Put another way, assuming it is somewhat fair to say that we (not I, but those who know better--part of my question is whether my stated assumption is at all warranted) have a qualitatively better handle on manifolds up to and including dimension 4 than we do for the case of 5+ dimensions, is it just a coincidence (i.e., 'non-mathematical' factors) that it seems 'natural' for us to think of space-time in terms a four-dimensional manifold?

Put another way, assuming it is somewhat fair to say that we (not I, but those who know better--part of my question is whether my stated assumption is in fact warranted) have in some sense a qualitatively better handle on manifolds up to and including dimension 4 than we do for the case of 5+ dimensions, is it just a coincidence (i.e., 'non-mathematical' factors) that it seems 'natural' for us to think of space-time in terms a four-dimensional manifold?

Edit: apologies for my naivety; I do not mean to imply that my assumption is solid. I probably overestimated the legitimacy of the question I was asking.

Put another way, assuming it is somewhat fair to say that we have a qualitatively better handle on manifolds up to and including dimension 4 than we do for the case of 5+ dimensions, is it just a coincidence (i.e., 'non-mathematical' factors) that it seems 'natural' for us to think of space-time in terms a four-dimensional manifold?

Put another way, assuming it is somewhat fair to say that we (not I, but those who know better--part of my question is whether my stated assumption is at all warranted) have a qualitatively better handle on manifolds up to and including dimension 4 than we do for the case of 5+ dimensions, is it just a coincidence (i.e., 'non-mathematical' factors) that it seems 'natural' for us to think of space-time in terms a four-dimensional manifold?