K. Janich, Vector Analysis, Chapter 3 Test In the text by Klaus Janich, "Vector Analysis," there is a Test at the end of Chapter 3, on Differential Forms.  Question #6 reads:

Let $M$ be a nonempty manifold with $\dim M = n, n > 0$ and $0 < k < n$. Then, $\dim \Omega^{k}M = {}$

*

*$\infty$,

*$\displaystyle\binom{n}{k}$,

*$k(k-1)/2$.


The answer key in the back of the book says #1 is the correct answer.  This makes no sense to me.  I believe the correct answer should be #2, $\binom{n}{k}$.
 A: Recall that $C^\infty(M)$ is a vector space over $\mathbb{R}$, in the usual way by pointwise operations. Given $f, g: M \to \mathbb{R}$ and $r \in \mathbb{R}$, the sum $f+g$ and the scalar multiple $rf$ are defined by
\begin{align}
(f+g)(m) &= f(m) + g(m), \\
(rf)(m) &= r \, f(m)
\end{align}
for all $m \in M$. This is an (uncountably) infinite dimensional as a vector space over $\mathbb{R}$.
Now, on a coordinate patch $U \subseteq M$, for $0 \leq k \leq n$, the space of $k$-forms $\Omega^k(U)$ is a (free) module over the ring $C^\infty(U)$ with a basis
$$
\bigl\{ dx^I = dx^{i_1} \wedge \cdots \wedge dx^{i_k} \bigr\}, 
$$
where the multi-index sets $I = (i_1, \dots, i_k)$ run over all $k$-subsets, which for our convenience, we can sort to satisfy
$$
1 \leq i_1 \leq \cdots \leq i_k \leq n.
$$
This basis set clearly has dimension $\displaystyle\binom{n}{k}$ as a module over $C^\infty(U)$.
Using bump functions and a partition of unity, at least in the case that $M$ is paracompact, these arguments extend to show that the space of forms over the whole manifold $M$ is finite-dimensional. Examples include: compact spaces, normal spaces, metric spaces, smooth CW complexes.
However as a vector space over $\mathbb{R}$, the space $\Omega^k(M)$ is infinite-dimensional. Even for $k=0$, where $\Omega^0(M) \cong C^\infty(M)$ it's infinite-dimensional.
