Fields of 2-planes and two kinds of almost complex structures on compact 4-dimensional manifolds
Fields of 2-planes and two kinds of almost complex structures on compact 4-dimensional manifolds
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DOI：
10.1007/bf02571388
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发表时间：
1991-05
影响因子：
0.8
通讯作者：
Y. Matsushita
中科院分区：
文献类型：
--
作者：
Y. Matsushita
在本文中，所谓4-流形上的2-平面场，是指紧致定向光滑4-流形上的定向切2-平面场的非奇异场，4-流形上存在2-平面场的条件是Hirzebruch和Hopf [HH]的一个基本结果。在作者的早期文章[M]中，证明了Atiyah条件[A]是紧致单连通4-流形允许2-平面域的充分条件。Saeki [Sa]在Hirzebruch和Hopf定理的基础上，用Euler特征线和Hirzebruch指标给出了一个条件，并指出Atiyah条件对于交形式不确定的4-流形是充分的，但在一般情况下是不成立的。
Throughout this paper, by a field of 2-planes on a 4-manifold we shall mean a nonsingular field of oriented tangent 2-planes on a compact oriented smooth 4-manifold.The condition for a 4-manifold to admit a field of 2-planes is a fundamental result of Hirzebruch and Hopf [HH]. In the author's earlier paper [M], it is shown that Atiyah's condition [A] is sufficient for a compact simply-connected 4-manifold to admit a field of 2-planes. On the basis of the theorem of Hirzebruch and Hopf, Saeki [Sa] recently obtained a condition explicitly in terms of the Euler characteristics and the Hirzebruch indices, and pointed out that Atiyah's condition is in fact sufficient for a 4-manifold whose intersection form is indefinite and that it is not true in general.