Title: On the Bogomolov-Positselski Conjecture Show Chinese title

Title: 关于波戈莫洛夫-波西捷尔斯基猜想

Abstract: Let $p$ be a prime, we say that a Kummerian oriented pro-$p$ group $(G,\theta)$ has the Bogomolov-Positselski property if $I_\theta(G)$ is a free pro-$p$ group. We give a new criterion for an oriented pro-$p$ group to have the Bogomolov-Positselski property based on previous work by Positselski (arXiv:1405.0965) and Quadrelli and Weigel (arXiv:2103.12438) linking their seemingly unrelated approaches and thereby answering a question posed by Quadrelli and Weigel. Under further assumptions, we derive two additional criteria. The first of which strongly resembles an analogue of the Merkujev-Suslin theorem. The second allows to relax the conditions given by Positselski in Theorem 2 of arXiv:1405.0965. In addition, we show how to make those weaker assumptions computationally effective in some special cases. Show Chinese abstract

Abstract: 设$p$为素数，我们说一个Kummer定向的拟有限$p$群$(G,\theta)$如果$I_\theta(G)$是自由的拟有限$p$群，则具有Bogomolov-Positselski性质。我们根据Positselski之前的工作（arXiv:1405.0965）以及Quadrelli和Weigel（arXiv:2103.12438）的研究，给出一个关于定向拟有限$p$群具有Bogomolov-Positselski性质的新准则，将他们看似无关的方法联系起来，从而回答了Quadrelli和Weigel提出的问题。在进一步的假设下，我们推导出两个额外的准则。其中第一个准则与Merkujev-Suslin定理的一个类比非常相似。第二个准则允许放松Positselski在arXiv:1405.0965中定理2所给的条件。此外，我们展示了如何在一些特殊情况下使这些较弱的假设在计算上有效。