S~1作用的4维流形的符号差
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摘要
设M为具有有限不动点集的4维连通闭G-流形,它的Q系数等变上同调环H_G~*(M;Q)是自由H~*(BG;Q)模,其中G=S~1为圆周群。本文给出了M的相交形式(intersection form)的符号差(signature)σ(M)与不动点集中元素个数|M~G|之间的关系。
In this paper we consider 4 dimensional connected closed S~1-manifolds M with a non-empty finite fixed point set,each equivariant cohomology rings of M,with coefficient Q,H_G~*(M;Q) is a free H~*(BG;Q) module.The main result is about the relationship between the signature of Mσ(M) and the number of the fixed points |M~G|.
In this paper we consider 4 dimensional connected closed S~1-manifolds M with a non-empty finite fixed point set,each equivariant cohomology rings of M,with coefficient Q,H_G~*(M;Q) is a free H~*(BG;Q) module.The main result is about the relationship between the signature of Mσ(M) and the number of the fixed points |M~G|.
引文
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[AH]M.Atiyah and F.Hirzebruch,The moment map and equivariant cohomology,Topology 23(1984),1-28.
[AP]C.Allday and V.Puppe,Cohornological Methods in Transformation Groups,Cambridge Studies in Advanced Mathematics,32(1993),Cambridge University Press.
[B]G.E.Bredon,Intrduction to compact transformation groups,Pure and Applied Mathematics,Vol.46(1972) Academic Press,New York-London.
[CL]Chen B,L(u|¨) Z,Equivariant cohomology and analytic descriptions of ring isomorphisms,Math Z,261(2009),891-908
[D]T.tom Dieck,Characteristic numbers of G-manifolds.I,Invent.Math.13(1971),213-224.
[F]Ronald Fintushcl,Classification of circle action on 4-manifolds,Transactions of the Amcrican Mathematical Socicty,242(1978),377-390
[F1]Ronald Fintushcl,Circle actions on simply connected 4-manifolds,Transactions of thc Amcrican Mathematical Society,230(1977),147-171
[G]Lconor Godinho,Equivariant cohomology of S~1-actions on 4-manifolds,Canad Math Bull,50(3)(2007),365-376
[GH]R.Goldin and T.S.Holm,The equivariant cohornology of Harniltonian Gspaces from residual S~1 actions,Math.Res.Lett.8(2001),67-77.
[GKM]M.Goresky,R.Kottwitz,and R.MacPherson Equivariant cohomology,Koszul duality,and the localization theorem,Invent.Math.131(1998),25-83.
[H]W.Y.Hsiang,Cohomology theory of topological transformation groups,Springer-Verlag,New York-Heidelbcrg,Ergebnisse der Mathematik und ihrer Grenzgcbietc,Band 85(1975).
[H1]Wolfgang Huck,Circle actions on 4-manifolds I,Manuscripts Math,87(1995),51-70
[H2]Dale Husemoller,Fiber Bundles(Third Edition),Springer-Verlag,New York,Inc.1994.
[H3]Herrera,Haydee,The signature of even 4-manifolds with S~1 actions,JP J.Geom.Topol.6,no.3(2006),237-243.
[K]Kawakubo,Katsuo,Global and local equivariant characteristic numbers of Gmanifolds,Math.Soc.Japan 32(1980),no.2,301-323.
[KS]Slawomir Kwasik and Reinhard Schultz,Fundamental groups of 4-manifolds with circle actions,Math.Proc.Camb.Phil.Soc,119(1996),645-655
[LL]R.Lee and T.-J.Li,Intersection forms of non-spin four manifolds,Math.Ann.319(2001),no.2,311-318.
[MB]J.W.Morgan and H.Bass,(eds):The Smith conjecture,Pure Appl.Math.112 Academic,Orlando(1984)
[SY]Osamu Saeki,Takahiro Yamamoto Singular fibers of stable maps and signatures of 4-manifolds,Geometry & Topology 10(2006) 359-399
[AH]M.Atiyah and F.Hirzebruch,The moment map and equivariant cohomology,Topology 23(1984),1-28.
[AP]C.Allday and V.Puppe,Cohornological Methods in Transformation Groups,Cambridge Studies in Advanced Mathematics,32(1993),Cambridge University Press.
[B]G.E.Bredon,Intrduction to compact transformation groups,Pure and Applied Mathematics,Vol.46(1972) Academic Press,New York-London.
[CL]Chen B,L(u|¨) Z,Equivariant cohomology and analytic descriptions of ring isomorphisms,Math Z,261(2009),891-908
[D]T.tom Dieck,Characteristic numbers of G-manifolds.I,Invent.Math.13(1971),213-224.
[F]Ronald Fintushcl,Classification of circle action on 4-manifolds,Transactions of the Amcrican Mathematical Socicty,242(1978),377-390
[F1]Ronald Fintushcl,Circle actions on simply connected 4-manifolds,Transactions of thc Amcrican Mathematical Society,230(1977),147-171
[G]Lconor Godinho,Equivariant cohomology of S~1-actions on 4-manifolds,Canad Math Bull,50(3)(2007),365-376
[GH]R.Goldin and T.S.Holm,The equivariant cohornology of Harniltonian Gspaces from residual S~1 actions,Math.Res.Lett.8(2001),67-77.
[GKM]M.Goresky,R.Kottwitz,and R.MacPherson Equivariant cohomology,Koszul duality,and the localization theorem,Invent.Math.131(1998),25-83.
[H]W.Y.Hsiang,Cohomology theory of topological transformation groups,Springer-Verlag,New York-Heidelbcrg,Ergebnisse der Mathematik und ihrer Grenzgcbietc,Band 85(1975).
[H1]Wolfgang Huck,Circle actions on 4-manifolds I,Manuscripts Math,87(1995),51-70
[H2]Dale Husemoller,Fiber Bundles(Third Edition),Springer-Verlag,New York,Inc.1994.
[H3]Herrera,Haydee,The signature of even 4-manifolds with S~1 actions,JP J.Geom.Topol.6,no.3(2006),237-243.
[K]Kawakubo,Katsuo,Global and local equivariant characteristic numbers of Gmanifolds,Math.Soc.Japan 32(1980),no.2,301-323.
[KS]Slawomir Kwasik and Reinhard Schultz,Fundamental groups of 4-manifolds with circle actions,Math.Proc.Camb.Phil.Soc,119(1996),645-655
[LL]R.Lee and T.-J.Li,Intersection forms of non-spin four manifolds,Math.Ann.319(2001),no.2,311-318.
[MB]J.W.Morgan and H.Bass,(eds):The Smith conjecture,Pure Appl.Math.112 Academic,Orlando(1984)
[SY]Osamu Saeki,Takahiro Yamamoto Singular fibers of stable maps and signatures of 4-manifolds,Geometry & Topology 10(2006) 359-399