whitney-trick-fails-dimension-4-exotic-r4

Summary

In four dimensions, the standard geometric technique for untangling and simplifying spaces falls apart, which is what allows bizarre smooth structures (exotic R^4s) to exist and why the powerful high-dimensional toolkit of surgery theory simply cannot be applied here. This makes four-dimensional topology fundamentally harder than anything in five dimensions or above, and the system treats this breakdown as an accepted starting fact that other geometric claims may depend on.