2024/06/19 by Cho, Sangbum, Koda, Yuya, Lee, Jung Hoon +1
#20F05 #57K20 #57K30 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2406.13309
In our previous version entitled ``The reducing sphere complexes for the 3-sphere are connected: a proof of the Powell Conjecture", we claimed to prove the Powell Conjecture, which states that the Goeritz group of the genus-g Heegaard splitting of the 3-sphere is finitely generated for any non-negative integer g. However, we have found a critical error in the proof of Lemma 2.5 in that version. In this note, we prove that the statement of Lemma 2.5 does not hold in general. This invalidates a key step in our argument and leaves the proof of the Powell Conjecture incomplete. Consequently, the Powell Conjecture remains an open problem in the case of g ≥ 4.