Markov proved that there exists an unrecognizable 4-manifold, that is, a 4-manifold for which the homeomorphism problem is undecidable. In this paper we consider the question how close we can get to S^4 with an unrecognizable manifold. One of our achievements is that we show a way to remove so-called Markov's trick from the proof of existence of such a manifold. This trick contributes to the complexity of the resulting manifold. We also show how to decrease the deficiency (or the number of relations) in so-called Adian-Rabin set which is another ingredient that contributes to the complexity of the resulting manifold. Altogether, our approach allows to show that the connected sum #_9(S^2 x S^2) is unrecognizable while the previous best result is the unrecognizability of #_12(S^2 x S^2) due to Gordon.

翻译：马尔可夫证明了存在不可识别的四维流形，即同胚问题不可判定的四维流形。本文探讨了在不可识别流形中能多接近S^4的问题。我们的成果之一，是展示了如何从这类流形存在性的证明中移除所谓的“马尔可夫技巧”。该技巧增加了所得流形的复杂度。我们还展示了如何降低所谓“阿迪安-拉宾集合”中的亏数（或关系数），这是影响所得流形复杂度的另一个要素。综合而言，我们的方法证明了连通和#_9(S^2 x S^2)是不可识别的，而此前最优结果来自戈登，他证明了#_12(S^2 x S^2)的不可识别性。