一点を除いたｌens空間のnon-submersibilityについて

全文

(1)Title. 一点を除いたｌens空間のnon-submersibilityについて. Author(s). 中村, 正人. Citation. 北海道教育大学紀要. 第二部. A, 数学・物理学・化学・工学編, 20(1) : 6-7. Issue Date. 1969-09. URL. http://s-ir.sap.hokkyodai.ac.jp/dspace/handle/123456789/5911. Rights. Hokkaido University of Education.

(2) Vol. 20, No. 1 Journal of Hokkaido University of Education (Section K A) September 1969. On the Non-submersibility of the Punctured Lens Space. Masato NAKAMURA Department of Mathematics, Hakodate Branch, Hokkaiclo University of Education. 4'N'IEA : —As^li^'-'fc lens SfRl^ non-subinersibility ^--5^^-C. § 1 Introduction In this note we consider the non-submersibility of the punctured lens space L"(p)—x, which is the lens space L" (p) with a point removed. The lens space is defined as follow : Let p be an integer > 1 and f be the rotation of (2n +1) — sphere S2"+l=[(zo, zi,. .... z,,)/S|zil2=l] i=«. of the complex (n+1)—space C"+l given by r(z», zi, . . . . . , z,,)=(e2I(l/Pzo, e2'ti/rzi, ,...., e^i'z,,).. Then f generates the topological transformation group F of S3"+l of order p, and the lens space L"(p) is defined to be the orbit space: L"(p)=s2"+l/r. This is the compact differentiable (2n+l)— manifold without boundary.. § 2 Non-submersing Theorem As is well known, the lens space L"(p) has a cell structure given by L"(p)=slL)ei!U ..... e2"+l. Since the punctured lens space L"(p)—x has Lg(p) :=slUe2U ..... e2" as deformation retract,. the study of the tangent bundle T(L"(p)-x) reduces to that of T(L"(p)-x)|L!)'(p). Let T represent T(L"(p)—x)|Lo*(p). This is the (2n+l)—dimensional bundle, so To=. T-(2n+l)6=KO(L'o(p)), where KO(Ln'(p)) is the reduced Grothendick ring of real vector bundles over Lo(p). If L"(p)—x sub merges in R2" +l-k, i.e. if T has a trivial summand of dimension (2n+l —k), then T|) has geometric dimension ^k. This implies for the Grothendick operators r* that r' (To)=0 for i>k. According of the result of T. Kambe, the structure of KO(Ln'(p)) is the following. Let i) be the canonical complex line bundle over the complex projective space CP". Consider the natural projection TT : L"(p)=s2n+1/^—>S2"+1/S1=CP" and the element. «=7r!0?-lc)eK(L"(p)). (6).

(3) Masato Nakamura. where Tt': K(CP")—>K(L"(p)) is the induced homomorphism of it. Consider the operator r:K(L"(p))—>KO(L"(p)) which send complex vector bundles to the corresponding naturally defined real vector bundles and the element <7=7'tfeKO(L"(p)). Let p be an odd prime, q=(p—l)/2 and n=s(p—l)+r(05ar<p—l). Then. n/ ^((ZpB+lyr'"+(Zp8)o-c-W (if n^O mod (4)). ,'y^P))^. '°^=lZ2+(Zp8+lyr/"+(Zp,,)<«-C'-? (if n=0 mod (4)) and the direct summand (Zp8+l)cr/2:l and (Zp8)<>-cr/2:l are additively generated by a, . . . . . , ffWal and tfcl'/2:l+l, ..... ,rfq respectively. Moreover its ring structure is given by. <?'+l=s(^2q+l)K+i-,rl?, ?"wi=o (=i ^1—1 \ Zl—. and the action of the r' by ri(tf)=l+tft-o:t2.. Since r (L'o (p)) =7r'r(CP")+1, then r, (Lo'(p)) = (n+l)ff. Thus rt(ro(U'(p))=(n(d))"+I=a+5(t-t2))"+l=i]fn+l')tf'(t-t2)1. 1^0 \ 1. •n—2i-. Since o'1 is of order p LprT-l, the condition rl(fo)=0 is equivalent to (nihl)=0 mod (pi+C(»-2D/(p-i)3). Considering ai-'s-1 =0, we define L(n,p) to be the integer given by. L(n,p)=max{i^Cn/2]|("ihl)^0 mod (pi+co-aD/fp-U)}, By the Theorem of Atiyah, we obtain the following theorem, Theorem. Let p be odd prime, then the punctured lens space L'\p')—x cannot be siibinersed il't 7?3"+l-{2JE("'i'>-1} =^2{»+1-^.(»,;1)}. Refference 1. Atiyah, M. F. 1961 : Immersions and embeddings of manifolds. Topology 1, pp. 125-132. Pergamon Press, Great Britain. 2. Kambe, T. 1966 : The structure of K-ring of the lens space and their applications. J. Math. Soc. Japan. 3. Phillips, A. 1966 : Submersions of open manifolds, Topology 6, pp. 171-206. Pergamon Press, Great Britain.. C 7 ').

(4)