{"title":"Monotone versus non-monotone projective operators","authors":"J. P. Aguilera, P. D. Welch","doi":"10.1112/blms.13194","DOIUrl":"https://doi.org/10.1112/blms.13194","url":null,"abstract":"<p>For a class of operators <span></span><math>\u0000 <semantics>\u0000 <mi>Γ</mi>\u0000 <annotation>$Gamma$</annotation>\u0000 </semantics></math>, let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 <mi>Γ</mi>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 <annotation>$|Gamma |$</annotation>\u0000 </semantics></math> denote the closure ordinal of <span></span><math>\u0000 <semantics>\u0000 <mi>Γ</mi>\u0000 <annotation>$Gamma$</annotation>\u0000 </semantics></math>-inductive definitions. We give upper bounds on the values of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 <msubsup>\u0000 <mi>Σ</mi>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>n</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mi>m</mi>\u0000 <mi>o</mi>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$|Sigma ^{1,mon}_{2n+1}|$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 <msubsup>\u0000 <mi>Π</mi>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>n</mi>\u0000 <mo>+</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mi>m</mi>\u0000 <mi>o</mi>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$|Pi ^{1,mon}_{2n+2}|$</annotation>\u0000 </semantics></math> under the assumption that all projective sets of reals are determined, significantly improving the known results. In particular, the bounds show that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>|</mo>\u0000 </mrow>\u0000 <msubsup>\u0000 ","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 1","pages":"256-264"},"PeriodicalIF":0.8,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13194","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143115892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Algebraically overtwisted tight 3-manifolds from \u0000 \u0000 \u0000 +\u0000 1\u0000 \u0000 $+1$\u0000 surgeries","authors":"Youlin Li, Zhengyi Zhou","doi":"10.1112/blms.13211","DOIUrl":"https://doi.org/10.1112/blms.13211","url":null,"abstract":"<p>We execute Avdek's algorithm to find many algebraically overtwisted and tight 3-manifolds by contact <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$+1$</annotation>\u0000 </semantics></math> surgeries. In particular, we show that a contact <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <mi>k</mi>\u0000 </mrow>\u0000 <annotation>$1/k$</annotation>\u0000 </semantics></math> surgery on the standard contact 3-sphere along any Legendrian positive torus knot with the maximal Thurston–Bennequin invariant yields an algebraically overtwisted and tight 3-manifold, where <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math> is a positive integer.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 2","pages":"534-550"},"PeriodicalIF":0.8,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143362383","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}