For generic fractal curves the topology induced on the curve from the surrounding space will typically prevent that curve from being a topological manifold, since locally it won't look like R^n. It will merely be a topological space.

If, however, the (image of) the curve is a topological manifold (with the induced topology), then the (integer) dimension of the manifold will agree with the Minkowski dimension that's explained in the article (and also with the more commonly used Hausdorff dimension[0]).

[0]: https://en.m.wikipedia.org/wiki/Hausdorff_dimension