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As mentioned in the Preface, knowledge of homotopy groups is not required to read Sect.  4.20 but the second half of Chap.  7 requires basic results for homotopy groups. We review here homotopy groups and characterize n-equivalences and weak homotopy equivalences in terms of homotopy groups. Moreover, we also prove some theorems which are required in the second half of Chap.  7 . In Sect.  7.9 , avoiding the use of cohomology groups, we defined the cohomological dimension \(\operatorname {cd}_\mathbb {Z}X\) by using the Eilenberg–MacLane complex \(K(\mathbb {Z},n)\) .

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Appendix: Homotopy Groups and Čech Cohomology Groups

  • 章节

  • 发表日期：

    2026年

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  • Katsuro Sakai

摘要

As mentioned in the Preface, knowledge of homotopy groups is not required to read Sect.  4.20 but the second half of Chap.  7 requires basic results for homotopy groups. We review here homotopy groups and characterize n-equivalences and weak homotopy equivalences in terms of homotopy groups. Moreover, we also prove some theorems which are required in the second half of Chap.  7 . In Sect.  7.9 , avoiding the use of cohomology groups, we defined the cohomological dimension \(\operatorname {cd}_\mathbb {Z}X\) by using the Eilenberg–MacLane complex \(K(\mathbb {Z},n)\) .