Exercise from Hirsch Differential Topology, Generalization of Fund. Theorem of Algebra

Question

This is problem 11, from Chapter 5 Section 1 (Degrees of Mappings) in Hirsch's differential topology.

Let $U \subset \mathbb{R}^n$ be a nonempty open set, and $F :U \to \mathbb{R}^n$ a $C^1$ map. Assume that

1. $F$ is proper (the preimage of a compact set is also compact)
2. Outside of some compact set $K$, we have that $\mathrm{det}(DF_x)$ is positive, i.e. $\mathrm{det}(DF_x)|_{U\setminus K}$ is positive.

Prove that $F$ is surjective.

My main idea is to transport the open set $U$ to a compact manifold without boundary (for example an $n$-sphere, and then somehow transport the map $F$ alongside it. Using the fact that $\mathrm{det} DF_x$ is positive at all points outside a compact set, we can compute the degree of the resultant map, and in particular show that it is positive. If we show that the degree is positive, the result follows by using the stereographic projection to transport the $n$ sphere back to Euclidean space, as any map with positive degree is surjective, and the stereographic projection is surjective as a function from the sphere minus the north pole to Euclidean space.

In this vein, consider the stereographic projection, $\phi: \mathbb{S}^n \setminus N \to \mathbb{R}^{n}$ where here $N$ denotes the north pole of the $n$ sphere, and consider its associated inverse on $\mathbb{R}^n$. I am not sure how one might use the properties of $F$ now. It seems as if we need to extend $F$ smoothly so that $F$ is defined at the north pole (point at infinity) and then compute the degree of the map on the sphere, but I'm not sure how to do this properly. Does anyone have any hints/suggestions? I'm also not sure how one would use the fact that $F$ is proper.

Answer 1

Edit: this is cleaner than my last attempt, although the basic ideas are the same.

Let $V \subset \mathbb{R}^n$ be a nonempty open subset. We denote as $W^n(V)$ the space of continuous compactly supported $n$-forms over $V$. We denote as $I^n(V)$ the subspace of the such $n$-forms the integral of which vanishes.

If $U,V \subset \mathbb{R}^n$ are nonempty open subsets, and $F: U \rightarrow V$ is $C^1$ and proper, then $F$ induces a linear map $F^*: W^n(V) \rightarrow W^n(U)$.

The main claim is that $F^*(I^n(V)) \subset I^n(U)$, in the case $V=\mathbb{R}^n$ (then we conclude as in the old answer, by defining $u$ and $v$ and noting $u-v \in I^n(V)$ but $F^*(u-v)=F^*(u) \notin I^n(U)$).

Let $u$ be a compactly supported continuous $n$-form on $V=\mathbb{R}^n$ with integral $0$.

Using convolutions, there is a sequence $u_n$ of smooth compactly supported $n$-forms converging to $u$ in the sense of $C^0_c(V’)$ for some $V’$ containing the support of $u$ and such that $V’ \subset \subset V$. (indeed, there is an isomorphism $f \in C^0_c(V) \longmapsto f(x)dx_1 \wedge \ldots \wedge \ldots dx_n \in W^n(V)$, that fact holding for any nonempty open subset $V \subset \mathbb{R}^n$)

As in the old answer, we can write $u_n=dv_n$, where $v_n$ is a smooth compactly supported $(n-1)$-form (and we can assume that all of their supports are compact subsets of some $V’’$ with $V’ \subset V’’ \subset \subset V_3 \subset \subset V_4 \subset \subset V$).

There exists a sequence of smooth $F_p: F^{-1}(V_4) \rightarrow V$ such that $\|F_p-F\|_{C^1} \rightarrow 0$.

Note that for large $p$, each $F_p^*v_n$ is compactly supported (the support is in $F_p^{-1}(V_3) \subset \subset F^{-1}(V_4)$) and thus the integral of $F_p^*u_n$ is the integral of $d(F_p^*v_n)$, where $F_p^*v_n$ is a smooth compactly supported $(n-1)$ form in $F^{-1}(V_4)$, so the $F_p^*u_n$ have integral $0$.

But for a fixed $n$, the $F_p^*u_n$ converge to $F^*u_n$ in $C^0_c$ so the integral of $F^*u_n$ is the limit of the integrals of the $F_p^*u_n$. It follows that $F^*u_n \in I^n(U)$. Taking limits again (but now, $n$ grows), it follows that $F^*u \in I^n(U)$ and we are done.

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Old answer:

That’s not a full proof, because I only deal with the case $F$ smooth. But it’s the best sketch so far, I think.

Because it is proper, $F$ induces a morphism $F^*: H^n_c(\mathbb{R}^n) \rightarrow H^n_c(U)$, where $H^n_c(X)$ is the quotient space of the smooth compactly supported $n$-forms on $X$ by the subspace of the differentials of the smooth compactly supported $(n-1)$-forms.

The first key fact is that the integral induces an isomorphism $H^n_c(\mathbb{R}^n) \rightarrow \mathbb{R}$.

Indeed, good definition and surjectivity are clear. Let us show injectivity. In the following proof, cohomology is de Rham cohomology.

Assume $f$ is a smooth compactly supported $n$-form with null integral, then $f=dg$, for some smooth $(n-1)$-form $g$. Let $B$ be a large ball containing a neighborhood of the support of $f$, let $S$ be its boundary, and let $A=\mathbb{R}^n \backslash B$. As $S \rightarrow A$ is a homotopy equivalence, $H^*(S) \equiv H^*(A)$. Now $g$ corresponds to a class of $H^*(S)$ with null integral, so it is zero, so $g$ is exact in $A$, we write $g=dh$.

Let $h_1=\chi h$, where $\chi$ vanishes on a neighborhood of $B$. Then $g-dh_1$ is a smooth compactly supported $(n-1)$-form with differential $f$.

$\,$

Next, take $x \notin F^{-1}(F(K))$. Then $F^{-1}(F(x))$ is discrete, so (nonvanishing determinants and inverse function theorem) we have a neighborhood $V$ of $F(x)$ such that $F: F^{-1}(V) \rightarrow V$ is a trivial $C^1$ covering map.

Let $\ell : \mathbb{R}^n \rightarrow [0,\infty)$ be smooth, nonzero, and supported in $V$. Let $u=\ell dx_1 \wedge \ldots \wedge dx_n$.

Assume $F$ is not surjective: $F$ is proper hence closed, so there is a smooth nonzero $n$-form $v$ in $\mathbb{R}^n$ supported outside $F(X)$, that is at each point a nonnegative multiple of $dx_1 \wedge \ldots \wedge dx_n$, such that $\int_{\mathbb{R}^n}{u-v}=0$.

Therefore, $u-v$ is zero in $H^n_c(\mathbb{R}^n)$, so $F^*(u-v)$ must be zero in $H^n_c(U)$. But $F^*v=0$. So $F^*u$ must be exact, in particular has a null integral. But using again the positive determinants, we get a contradiction.

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I am not too sure about the nonsmooth case: then, even when changing the definition of the $H^n_c$ to include nonsmooth forms, it’s not obvious that $F$ will work functiorially with the cohomology (because $F^*$ causes a loss of regularity so it doesn’t exactly map exact forms to exact forms). Maybe singular cohomology becomes the best tool, but then the “positive determinant” part becomes harder to use. Maybe instead we can use distributions but I’m not sure.