The Dixmier-Douady classes and abelian extensions of groups of homeomorphisms
Let X be a connected topological space and c∈H2(X;Z) be a non-zero cohomology class. A Homeo(X,c)-bundle is a fiber bundle with fiber X whose structure group reduces to the group Homeo(X,c) of c-preserving homeomorphisms of X. If H1(X;Z)=0, then a characteristic class for Homeo(X,c)-bundles called t...
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| Published in | Topology and its applications Vol. 336; p. 108600 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier B.V
15.08.2023
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| Abstract | Let X be a connected topological space and c∈H2(X;Z) be a non-zero cohomology class. A Homeo(X,c)-bundle is a fiber bundle with fiber X whose structure group reduces to the group Homeo(X,c) of c-preserving homeomorphisms of X. If H1(X;Z)=0, then a characteristic class for Homeo(X,c)-bundles called the Dixmier-Douady class is defined via the Serre spectral sequence. We show the relation between the universal Dixmier-Douady class for foliated Homeo(X,c)-bundles and the gauge group extension of Homeo(X,c). Moreover, under some assumptions, we construct a central S1-extension and a group two-cocycle on Homeo(X,c) corresponding to the Dixmier-Douady class. |
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| AbstractList | Let X be a connected topological space and c∈H2(X;Z) be a non-zero cohomology class. A Homeo(X,c)-bundle is a fiber bundle with fiber X whose structure group reduces to the group Homeo(X,c) of c-preserving homeomorphisms of X. If H1(X;Z)=0, then a characteristic class for Homeo(X,c)-bundles called the Dixmier-Douady class is defined via the Serre spectral sequence. We show the relation between the universal Dixmier-Douady class for foliated Homeo(X,c)-bundles and the gauge group extension of Homeo(X,c). Moreover, under some assumptions, we construct a central S1-extension and a group two-cocycle on Homeo(X,c) corresponding to the Dixmier-Douady class. |
| ArticleNumber | 108600 |
| Author | Maruyama, Shuhei |
| Author_xml | – sequence: 1 givenname: Shuhei orcidid: 0000-0003-2491-2728 surname: Maruyama fullname: Maruyama, Shuhei email: smaruyama1001b@g.chuo-u.ac.jp organization: Department of Mathematics, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo, 112-8551, Japan |
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| References | Dobbins (br0050) 2021 Neukirch, Schmidt, Wingberg (br0120) 2008; vol. 323 Davis, Kirk (br0040) 2001; vol. 35 Savelyev, Shelukhin (br0130) 2020; 18 Ismagilov, Losik, Michor (br0090) 2006; 6 Brown (br0010) 1982; vol. 87 Shuhei Maruyama, The dixmier-douady class, the action homomorphism, and group cocycles on the symplectomorphism group. Frigerio (br0060) 2017; vol. 227 Gal, Kędra (br0070) 2011 Brylinski (br0020) 1993; vol. 107 Yamanoshita (br0140) 1995; 31 Hochschild, Serre (br0080) 1953; 74 Carey, Crowley, Murray (br0030) 1998; 193 Milnor (br0110) 1983; 58 Brylinski (10.1016/j.topol.2023.108600_br0020) 1993; vol. 107 10.1016/j.topol.2023.108600_br0100 Brown (10.1016/j.topol.2023.108600_br0010) 1982; vol. 87 Neukirch (10.1016/j.topol.2023.108600_br0120) 2008; vol. 323 Gal (10.1016/j.topol.2023.108600_br0070) Milnor (10.1016/j.topol.2023.108600_br0110) 1983; 58 Yamanoshita (10.1016/j.topol.2023.108600_br0140) 1995; 31 Frigerio (10.1016/j.topol.2023.108600_br0060) 2017; vol. 227 Hochschild (10.1016/j.topol.2023.108600_br0080) 1953; 74 Ismagilov (10.1016/j.topol.2023.108600_br0090) 2006; 6 Dobbins (10.1016/j.topol.2023.108600_br0050) Savelyev (10.1016/j.topol.2023.108600_br0130) 2020; 18 Davis (10.1016/j.topol.2023.108600_br0040) 2001; vol. 35 Carey (10.1016/j.topol.2023.108600_br0030) 1998; 193 |
| References_xml | – volume: 74 start-page: 110 year: 1953 end-page: 134 ident: br0080 article-title: Cohomology of group extensions publication-title: Trans. Am. Math. Soc. contributor: fullname: Serre – volume: 6 start-page: 307 year: 2006 end-page: 315 ident: br0090 article-title: A 2-cocycle on a symplectomorphism group publication-title: Mosc. Math. J. contributor: fullname: Michor – volume: vol. 323 year: 2008 ident: br0120 article-title: Cohomology of Number Fields publication-title: Grundlehren der mathematischen Wissenschaften contributor: fullname: Wingberg – volume: 31 start-page: 953 year: 1995 end-page: 958 ident: br0140 article-title: On the group of publication-title: Publ. Res. Inst. Math. Sci. contributor: fullname: Yamanoshita – volume: vol. 107 year: 1993 ident: br0020 article-title: Loop Spaces, Characteristic Classes and Geometric Quantization publication-title: Progress in Mathematics contributor: fullname: Brylinski – year: 2021 ident: br0050 article-title: A strong equivariant deformation retraction from the homeomorphism group of the projective plane to the special orthogonal group contributor: fullname: Dobbins – year: 2011 ident: br0070 contributor: fullname: Kędra – volume: vol. 35 year: 2001 ident: br0040 article-title: Lecture Notes in Algebraic Topology publication-title: Graduate Studies in Mathematics contributor: fullname: Kirk – volume: 58 start-page: 72 year: 1983 end-page: 85 ident: br0110 article-title: On the homology of Lie groups made discrete publication-title: Comment. Math. Helv. contributor: fullname: Milnor – volume: 193 start-page: 171 year: 1998 end-page: 196 ident: br0030 article-title: Principal bundles and the Dixmier Douady class publication-title: Commun. Math. Phys. contributor: fullname: Murray – volume: vol. 227 year: 2017 ident: br0060 article-title: Bounded Cohomology of Discrete Groups publication-title: Mathematical Surveys and Monographs contributor: fullname: Frigerio – volume: vol. 87 year: 1982 ident: br0010 article-title: Cohomology of Groups publication-title: Graduate Texts in Mathematics contributor: fullname: Brown – volume: 18 start-page: 251 year: 2020 end-page: 289 ident: br0130 article-title: K-theoretic invariants of Hamiltonian fibrations publication-title: J. Symplectic Geom. contributor: fullname: Shelukhin – volume: vol. 35 year: 2001 ident: 10.1016/j.topol.2023.108600_br0040 article-title: Lecture Notes in Algebraic Topology contributor: fullname: Davis – volume: 31 start-page: 953 issue: 5 year: 1995 ident: 10.1016/j.topol.2023.108600_br0140 article-title: On the group of S1-equivariant homeomorphisms of the 3-sphere publication-title: Publ. Res. Inst. Math. Sci. doi: 10.2977/prims/1195163725 contributor: fullname: Yamanoshita – volume: vol. 227 year: 2017 ident: 10.1016/j.topol.2023.108600_br0060 article-title: Bounded Cohomology of Discrete Groups contributor: fullname: Frigerio – volume: 193 start-page: 171 issue: 1 year: 1998 ident: 10.1016/j.topol.2023.108600_br0030 article-title: Principal bundles and the Dixmier Douady class publication-title: Commun. Math. Phys. doi: 10.1007/s002200050323 contributor: fullname: Carey – ident: 10.1016/j.topol.2023.108600_br0100 – volume: vol. 323 year: 2008 ident: 10.1016/j.topol.2023.108600_br0120 article-title: Cohomology of Number Fields contributor: fullname: Neukirch – ident: 10.1016/j.topol.2023.108600_br0050 contributor: fullname: Dobbins – ident: 10.1016/j.topol.2023.108600_br0070 contributor: fullname: Gal – volume: 6 start-page: 307 issue: 2 year: 2006 ident: 10.1016/j.topol.2023.108600_br0090 article-title: A 2-cocycle on a symplectomorphism group publication-title: Mosc. Math. J. doi: 10.17323/1609-4514-2006-6-2-307-315 contributor: fullname: Ismagilov – volume: 18 start-page: 251 issue: 1 year: 2020 ident: 10.1016/j.topol.2023.108600_br0130 article-title: K-theoretic invariants of Hamiltonian fibrations publication-title: J. Symplectic Geom. doi: 10.4310/JSG.2020.v18.n1.a7 contributor: fullname: Savelyev – volume: 58 start-page: 72 issue: 1 year: 1983 ident: 10.1016/j.topol.2023.108600_br0110 article-title: On the homology of Lie groups made discrete publication-title: Comment. Math. Helv. doi: 10.1007/BF02564625 contributor: fullname: Milnor – volume: vol. 87 year: 1982 ident: 10.1016/j.topol.2023.108600_br0010 article-title: Cohomology of Groups contributor: fullname: Brown – volume: vol. 107 year: 1993 ident: 10.1016/j.topol.2023.108600_br0020 article-title: Loop Spaces, Characteristic Classes and Geometric Quantization contributor: fullname: Brylinski – volume: 74 start-page: 110 year: 1953 ident: 10.1016/j.topol.2023.108600_br0080 article-title: Cohomology of group extensions publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1953-0052438-8 contributor: fullname: Hochschild |
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| Title | The Dixmier-Douady classes and abelian extensions of groups of homeomorphisms |
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