Rational Homotopy Theory
Rational Homotopy Theory
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DOI：
10.1007/978-1-4613-0105-9
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发表时间：
2000-12
期刊：
影响因子：
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通讯作者：
Y. Félix;S. Halperin;Jean-Claude Thomas
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文献类型：
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作者：
Y. Félix;S. Halperin;Jean-Claude Thomas
以及本专著最后一节中的未决问题列表。有理同伦理论的计算能力是由于Quillen [135]和Sullivan [144]发现了一个显式的代数公式。在每种情况下，拓扑空间的有理同伦类型与其代数模型的同构类相同，连续映射的有理同伦类型与模型之间对应的态射的代数同伦类相同。这些模型使得空间的有理同调和同伦变得透明。它们也（在原则上总是，在实践中有时）使其他同伦不变量的计算，如上同调中的杯积，同伦中的Whitehead积和有理Lusternik-Schnirelmann范畴。在其初始阶段，有理同伦理论的研究集中在这些模型的恒等式上。其中包括有理同伦不变量在同伦李代数中的形式化（将Whitehead积转化为同构11 '+1（X）~ 1 I下的圈空间OX的同伦群）。(OX、LS类别和锥长度。然而，从那时起，工作集中在这些变体的性质上，并发现了一些真正值得注意的，以前未被怀疑的现象。如果X是一个n维单连通有限CW复形，那么它的有理同伦群要么在2 '：2n度上消失，要么指数增长。
as well as by the list of open problems in the final section of this monograph. The computational power of rational homotopy theory is due to the discovery by Quillen [135] and by Sullivan [144] of an explicit algebraic formulation. In each case the rational homotopy type of a topological space is the same as the isomorphism class of its algebraic model and the rational homotopy type of a continuous map is the same as the algebraic homotopy class of the correspond ing morphism between models. These models make the rational homology and homotopy of a space transparent. They also (in principle, always, and in prac tice, sometimes) enable the calculation of other homotopy invariants such as the cup product in cohomology, the Whitehead product in homotopy and rational Lusternik-Schnirelmann category. In its initial phase research in rational homotopy theory focused on the identi of these models. These included fication of rational homotopy invariants in terms the homotopy Lie algebra (the translation of the Whitehead product to the homo topy groups of the loop space OX under the isomorphism 11'+ 1 (X)~ 1I.(OX», LS category and cone length. Since then, however, work has concentrated on the properties of these in variants, and has uncovered some truly remarkable, and previously unsuspected phenomena. For example• If X is an n-dimensional simply connected finite CW complex, then either its rational homotopy groups vanish in degrees 2': 2n, or else they grow exponentially.