我們先介紹一些同調代數的相關知識、CW-複形以及胞腔同調群。接著以顏東勇教授
的代數拓樸教材為基礎，推導並重新定義出Hn(Xn,Xn−1) 的標準生成元，這些生成元使我
們可以用比一般課本教材更為精確且具幾何意義的方式，描述胞腔鏈複形邊界函數公式。有
了函數的degree 之概念後，我們就可以用胞腔鏈複形邊界函數公式來計算一些CW-複形的同
調群。

In this thesis, we first go through some facts in homological algebra, CW-complexes, and
cellular homology groups. Then by the algebraic topology course materials written by professor
Yan Dung-Yung, we can derive and redefine the canonical generators of Hn(Xn,Xn−1). These
generators provide a more precise and geometrical way to describe the cellular boundary formula
than the way in common textbooks. After introducing the concept of degree of a map,
we compute the homology groups of some CW complexes by cellular boundary formula.

Contents
Abstract (Chinese) I
Acknowledgements (Chinese) II
Abstract IV
Contents V
1 Introduction 1
2 Preliminaries 3
2.1 Settings and notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 A quick review on homological algebra . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 CW complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Cellular homology groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 The canonical generators for Hn(Xn,Xn−1) 20
3.1 The canonical generators for Hn(Dn, Sn−1) and Hn−1(Sn−1) . . . . . . . . . . . . 20
3.2 Geometrical interpretation of Hn(Xn,Xn−1) . . . . . . . . . . . . . . . . . . . . 25
3.3 The canonical generators for Hn(Xn/Xn−1) . . . . . . . . . . . . . . . . . . . . 29
4 Cellular boundary formula and its applications 36
4.1 Cellular boundary formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2 Cellular space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3 Degree of a map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.4 Computation of homology groups . . . . . . . . . . . . . . . . . . . . . . . . . . 49
References 55

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[3] Yan, Dung Yung. Lecture note of algebraic topology, National Tsing Hua University, Hsinchu, Taiwan, 2013.
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Hua University, Hsinchu, Taiwan, 2015.
[5] Chen, Jiun Jia. The three-dimensional animation of the geometric description of the boundary map of the
cellular chain complex, National Tsing Hua University, Hsinchu, Taiwan, 2018.
[6] Degiorgi, Paolo. Cellular Homology and the Cellular Boundary Formula, 2016.