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On Cross Sections to Fiber SpacesAuthor(s): Emery ThomasSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 54, No. 1 (Jul. 15, 1965), pp. 40-41Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/72987 .
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40 MATHEMATICS: E. THOMAS PROC. N. A. S.
Pn /(X, . ,Xn-1,. ,B) = X,3(Xn_-I).
Again we have En'(h) = h*(xo). But then
0 = En(h) - En'(h)
= 21-"[(1 - d)h(x') + dh(x") - h(x,,_-)],
so h is A-linear on xo + sp(A). By analogy with ordinary convex functions, one might conjecture that h, is the
supremum of A-linear functions g < h. This is false. If S is the Euclidean plane and A consists of all vectors (x,y) with x = 0 or y = 0, the A-linear functions are those of the form a + bx + cy + dxy. The function h(x,y) = [max (x,y,0)]2 {2\x - y - x - y} is A-convex, so h* = h; but there is no A-linear function g withg < h.
ON CROSS SECTIONS TO FIBER SPACES*
BY EMERY THOAMAS
I)EPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, BERKELEY
Communicated by S. S. Chern, April 21, 1965
A classical problem in Algebraic Topology is to determine whether a given fiber
space has a cross section. Beginning with Postnikov,l the problem is usually treated in the following way. The fiber space is factored into a sequence of fiber spaces, and at each stage the obstruction to lifting a map to the next stage is given by one or more cohomology classes, the Postnikov invariants of the fibration. Recently, iMIahowald' has given a detailed description of how one decides precisely which cohomology classes at each stage can be taken as the Postnikov invariants for the next stage. In a forthcoming paper I shall describe a method whereby, for certain fiber spaces, one or more of the Postnikov invariants can be expressed in terms of secondary (or tertiary) cohomology operations applied to classes coming from the base of the fibration. Since we now have a variety of methods for computing higher order operations, the calculation of the Postnikov invariants is sometimes possible. The following theorems illustrate the kind of results one can obtain by this method.
Let M11 be a compact, connected, smooth manifold. We define the span of M to be the maximal number of linearly independent tangent vector fields on Ml. By the theorem3 of H. Hopf, span M is positive if and only if the Euler characteristic of 1M, X(M), is zero. In particular every odd-dimensional manifold has positive span.
Let wi(M) e Hi(MlZ;Z2) denote the ith Stiefel-Whitney class of M,11 i > 0, and let 8 denote the Bockstein cohomology coboundary from mod 2 coefficients to integer coefficients.
THEOREM 1. Let 111 be an orientable manifold, and let k be a nonnegative integer. If dim M = 4k + 3, or if dim lM = 4k + 2 and bw4, (M) = X (M) = 0, then span M11 > 2.
Following Borel-Hirzebruch, we say that M is a spin manifold if '{l(11) =
W2(M) = 0.
40 MATHEMATICS: E. THOMAS PROC. N. A. S.
Pn /(X, . ,Xn-1,. ,B) = X,3(Xn_-I).
Again we have En'(h) = h*(xo). But then
0 = En(h) - En'(h)
= 21-"[(1 - d)h(x') + dh(x") - h(x,,_-)],
so h is A-linear on xo + sp(A). By analogy with ordinary convex functions, one might conjecture that h, is the
supremum of A-linear functions g < h. This is false. If S is the Euclidean plane and A consists of all vectors (x,y) with x = 0 or y = 0, the A-linear functions are those of the form a + bx + cy + dxy. The function h(x,y) = [max (x,y,0)]2 {2\x - y - x - y} is A-convex, so h* = h; but there is no A-linear function g withg < h.
ON CROSS SECTIONS TO FIBER SPACES*
BY EMERY THOAMAS
I)EPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, BERKELEY
Communicated by S. S. Chern, April 21, 1965
A classical problem in Algebraic Topology is to determine whether a given fiber
space has a cross section. Beginning with Postnikov,l the problem is usually treated in the following way. The fiber space is factored into a sequence of fiber spaces, and at each stage the obstruction to lifting a map to the next stage is given by one or more cohomology classes, the Postnikov invariants of the fibration. Recently, iMIahowald' has given a detailed description of how one decides precisely which cohomology classes at each stage can be taken as the Postnikov invariants for the next stage. In a forthcoming paper I shall describe a method whereby, for certain fiber spaces, one or more of the Postnikov invariants can be expressed in terms of secondary (or tertiary) cohomology operations applied to classes coming from the base of the fibration. Since we now have a variety of methods for computing higher order operations, the calculation of the Postnikov invariants is sometimes possible. The following theorems illustrate the kind of results one can obtain by this method.
Let M11 be a compact, connected, smooth manifold. We define the span of M to be the maximal number of linearly independent tangent vector fields on Ml. By the theorem3 of H. Hopf, span M is positive if and only if the Euler characteristic of 1M, X(M), is zero. In particular every odd-dimensional manifold has positive span.
Let wi(M) e Hi(MlZ;Z2) denote the ith Stiefel-Whitney class of M,11 i > 0, and let 8 denote the Bockstein cohomology coboundary from mod 2 coefficients to integer coefficients.
THEOREM 1. Let 111 be an orientable manifold, and let k be a nonnegative integer. If dim M = 4k + 3, or if dim lM = 4k + 2 and bw4, (M) = X (M) = 0, then span M11 > 2.
Following Borel-Hirzebruch, we say that M is a spin manifold if '{l(11) =
W2(M) = 0.
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VoL. 54, 1965 MATHEMATICS: E. THOMAS 41
THEOREM 2. Let M be a spin manifold of dim 4k +3, k > 0. If k is odd or if 6W41k(M) = 0, then span M > 3.
THEOREM 3. Let dim M = 8k + 7, k > 1. If Hi(M;Z2) = 0, 1 < i < 4, then span M > 6.
If M is a complex manifold, we define, in an analogous way, the span of M1 in terms of (differentiable) complex vector fields. We denote by c,(M) e H2i(Mj ;Z) the ith Chern class of Al.
THEOREM 4. Let M be a comiplex manifold of dim n > 3 such that Cn_,(M/) = C (M) = 0. If n is odd, then span Ml > 2. Suppose that n - 0 mood 4 and that cl(M) = c2(j) = 0. If either Cn_2(M) - 0 mod 2 or H'l(MIZ;Z2) = 0, then span 1M > 2.
The result for n odd is ilmmediate and does not require secondary operations. A drawback to the theorem is that we can say nothing about the analyticity of the vector fields.
As a final example, we consider the problem of whether a (real) manifold has a stable almost-complex structure, i.e., whether the classifying map for its stable
tangent bundle can be factored through the classifying space for the stable unitary group. Massey4 has shown that for an 8-dim orientable manifold, with w2(M) and w6(M) the mod 2 reduction of integral classes, there is a single remaining obstruc- tion in Hs(M;Z2). Using the method referred to above, one can express this obstruc- tion in terms of a secondary operation; this yields in particular the following re- sult.
THEOREM 5. Let M be an 8-dim spin manifold such that W6(M) is the mod 2 re- duction of an integral class. If w4(1M) = 0, then Mi admits a stable almost-complex structure. If w4(M) # 0 and if Ml admits a stable almost-complex structure with
vanishing first Chern class, then ws(Ml) = 0. As an example, the theorem shows that the homogeneous space G2/SO(4) does
not admit a stable almost-complex structure, since' H2(jM';Z) = 0; w2(M) = 0; w4(M), ws(M) r 0 [M = G/S0 (4) .
Remark: In a recent paper,6 M\ahowald and Peterson computed certain Post- nikov invariants by applying secondary cohomology operations to the Thom class of a bundle. The method referred to here is different from theirs.
* Research supported in part by the U.S. Air Force Office of Scientific Research. 1 Postnikov, M., "Investigations in the homotopy theory of continuous mappings," Amer. Math.
Soc. Transl., 7, 1-134 (1957). 2 Mahowald, M. E., "On obstruction theory in orientable fibre bundles," Trans. Amer. Math.
Soc., 110, 315-349 (1964). 3 Hopf, H., "Vektorfelder in n-dimensionalen Mannigfaltigkeiten," Math. Ann., 96, 225-260
(1927). 4 Massey, W., "Obstruutions to the existence of almost complex structures," Bull. Amer. Math.
Soc., 67, 559-564 (1961). 5 Borel, A., and F. Hirzebruch, "Characteristic classes and homogeneous spaces, Part II," Amer.
J. Math., 81, 315-382 (1959). 6 Mahowald, M., and F. Peterson, "Secondary cohomology operations on the Thom class,"
Topology, 2, 367-377 (1964).
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