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Conclutions of Two Diemension Torus

Author: ZhangHao
Tutor: YangMing
School: Nanjing Agricultural College
Course: Applied Mathematics
Keywords: fundamental group covering space intersection number
CLC: O186.12
Type: Master's thesis
Year: 2008
Downloads: 21
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Abstract


The properties of fundamental group of T2 are discussed deeply in this paper. The formula of intersection numbers of sub-manifolds on T2 and some conclusions on the fundamental group of T2 are derived.By summarizing the concept and characteristic of fundamental group, Structure of the fundamental group of T2 will be discussed in chapter 1.In chapter 2,the concept and properties of covering space are quoted. One method to judge the homotopic relations on closed paths is obtained.In chapter 3, we give some explanation on the concept and characteristic of intersection numbers in differential topology.In chapter 4, by covering space, intersection numbers and fundamental group of T2, the author makes some conclusions on how closed paths on T2 is being the necessary and sufficient condition of 1-submanifold and how 1-submanifold of T2 is being the necessaryand sufficient condition ofπ1（T2）’s generators.Theorem 4.5 Supposeα,β∈π1（T2）,e1,e2∈π1（T2）,and （?）=A（?）,hereA=(aij)2×2,then #（α,β）=det（A）#（e1,e2）.Theorem 4.6 Letα,βbe two 1-manifolds in T2,then [α],[β] are two generators ofπ1（T2） if and only if #（α,β）=±1.Theorem 4.7 Suppose [α]∈π1（T2） andα=me1+ne2,thenαis homotopic to 1-submanifold in T2 if m and n are prime to each other.Theorem 4.8 suppose [α]∈π1（T2）andαis not null homotopy in T2,thenαis homotopic to 1-submanifold in if and only if that there exists an elementβinπ1（T2） such than #（α,β）=±1.

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CLC: > Mathematical sciences and chemical > Mathematics > Geometry, topology > Differential geometry,integral geometry > Differential Geometry > Riemannian geometry
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