Both the gauge groups and 5-manifolds are important in physics and mathematics. In this paper,we combine them to study the homotopy aspects of gauge groups over 5-manifolds. For principal bundles over non-simply conne...Both the gauge groups and 5-manifolds are important in physics and mathematics. In this paper,we combine them to study the homotopy aspects of gauge groups over 5-manifolds. For principal bundles over non-simply connected oriented closed 5-manifolds of a certain type, we prove various homotopy decompositions of their gauge groups according to different geometric structures on the manifolds, and give the partial solution to the classification of the gauge groups. As applications, we estimate the homotopy exponents of their gauge groups, and show periodicity results of the homotopy groups of gauge groups analogous to the Bott periodicity.Our treatments here are also very effective for rational gauge groups in the general context, and applicable for higher dimensional manifolds.展开更多
We give an overview of five rationalization theories for spaces(Bousfield-Kan’s Q-completion;Sullivan’s rationaliz ation;Bousifeld’s homology rationaliz ation;Casacuberta-Peschke’s Ω-rationalization;Gomez-Tato-Ha...We give an overview of five rationalization theories for spaces(Bousfield-Kan’s Q-completion;Sullivan’s rationaliz ation;Bousifeld’s homology rationaliz ation;Casacuberta-Peschke’s Ω-rationalization;Gomez-Tato-Halperin-Tanré’s π-fiberwise rationalization)that extend the classical rationalization of simply connected spaces.We also give an overview of the corresponding rationalization theories for groups(Q-completion;HQ-localization;Baumslag rationalization)that extend the classical Malcev completion.展开更多
基金supported by Postdoctoral International Exchange Program for Incoming Postdoctoral Students under Chinese Postdoctoral Council and Chinese Postdoctoral Science Foundation, Chinese Postdoctoral Science Foundation (Grant No.2018M631605)National Natural Science Foundation of China (Grant No.11801544)。
文摘Both the gauge groups and 5-manifolds are important in physics and mathematics. In this paper,we combine them to study the homotopy aspects of gauge groups over 5-manifolds. For principal bundles over non-simply connected oriented closed 5-manifolds of a certain type, we prove various homotopy decompositions of their gauge groups according to different geometric structures on the manifolds, and give the partial solution to the classification of the gauge groups. As applications, we estimate the homotopy exponents of their gauge groups, and show periodicity results of the homotopy groups of gauge groups analogous to the Bott periodicity.Our treatments here are also very effective for rational gauge groups in the general context, and applicable for higher dimensional manifolds.
基金Supported by the Ministry of Science and Higher Education of the Russian Federation,agreement 075-15-2019-1619。
文摘We give an overview of five rationalization theories for spaces(Bousfield-Kan’s Q-completion;Sullivan’s rationaliz ation;Bousifeld’s homology rationaliz ation;Casacuberta-Peschke’s Ω-rationalization;Gomez-Tato-Halperin-Tanré’s π-fiberwise rationalization)that extend the classical rationalization of simply connected spaces.We also give an overview of the corresponding rationalization theories for groups(Q-completion;HQ-localization;Baumslag rationalization)that extend the classical Malcev completion.
