环面拓扑中闭流形及矩角复形性质的研究
|
摘要
本文共分为三个部分：
第一章,讨论带有(z2)κ作用且不动点集为常余维数2k+2v+1的一类特殊闭流形的上协边分类问题.设φ:(Z2)k×Mn→Mn是群(Z2)κ在给定的n维光滑闭流形Mn上的光滑作用,其中(Z2)κ表示由κ个可交换对合生成的群,即(Z2)k={T1,T2,...,Tk｜Ti2=1,TiTj=TjTi}.此时(Z2)κ作用的不动点集F是有限个Mn的闭子流形的不交并.如果F的每个分支均为n-γ维,则称F具有常余维数γ.记Jn,κγ是具有以下性质的n维光滑闭流形Mn所在的上协边类构成的集合：Mn上具有光滑的(Z2)κ作用,且作用的不动点集为常余维数r.则Jn,κγ为未定向上协边群MO。的子群,J*,kr=∑n≥rJn,kr为未定向上协边环MO*=∑n≥0MOn的理想.本文通过数学归纳法构造MO*的生成元,讨论了γ=2κ+2v+1时Jn,κγ的代数结构,从而决定了这类特殊流形的上协边分类.
第二章,讨论不动点集为偶数维实射影空间与Dold流形不交并的对合的等变协边分类.设(M,T)是带有光滑对合T的光滑闭流形,我们证明了当T在Mn上的不动点集F=RP(2m)(?)P(2m,2n+1)时,(M,T)等变协边于(P(2m,RP(2n+2)),T0)或者(RP(2m)×RP(2m),twist).
第三章,我们主要讨论带有环面作用的拓扑空间的性质.环面作用的轨道空间常常具有丰富的组合结构(例如凸多胞形),因而可以通过轨道空间的组合性质来研究全空间的拓扑性质.另一方面,环面作用的等变拓扑有时也有助于从拓扑的角度解释和证明一些精巧的组合结果.本章我们首先通过考虑轨道空间的组合性质计算了其中一类特殊空间等变同胚类的个数,即△n1×△2×P(m)上小覆盖的等变同胚类的个数,其中△ni表示ni维的单形,P(m)表示m边形,n1≥2,n2≥1,m≥3.其次,我们讨论了矩角复形的相关性质,利用单纯复形K的f向量计算了K的矩角复形轨道构型空间的欧拉示性数.
This dissertation consists of three chapters.
In the first chapter, we discuss the classification of manifolds admitting (Z_2)~k-actions withfixed point set of constant codimension2k+2v+1up to cobordism. Let φ:(Z_2)~k×Mn→Mndenote a smooth action of the group (Z_2)~k={T_1, T_2,..., T_k|T_2~i=1, T_iT_j=T_jT_i} on a closedmanifold M~n. The fixed point set F of the action is the disjoin union of closed submanifolds ofMn, which are finite in number. If each component of F is of constant dimension n r, we saythat F is of constant codimension r. Let J_(n,k)~rbe the set of unoriented cobordism class of Mnthat admits a (Z_2)~k-action with fixed point set of constant codimension r. J_(n,k)~ris a subgroup ofunoriented cobordism group MOnand J_(n,k)~r=(?)J_(n,k)~r is an ideal of the unoriented cobor-dism ring MO_+=(?). In this paper, we use the mathematical induction to constructgenerators of MO and determine the algebraic structure of J_(n,k)~r, where r=2~k+2v+1. So,we complete the classification of such manifolds up to cobordism.
In the second chapter, we discuss the classification up to equivariant bordism of smoothmanifolds with involution whose the fixed point set is the disjoint union of an real projectivespace of even dimension and a Dold manifold. Let (M, T) be a smooth closed manifold witha smooth involution T, when the fixed point set of T on M is F=RP (2m) P (2m,2n+1), we prove that (M~(2m+4n+k+2), T) is equivariantly bordant to (P (2m, RP (2n+2)), T_0) or(RP (2m)×RP (2m), twist).
In the third chapter, we mainly discuss the properties of topological space with torus ac-tions. In some cases, the orbit space of torus action carries a rich combinatorial structure(suchas convex polytope). So we can study the topology of the toric space through the combinatoricsof the orbit space. On the other hand, the equivariant topology of a torus action sometimes helpsto interpret and prove the most subtle combinatorial results topologically. In this chapter, firstly,by using the combinatorics of the orbit space, we calculate the number of classes of a kind oftoric space up to equivariant homeomorphism, i.e. small covers over△~(n1)×△~(n2)×P(m), where△~(ni)is the simplex of dimension n_i and P (m) is an m-polygon, n_1≥2, n_2≥1, m≥3.Secondly, we calculate the Euler characteristic for orbit configuration space of moment-anglecomplex of simplicial complex K in terms of the f-vector of K.
第一章,讨论带有(z2)κ作用且不动点集为常余维数2k+2v+1的一类特殊闭流形的上协边分类问题.设φ:(Z2)k×Mn→Mn是群(Z2)κ在给定的n维光滑闭流形Mn上的光滑作用,其中(Z2)κ表示由κ个可交换对合生成的群,即(Z2)k={T1,T2,...,Tk｜Ti2=1,TiTj=TjTi}.此时(Z2)κ作用的不动点集F是有限个Mn的闭子流形的不交并.如果F的每个分支均为n-γ维,则称F具有常余维数γ.记Jn,κγ是具有以下性质的n维光滑闭流形Mn所在的上协边类构成的集合：Mn上具有光滑的(Z2)κ作用,且作用的不动点集为常余维数r.则Jn,κγ为未定向上协边群MO。的子群,J*,kr=∑n≥rJn,kr为未定向上协边环MO*=∑n≥0MOn的理想.本文通过数学归纳法构造MO*的生成元,讨论了γ=2κ+2v+1时Jn,κγ的代数结构,从而决定了这类特殊流形的上协边分类.
第二章,讨论不动点集为偶数维实射影空间与Dold流形不交并的对合的等变协边分类.设(M,T)是带有光滑对合T的光滑闭流形,我们证明了当T在Mn上的不动点集F=RP(2m)(?)P(2m,2n+1)时,(M,T)等变协边于(P(2m,RP(2n+2)),T0)或者(RP(2m)×RP(2m),twist).
第三章,我们主要讨论带有环面作用的拓扑空间的性质.环面作用的轨道空间常常具有丰富的组合结构(例如凸多胞形),因而可以通过轨道空间的组合性质来研究全空间的拓扑性质.另一方面,环面作用的等变拓扑有时也有助于从拓扑的角度解释和证明一些精巧的组合结果.本章我们首先通过考虑轨道空间的组合性质计算了其中一类特殊空间等变同胚类的个数,即△n1×△2×P(m)上小覆盖的等变同胚类的个数,其中△ni表示ni维的单形,P(m)表示m边形,n1≥2,n2≥1,m≥3.其次,我们讨论了矩角复形的相关性质,利用单纯复形K的f向量计算了K的矩角复形轨道构型空间的欧拉示性数.
This dissertation consists of three chapters.
In the first chapter, we discuss the classification of manifolds admitting (Z_2)~k-actions withfixed point set of constant codimension2k+2v+1up to cobordism. Let φ:(Z_2)~k×Mn→Mndenote a smooth action of the group (Z_2)~k={T_1, T_2,..., T_k|T_2~i=1, T_iT_j=T_jT_i} on a closedmanifold M~n. The fixed point set F of the action is the disjoin union of closed submanifolds ofMn, which are finite in number. If each component of F is of constant dimension n r, we saythat F is of constant codimension r. Let J_(n,k)~rbe the set of unoriented cobordism class of Mnthat admits a (Z_2)~k-action with fixed point set of constant codimension r. J_(n,k)~ris a subgroup ofunoriented cobordism group MOnand J_(n,k)~r=(?)J_(n,k)~r is an ideal of the unoriented cobor-dism ring MO_+=(?). In this paper, we use the mathematical induction to constructgenerators of MO and determine the algebraic structure of J_(n,k)~r, where r=2~k+2v+1. So,we complete the classification of such manifolds up to cobordism.
In the second chapter, we discuss the classification up to equivariant bordism of smoothmanifolds with involution whose the fixed point set is the disjoint union of an real projectivespace of even dimension and a Dold manifold. Let (M, T) be a smooth closed manifold witha smooth involution T, when the fixed point set of T on M is F=RP (2m) P (2m,2n+1), we prove that (M~(2m+4n+k+2), T) is equivariantly bordant to (P (2m, RP (2n+2)), T_0) or(RP (2m)×RP (2m), twist).
In the third chapter, we mainly discuss the properties of topological space with torus ac-tions. In some cases, the orbit space of torus action carries a rich combinatorial structure(suchas convex polytope). So we can study the topology of the toric space through the combinatoricsof the orbit space. On the other hand, the equivariant topology of a torus action sometimes helpsto interpret and prove the most subtle combinatorial results topologically. In this chapter, firstly,by using the combinatorics of the orbit space, we calculate the number of classes of a kind oftoric space up to equivariant homeomorphism, i.e. small covers over△~(n1)×△~(n2)×P(m), where△~(ni)is the simplex of dimension n_i and P (m) is an m-polygon, n_1≥2, n_2≥1, m≥3.Secondly, we calculate the Euler characteristic for orbit configuration space of moment-anglecomplex of simplicial complex K in terms of the f-vector of K.
引文
[1] Eilenberg S, Steenrod N. Foundations of algebraic topology[M]. Princeton University Press,Princeton Press,1952.
[2] Whitney H. Diffferentiable manifolds[J]. Ann.of Math.,1936,37:645-680.
[3] Milnor J. On manifolds homeomorphic to the7-sphere[J]. Ann. of Math.,1956,64:399-405.
[4] Kervaire M. A manifold which does not admit any differentiable sructure[J]. Comment. Math.Helv.,1960,34:257-270.
[5] Davis M, Januszkiewicz T. Convex polytope, Coxeter orbifolds and torus actions[J]. Duke Math.J.1991,62:417-451.
[6] Thom R. Quelques proprie′te′s globales des varie′te′s diffe′rentiables[J]. Comment. Math. Helv.,1954,28:17-86.
[7] Rohlin V A. A three-dimensional manifold is the boundary of a four-dimensional one[J]. Dokl.Akad. Nauk USSR,1951,81:355-357.
[8] Dold A. Erzeugende der Thomschen Algebra N[J]. Math. Zeitschr.,1956,65:25-35.
[9] Conner P E, Floyd E E. Fibering within a cobordsim class[J]. Michigan Math. J.1965,12:33-47.
[10] Stong R E. On fibering of cobordism classes[J]. Trans. Amer. Math. Soc,1973,178:431-447.
[11] Capobianco F L. Cobordism classes represented by fibering with fiber RP(2k+1)[J]. Michigan.Math. J.,1977,24:185-192.
[12] Capobianco F L. Manifold with involution whose fixed set has codimension four[J]. Proc. Amer.Math. Soc.1976,61:157-162.
[13] K. Iwata, Involutions with fixed point set of constant codimension[J], Proc. Amer. Math. Soc.1981,83:829-832.
[14] Wada T. Manifolds with involution whose fixed point set has codimension six[J], Bull. YamagataUniv.1981,10:193-205.
[15].:8~{‘ê ‘kéü6/[J].￥I‰(A6)1989,12:1250-1256.
[16] Kikuchi S. Involution with fixed point set of codimension eight[J], Sci. Rep. Hirosaki Univ.1986,33:829-832.
[17] Pergher P L Q.(Z2)k-actions with fixed point set of constant codimension[J]. Topology Appl.1992,46:55-64.
[18] Shaker R J. Constant codimension fixed point sets of commuting involutions[J]. Proc. Amer.Math. Soc.,1994,121:275-281.
[19] Shaker R J. Dold manifolds with (Z2)k-Action[J]. Proc. Amer. Math. Soc.,1995,123:955-958.
[20] Wang Y, Wu Z, Ding Y. Commuting involutions with fixed point set of constant codimension[J].Acta Math. Sinica, English Series,1999,15:181-186.
[21] Wang Y, Wu Z, Ma K.(Z2)k-actions with fixed point set of constant codimension2k+1[J].Proc. Amer. Math. Soc.,1999,128:1515-1521.
[22] Liu Z, Wu Z.(Z2)k-actions with fixed point set of constant codimension2k+2[J]. Acta. Math.Sinica, English Series,2005,21:409-412.
[23] Stong R E, Involutions fixing the projective spaces[J]. Michigan Math. J.,1966,13:445-447.
[24].:8Dold6/P (2m,2n)‘kéü6/[J],ê,1988,1:72-82.
[25] Lu¨ Z. Involutions fixing RPoddP (h, i)(I)[J]. Trans. Amer. Math. Soc.,2002,354:4539-4570.
[26] Lu¨ Z. Involutions fixing RPoddP (h, i)(II)[J]. Trans. Amer. Math. Soc.,2004,356:1291-1314.
[27] Kosniowski C, Stong R E. Involutions and characteristic numbers[J]. Topology,1973,17(4):309-330.
[28] Garrison A, Scott R. Small covers of the dodecahedron and the120-cell[J]. Proc. Amer. Math.Soc.,2003,131(3):963–971.
[29] Cai M, Chen X, Lu¨ Z. Small covers over prisms, Topology and its Applications[J].2007,154:2228-2234.
[30] Lu¨ Z, Masuda M. Equivariant classification of2-torus manifolds[J]. Colloq. Math.,2009,115:171–188.
[31] Choi S. The number of small covers over cubes[J]. Algebraic and Geometric Topology,2008,8:2391-2399.
[32] Choi S. The number of orientable small covers over cubes[J]. Proc. Japan Acad., Ser.2010,86(A):97-100.
[33],=.c ú>/t CX[J].ê cr,2010,31A(5):541–548.
[34] Lu¨ Z, Yu L. On3-manifols with locally standard (Z2)3-actions[J]. arXiv:0807.3062.
[35] Lu¨ Z, Tan Q. A differential operator and Tom Dieck-Kosniowski-Stong localization theorem[J].arXiv:1008.2166v3
[36] Buchstaber V M, Panov T E. Torus Action and their applications in topology and combina-torics[M]. University Lecture Series, vol.24. Amer. Math. Soc., Providence. RI.2002.
[37] Farber E. Invitation to topological robotics. Zu¨rich Lectures in Advanced Mathematics[J]. Eu-ropean Mathematical Socitey (EMS), Zu¨rich,2008. ISBN:978-3-03719-054-8.
[38] Ghrist R. Configuration spaces and braid groups on graphs in robotics. Knots, braids, and map-ping class groups[J], papers dediceted to Joan S.Birman (New York,1998),29-40, AMS/IPStud. Adv. Math.,24, Amer. Math. Soc., Providence, RI,2001.
[39] Chen J, Lu¨ Z, Wu J. Orbit configuration spaces of small covers and quasi-toric manifolds[J],arXiv:1111.6699v1[math.AT]29Nov2011.
[40]oF¤,,. J2k+2k1n,k([J].ê,2006,49:249-254.
[41]oéé. k~{‘ê2k+2v:8(Z2)k^[J].à “‰a,2007:1-42.
[42] Conner P E. Differentiable periodic maps[M].2nd ed. Springer Verlag, Berlin and New York,1979.
[43] Brown R L W. Immersions and embeddings up to cobordism[J]. Canad. J. Math,1971,23:1102-1115.
[44] Steenrod N E, Epstein D B A. Cohomology operations[M]. Princetone University Press,1962.
[45] Yue Q. Manifolds with constant codimensional involution[J]. Science in China Series A.1992,35:837-841.
[46] Kosniowski C, Stong R E.(Z2)k-actions and characteristic numbers[J]. Indiana Univ. Math. J,1973,28:725-743.
[47] Wang Y. Characteristic numbers and cobordism classes of fiberings with fiber RP (2k)[J]. Proc.Amer. Math. Soc.,2001,129:899-911.
[48] Stong R E. Vector bundles over Dold manifolds[J]. Fund. Math.2001,169:85-95.
[49] Fujii M, Yasui T. KO-cohomologies of Dold manifold[J]. Math. J Okayama Univ.,1973,16:55-84.
[50] Milnor J W, Stasheff J D. Characteristic class[M]. Princeton Univ. Press and Univ. of TokyoPress, Princeton New Jersey,1974.
[51] Torrence B F. Bordism classes of vector bundles over real projective spaces[J]. Proc. Amer.Math. Soc,1993,118:963-969.
[52] Chen D, Wang Y. Involution fixing RP (4) P (4,2n1)[J]. Advances in Mathematics,2007,36:89-93.
[53] Alperin J. L, Bell R. B., Groups and representations[M]. Graduate Texts in Mathematics,Springer-Verlag,1995.
[54] Davis M. Groups generated by reflections and aspherical manifolds not covered by Euclideanspace[J]. Ann. Math.,1983,117:293-324.
[55] Ziegler G M. Lectures on polytopes[M]. Graduate Texts in Math., Springer-Verlag, Berlin,1991.
[56] Stong R E. Equivariant bodism and (Z2)k-actions[J]. Duke Math.J.,1970,37:779-785.
[57] Lu¨ Z.2-torus manifold, cobodism and small covers[J]. Pacific J. Math.,2009,241(2):285-308.
[58] Husemoller D. Fiber bundles[M]. Springer-Verlag,1975.
[59] Lu¨ Z, Liu X. Involutions fixing the disjoint union of3-real projective space with Dold mani-fold[J]. Kodai. Math.,2000,23:187-213.
[60] Royster D C. Involutions fixing the disjiont union of two projective spaces[J]. Indiana univ.,Math. J.,1980,29:267-276.
[61],4mL.:8RP (2m) RP (2n)‘éü6/[J].ê Gr,1986,1:19-41.
[62] Hou D, Torrence B F. Involutions fixing the disjoint union of odd-dimensional projectivespaces[J]. Canad. Math. Bull.,1994,37:66-74.
[63] Pergher P L Q. Involutions fixing an arbitrary product of spheres and a point[J]. ManuscriptaMath.,1996,89:471-474.
[64] Stong R E. Involutions fixing products of circles[J]. Proc. Amer. Math. Soc,1993,119:1005-1008.
[65] Bredon G E. Introduction to compact transformation groups[M]. Pure and Applied Mathematics,46Academic Press, New York-London,1972.
[66] Allday C, Puppe V. Cohomological methods in transformation group[M]. Cambridge Studies inAdvanced Mathematics,32, Cambridge University Press,1993.
[2] Whitney H. Diffferentiable manifolds[J]. Ann.of Math.,1936,37:645-680.
[3] Milnor J. On manifolds homeomorphic to the7-sphere[J]. Ann. of Math.,1956,64:399-405.
[4] Kervaire M. A manifold which does not admit any differentiable sructure[J]. Comment. Math.Helv.,1960,34:257-270.
[5] Davis M, Januszkiewicz T. Convex polytope, Coxeter orbifolds and torus actions[J]. Duke Math.J.1991,62:417-451.
[6] Thom R. Quelques proprie′te′s globales des varie′te′s diffe′rentiables[J]. Comment. Math. Helv.,1954,28:17-86.
[7] Rohlin V A. A three-dimensional manifold is the boundary of a four-dimensional one[J]. Dokl.Akad. Nauk USSR,1951,81:355-357.
[8] Dold A. Erzeugende der Thomschen Algebra N[J]. Math. Zeitschr.,1956,65:25-35.
[9] Conner P E, Floyd E E. Fibering within a cobordsim class[J]. Michigan Math. J.1965,12:33-47.
[10] Stong R E. On fibering of cobordism classes[J]. Trans. Amer. Math. Soc,1973,178:431-447.
[11] Capobianco F L. Cobordism classes represented by fibering with fiber RP(2k+1)[J]. Michigan.Math. J.,1977,24:185-192.
[12] Capobianco F L. Manifold with involution whose fixed set has codimension four[J]. Proc. Amer.Math. Soc.1976,61:157-162.
[13] K. Iwata, Involutions with fixed point set of constant codimension[J], Proc. Amer. Math. Soc.1981,83:829-832.
[14] Wada T. Manifolds with involution whose fixed point set has codimension six[J], Bull. YamagataUniv.1981,10:193-205.
[15].:8~{‘ê ‘kéü6/[J].￥I‰(A6)1989,12:1250-1256.
[16] Kikuchi S. Involution with fixed point set of codimension eight[J], Sci. Rep. Hirosaki Univ.1986,33:829-832.
[17] Pergher P L Q.(Z2)k-actions with fixed point set of constant codimension[J]. Topology Appl.1992,46:55-64.
[18] Shaker R J. Constant codimension fixed point sets of commuting involutions[J]. Proc. Amer.Math. Soc.,1994,121:275-281.
[19] Shaker R J. Dold manifolds with (Z2)k-Action[J]. Proc. Amer. Math. Soc.,1995,123:955-958.
[20] Wang Y, Wu Z, Ding Y. Commuting involutions with fixed point set of constant codimension[J].Acta Math. Sinica, English Series,1999,15:181-186.
[21] Wang Y, Wu Z, Ma K.(Z2)k-actions with fixed point set of constant codimension2k+1[J].Proc. Amer. Math. Soc.,1999,128:1515-1521.
[22] Liu Z, Wu Z.(Z2)k-actions with fixed point set of constant codimension2k+2[J]. Acta. Math.Sinica, English Series,2005,21:409-412.
[23] Stong R E, Involutions fixing the projective spaces[J]. Michigan Math. J.,1966,13:445-447.
[24].:8Dold6/P (2m,2n)‘kéü6/[J],ê,1988,1:72-82.
[25] Lu¨ Z. Involutions fixing RPoddP (h, i)(I)[J]. Trans. Amer. Math. Soc.,2002,354:4539-4570.
[26] Lu¨ Z. Involutions fixing RPoddP (h, i)(II)[J]. Trans. Amer. Math. Soc.,2004,356:1291-1314.
[27] Kosniowski C, Stong R E. Involutions and characteristic numbers[J]. Topology,1973,17(4):309-330.
[28] Garrison A, Scott R. Small covers of the dodecahedron and the120-cell[J]. Proc. Amer. Math.Soc.,2003,131(3):963–971.
[29] Cai M, Chen X, Lu¨ Z. Small covers over prisms, Topology and its Applications[J].2007,154:2228-2234.
[30] Lu¨ Z, Masuda M. Equivariant classification of2-torus manifolds[J]. Colloq. Math.,2009,115:171–188.
[31] Choi S. The number of small covers over cubes[J]. Algebraic and Geometric Topology,2008,8:2391-2399.
[32] Choi S. The number of orientable small covers over cubes[J]. Proc. Japan Acad., Ser.2010,86(A):97-100.
[33],=.c ú>/t CX[J].ê cr,2010,31A(5):541–548.
[34] Lu¨ Z, Yu L. On3-manifols with locally standard (Z2)3-actions[J]. arXiv:0807.3062.
[35] Lu¨ Z, Tan Q. A differential operator and Tom Dieck-Kosniowski-Stong localization theorem[J].arXiv:1008.2166v3
[36] Buchstaber V M, Panov T E. Torus Action and their applications in topology and combina-torics[M]. University Lecture Series, vol.24. Amer. Math. Soc., Providence. RI.2002.
[37] Farber E. Invitation to topological robotics. Zu¨rich Lectures in Advanced Mathematics[J]. Eu-ropean Mathematical Socitey (EMS), Zu¨rich,2008. ISBN:978-3-03719-054-8.
[38] Ghrist R. Configuration spaces and braid groups on graphs in robotics. Knots, braids, and map-ping class groups[J], papers dediceted to Joan S.Birman (New York,1998),29-40, AMS/IPStud. Adv. Math.,24, Amer. Math. Soc., Providence, RI,2001.
[39] Chen J, Lu¨ Z, Wu J. Orbit configuration spaces of small covers and quasi-toric manifolds[J],arXiv:1111.6699v1[math.AT]29Nov2011.
[40]oF¤,,. J2k+2k1n,k([J].ê,2006,49:249-254.
[41]oéé. k~{‘ê2k+2v:8(Z2)k^[J].à “‰a,2007:1-42.
[42] Conner P E. Differentiable periodic maps[M].2nd ed. Springer Verlag, Berlin and New York,1979.
[43] Brown R L W. Immersions and embeddings up to cobordism[J]. Canad. J. Math,1971,23:1102-1115.
[44] Steenrod N E, Epstein D B A. Cohomology operations[M]. Princetone University Press,1962.
[45] Yue Q. Manifolds with constant codimensional involution[J]. Science in China Series A.1992,35:837-841.
[46] Kosniowski C, Stong R E.(Z2)k-actions and characteristic numbers[J]. Indiana Univ. Math. J,1973,28:725-743.
[47] Wang Y. Characteristic numbers and cobordism classes of fiberings with fiber RP (2k)[J]. Proc.Amer. Math. Soc.,2001,129:899-911.
[48] Stong R E. Vector bundles over Dold manifolds[J]. Fund. Math.2001,169:85-95.
[49] Fujii M, Yasui T. KO-cohomologies of Dold manifold[J]. Math. J Okayama Univ.,1973,16:55-84.
[50] Milnor J W, Stasheff J D. Characteristic class[M]. Princeton Univ. Press and Univ. of TokyoPress, Princeton New Jersey,1974.
[51] Torrence B F. Bordism classes of vector bundles over real projective spaces[J]. Proc. Amer.Math. Soc,1993,118:963-969.
[52] Chen D, Wang Y. Involution fixing RP (4) P (4,2n1)[J]. Advances in Mathematics,2007,36:89-93.
[53] Alperin J. L, Bell R. B., Groups and representations[M]. Graduate Texts in Mathematics,Springer-Verlag,1995.
[54] Davis M. Groups generated by reflections and aspherical manifolds not covered by Euclideanspace[J]. Ann. Math.,1983,117:293-324.
[55] Ziegler G M. Lectures on polytopes[M]. Graduate Texts in Math., Springer-Verlag, Berlin,1991.
[56] Stong R E. Equivariant bodism and (Z2)k-actions[J]. Duke Math.J.,1970,37:779-785.
[57] Lu¨ Z.2-torus manifold, cobodism and small covers[J]. Pacific J. Math.,2009,241(2):285-308.
[58] Husemoller D. Fiber bundles[M]. Springer-Verlag,1975.
[59] Lu¨ Z, Liu X. Involutions fixing the disjoint union of3-real projective space with Dold mani-fold[J]. Kodai. Math.,2000,23:187-213.
[60] Royster D C. Involutions fixing the disjiont union of two projective spaces[J]. Indiana univ.,Math. J.,1980,29:267-276.
[61],4mL.:8RP (2m) RP (2n)‘éü6/[J].ê Gr,1986,1:19-41.
[62] Hou D, Torrence B F. Involutions fixing the disjoint union of odd-dimensional projectivespaces[J]. Canad. Math. Bull.,1994,37:66-74.
[63] Pergher P L Q. Involutions fixing an arbitrary product of spheres and a point[J]. ManuscriptaMath.,1996,89:471-474.
[64] Stong R E. Involutions fixing products of circles[J]. Proc. Amer. Math. Soc,1993,119:1005-1008.
[65] Bredon G E. Introduction to compact transformation groups[M]. Pure and Applied Mathematics,46Academic Press, New York-London,1972.
[66] Allday C, Puppe V. Cohomological methods in transformation group[M]. Cambridge Studies inAdvanced Mathematics,32, Cambridge University Press,1993.