Geodesic
Structure Theory for Montgomery-Samelson Fiberings between Manifolds, II
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 180-186

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Let f: Mn → Np be the projection map of an MS-fibering of manifolds β with finite non-empty singular set Aand simply connected total space (see 1). Results of Timourian (10) imply that (n, p) = (4, 3), (8, 5) or (16, 9), while a theorem of Conner (2) yields that #(A), the cardinality of the singular set, is equal to the Euler characteristic of Mn . We give an elementary proof of this fact and, in addition, prove that #(A) is actually determined by b n/2(Mn ), the middle betti number of Mn , or what is the same, by b n/2(Np – f(A)). It is then shown that β is topologically the suspension of a (Hopf) sphere bundle when Np is a sphere and b n /2(Mn ) = 0. It follows as a corollary that β must also be a suspension when Mn is n/4-connected with vanishing bn /2. Examples where bn /2 is not zero are constructed and we state a couple of conjectures concerning the classification of such objects. 
Antonelli, Peter L. Structure Theory for Montgomery-Samelson Fiberings between Manifolds, II. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 180-186. doi: 10.4153/CJM-1969-017-1
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