Unrecognizability of manifolds

We present a modernized proof, with a modification by M.A. Shtan’ko, of the Markov theorem on the unsolvability of the homeomorphy problem for manifolds. We then discuss a proof of the S.P. Novikov theorem on the unrecognizability of spheres S n for n ≥ 5 , from which we obtain a corollary about unrecognizability of all manifolds of dimension at least five. An analogous argument then proves the unrecognizability of stabilizations (i.e. the connected sum with 14 copies of S 2 × S 2 ) of all four-dimensional manifolds. We also give a brief overview of known results concerning algorithmic recognizability of three-dimensional manifolds.