Characteristic number obstructions to fibering oriented and complex manifolds over surfaces

Abstract

WE CONSIDER here the constraints on the Chern, Pontrjagin and Stiefel-Whitney numbers of a manifold imposed by the geometric property of being the total space of a fibration. There are of course universal (Wu, Riemann-Roth) relations among characteristic numbers, and to avoid having to mention these, the results are best expressed in terms of appropriate cobordism rings. Accordingly, let a, be one of the standard cobordism rings and let w E R,. We say w fibers over a manifold B if there is a representative A4 of w which fibers over B. For fixed B, the set of w which fiber over I? is an ideal in II, (this depends on the fact' that if M represents w, a representative for -w can be realized by changing the R structure on M), and the description of this ideal completely describes the conditions imposed on the appropriate characteristic numbers. Note that no structure is imposed on I?, nor does the fiber map have to preserve any structure. Presumably for our results, the fibrations need not be differentiable, but only continuous. All characteristic numbers are tangental. A general reference for cobordism theory is Stong‘s book[l]. The first results of this type are due to Conner and Floyd[2]. For example, they showed the only condition imposed on the total space of a fibration over a circle S’ is the obvious one imposed by the fact that the Euler characteristic x is multiplicative for fibrations. That is, a class w in the unoriented cobordism ring 8, fibers over S’ if and only if the Stiefel-Whitney number w,(o) = 0. Conner continued the program in [3]. Meanwhile Burdick [4] investigated which oriented bordism classes fiber over S’ and obtained partial results. Neumann[5] completed the answer: a class o E R,So fibers over S’ if and only if the signature u(w) is zero, again an obvious necessary condition. The determination of which unitary bordism classes fiber over S’ is found in[6, p. 681; again the only condition is that the signature be zero. Brown[7] investigated which classes in (n, fiber over higher dimensional spheres. He showed that w E ‘%* fibers over S2 if and only if the Stiefel-Whitney number w,(w) = 0 if n is even and w~w,_~(w) = 0 if n is odd. Stong[8] considered a number of fibration problems, and in particular showed that for surfaces B other than S2 (orientable or not), the only condition for a class w E !J17, to fiber over B is the one imposed on w,(o) by the multiplicativity of the Euler characteristic x if x(B) is even. Nelson [9, Theorem 3.81 investigated which classes in unitary cobordism R,” fiber over complex projective spaces CP’ and determined that the signature u is the only obstruction to fibering a unitary cobordism class over S2. Again this is an obvious necessary condition since the signature is multiplicative for fibrations over simplyconnected base manifolds [ 101. We take the next step in the program and completely determine which classes in flz” and ReU fiber over which surfaces B. Let us recall that there are four surfaces with non-negative Euler characteristic, the sphere S2, the real projective plane P, the