不动点集为Dold流形不交并的对合
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摘要
设(M，T)是一个带有光滑对合T的光滑闭流形，T在M上的不动点集为F={x|T(x)=x，x∈M}．则F为M的闭子流形的不交并．
本文讨论了：
(1)当F=P(2m，2l+1)(?)P(2m，2n+1)时，其中n>l≥m，m≠1，3，(M，T)协边于零．
(2)当F=P(2~m，2~m)(?)P(2~m，2~m+1)，且m≥3时，完全决定了带有对合的流形(M，T)的等变协边分类．
Let(M，T)be a smooth closed manifold with a smooth involution Twhose fixed point set is F={x|T(x)=x，x∈M}，then F is the disjoint union ofsmooth closed submaniflod of M．
In this paper，we discuss：
(1) For F=P(2m，2l+1)(?)P(2m，2n+1)，n>l≥m，m≠1，3，them(M，T)isbounded．
(2) For F=P(2~m，2~m)(?)P(2~m，2~(m)+1)，m≥3，the all bordism classes of(M，T)are decided．
本文讨论了：
(1)当F=P(2m，2l+1)(?)P(2m，2n+1)时，其中n>l≥m，m≠1，3，(M，T)协边于零．
(2)当F=P(2~m，2~m)(?)P(2~m，2~m+1)，且m≥3时，完全决定了带有对合的流形(M，T)的等变协边分类．
Let(M，T)be a smooth closed manifold with a smooth involution Twhose fixed point set is F={x|T(x)=x，x∈M}，then F is the disjoint union ofsmooth closed submaniflod of M．
In this paper，we discuss：
(1) For F=P(2m，2l+1)(?)P(2m，2n+1)，n>l≥m，m≠1，3，them(M，T)isbounded．
(2) For F=P(2~m，2~m)(?)P(2~m，2~(m)+1)，m≥3，the all bordism classes of(M，T)are decided．
引文
[1]Stong R. M., Involutions fixing procjective spaces, Michigan Math. J., 1966, 13: 445-447.
[2]Royster D. C., Involutions fixing the disjoint union of two projective spaces, Indiana univ., Math. J., 1980, 29:267-276.
[3]吴振德，刘宗泽，不动点集为RP(2m)(?)RP(2n)的带有对合的流形，数学季刊，1986，1：19-41．
[4]Yue Q．Manifolds with constant codimensional involution，Science in China(Series A)，1992，35：837-841．
[5]吴振德，不动点集为常余维数的带有对合的流形，中国科学(A辑)，1989，12：1250-1256．
[6]吴振德，刘宗泽，群J~(6,k_1,…,k_6)的决定，数学学报，1994，3：296-300．
[7]Wang Y. Y., Wu Z. D. et al, Commuting involutions with fixed point set of variable codimen-sion, Acta Math. Sinica, English Series, 2001, 17: 425-430.
[8]Wang Y. Y., Wu Z. D. et al, Commuting involutions with fixed point set of constant codi-mension, Acta Math. Sinica, English Series, 1999, 15: 181-186.
[9]Yang H. J., Wu Z. D. et al, Involution number sequence and dimensions of the components in fixed point set of an involution Ⅱ, Science in China (Series A), 1992, 35: 819-825.
[10]Yang H. J., Wu Z. D. et al, Classifying involution on RP(2k) up to equivariant cobordism, Chin. Ann. Math. 1993, 14B: 55-58.
[11]Wu Z. D. and Liu Z. Z., Classifying of involution on Dold manifold P(1, 2l), Acta Math. Sinica, English Series, 1997, 1: 9-20.
[12]Meng Z. J. and Wang Y. Y., Bordism classes of involutions RP(2n)、CP(2n) and HP(2n), Chinese Quarterly J. Math., 1999, 14: 33-37.
[13]刘宗泽，吴振德，带有对合流形的协边类Ⅱ，数学学报，1993，5：577-582．
[14]吴振德，不动点集为Dold流形P(2m，2n)的带有对合的流形，数学学报，1988，1：72-82．
[15]刘秀贵，不动点集是Dold流形P(2m+1，2n+1)的对合，数学年刊， 2002，6：779-788．
[16]刘秀贵，不动点集是Dold流形P(2m，2n+1)的对合，四川大学学报，2003，5：798-797．
[17]Conner P. E., Differentiable Periodic Maps(2nd ed), Lecture Notes in Math. 738, Spring Berlin and New York, 1979.
[18]Kosniowski C. and Stong R. E., Involutions and characteristic numbers, Topology, 1978, 17: 309-330.
[19]Fujii M. and Yasui T., KO-cohomologies of Dold manifold, Math. J. Okayama Univ., 1973, 16: 55-84.
[20]Stong R. E., Vector bundles over Dold manifolds, Fundamenta Mathematicae, 2001, 169: 85-95.
[2]Royster D. C., Involutions fixing the disjoint union of two projective spaces, Indiana univ., Math. J., 1980, 29:267-276.
[3]吴振德，刘宗泽，不动点集为RP(2m)(?)RP(2n)的带有对合的流形，数学季刊，1986，1：19-41．
[4]Yue Q．Manifolds with constant codimensional involution，Science in China(Series A)，1992，35：837-841．
[5]吴振德，不动点集为常余维数的带有对合的流形，中国科学(A辑)，1989，12：1250-1256．
[6]吴振德，刘宗泽，群J~(6,k_1,…,k_6)的决定，数学学报，1994，3：296-300．
[7]Wang Y. Y., Wu Z. D. et al, Commuting involutions with fixed point set of variable codimen-sion, Acta Math. Sinica, English Series, 2001, 17: 425-430.
[8]Wang Y. Y., Wu Z. D. et al, Commuting involutions with fixed point set of constant codi-mension, Acta Math. Sinica, English Series, 1999, 15: 181-186.
[9]Yang H. J., Wu Z. D. et al, Involution number sequence and dimensions of the components in fixed point set of an involution Ⅱ, Science in China (Series A), 1992, 35: 819-825.
[10]Yang H. J., Wu Z. D. et al, Classifying involution on RP(2k) up to equivariant cobordism, Chin. Ann. Math. 1993, 14B: 55-58.
[11]Wu Z. D. and Liu Z. Z., Classifying of involution on Dold manifold P(1, 2l), Acta Math. Sinica, English Series, 1997, 1: 9-20.
[12]Meng Z. J. and Wang Y. Y., Bordism classes of involutions RP(2n)、CP(2n) and HP(2n), Chinese Quarterly J. Math., 1999, 14: 33-37.
[13]刘宗泽，吴振德，带有对合流形的协边类Ⅱ，数学学报，1993，5：577-582．
[14]吴振德，不动点集为Dold流形P(2m，2n)的带有对合的流形，数学学报，1988，1：72-82．
[15]刘秀贵，不动点集是Dold流形P(2m+1，2n+1)的对合，数学年刊， 2002，6：779-788．
[16]刘秀贵，不动点集是Dold流形P(2m，2n+1)的对合，四川大学学报，2003，5：798-797．
[17]Conner P. E., Differentiable Periodic Maps(2nd ed), Lecture Notes in Math. 738, Spring Berlin and New York, 1979.
[18]Kosniowski C. and Stong R. E., Involutions and characteristic numbers, Topology, 1978, 17: 309-330.
[19]Fujii M. and Yasui T., KO-cohomologies of Dold manifold, Math. J. Okayama Univ., 1973, 16: 55-84.
[20]Stong R. E., Vector bundles over Dold manifolds, Fundamenta Mathematicae, 2001, 169: 85-95.