Independent Vector Fields Research Articles

Span of lens spaces

We prove that the span (i.e., maximal number of linearly independent vector fields) of ( 2 n + 1 ) (2n + 1) -dimensional lens spaces L n ( p ) {L^n}(p) , where p p is an odd prime and where n + 1 = m ⋅ 2 t n + 1 = m \cdot {2^t} ( m m odd), is equal to the span of ( 2 n + 1 ) (2n + 1) -dimensional sphere if t + 1 ≡ 0 , 1 , 2 ( mod 4 ) t + 1 \equiv 0,1,2\pmod 4 or if n > 3 , t + 1 ≡ 3 ( mod 4 ) n > 3,t + 1 \equiv 3\pmod 4 , and p ≧ t + 3 p \geqq t + 3 . This result is an improvement of a theorem given by T. Yoshida.

On the fibre homotopy type of normal bundles.

ON THE FIBRE HOMOTOPY TYPE OF NORMAL BUNDLES Morris W. Hirsch 1 . INTRODUCTION It was proved by Atiyah [1] that the fibre homotopy type of the stable normal sphere bundle of a manifold M is an invariant of the homotopy type of M. Theorem A below (discovered before I learned of Atiyah’s proof) gives an elementary proof of this fact, and also applies to nonstable cases. (See [5], for example. One purpose of the present work is to supply item 6 in the bibliography of [5].) The situation con- sidered is a homotopy—commutative diagram 1' M0 '—‘”M1 8o /31 U1 with f a homotopy equivalence, gi: Mi —g V embeddings (i = 0, 1), and U1 a closed tubular neighborhood of g1(M1). Theorem A implies that the normal sphere bundles of go and g1 are fibre-homotopically equivalent. Theorem B applies Theorem A to the problem of choosing go (given f and g) so that it will have as many independent normal vector fields as g1 . The proof of Theorem A in the case dim V 2 dim M + 3 depends on Lemma 2, due to Milnor, which states that if U0 is a closed tubular neighborhood of g0(M 0) inside int U1 , then U1 — int U0 is an h-cobordism between the boundaries bU1 and bU0 . This Lemma is no longer universally true if dim V = dim M + 2; Theorem C (which is independent from Theorems A and B) exhibits a special case where it is true. An immediate corollary is that if M0 X R1‘ is diffeomorphic to M1 gl Rk, then M0 gl Sk-1 is h— cobordant to M1 X Sk'1 . (The interesting case is k = 2.) All manifolds, immersions, and embeddings are smooth. Throughout the paper, Mo and M1 are compact unbounded manifolds of dimen- sion m, and V is a Riemannian manifold of dimension v. 2. FIBRE HOMOTOPY TYPE If at and B are bundles, then oz ~ 3 indicates that a and /3 are isomorphic, while 0: g 3 means that oz and B have the same fibre homotopy type. For this con- cept, the reader is referred to Dold Let f: M -g V be an immersion. If V is the normal vector space bundle of f, then 13‘ will denote the normal sphere bundle of f, and conversely. THEOREM A. Let gi: Mi —g V be embeddings (i = O, 1). Let U1 C V be a closed tubular neighborhood of g1(M1) such that g0(M0) C U1 . Let f: MO —g MI be a homotopy equivalence making a homotopy—commutative diagram Received July 31, 1964. 225

Vector Fields on Spheres

The question of vector fields on spheres arises in homotopy theory and in the theory of fibre bundles, and it presents a classical problem, which may be explained as follows. For each n, let Sn-I be the unit sphere in euclidean n-space Rn. A vector field on Sn-1 is a continuous function v assigning to each point x of Sn-1 a vector v(x) tangent to Sn-1 at x. Given r such fields v1, v2, ..., Vr, we say that they are linearly independent if the vectors v1(x), v2(x), *--, vr(x) are linearly independent for all x. The problem, then, is the following: for each n, what is the maximum number r of linearly independent vector fields on Sn-i? For previous work and background material on this problem, we refer the reader to [1, 10, 11, 12, 13, 14, 15, 16]. In particular, we recall that if we are given r linearly independent vector fields vi(x), then by orthogonalisation it is easy to construct r fields wi(x) such that w1(x), w2(x), *I * , wr(x) are orthonormal for each x. These r fields constitute a cross-section of the appropriate Stiefel fibering. The strongest known positive result about the problem derives from the Hurwitz-Radon-Eckmann theorem in linear algebra [8]. It may be stated as follows (cf. James [13]). Let us write n = (2a + 1)2b and b = c + 4d, where a, b, c and d are integers and 0 < c < 3; let us define p(n) = 2c + 8d. Then there exist p(n) 1 linearly independent vector fields on Sn-1. It is the object of the present paper to prove that the positive result stated above is best possible.

Vector fields on spheres

The question of vector fields on spheres arises in homotopy theory and in the theory of fibre bundles, and it presents a classical problem, which may be explained as follows. For each n, let Sn-I be the unit sphere in euclidean n-space Rn. A vector field on Sn-1 is a continuous function v assigning to each point x of Sn-1 a vector v(x) tangent to Sn-1 at x. Given r such fields v1, v2, ..., Vr, we say that they are linearly independent if the vectors v1(x), v2(x), *--, vr(x) are linearly independent for all x. The problem, then, is the following: for each n, what is the maximum number r of linearly independent vector fields on Sn-i? For previous work and background material on this problem, we refer the reader to [1, 10, 11, 12, 13, 14, 15, 16]. In particular, we recall that if we are given r linearly independent vector fields vi(x), then by orthogonalisation it is easy to construct r fields wi(x) such that w1(x), w2(x), *I * , wr(x) are orthonormal for each x. These r fields constitute a cross-section of the appropriate Stiefel fibering. The strongest known positive result about the problem derives from the Hurwitz-Radon-Eckmann theorem in linear algebra [8]. It may be stated as follows (cf. James [13]). Let us write n = (2a + 1)2b and b = c + 4d, where a, b, c and d are integers and 0 < c < 3; let us define p(n) = 2c + 8d. Then there exist p(n) 1 linearly independent vector fields on Sn-1. It is the object of the present paper to prove that the positive result stated above is best possible.

The Topology of Rotation Groups

The Betti numbers of the rotation group, R(n), in n real variables were first determined by R. Brauer [5].2 Later Pontrjagin [12] extended slightly these results by computing the integral homology groups of R(n). The Betti numbers combined with Hopf's Theorem on group-manifolds [9], which states that the rational cohomology ring of a group-manifold is isomorphic to that of a product of odd dimensional spheres, can be used to determine abstractly the rational cohomology ring of R(n). Pontrjagin's results can be used to find the additive structure of the cohomology groups of R(n) over an arbitrary coefficient domain, but there are no general theorems such as Hopf's Theorem available for the determination of the cup product over coefficient rings other than the rational numbers. This paper gives a treatment of the cohomology structure of R(n) which links that structure more intimately with the space R(n) than does Pontrjagin's treatment and extends the results mentioned above to other coefficient domains. The basic tool used is the notion of a cell complex. The particular cell complex used on R(n) gives rise to a cell complex on the Stiefel manifold V(n, r) of r-frames in euclidian n-space, which J. H. C. Whitehead used [17] to compute some of the higher homotopy groups of V(n, r). In Section 2 we give a description of this cell complex, rephrasing Whitehead's definitions slightly in order to facilitate later computations. Sections 3, 5, and 7 are devoted to the computation of the homology and cohomology groups and the calculation of the cup product. In Section 6 the Steenrod squaring homomorphisms are determined for V(n, r). These are then used in Section 9 to give a proof of the Steenrod-Whitehead Theorem that 2k independent vector fields do not exist on S'-', where 2k is the largest power of 2 dividing n [16], and are also used in Section 11 to show that, unless 2n = 2-k 2, S2n has no quasi-complex structure.3 In each case the desired result is obtained by employing the squaring homomorphism to show the nonexistence of cross sections in certain fibre bundles. The appendix gives a brief account of a technique for finding the integral cup

On Families of Continuous Vector Fields Over Spheres

It has long been known that no continuous field of unit tangent vectors can exist on the 2n-dimensional sphere S2. On the other hand, if n = 2k 1, 4k 1, or 8k 1, there exist 1, 3, 7 everywhere independent continuous vector fields over S'. It was recently proved independently by B. Eckmann2 and the author3 that if n 1 (mod 4) every two vector fields over St are somewhere dependent. In the present paper an analogous result is obtained for the case n 3 (mod 8): every four vector fields over Sn are dependent at some point. The principal tools are the notion of fibre space developed by Huxewicz and Steenrod4 andthe results of Freudenthal5 on the homotopy groups of spheres. As a corollary to the above results and a result of C. Ehresmann,6 it is shown that no sphere of dimension > k can be a k-sphere bundle over any complex B if k = 2m, 4m + 1, or 8m + 3 with m > O.6