

1. $f(x)\notin W$, or
2. $f(x)\in W$, and $T_{f(x)}W⋔(df{)}_x(T_{x}X)$ (as vector spaces).

We write $f⋔W$ on $A$ if $f⋔W$ at $x,\forall x\in A$. We leave the “on …” if $f$ intersects $W$ transversally on the entire $X$.

Note that for the case were $\dim (T_{f(x)})+\dim ((df{)}_x(T_{x}X))<T_{f(x)}Y$, the only way the function can intersect $W$ transversally is if $f(X)\cap W=\varnothing$.

For a definition when $X$ has a boundary, see Wikipedia.

Definition of corank

Let $X,Y$ be differentiable manifolds and $f:X\to Y$ a $C^1$ map. Then we define:

$$\text{corank}(df{)}_p=\min (\dim (X),\dim (Y))-\text{rank}(df{)}_p.$$

We call a point $p\in X$ a critical point of $f$ if $\text{corank}(df{)}_p>0$, and we denote the set of critical points of $f$ by ${\Sigma }_{\text{crit},f}$.

Definition 3.2 (p.44) residual set

Say we have a space $X$. We then call a subset $A\subset X$ a residual set if $A$ is a countable intersection of dense open subsets of $X$.

Exercise

1. Assume we have two (k-differentiable) maps $(f,g)\in C^k({\mathbb{D}}^2,{\mathbb{R}}^2)\times C^k({\mathbb{D}}^2,{\mathbb{R}}^2)$. We identify $\partial {\mathbb{D}}^2$ with ${\mathbb{S}}^1$. Show that $f({\mathbb{S}}^1)⋔g({\mathbb{S}}^1)$ (as manifolds) is a generic property of $C^k({\mathbb{D}}^2,{\mathbb{R}}^2)\times C^k({\mathbb{D}}^2,{\mathbb{R}}^2)$.

We note that $f({\mathbb{S}}^1)⋔g({\mathbb{S}}^1)$ means that $\forall q\in f({\mathbb{S}}^1)\cap g({\mathbb{S}}^1),T_{q}f({\mathbb{S}}^1)+T_{q}g({\mathbb{S}}^1)=T_q{\mathbb{R}}^2\cong {\mathbb{R}}^2$.

Proof:

The most straightforward way to do this is with multijets.

2. Show this implies the images of $f$ and $g$ (along the boundary) intersect in finitely many points, generically.

3. Show that for three functions $f,g,h$ the generic case is no triple intersection.

Questions

1. Is it true that for two smooth ($k$-smooth?) manifolds that $C^k(X,Y)$ with the Whitney $C^k$ topology is a baire space? Or only in the $\infty$ case? (We discussed things on the board for $C^k$, but in the book it only mentions $C^{\infty }$)
2. What is the intuition for allowing a countable amount of intersections in the definition of a residual set? This is fine in a Baire space, but does this break in other spaces? As in does this allow you to construct a generic property that does not hold on a dense subset of the space?