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Explore the intriguing genus-one question for open book decomposition in this 1-hour 6-minute lecture by Pierre Dehornoy from Institut Fourier, presented at Institut Henri Poincaré. Delve into the world of 3-dimensional contact structures and their relationship with open book decompositions. Learn about the distinction between overtwisted and tight structures, and how they relate to the genus of supporting open book decompositions. Examine Etnyre's 2006 question regarding the existence of genus-one open book decompositions for tight structures, which remains unsolved. Investigate the parallel question for Anosov flows and discover various constructions of genus-one open book decompositions. Gain insights into a partial result involving genus-two pages, achieved with computer assistance. This lecture offers a deep dive into an open problem in contact geometry and topology, suitable for those with a background in differential geometry and low-dimensional topology.
