Andrew Ranicki has an excellent webpage devoted to the the Hauptvermutung; the conjecture, now known to be false, that for a triangulable space, all triangulations are equivalent. Even more surprising, it’s false even if you restrict yourself to only manifolds. The discovery spelled the end of the original combinatorial approach to algebraic topology (though I think the approach was largely superceded by the time the falsity of the conjecture was discovered.) Ranicki includes a link to a PDF of The Hauptvermutung Book, an introductory collection of papers on the subject that he edited.

I also came across these lecture notes that describe Milnor’s counterexample in detail.