一個H.Hopf推測的證明

<正> 假設一個有限複合形K上的定向S~2叢在K~3上有截面,那末Hopf從他的第二阻礙公式獲得了一個叢不變量△~4()∈H~4(K).Hopf曾經猜測過這個叢不變量與叢的4維示性類有關.本文的目的在證明這個推測是對的,更明確言之,應有

PROOF OF A CERTAIN CONJECTURE OF H.HOPF

For an oriented S~2-bundle with trivial 3-dim. Stiefel-Whitney class H. Hopf has deduced from his formula about second obstruction in this bundle a. certain bundle invariant ⊿~4.He has conjectured that this invariant coincides essentially with the 4-dim. Pontrjagin class of. Without assuming a priori the invariance of ⊿~4,we prove directly that Hopf’s conjecture is true.