[enasterosophes]

It's a good question. It's easy to assume they're talking about R^126 (where R is the reals) but digging a bit deeper I don't think it's true.

The Kervaire invariant is a property of an "n-dimensional manifold", so the paper is likely about 126-dimensional manifolds. That in turn has a formal definition, and although it's not my specialization, I think means it can be locally represented as an n-dimensional Euclidean space.

A simple example would be a circle, which I guess would be a 1-dimensional manifold, because every point on a circle has a tangent where the circle can be approximated by a line passing through the same point.

So they're saying that there are these surfaces which can be locally approximated by 126-dimensional Euclidean spaces. This in turn probably requires that the surface itself is embedded in some higher-dimensional space such as R^127.

[DevelopingElk]

Manifolds are generally considered objects of themselves, and it may be difficult to embed then in higher dimensional objects. This is especially the case for tricky manifolds like those with a Kervaire invariant of 1.

[red_trumpet]

At this point it might be worthy to note the Whitney Embedding Theorem[1], which states:

> Every smooth n-dimensional manifold can be embedded into R^{2n}.

[1] https://en.wikipedia.org/wiki/Whitney_embedding_theorem