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.
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.
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.