Some Notes on Orientation Preserving Involutions of Manifold Li-Jiang ZENG Research Centre of Zunyi Normal College, Zunyi 563000, Guizhou, China [email protected]
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2016 2nd International Conference on Education Science and Human Development (ESHD 2016) ISBN: 978-1-60595-405-9 Some Notes on Orientation Preserving Involutions of Manifold Li-Jiang ZENG Research Centre of Zunyi Normal College, Zunyi 563000, GuiZhou, China [email protected] Keywords: Bordism group An(2 k), Orthogonal 2 k-plane, Orthogonal bundle, Whitney sum. Abstract. In many applied sciences, the theory of manifold plays an extremely important role, such as mechanics, electromagnetism, and field theory, etc. and in applied science derived from these sciences, manifold has many applications, it often acts as important research tools. In this article, we introduces the derivation process on involutions of the manifold, from these process, some skills are considered when the tool (i.e. manifold) can be applied. Introduction In mathematics, the manifold and roughly speaking simply, is a kind of abstract linear space. In a lot of applications in many areas of study in science and its wide application [1-6], manifold has its widespread application, for example, the theory of mechanics, electromagnetism, and field theory [7-10], etc. In applied science derived from these science manifold and has many applications, for example, ballistic engineering, aerospace engineering, a lot of projects related to electromagnetic radiation, water wave, electronic waves or magnetic wave and so on. Because of the manifold is closely linked with bordism group An(2 k) [11-17], this paper mainly introduces a derivation process on involutions of the manifold, since this process is always with bordism group An(2 k) close connection, also can be seen as bordism group An(2 k), from these derivation process, some skills are considered when the tool (i.e. manifold) can be applied. On the Bordism Group An(2 k) Our basic object to consider is a pair (ξ→ Bn , σ ) wherein ξ → Bn is an orthogonal 2k-plane bundle over a compact n-manifold and σ is an orientation on the Whitney sum(In fact, it is called in inner direct sum and or is referred to as direct sum) [1-6] ξ⊕ τ → Bn where τ → Bn is the tangent bundle. Let −→(ξBn ,)( σξ =→ B n ,) − σ . We identify (ξ '→W n , σ ') with (ξ→ Bn , σ ) if and only if there is an orthogonal bundle equivalence Φ ξ' → ξ ↓ ↓ n n W→ B for which 1) ϕ is a diffeomorphism 2) The induced bundle equivalence Φ⊕ dϕ ξτ'⊕ ' → ξτ ⊕ ↓ ↓ ϕ Wn B n → preserves the orientation. n n A boundary operator ∂→(ξB ,)( σξ =→∂ B ,()) ∂ * σ can be defined as follows. Along ∂Bn identify ξ⊕ τ with ξ⊕ τ' ⊕ η , where τ ' → ∂ Bn is the tangent bundle to the boundary and η → ∂ Bn is the trivial normal line bundle. Now ξ⊕ τ' ⊕ η →∂ Bn in here its an orientation from that of (ξ⊕ τ ) , while, at each point of ∂Bn , η is given its orientation by the outward pointing unit normal vector. There is a unique orientation of n n ξ⊕ τ' ⊕ η →∂ B compatible with those of η and of ξ⊕ τ' ⊕ η →∂ B , and this is ∂* (σ ) . n n The bordism group An(2 k) can now be defined. If M and V are closed n-manifolds n n n+1 then (ξo→ M , σ o ) is bordant to (ξ1→V , σ 1 ) if and only if there is a (ξ→ B , σ ) for which ∂→(ξBn+ ,)( σξ =→ M n ,)( σξ → V n ,) σ O o ∪ 1 1 The symbol ∪ denotes disjoint union. Denote a bordism class by [ ξ→ M n , σ ], and the collection of all such bordism classes by An(2 k). As usual, an abelian group structure is imposed on An(2 k) by disjoint union. We agree that An( o ) = Ω n the co- bordism group of closed oriented n-manifolds regarded as carrying "o-plane bundles". The Connection between Bordism Group with o-Plane Bundles ∞ Set αm= ∑ A n (2 k ) and α = ∑ An(2 k ) . We can define in α the structure of a n+2 k = m 0 graded commutative algebra with unit over Ω , the oriented co-bordism ring. We define the product [ ξ→ M n , σ ][ ξ'→V m , σ ' ] as follows. From the external whitney sum (ξτ⊕× ) ( ξ ' ⊕ τ ') →Mn × V m . This is given the product orientation σ× σ ' . The canonical equivalence (ξτ⊕× ) ( ξτ ' ⊕ ') = ( ξξ × ') ⊕×× ( ττ ') then induces the desired orientation. Since An( o ) = Ω n , Ω ⊂ α , hence α is also an Ω -algebra with unit. To Compute An(2 k) To compute An(2 k) we must show that it is naturally isomorphic to the Atiyah bordism group [14] of the covering involution (T, BSO (2 k)) over BO (2 k) and then apply results [14]. An element of BSO (2 k) is a 2 k-plane together with an orientation. The involution T is to reverse that orientation. Briefly, the Atiyah group An(T, BSO (2 k)) is defined as follows. The ~ ~ ~ basic object is a pair ((t , Bn ), f ) , where (t , B n ) is a fixed point free orientation reversing diffeomorphism of period 2 on a compact oriented manifold together with an equivariant map ~ ~ f:(, tBn )→ (, TBSOk (2)) . ~~ ~ ~ ~ ~ There is ∂((tB ,n ), f ) =∂ (( tB , n ), fB / ∂ n ) , where ∂Bn receives the induced orientation from ~ n that of B . We can go on to complete the definition of An(T, BSO (2 k)). We are ~ given (ξ→ Bn , σ ) . Let Bn be the set of all pair s( b , o ) , where b∈ B n and o is an orientation of the fibre of ξ at b, that is, of the linear space ξb . There is an obvious fixed ~ point free involution t(b , o) = ( b, -o) and a projection γ : Bn→ B n given by γ (b , o ) = b . ~ ~ n n n Topologize B so that γ : B→ B is the principal z2 -covering associated with the ~ 1 n n Whitney class v1()ξ ∈ HBz (,) 2 . Then B receives a unique differential structure in which ~ t is a diffeomorphism and for which γ : Bn→ B n is a local diffeomorphism. ~ Now the induced bundle ξ= γ−1( ξ ) consists of all triples. Conclusions ~ In the above, it induced bundle ξ= γ−1( ξ ) consists of all triples. It is the derivation process on involutions of the manifold, from these process, we got to know of bordism group An(2 k), orthogonal 2 k-plane, orthogonal bundle, Whitney sum and so on, they have good application in many since, for example, ballistic engineering, aerospace engineering, a lot of projects related to electromagnetic radiation, water wave, electronic waves or magnetic wave and so on. Author Introduction Lijiang Zeng (1962 - ), male, born in Chishui of Guizhou Province, Professor of Zunyi Normal College, major research field: mathematics and applied mathematics, research direction: algebra and its application, number theory and its application, function theory and application. Until now, have existed search results: CPCI-S(ISTP), CPCI-SSH(ISSHP), and EI 17 articles published, 25 English papers published, Email: [email protected]. References [1] Potvin Jean Yves, Rousseau Jean Marc. Aparall elroute building algorithm for the vehicle routing and scheduling problem with time windows [J]. European Journal of Operational Research. 1993(03), 40-45. [2] Wang Y, Wei H, Li Y. A generalization of a theorem of Kramer and its applications. Bulletin of the Australian Mathematical Society. 2002(02), 39-45. [3] P. Schmid. Subgroups permutable with all Sylow subgroups. Journal of Algebra. 1998(01), 40-44. [4] Zeng Lijiang, Reducible property of a finitely generated module [J]. Advances in Natural Science (Canada). 2010: 3(2), 270-276. [5] Lijiang Zeng, On the algebraic integers in cyclotomic fields [C], Proceedings of International Conference on Engineering and Business Management (EBM2011), March, 22-24, 2011 (Wuhan) 2293-2296 (ISTP and ISSHP). [6] Tam Bitshum, Yang Shangjun. On matrices whose numerical ranges have circular or weak circular symmetry [J]. Linear Algebra its Applications. 1999(02), 76-82. [7] Beylkin G., Coifman R., Rokhlin V. Fast wavelet transforms and numerical algorithms I [J]. Communications in Pure Applied Mathematics. 1991(02), 42-46. [8] Yangming Li, Yanming Wang, Huaquan Wei. On p-nilpotency of finite groups with some subgroups π-quasinormally embedded [J]. Acta Mathematica Hungarica. 2005 (02), 40-46. [9] S. Srinivasan. Two sufficient conditions for supersolvability of finite groups [J]. Israel Journal of Mathematics. 1980 (3), 30-36. [10] Wanzhexian. Geometry of Hamilton matrices and its application II [J]. Algebra Colloquium. 1996(01), 66-69. [11] Xiuyun Guo, K.P. Shum. On c-normal maximal and minimal subgroups of Sylow p-subgroups of finite groups [J]. Archiv der Mathematik. 2003 (6), 64-68. [12] Prof. Joseph Buckley. Finite groups whose minimal subgroups are normal [J]. Mathematische Zeitschrift. 1970 (1), 50-56. [13] Lijiang Zeng, Equivalence on finitely generated R[G] module [C], Proceedings of 2011 Asia-Pacific Youth Conference on Communication (2011APYCC), April, 4-6, 2011 (Hangzhou) pp434-436. [14] Skiba N. On weaklys-permutable subgroups of finite groups. Journal of Algebra. 2007(01), 44-50. [15] Lijiang Zeng, Note on coprime action of two groups with finite orders [C], 2013 International Conference on Economic, Business Management and Education Innovation (EBMEI 2013), May 22-23, 2013, Beijing, China, 166-172 (ISSHP). [16] Lijiang Zeng, On the classification of the finite subgroups of SO(3) [C], Proceedings of International Conference on Engineering and Business Management(EBM2011), March, 22-24, 2011 (Wuhan) 2289-2292 (ISTP and ISSHP). [17] N.Y. Deng, Y. Xiao, F. J. Zhou. Nonmonotonic trust region algorithm [J]. Journal of Optimization Theory and Applications. 1993 (02). 70-76. .