ON HIGHER ORDER NONSINGULAR IMMERSIONS OF DOLD MANIFOLDS

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1 PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 39, Number 1, June 1973 ON HIGHER ORDER NONSINGULAR IMMERSIONS OF DOLD MANIFOLDS WEI-LUNG TING1 Abstract. In this paper we employ y-operations and characteristic classes to study nonexistence of higher order nonsingular immersions of a Dold manifold into a Euclidean space. 1. Introduction. In [2] and [3], W. F. Pohl and E. A. Feldman have considered higher order tangent bundles of a smooth manifold M and the higher order nonsingular immersion of M into euclidean spaces. In [4], [5], and [6], H. Suzuki obtained some higher order nonimmersion theorems of projective spaces into euclidean spaces or projective spaces by means of characteristic classes, y-operations and spin operations. In [8] C. Yoshioka obtained complete formulas of Stiefel-Whitney classes of higher order tangent bundles of complex projective spaces and Dold manifolds and he applied his results to higher order nonimmersions of these spaces. The purpose of this paper is to prove a higher order nonimmersion theorem for Dold manifolds, using y-operations and characteristic classes. 2. Preliminaries. Let M be an «-dimensional smooth monifold. Let r.p(m) be the pth order tangent bundle of M, then rv(m) is a smooth v(n,p)-\ector bundle, where v(n,p) = ("J") 1. Set rl(m) = v(n, p) rp(m) in [KO]~(M). Let X1, y' and g dim be as defined in [1]. Let WP(M), WV(M), Wf(M) and Wf(M) be total, dual total, /-dimensional and dual -dimensional Stiefel-Whitney classes of tp(m), respectively. Let v denote pt\\ order nonsingular immersion and $ v its negative. We have the following theorem. Theorem 1 ([4]). (i) If M v R"^^+k, then W*=0for ;>A;=0; (ii) If M S, pj<».î>>+*, then W^ =0for 0=Ä;> -/; (iii) If M Ç *<»»>+*, then yi(r JM))=0for >&=Q; (iv) If M Pv<"^>+*, then yi(-r ]1(M))=0for 0^k>-i. Received by the editors August 24, AMS (MOS) subject classifications (1970). Primary 57D40. Key words and phrases. Dold manifold, pth. order nonsingular immersion, real and complex AT-theory, y-operations, Stiefel-Whitney classes. 1 The author wishes to express his thanks to the referee for his many helpful suggestions. American Mathematical Society

2 196 WEI-LUNG TING [June Let Oi:KO(M)^KO(M) (or 0{:K(M)^K(M)) (i=l,2,---) be the symmetric ith power operation which has the following properties [4]: (i) 0 x=l, (ii) Olx<=x, (iii) 0%x+y) = 2i O'x C-t-y for x,ye KO(M). We have Theorem 2 ([4]). rv(m)=ov(r(m)+\)-l. 3. Nonimmersion theorem and its proof. Let RPm, CPn and P(m, n) be /n-dimensional real, n-dimensional complex projective spaces and a Dold manifold of type (m, n) respectively. In this section we will prove the higher order nonimmersion theorem for P(m, n). Let r:k(m)^>-ko(m) be the realification. Let î and fj be the canonical line bundles over RPm and CPn, respectively. Let f and rj be the bundles over P(m, n) which are defined in [7]. We have the following: Proposition 1 ([7]). There exist a l-plane bundle and a 2-plane bundle r\ over P(m, ri) such that (i) i* =i,j*r =r(f ), i*r = l Í; (ii) f<g>f=l, C r = r) where i:rpm->p(m, n),j: CPn-^-P(m, n) are inclusions. Remark. It is easy to prove thaty'* =l. Theorem 3 ([7]). T(P(m,n)) C 2=(m+l)C (n + l)r. Now let i:rpm->-p(m, n),j: CPn >-P(m, n) be inclusions. From Theorem 2, Theorem 3 and the natural property of 0\ Theorem 4. (i) we have: (dí w V /n + j? - i - 1\ /m + n + A f v(p(m,n))= Z \. If (KouniSu \ P ' / V ' /, y in + p i 1\ Im + n + i\ _ OSevcntSp \ P ' / V ' ' (ii) j*rv(p(m, n)) = t (m + P *! - 2)oX(n + l)r(fj)) - 1. tfo \ P - i I From Theorem 4 we have, * o/o/ w V in + p i l\ Im + n + i\ i*rl(p(m,n)) = - Z \ )x where x= 1. By Atiyah [1], we obtain y\i*rl(p(m, n))) - ±2^^ + '. "!)x

3 1973] HIGHER ORDER NONSINGULAR IMMERSIONS OF DOLD MANIFOLDS 197 and where Hence and /(- *rp (P(«i,«)))= ±2i-1^\x, A = V (n -Y p i l\ m -Y n + i\ 0<ondiSi> \ P ' I \ i ' y%i*tl(p(m, «))) = 0o2M(A + ' ~ 1 ) = 0 mod 2*(m) /(- *T (P(m, «))) = 0o2 -.(;),o mod 24,{m) where <f>(m) is defined as the number of integers t with 0<r_^ and t = 0, 1, 2, or 4 mod 8. By using the same method as in [8], we can obtain and Wi(j*rp(P(m,n)))= {^ß* Wi(j*Tv(P(m, «))) = (J (B + i 1 where ß is the generator of H2(CPn;Z2) and 1 i I )r/n m -Y p i 2 * (XoddiSp P - i Let us now define ax maxji >0,2i~1(A + i 'Wo mod 2*(r a2 = maxi; i > 0, 2{-1(A\ 0 mod 2*(m)], ax = maxi ; a2 = maxíí 4 bx = maxji b2 = max i IA +! 11 0 < / = m, /^ J =é 0 mod 2J, 0 < i = n, (B + ' ~" l j =É 0 mod 2 0 < i = «, BJ = É 0 mod 2, <7X = max{a!, a'x, bx}, a2 = max{a2, a2, b2).

4 198 WEI-LUNG TING [June Theorem 5. If -at<k<au then P(m, n) $ /r<m+2"-*»+fc. Proof. By the natural properties of y'-operations and Stiefel-Whitney classes, we have that y\i*rl(pim, n))) * 0, yx-i*t%p(m, n))) * 0, W j*rpipim, n)))^0 and W j*rj(pim, n)))^0 imply y\rl(p(m, «))) * 0, y\-rl(p(m, n))) * 0, IF^Piw.n^^O and Wf(P(m, n))^0 respectively. Thus Theorem 5 follows from Theorem 1. Remarks. (I) In Theorem 5, if we use Pontrjagin classes instead of Stiefel-Whitney classes, then when/?=l we recover the main results of J. J. Ucci [7]. (II) In [8] C. Yoshioka obtained the following formula W\P(m, n))=(l+c)a'(l+c+d)b', where c, d are the classes which are defined in [7]. From this formula he obtained the following results. Theorem 6. Let b[ = max! i 0 < i = a + 2ß ^ m + 2n, y (oi,b'-ß)( Ä' ) - -^0mod2), OSySmin V - y) (B' - ß - y)\ ylßl f í>2 = max i 0<í = a + 2/?^m + 2n, where 2 («,2<-B--ß)(A'-l+«-A OSySmin V a. y I (2'-B')l ) X- -^ 0 mod 2, (2l - B'- ß-y)\y\ß\ í A, = i y í/2n + i + l\ im + Ijp -j)-\ln + 2-1i\ <m + p-i-2y \ m 2! 2o<o4ä/ ' A m-2 /'

5 1973] HIGHER ORDER NONSINGULAR IMMERSIONS OF DOLD MANIFOLDS 199 t is an integer such that 2i>max{«2, «, B' l}. If k is an integer such that -b'x<k<b'2, then P(m, «) $»<»+»+*. Here we will give some examples to show that in some cases our Theorem 5 can give sharper nonimmersion results than the above theorem. (1) When p=\, then A' m, B'=n+\, A=m+n+\, P=«+l. Let (w,«) = (14, 1). Then W(P(m,n)) = (\+c)li(\+c+d)2=\ and Theorem 6 gives no information. By direct calculations we have ax = 4. So we have: Corollary 1. P(14, 1)$ P16+*, k<:3. In general, let (m, n) = (2t-2s, 2s-1), /^j=0. Then W(P(m,n)) = (l-fc)2~2*(l-r-c-(-ûi)2l'=l and Theorem 6 gives no information. By direct calculations we have o-1^2s-1(2(-s l) i if í=1,ct1=2(-2 if r^4 and í=0. So we have : Corollary 2. (a) If k<2s-\2l-3-\)-t, then P(m, n)$rm+2m+k where (m, «) = (2<-2s, 2s-1), f^ssgl. (b) ([1]). Ifk<2t~2, thenp(m,0)^rm+k, where «7=2'-1, r^4. (2) When p=2, then A' = (n+ l)2-m, B' = (n+\)(m-\), A = n(m-yn-yi). Let (m, «) = (12, 3). Then W2(P(m, «)) = (1 + c)4(l + c + d)m = 1 and Theorem 6 gives no information. By direct calculations we have ox=4, g2=4. Thus we obtain: Corollary 3. P(12, 3) $ 2 R +k, -3^k^3. In general, let (m, «) = (2<-2s, 2S-1), t^s^l. Then W2(P(m,n)) = (l+c)2'-2,+2'(1+c+ê?-)2'(2!-2s-i) = (1+c.)2'(2i-i) = 1 and Theorem 6 gives no information. By direct calculations we have o-1^2s_1(2<_s 1) r, a2>2'^1(2t~s-\)-t. Thus we obtain: Corollary 4. If-2s-1(2t-s-\) + t<k<2'-1(2t-'-\)-t, then P(m,n) where (m, «) = (2( 2s, 2s-1), r^s^l. ^2R^+2n.2)+k References 1. M. F. Atiyah, Immersions and embeddings of manifolds, Topology 1 (1962), MR 26 # E. A. Feldman, The geometry of immersions. I, Trans. Amer. Math. Soc. 120 (1965), MR 32 # W. F. Pohl, Differential geometry of higher order, Topology 1 (1962), MR 27 #4242.

6 200 WEI-LUNG TING 4. H. Suzuki, Bounds for dimensions of odd order nonsingular immersions of RP", Trans. Amer. Math. Soc. 121 (1966), MR 33 # , Characteristic classes of some higher order tangent bundles of complex projective spaces, J. Math. Soc. Japan 18 (1966), MR 35 # , Higher order non-singular immersions in projective spaces, Quart. J. Math. Oxford Ser. (2) 20 (1969), 33-^4. MR 39 # J. J. Ucci, Immersions and embeddings of Dold manifolds, Topology 4 (1965), MR 32 # C. Yoshioka, On the higher order non-singular immersion, Sei. Rep. Niigata Univ. Ser. A No. 5 (1967), MR 38 #1690. Department of Mathematics, State University of New York, College at Plattsburgh, Plattsburgh, New York 12901