As I understand it, string theory (incorporating bosons and fermions) "works" in $9+1=10$ spacetime dimensions. In the context of dual resonance theory, I've read descriptions of why that is "physically", i.e., how in the process of formulating a quantum theory of relativistic interacting (closed and open) strings, the only way one can do things like get rid of ghosts, for example, is to have ten dimensions.
I'm looking for a way to appreciate more intuitively this restriction on dimensionality. Is there something special about ten-dimensional spaces purely from the point of view of mathematics? Without reference to the particular detailed exigencies of building a physical theory based on strings, does ten-dimensional topology have some unique features that are lost when one dimension is added or subtracted?
Another way to pose the question: For purposes of this discussion, let's stipulate by fiat that the universe has $8+1=9$ dimensions. Would we humans simply be back to the drawing board in terms of coming up with a unified theory of the fundamental forces? Would anything in string theory be salvagable?
I realize that M-theory adds a dimension. So if someone wants to answer the question by telling me what's unique or special about eleven dimensions, that's fine.