Chain homotopy and the de Rham theory
Chain homotopy and the de Rham theory
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链同伦和德拉姆理论
DOI：
10.1090/s0002-9939-1956-0087150-0
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发表时间：
1956
影响因子：
1.7
通讯作者：
D. Spencer
中科院分区：
文献类型：
--
作者：
V. Gugenheim;D. Spencer
介绍。本文给出了一种构造适合于de Rham上同调理论的链同伦算子的方法。特别地，证明了微同伦映射在微分形式的外代数上可以推导出链式同伦链映射（下面的公式13；参见[1]的80-81页，得到了相同的公式）。这表明de Rham理论满足S. Eilenberg和N. E. Steenrod意义上的“同伦公理”（cf. [2]）；因此可微可缩流形的de Rham上同调群是平凡的。这个基本结果通常被称为“庞加莱引理”。对概积结构给出了一个简单的推广。§5研究了几乎复杂和复杂的结构；在§6中给出了一个例子，证明了d上同伦不满足同伦公理，即使在复流形和解析同伦的情况下也是如此；这个例子是由K. Kodaira教授提出的。
Introduction. This note contains a method for constructing chainhomotopy operators suitable for the de Rham cohomology theory. In particular, it is proved that differentiably homotopic maps induce chain homotopic chain-mappings in the exterior algebra of differential forms (Formula 13 below; cf. pp. 80-81 of [l], where the same formula is obtained). This shows that the de Rham theory satisfies the "homotopy axiom" in the sense of S. Eilenberg and N. E. Steenrod (cf. [2]); hence the de Rham cohomology groups of a differentiably contractible manifold are trivial. This fundamental result is often referred to as the "Poincaré Lemma." A simple generalization is given in the case of an almost product structure. Almost complex and complex structures are investigated in §5 ; no genuine chain-homotopies are obtained, and in §6 is given an example which shows that d-cohomology does not satisfy the homotopy axiom, even in the case of complex manifolds and analytic homotopies; this example is due to Professor K. Kodaira.