{"title":"透镜空间上的环束","authors":"Oliver H. Wang","doi":"10.1515/forum-2022-0279","DOIUrl":null,"url":null,"abstract":"Let <jats:italic>p</jats:italic> be an odd prime and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>ρ</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:mi>ℤ</m:mi> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> <m:mo>→</m:mo> <m:mrow> <m:msub> <m:mi>GL</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>ℤ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0655.png\"/> <jats:tex-math>{\\rho:\\mathbb{Z}/p\\rightarrow\\operatorname{{GL}}_{n}(\\mathbb{Z})}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be an action of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>ℤ</m:mi> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0555.png\"/> <jats:tex-math>{\\mathbb{Z}/p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> on a lattice and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"normal\">Γ</m:mi> <m:mo>:=</m:mo> <m:mrow> <m:mrow> <m:msup> <m:mi>ℤ</m:mi> <m:mi>n</m:mi> </m:msup> <m:msub> <m:mo>⋊</m:mo> <m:mi>ρ</m:mi> </m:msub> <m:mi>ℤ</m:mi> </m:mrow> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0490.png\"/> <jats:tex-math>{\\Gamma:=\\mathbb{Z}^{n}\\rtimes_{\\rho}\\mathbb{Z}/p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the corresponding semidirect product. The torus bundle <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>M</m:mi> <m:mo>:=</m:mo> <m:mrow> <m:msubsup> <m:mi>T</m:mi> <m:mi>ρ</m:mi> <m:mi>n</m:mi> </m:msubsup> <m:msub> <m:mo>×</m:mo> <m:mrow> <m:mi>ℤ</m:mi> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msub> <m:msup> <m:mi>S</m:mi> <m:mi mathvariant=\"normal\">ℓ</m:mi> </m:msup> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0440.png\"/> <jats:tex-math>{M:=T^{n}_{\\rho}\\times_{\\mathbb{Z}/p}S^{\\ell}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> over the lens space <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>S</m:mi> <m:mi mathvariant=\"normal\">ℓ</m:mi> </m:msup> <m:mo>/</m:mo> <m:mi>ℤ</m:mi> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0463.png\"/> <jats:tex-math>{S^{\\ell}/\\mathbb{Z}/p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> has fundamental group Γ. When <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>ℤ</m:mi> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0555.png\"/> <jats:tex-math>{\\mathbb{Z}/p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> fixes only the origin of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℤ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0571.png\"/> <jats:tex-math>{\\mathbb{Z}^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, Davis and Lück (2021) compute the <jats:italic>L</jats:italic>-groups <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msubsup> <m:mi>L</m:mi> <m:mi>m</m:mi> <m:mrow> <m:mo stretchy=\"false\">〈</m:mo> <m:mi>j</m:mi> <m:mo stretchy=\"false\">〉</m:mo> </m:mrow> </m:msubsup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>ℤ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mi mathvariant=\"normal\">Γ</m:mi> <m:mo stretchy=\"false\">]</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0421.png\"/> <jats:tex-math>{L^{\\langle j\\rangle}_{m}(\\mathbb{Z}[\\Gamma])}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and the structure set <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi mathvariant=\"script\">𝒮</m:mi> <m:mrow> <m:mi>geo</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> </m:mrow> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0610.png\"/> <jats:tex-math>{\\mathcal{{S}}^{{\\rm geo},s}(M)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we extend these computations to all actions of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>ℤ</m:mi> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0555.png\"/> <jats:tex-math>{\\mathbb{Z}/p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℤ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0571.png\"/> <jats:tex-math>{\\mathbb{Z}^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In particular, we compute <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msubsup> <m:mi>L</m:mi> <m:mi>m</m:mi> <m:mrow> <m:mo stretchy=\"false\">〈</m:mo> <m:mi>j</m:mi> <m:mo stretchy=\"false\">〉</m:mo> </m:mrow> </m:msubsup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>ℤ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mi mathvariant=\"normal\">Γ</m:mi> <m:mo stretchy=\"false\">]</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0421.png\"/> <jats:tex-math>{L^{\\langle j\\rangle}_{m}(\\mathbb{Z}[\\Gamma])}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi mathvariant=\"script\">𝒮</m:mi> <m:mrow> <m:mi>geo</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> </m:mrow> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0610.png\"/> <jats:tex-math>{\\mathcal{{S}}^{{\\rm geo},s}(M)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> in a case where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:munder accentunder=\"true\"> <m:mi>E</m:mi> <m:mo>¯</m:mo> </m:munder> <m:mo></m:mo> <m:mi mathvariant=\"normal\">Γ</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2022-0279_eq_0680.png\"/> <jats:tex-math>{\\underline{E}\\Gamma}</jats:tex-math> </jats:alternatives> </jats:inline-formula> has a non-discrete singular set.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Torus bundles over lens spaces\",\"authors\":\"Oliver H. Wang\",\"doi\":\"10.1515/forum-2022-0279\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let <jats:italic>p</jats:italic> be an odd prime and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>ρ</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:mi>ℤ</m:mi> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> <m:mo>→</m:mo> <m:mrow> <m:msub> <m:mi>GL</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>ℤ</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2022-0279_eq_0655.png\\\"/> <jats:tex-math>{\\\\rho:\\\\mathbb{Z}/p\\\\rightarrow\\\\operatorname{{GL}}_{n}(\\\\mathbb{Z})}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be an action of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>ℤ</m:mi> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2022-0279_eq_0555.png\\\"/> <jats:tex-math>{\\\\mathbb{Z}/p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> on a lattice and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi mathvariant=\\\"normal\\\">Γ</m:mi> <m:mo>:=</m:mo> <m:mrow> <m:mrow> <m:msup> <m:mi>ℤ</m:mi> <m:mi>n</m:mi> </m:msup> <m:msub> <m:mo>⋊</m:mo> <m:mi>ρ</m:mi> </m:msub> <m:mi>ℤ</m:mi> </m:mrow> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2022-0279_eq_0490.png\\\"/> <jats:tex-math>{\\\\Gamma:=\\\\mathbb{Z}^{n}\\\\rtimes_{\\\\rho}\\\\mathbb{Z}/p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the corresponding semidirect product. The torus bundle <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>M</m:mi> <m:mo>:=</m:mo> <m:mrow> <m:msubsup> <m:mi>T</m:mi> <m:mi>ρ</m:mi> <m:mi>n</m:mi> </m:msubsup> <m:msub> <m:mo>×</m:mo> <m:mrow> <m:mi>ℤ</m:mi> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msub> <m:msup> <m:mi>S</m:mi> <m:mi mathvariant=\\\"normal\\\">ℓ</m:mi> </m:msup> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2022-0279_eq_0440.png\\\"/> <jats:tex-math>{M:=T^{n}_{\\\\rho}\\\\times_{\\\\mathbb{Z}/p}S^{\\\\ell}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> over the lens space <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:msup> <m:mi>S</m:mi> <m:mi mathvariant=\\\"normal\\\">ℓ</m:mi> </m:msup> <m:mo>/</m:mo> <m:mi>ℤ</m:mi> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2022-0279_eq_0463.png\\\"/> <jats:tex-math>{S^{\\\\ell}/\\\\mathbb{Z}/p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> has fundamental group Γ. When <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>ℤ</m:mi> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2022-0279_eq_0555.png\\\"/> <jats:tex-math>{\\\\mathbb{Z}/p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> fixes only the origin of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mi>ℤ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2022-0279_eq_0571.png\\\"/> <jats:tex-math>{\\\\mathbb{Z}^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, Davis and Lück (2021) compute the <jats:italic>L</jats:italic>-groups <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:msubsup> <m:mi>L</m:mi> <m:mi>m</m:mi> <m:mrow> <m:mo stretchy=\\\"false\\\">〈</m:mo> <m:mi>j</m:mi> <m:mo stretchy=\\\"false\\\">〉</m:mo> </m:mrow> </m:msubsup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mrow> <m:mi>ℤ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">[</m:mo> <m:mi mathvariant=\\\"normal\\\">Γ</m:mi> <m:mo stretchy=\\\"false\\\">]</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2022-0279_eq_0421.png\\\"/> <jats:tex-math>{L^{\\\\langle j\\\\rangle}_{m}(\\\\mathbb{Z}[\\\\Gamma])}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and the structure set <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:msup> <m:mi mathvariant=\\\"script\\\">𝒮</m:mi> <m:mrow> <m:mi>geo</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> </m:mrow> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2022-0279_eq_0610.png\\\"/> <jats:tex-math>{\\\\mathcal{{S}}^{{\\\\rm geo},s}(M)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we extend these computations to all actions of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>ℤ</m:mi> <m:mo>/</m:mo> <m:mi>p</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2022-0279_eq_0555.png\\\"/> <jats:tex-math>{\\\\mathbb{Z}/p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msup> <m:mi>ℤ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2022-0279_eq_0571.png\\\"/> <jats:tex-math>{\\\\mathbb{Z}^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In particular, we compute <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:msubsup> <m:mi>L</m:mi> <m:mi>m</m:mi> <m:mrow> <m:mo stretchy=\\\"false\\\">〈</m:mo> <m:mi>j</m:mi> <m:mo stretchy=\\\"false\\\">〉</m:mo> </m:mrow> </m:msubsup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mrow> <m:mi>ℤ</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">[</m:mo> <m:mi mathvariant=\\\"normal\\\">Γ</m:mi> <m:mo stretchy=\\\"false\\\">]</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2022-0279_eq_0421.png\\\"/> <jats:tex-math>{L^{\\\\langle j\\\\rangle}_{m}(\\\\mathbb{Z}[\\\\Gamma])}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:msup> <m:mi mathvariant=\\\"script\\\">𝒮</m:mi> <m:mrow> <m:mi>geo</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> </m:mrow> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2022-0279_eq_0610.png\\\"/> <jats:tex-math>{\\\\mathcal{{S}}^{{\\\\rm geo},s}(M)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> in a case where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:munder accentunder=\\\"true\\\"> <m:mi>E</m:mi> <m:mo>¯</m:mo> </m:munder> <m:mo></m:mo> <m:mi mathvariant=\\\"normal\\\">Γ</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2022-0279_eq_0680.png\\\"/> <jats:tex-math>{\\\\underline{E}\\\\Gamma}</jats:tex-math> </jats:alternatives> </jats:inline-formula> has a non-discrete singular set.\",\"PeriodicalId\":12433,\"journal\":{\"name\":\"Forum Mathematicum\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-05-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Forum Mathematicum\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/forum-2022-0279\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Forum Mathematicum","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/forum-2022-0279","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
让 p 是奇素数,让 ρ : ℤ / p → GL n ( ℤ ) {\rho:\mathbb{Z}/p\rightarrow\operatorname{{GL}}_{n}(\mathbb{Z})} } 是 ℤ / p {\mathbb{Z}/p} 在网格上的作用,让 Γ := ℤ n ⋊ ρ ℤ / p {\Gamma:=\mathbb{Z}^{n}\rtimes_\{rho}\mathbb{Z}/p} 是相应的半间接积。透镜空间 S ℓ / ℤ / p {S^\{ell}/\mathbb{Z}/p} 上的环束 M := T ρ n × ℤ / p S ℓ {M:=T^{n}_{\rho}\times_{\mathbb{Z}/p}S^{\ell}} 具有基群 Γ。当ℤ / p {\mathbb{Z}/p} 只固定了ℤ n {\mathbb{Z}^{n} 的原点时} Davis 和 Lück (2021) 计算了 L 群 L m 〈 j 〉 ( ℤ [ Γ ] ) {L^{langle j\rangle}_{m}(\mathbb{Z}[\Gamma])} 和结构集 𝒮 geo , s ( M ) {\mathcal{S}}^{\rm geo},s}(M)} 。在本文中,我们将这些计算扩展到ℤ / p {mathbb{Z}/p} 对ℤ n {mathbb{Z}^{n} 的所有作用。} .具体而言,我们计算 L m 〈 j 〉 ( ℤ [ Γ ] ) {L^{langle j\rangle}_{m}(\mathbb{Z}[\Gamma])} 和 𝒮 geo 、s ( M ) {\mathcal{S}}^{\rm geo},s}(M)} 在 E ¯ Γ {\underline{E}\Gamma} 有一个非离散奇异集的情况下。
Let p be an odd prime and let ρ:ℤ/p→GLn(ℤ){\rho:\mathbb{Z}/p\rightarrow\operatorname{{GL}}_{n}(\mathbb{Z})} be an action of ℤ/p{\mathbb{Z}/p} on a lattice and let Γ:=ℤn⋊ρℤ/p{\Gamma:=\mathbb{Z}^{n}\rtimes_{\rho}\mathbb{Z}/p} be the corresponding semidirect product. The torus bundle M:=Tρn×ℤ/pSℓ{M:=T^{n}_{\rho}\times_{\mathbb{Z}/p}S^{\ell}} over the lens space Sℓ/ℤ/p{S^{\ell}/\mathbb{Z}/p} has fundamental group Γ. When ℤ/p{\mathbb{Z}/p} fixes only the origin of ℤn{\mathbb{Z}^{n}}, Davis and Lück (2021) compute the L-groups Lm〈j〉(ℤ[Γ]){L^{\langle j\rangle}_{m}(\mathbb{Z}[\Gamma])} and the structure set 𝒮geo,s(M){\mathcal{{S}}^{{\rm geo},s}(M)}. In this paper, we extend these computations to all actions of ℤ/p{\mathbb{Z}/p} on ℤn{\mathbb{Z}^{n}}. In particular, we compute Lm〈j〉(ℤ[Γ]){L^{\langle j\rangle}_{m}(\mathbb{Z}[\Gamma])} and 𝒮geo,s(M){\mathcal{{S}}^{{\rm geo},s}(M)} in a case where E¯Γ{\underline{E}\Gamma} has a non-discrete singular set.
期刊介绍:
Forum Mathematicum is a general mathematics journal, which is devoted to the publication of research articles in all fields of pure and applied mathematics, including mathematical physics. Forum Mathematicum belongs to the top 50 journals in pure and applied mathematics, as measured by citation impact.