Differential forms in algebraic topology
Differential forms in algebraic topology
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DOI：
10.1007/978-1-4757-3951-0
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发表时间：
1982-05
期刊：
影响因子：
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通讯作者：
R. Bott;L. Tu
中科院分区：
文献类型：
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作者：
R. Bott;L. Tu
这本书的指导原则是使用微分形式作为辅助来探索代数拓扑学中一些不太容易理解的方面。相应地，我们主要移动在光滑流形的领域，并使用德罗姆理论作为所有上同调的原型。对于同伦理论的应用，我们还讨论了任意系数上同调的类比上同调。虽然我们的读者以前接触过代数或微分拓扑学，但在大多数情况下，良好的线性代数、高等微积分和点集拓扑学知识应该就足够了。对流形、单纯复形、奇异同调和上同调以及同伦群有一定的了解是有帮助的，但不是真正必要的。在文本本身中，我们谨慎地陈述了所需的更高级的结果，以便一个数学成熟的读者如果相信这些背景材料，应该能够在最低限度的先决条件下阅读整本书。这里的材料比一学期的课程所合理涵盖的要多得多。某些部分在第一次阅读时可能会被省略，而不会失去连续性。我们已经在下面的示意图中指出了这些。这本书的目的不是为了成为基础；相反，它只是为了打开现代代数拓扑学的强大大厦的一些大门。我们提供它是希望在半入门水平上对这个主题进行这样的非正式描述，以填补文献中的空白。
The guiding principle in this book is to use differential forms as an aid in exploring some of the less digestible aspects of algebraic topology. Accord ingly, we move primarily in the realm of smooth manifolds and use the de Rham theory as a prototype of all of cohomology. For applications to homotopy theory we also discuss by way of analogy cohomology with arbitrary coefficients. Although we have in mind an audience with prior exposure to algebraic or differential topology, for the most part a good knowledge of linear algebra, advanced calculus, and point-set topology should suffice. Some acquaintance with manifolds, simplicial complexes, singular homology and cohomology, and homotopy groups is helpful, but not really necessary. Within the text itself we have stated with care the more advanced results that are needed, so that a mathematically mature reader who accepts these background materials on faith should be able to read the entire book with the minimal prerequisites. There are more materials here than can be reasonably covered in a one-semester course. Certain sections may be omitted at first reading with out loss of continuity. We have indicated these in the schematic diagram that follows. This book is not intended to be foundational; rather, it is only meant to open some of the doors to the formidable edifice of modern algebraic topology. We offer it in the hope that such an informal account of the subject at a semi-introductory level fills a gap in the literature.