Do 2-forms and higher live in the cotangent space?

Question

Trying to self-study differential forms from a roughly high school maths level. I've got the concept (hopefully) that each point of a manifold is home to vectors (the tangent space) and 1-forms (the cotangent space) and if you have a metric you can flip between them. But how about higher forms such as 2-forms, 3-forms etc? Do they houseshare with 1-forms in the same cotangent space or do they have their own space?

Answer 1

The 1-forms are covectors. They eat vectors and spit out scalars. However, a 2-form is not a covector, as feeding it a vector gives you a 1-form, not a scalar. So they aren't the same kind of animal, and as such, they don't live in the same space. At least not in such a simple manner.

You can make a space that contains all of them, though. It's called the exterior algebra on the vector space of covectors. Put simply, it's the direct sum of all the spaces of $n$-forms for different $n$ (including $n=0$, the scalar functions). Kind of like how a polynomial ring over, say, $\Bbb R$ is a direct sum of different copies of $\Bbb R$ (at least when only considering the structure as an $\Bbb R$-vector space).

In this exterior algebra, you have a product called the wedge product which, at least in notation, looks a lot like polynomial multiplication. And when adding, like with polynomials, different degree terms do not really interact with one another.