論Понтрягин示性類Ⅴ

<正> 本文是這系列著作中Ⅱ的一個補充.在Ⅱ中(參閱Ⅱ的更正)我們證明了可微分閉流形的某些示性類特別是法3示性類的拓撲不變性.它的證明是隱合的(implicit).本文目的在進一步求得這些示性類用流形同調構造來表示的顧谿(explicit)公式,使我們能就任意可定向的可微分閉流形的這些示性類進行具體的計算.特別可以獲得下述結果:

ON PONTRJAGIN CLASSES V.

This paper gives the explicit formula for classes introduced in Ⅱ of this series of papers as follows. Let M be an oriented closed differentiable manifold of dimension m. For any odd prime p, let S_p~(2i(p-1)), T_p~(2i(p-1)) ∈H~(2i)(p-1))(M, I_p) be defined by S_p~(2i)(p-1)) U X~(m-2i(p-1))=X~(m-2i(p-1)),X~(m-2i(p-1)) ∈H~(m-2i(p-1)) (M, I_p), and in which are Steenrod powers in the new notations of J. P. Serre. Then the classes are given by In particular, for p=3, we deduce for the mod 3 Pontrjagin classes P_3~(4j) of the oriented manifold M: P_3~(4j)=T_3~(4j).As a corollary, we get the following result: In an oriented closed differentiable manifold M of dimension 2(p-l) or 2p-1, Q_p~(2(p-1) = 0. In particular, we have p_3~4 = 0 in an oriented closed differentiable manifold of dimension 4 or 5.