指導教授：王藹農臺灣大學：數學研究所闕昌漢Chueh, Chang-HanChang-HanChueh2014-11-302018-06-282014-11-302018-06-282014http://ntur.lib.ntu.edu.tw//handle/246246/264020這篇論文主要是整理[2]、[3]和[10]的結果，介紹定義在路徑上的同調群此一觀念，並討論它在有向圖上的一些應用，最後以此方法重新證明Brouwer’s fixed point theorem。The main content of this thesis is a reorganization of [2], [3], and [10]. We introduce the notion of path homology and discuss some applications on digraphs; finally we use the method to prove Brouwer’s fixed point theorem in an alternative way.Contents 1 Introduction 1 2 Basic definitions and properties 1 2.1 Preliminary. . . . . . . . . . . . . . . . . . . . . 1 2.2 Subspaces to develop the homologies of path complexes . . . . . 3 2.3 The relation between path complexes, simplicial complexes and digraphs . 7 2.4 Form and exterior differential . . . . . . . . . . . . . . . 10 2.5 ∂-invariant paths on digraphs . . . . . . . . . . . . . . . 14 2.6 Homologies of subgraphs. . . . . . . . . . . . . . . . 19 3 Sperner’s lemma and Brouwer’s fixed point theorem 26 3.1 Sperner’s lemma . . . . . . . . . . . . . . . . . . . 26 3.2 Brouwer’s fixed point theorem . . . . . . . . . . . . . . . 28 References 28548664 bytesapplication/pdf論文使用權限：不同意授權路徑同調有向圖thesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/264020/1/ntu-103-R95221030-1.pdf