一点を除いたDold多様体の non-submersibility について
4
0
0
全文
(2) Vol. 19, No. 1 Journal of Hokkaido University of Education (Section II A) September, 1968. On the Non-submersibility of the Punctured. Dold Manifold Masato NAKAMITRA. Department of Mathematics, Hakodate Branch,. Hokkaido University of Education.. 4]'14iEA : —^?^^k Bold ^"figfto non-submersibility K^i'-C. § 1. Introduction. In this note we consider the non-submersibility of the punctured Dold Manifold and the punctured total space of sphere bundles over the real projective space into Euclidean spaces. P(jiz, n) is defined as follows. Let Sm be the unit sphere in 7?m+l and CPn the complex w-dimensional projective space. Now P^m, n) is the manifold obtained from 5mxCP,, by identifying (.v, z) with (-;v. 2), where -.v denotes the antipodal of x and z the conjugate of z. Then the dimension of P<^m, n~) is m+2n. Let y(%)=the number of integers s with Q<s^n and s==Q, 1, 2 or 4 mod 8 ;. s for for which which2S~]['" y-f(m+n-s} {gnot divisible by 2<fw ; o-(w. w)=the largest integer integer s ' " "| is \ s. max(a(^n, n), 2\VL[) if m>0,. (T*OK, n~)= \ ^.n.. 2|y| if m=0,. where | -^-1 denotes the integral part of -^,-.. We shall prove the following. Theorem. If ;' : KO^P^m, %))->/fO(P(w, %)o) is an isomorphism, P(m, n~)a cannot be submersed in .Rm+zn-°-*(mm)+^ where PQn, n~)»=P<im, n)-x and i: P<im, n~)a. —>P^m, w) is the inclusion map.. § 2. The Tangent Bundle Define a line bundle $ over P^m, n~) whose total space £'(^) is Smx C73i,x.ffmod the identification (.v, z, <)~(--t;, z, -t~) and a real 2-plane bundle i] over P(»<, %) whose total space .E'(iy) is SmxS2n+1 X C mod the identifications (x,u, w)~(.v, }.u, <<w). ( 7 ).
(3) On the Non-submersibillty of the Punctured Dold Manifold. ~(-A;, 7%, JMO~(.-V, M, ?y), where us S;!"+l, and ^eC with |-<|=1. Let T denote the tangent bundle of P^m, n~).. Proposition. T©$©2=(w+l)^©(w+l)y. Proof. Write C=(m+l)^©(%+l)-<? and X=6'mxSzn+t x7?m+'xC»+i. Then £'(€) is the set of all (.v, u, y, v") sX mod the .identifications Ov, u, y, y)~(-v, ^u, y, ^v)~(-.v, ^{, -y, Zv)—-(-A;, u, -y, v), where /teC with |-1|:=1, Let < > and ( ) denote the real and complex inner products of Rm+1 and Cn+i respectively. Then £'CT) is the subset of E^-J of all (^x, u, y, w) satisfying (x, y~)=0 and (%, y)=0. Since £'(f)c:£'(c), we have r@v3=^, where v3 is the orthogonal complement of r in C. Now, v3 is equivalent to 2®;'.. § 3. Additive and Multiplicative Structures of KO (P(w<, »)) Write x=S-l, z==iy-2, and y=z-x.. Proposition. KO^P^m, %)) contains a summand isomorphic to Z2'y("I:> 4-Z'LTJ, n. generated by x, y, ys...., ,yL~^J, where Zsvw is the cyclic group of order 2(°(m) and -?-~1. .. .,. ^. ... ,. [. n. Z'L2J is the free abelian group of rank |-^-|.The multiplicative structure of Z^f(m'> + 2'LTJis given by Xs=x-2,x and xy=Q.. Proof. Let S be the canonical line bundle over the real projective space P,n and x=^-l. Then p'x=x is a generator of order 2<°<"'), where p : P(m, n)-^P,,> is the projection. We have j'c(y')=cr(,yi')=yi+yilr', where y':P(o,n)->P(m,n) is the inclusion map, c and r denote the operations of complexification and decomplexification, ••••' denotes con]ugation and yi='/j—la (^ : the canonical complex line bundle over the complex projective space CPn). Since (.Vi+.Vi*),. ..... (.>'i+^i!l!)LTJ is a subset of a basis for '"-1. .. .. T"'. Zft7(CPn), we see that yi,..... ,yiL'l'-i generate a summand isomorphic to ZL~s~J and. x2=^-iy=S®S -2^+1= -2(^-1)= -2A- andA-.)'=(?-l)(7?-2)-A-2=?(g)i?---?-2(?l)+2x= -2x+2x=Q.. § 4. n-Structure of Z^w+Z^ We have r©I:©2=(m+l)$©(w+l)i? so that in KO^P^m, w))we have ro=r-(w/ +2w)=m.v+(w+l)z=(w+%+l)A'+(%+l).)', where T=r(P(m, %)), -K=^-l, z=i?-2, and y=z—x. Since ri is a homomorphism, we have rt(ro)=rt(^)l:m+n+l)rt(.)')<"+I). Geometrical dimensions of x and y are 1 and 2, respectively, so rt(.x^=l+xt and rtW=l+yi+rsW- Hence we have rt(7o)=(l+.v0(m+n+) ^l+yt-yt2YT'+^ and the coefficient of il is. ( 8 ).
(4) Masato NAEAMUBA. r.(,0 =(m+rl)^i+s_ ._"..^'=±2i-'(w+?+l).v+2 ^.1 I '^[-^1],^ -~ v ' / '^[-^f where the aij is non-zero integers (the relation xy=0 kills all mixed product terms). n. We see that ffnr"-i r".-i-)'L'2'J appears in the coefficient of ^L^J, The coefficient -TJ) Lz of <"L'2-J is non-zero since anrn.-i r".-i:i?0.. .YJ> LY. We use the following lemma (Atiyah). Lemma. If P(m; n')a submersed in Rn~k, i. e. if r(P(w, %)o) has a trivial summand of dimension n-k, then To has the geometric dimension ^k. This implies for the operators r' that r' (?o)=0 for i<k. Using the function av\m, n~) defined in the. introduction we obtain the following Theorem. If i' : 7<0(P(m, n~)^-^KO(^P^m, %)o) is an isomorphism, P^mn~)s cannot be submerged in 7?m+zn-°'*(min)+1^ where P^m, n')a=P^in, n~)-x and ;' : P(m, w)o->. P(»2, n~) is the inclusion map.. § 5. The Non-submersibility of the Punctured Total Space of Sphere Bundles over the Real Projective Space into Euclidean Spaces Let ? be a k-vector bundle over the projective space Pn and 5(?) the total space of the associated (^—1) sphere bundle of ?. Let x denote the canonical line bundle over -Pn. We denote the stable class of a vector bundle S by So. Let the stable class of ^ be lXo(,0^l<2^M.') We have ro(5(?))=77'-{ro(Pn)+?o}=-?7'-[(w+m).Vo}, where. IT : 5(?)-^Pn is the projection.. Since r'((%+^+l):Vo)= ±2-'(%+J+l)-ro, we haveri(roCB(0)==77I(±2i-'(w+^.+l) Xa by the naturality of r". We define a by o'=max{?"|2'-1(%+I+l')^0 mod 2v(n)}. Then r<r((%+^+l).ro)-^0.. Theorem. If 77': 7TO(PJ-^If(5(?)) and ;••: 7ii;0(5(0)-^ff'0(5(0o) are both isomorphism, then -6(0o cannot be submersed in ^"+i<-i-(°-')=l?"+k-°-, where 77 : -B(?)-^. -Pi, is the projection, i : 5(^)o-^5(^) is the inclusion map and 5(?)o=S(^)-.i;. Reference 1. Atiyah, M. F. (1961), Immersions and embeddings of manifolds. Topology, vol. 1, p. 125-132. Pergamon Press. 2. Phillips, A, (1966), Submersions. of open Manifolds. Topology, Vol. 6, p. 171-206. Pergamon Press.. 3, Ucci, J. J. (1965), Immersions and embeddings of Dold manifolds. Topology, Vol. 4, p. 84-293. Pergamon Press.. ( 9 ).
(5)
関連したドキュメント
The only thing left to observe that (−) ∨ is a functor from the ordinary category of cartesian (respectively, cocartesian) fibrations to the ordinary category of cocartesian
In [6], Chen and Saloff-Coste compare the total variation cutoffs between the continuous time chains and lazy discrete time chains, while the next proposition also provides a
We list in Table 1 examples of elliptic curves with minimal discriminant achieving growth to each possible torsion group over Q
What is special about the once-punctured torus and the 4–times-punctured sphere is that there is a relatively simple classification of their simple closed curves, or more generally
In Section 3 the extended Rapcs´ ak system with curvature condition is considered in the n-dimensional generic case, when the eigenvalues of the Jacobi curvature tensor Φ are
Later, in [1], the research proceeded with the asymptotic behavior of solutions of the incompressible 2D Euler equations on a bounded domain with a finite num- ber of holes,
Then it follows immediately from a suitable version of “Hensel’s Lemma” [cf., e.g., the argument of [4], Lemma 2.1] that S may be obtained, as the notation suggests, as the m A
[9, 28, 38] established a Hodge- type decomposition of variable exponent Lebesgue spaces of Clifford-valued func- tions with applications to the Stokes equations, the
