射影空间乘积上向量丛的示性类与对合的协边分类
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摘要
本论文共分两章。
第一章,利用Steenrod上同调运算及吴公式决定了复射影空间CP(j)乘四元数射影空间HP(k)上的向量丛的全Stiefel-Whitney类。
第二章,作为第一章结论的应用,我们讨论了不动点集为CP(2m+1)×HP(k)的对合的协边分类,并证明每一个以CP(2m+1)×HP(k)为不动点集的对合均协边。
This thesis consists of two chapters.
In the first chapter,we determine the total Stiefel-Whitney classes of vector bundles over the product of the complex projective space CP(j)with the quaternionic projective space HP(k).
In the second chapter,as an application of the conclusion of the first chapter,we discuss the involutions fixing CP(2m+1)×HP(k)and prove that every involution fixing CP(2m+1)×HP(k)bounds.
第一章,利用Steenrod上同调运算及吴公式决定了复射影空间CP(j)乘四元数射影空间HP(k)上的向量丛的全Stiefel-Whitney类。
第二章,作为第一章结论的应用,我们讨论了不动点集为CP(2m+1)×HP(k)的对合的协边分类,并证明每一个以CP(2m+1)×HP(k)为不动点集的对合均协边。
This thesis consists of two chapters.
In the first chapter,we determine the total Stiefel-Whitney classes of vector bundles over the product of the complex projective space CP(j)with the quaternionic projective space HP(k).
In the second chapter,as an application of the conclusion of the first chapter,we discuss the involutions fixing CP(2m+1)×HP(k)and prove that every involution fixing CP(2m+1)×HP(k)bounds.
引文
[1]D.C.Royster,Involutions fixing the disjoint union of projective spaces,Indiana Univ.Math.J.,1980,29:267-276
[2]R.E.Stong,Involutions fixing projective spaces,Michigan Math.J.,1966,13:445-447.
[3]D.Hou and B.Torrance,Involutions fixing the disjoint union of even projective spaces,Acta.Math.Sinica,New Series,1996;12:162-166:
[4]Z.Lu,Involutions fixing RP~(odd)∪P(h,i)I,Trans.Amer.Math.Soc.,2002,354:4539-4570.
[5]R.E.Stong,Vector Bundles over Dold Manifolds,Fundsmenta Mathematicae,2001,169:85-95.
[6]Wu Zhende,Liu Zongze and Bai Ruipu,The cobordism classification of involution on Dold manifold p(1,21),Acta.Mathematica,Sinica,1997,13(1):9-20.
[7]R.E.Stong,Involutions fixing RP~(even)×RP~(even),University of Virginia,Manuscript.
[8]K.H.Weiss,Involutions with point set RP~(2m)×RP~(2n),Dissertation,University of Virginia,1997.
[9]N.Saiers,Involutions fixing products of projective spaces,Dissertation,University of Virginia,1997.
[10]李日成,具有常余维数不动点集的(Z_2)~k作用,博士学位论文,河北师范大学,2005.
[11]Jingyan Li and Yangying Wang,Characteristic classes of vector bundles over RP(h)×HP(k),Topology Appl.,2007,154:1778-1793.
[12]P.E.Conner,Differentiable periodic maps(2nd ed.),Lecture Notes in Math.738,Springer Verlag Berlin and New York,1979.
[13]J.W.Milnor and J.D.Stasheff,Characteristic classes,Princeton Press and Univ.of Tokyo Press,Princeton,New Jersey,1974.
[14]吴振德,刘宗泽,不动点集为RP(2m)∩RP(2n)的带对合的流形,数学学报,1986,1,19-41.
[15]吴振德,不动点集为Dold流形P(2m,2n)的带对合的流形,数学学报,1988,1:72-82.
[16]Z.Lu,Involutions fixing RP~(odd)∪P(h,i)I,Trans.Amer.Math.Soc.,2004,356:1291-1314.
[17]陈德华,有限群在微分流形上作用的研究,博士学位论文,河北师范大学,2006.
[18]R.E.Stong,Involutions fixing products of circles,Proc.Amer.Math.Soc.,1993,119:1005-1008.
[19]K.Ma,Z.Z.Liu et al,Involutions fixing(?)(RP(1))~(mi)×(UP(1))~(8i),Northeast Math.J.,1999,1:46-52
[20]Y.H.Ding,Z.Z.Liu et al,Involutions fixing products of projective spaces,Northeast Math.J.,1999,1:97-102
[21]R.E.Stong,Involutions fixing RP~(odd)×RP(odd),University of Virginia,Manuscript.
[22]R.E.Stong,Involutions fixing RP~(2m+1)×RP~(2m),University of Virginia,Manuscript.
[23]P.E.Conner,The bordism class of a bundie space,Michigan Math.J.,1967,14:289-303.
[24]Czes Kosniowski and R.E.Stong,Involutions and characteristic numbers,Topology,1978,17:309-330.
[25]Z.Lu,(Z_2)~k-actions with w(F)=1,Proc.Amer.Math.Soc.2005,133:3721-3733
[26]Czes Kosniowsld and R.E.Stong,(Z_2)~k-actions and characteristic numbers,Indiana Univ.Math.J.,1979,28:725-743.
[27]R.E.Stong,Equivariant bordism and(Z_2)~k-actions,Duke Math.J.1970,37:779-785.
[28]Andreas Gathmann,Algebraic Geometry,Notes for a class,taught at the University of Kaiserslautern 2002/2003.
[2]R.E.Stong,Involutions fixing projective spaces,Michigan Math.J.,1966,13:445-447.
[3]D.Hou and B.Torrance,Involutions fixing the disjoint union of even projective spaces,Acta.Math.Sinica,New Series,1996;12:162-166:
[4]Z.Lu,Involutions fixing RP~(odd)∪P(h,i)I,Trans.Amer.Math.Soc.,2002,354:4539-4570.
[5]R.E.Stong,Vector Bundles over Dold Manifolds,Fundsmenta Mathematicae,2001,169:85-95.
[6]Wu Zhende,Liu Zongze and Bai Ruipu,The cobordism classification of involution on Dold manifold p(1,21),Acta.Mathematica,Sinica,1997,13(1):9-20.
[7]R.E.Stong,Involutions fixing RP~(even)×RP~(even),University of Virginia,Manuscript.
[8]K.H.Weiss,Involutions with point set RP~(2m)×RP~(2n),Dissertation,University of Virginia,1997.
[9]N.Saiers,Involutions fixing products of projective spaces,Dissertation,University of Virginia,1997.
[10]李日成,具有常余维数不动点集的(Z_2)~k作用,博士学位论文,河北师范大学,2005.
[11]Jingyan Li and Yangying Wang,Characteristic classes of vector bundles over RP(h)×HP(k),Topology Appl.,2007,154:1778-1793.
[12]P.E.Conner,Differentiable periodic maps(2nd ed.),Lecture Notes in Math.738,Springer Verlag Berlin and New York,1979.
[13]J.W.Milnor and J.D.Stasheff,Characteristic classes,Princeton Press and Univ.of Tokyo Press,Princeton,New Jersey,1974.
[14]吴振德,刘宗泽,不动点集为RP(2m)∩RP(2n)的带对合的流形,数学学报,1986,1,19-41.
[15]吴振德,不动点集为Dold流形P(2m,2n)的带对合的流形,数学学报,1988,1:72-82.
[16]Z.Lu,Involutions fixing RP~(odd)∪P(h,i)I,Trans.Amer.Math.Soc.,2004,356:1291-1314.
[17]陈德华,有限群在微分流形上作用的研究,博士学位论文,河北师范大学,2006.
[18]R.E.Stong,Involutions fixing products of circles,Proc.Amer.Math.Soc.,1993,119:1005-1008.
[19]K.Ma,Z.Z.Liu et al,Involutions fixing(?)(RP(1))~(mi)×(UP(1))~(8i),Northeast Math.J.,1999,1:46-52
[20]Y.H.Ding,Z.Z.Liu et al,Involutions fixing products of projective spaces,Northeast Math.J.,1999,1:97-102
[21]R.E.Stong,Involutions fixing RP~(odd)×RP(odd),University of Virginia,Manuscript.
[22]R.E.Stong,Involutions fixing RP~(2m+1)×RP~(2m),University of Virginia,Manuscript.
[23]P.E.Conner,The bordism class of a bundie space,Michigan Math.J.,1967,14:289-303.
[24]Czes Kosniowski and R.E.Stong,Involutions and characteristic numbers,Topology,1978,17:309-330.
[25]Z.Lu,(Z_2)~k-actions with w(F)=1,Proc.Amer.Math.Soc.2005,133:3721-3733
[26]Czes Kosniowsld and R.E.Stong,(Z_2)~k-actions and characteristic numbers,Indiana Univ.Math.J.,1979,28:725-743.
[27]R.E.Stong,Equivariant bordism and(Z_2)~k-actions,Duke Math.J.1970,37:779-785.
[28]Andreas Gathmann,Algebraic Geometry,Notes for a class,taught at the University of Kaiserslautern 2002/2003.