Compactly Supported Forms and Integration

Let \(\omega \) be a compactly supported \(n\)-form on \(\mathbb {R}^n\), \(n\ge 1\). Show that the following statements are equivalent:

1. \[ \int _{\mathbb {R}^n}\omega =0. \]

2. There exists a compactly supported \((n-1)\)-form \(\eta \) on \(\mathbb {R}^n\) such that \(\omega =d\eta \).

   You are allowed to use computations of de Rham cohomology of Euclidean spaces and spheres but not of the compactly supported cohomology of \(\mathbb {R}^n\).

1. Construct an explicit \(n\)-form \(\omega \) on \(\mathbb {R}^n\) with compact support such that \(\omega \) is closed but there does not exists an \((n-1)\)-form \(\eta \) with compact support and \(d\eta =\omega \). You can use the existence of bump functions in your construction but you need to prove that your construction works.
2. Let \(M\) be a smooth, compact orientable \(n\)-manifold without bound- ary and let \((U,\phi )\) be a chart for \(M\). Show that there exists a closed \(n\)-form on \(M\) with support in \(U\) that is not exact.

1. State Stokes theorem.
2. If \(\omega \in \Omega ^n(\mathbb {R}^n)\) has compact support and \(\int _{\mathbb {R}^n}\omega \ne 0\) show that there does not exist an \(\alpha \in \Omega ^{n-1}(\mathbb {R}^n)\) with compact support and \(d\alpha =\omega \).
3. Now assume that \(n=1\) and that \(\int _{\mathbb {R}}\omega =0\). Find an \(\alpha \in \Omega ^0(\mathbb {R})\) with compact support and \(d\alpha =\omega \).

State Stokes’ theorem and use it to give an example (with proof) of a manifold with a form that is closed but not exact.

Let \(W\) be a vector field on a smooth manifold \(M\) and assume that \(V\) has a flow that is defined on all of \(M\) and for all time. Let \(V\) be another vector field on \(M\) such that \(V-W\) has compact support. Show that \(V\) has a flow on all of \(M\) defined for all time.

Suppose that \(a\) is a smooth, real valued, compactly supported function on \(\mathbb {R}^n\) so that \(\theta =a\,dx_1\wedge \cdots \wedge dx_n\) is an \(n\)-form on \(\mathbb {R}^n\). What is the definition of \(\int _{\mathbb {R}^n}\theta \)? What’s the definition of \(\int _M\omega \) where \(\omega \) is a compactly supported \(k\)-form on an oriented \(k\)-dimensional smooth manifold \(M\)? Show that this definition is well-defined.

State Stokes’ Theorem.

1. State Stokes’ Theorem.
2. Given two 3-forms on \(S^7\), denoted \(\theta \) and \(\omega \), prove that \[ \int _{S^7}d\theta \wedge \omega =\int _{S^7}\theta \wedge d\omega . \]

1. What theorem from multivariable calculus is the proof of the regular value theorem dependent on?
2. What theorem from single variable calculus is the proof of Stokes’ theorem dependent on?
3. What theorem from differential topology is used to prove that transverse intersection of manifolds is a manifold?

Show that the group of diffeomorphisms of a compact manifold \(M\) acts transitively on \(M\). That is, if \(p,q\in M\) there exists a diffeomorphism \(\phi :M\to M\) such that \(\phi (p)=q\). Hint: First assume \(p\) and \(q\) are in the same coordinate neighborhood \(U\) and construct \(\phi \) as the flow of a vector field supported in \(U\).

Let \(M\) be an orientable manifold of dimension \(n\). Prove that the top dimensional de Rham cohomology with compact supports \(H_c^n(m)\) is nontrivial.

Let \(\omega \) be an \(n\)-form on \(\mathbb {R}^n\) with compact support. Show that there exists a compactly supported \((n-1)\) form \(\eta \) with \(\omega =d\eta \) if and only if \[ \int _{\mathbb {R}^n}\omega =0. \] You are allowed to use the standard results about deRham cohomology of \(\mathbb {R}^n\) and spheres.

Prove that for any \(x,y\in \mathbb {R}^n\) there a compactly supported isotopy taking \(x\) to \(y\) (i.e. there is an isotopy \(f_t:\mathbb {R}^n\to \mathbb {R}^n\) such that \(f_0=\id \), \(f_1(x)=y\) and \(f_t=\id \) outside some ball).

Show that for any two points \(x,y\in \mathbb {R}^n\) there is a compactly supported isotopy \(\phi _t\) (i.e. it is identity outside a compact set) such that \(\phi _0=id\) and \(\phi _1(x)=y\).

Let \(\omega \) be a compactly supported smooth \(n\)-form on \(\mathbb {R}^n\). Prove that the following two statements are equivalent.

1. There exists a compactly supported smooth \((n-1)\)-form \(\eta \) such that \(\omega =d\eta \).
2. \(\displaystyle \int _{\mathbb {R}^n}\omega =0\).

Note: You are allowed to use the fact that \(H^n(S^n)=\mathbb {R}\), but you are not allowed to quote the fact that \(H_c^n(\mathbb {R}^n)=\mathbb {R}\).

Let \(\omega \) be a compactly supported smooth \(n\)-form on \(\mathbb {R}^n\). Show that there exists a compactly supported smooth \((n-1)\)-form \(\eta \) with \(\omega =d\eta \) if and only if \[ \int _{\mathbb {R}^n}\omega =0. \] You are allowed to quote facts about \(H^k(X)\) but not about \(H_c^k(X)\).

Let \(M\) be a smooth, orientable \(n\)-dimensional manifold. Prove that the \(n\) dimensional DeRham cohomology with compact supports, \(H_c^n(M)\) is nonzero.