同倫論
正文
同倫的概念,直觀上不難理解,同倫就是連續形變。以“形變收縮”為例,圖![同倫論](/img/0/0fe/nBnauM3X3IjMwEDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL3IzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![同倫論](/img/b/7f0/ml2ZuM3XzQzMwMDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzLzQzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
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![同倫論](/img/b/7f0/ml2ZuM3XzQzMwMDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzLzQzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
一般,設ƒ,g:x→Y為拓撲空間x到Y的兩個連續映射。如果有連續映射H:x×I→x,I=【0,1】,使得h(x,0)=ƒ(x),h(x,1)=g(x),則稱ƒ同倫於g,記作ƒ埍g。h是從ƒ到g的一個倫移,令ht(x)=h(x,t),人們也說連續依賴於參數t的一族映射ht:X→Y是從ƒ到g的一個倫移。若倫移ht在x的某個子集A上是靜止的,即h(x,t)=h(x,0),0≤t≤1,則說 ƒ相對於 A同倫於g,記作ƒ埍g(relA)前一段舉出的空間 H嶅K是空間K的形變收縮核,意思是指存在倫移h:K×I→K,使得對於x∈K有h(x,0)=x,h(x,1)∈H,而h(y,t)=y當y∈H,0≤t≤1。按照同倫關係埍,從x到Y的連續映射分成了同倫類。同倫類的集合記作【x,Y】。
在同倫論里,空間按同倫型而分類。若存在連續映射ƒ:x→Y,g:Y→x使得g。ƒ埍
![同倫論](/img/f/094/ml2ZuM3X5QjNxMDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL5QzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![同倫論](/img/8/f79/ml2ZuM3X5UDMzMDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL5UzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
同倫論的典型問題大體上有下列幾個,以下映射均指連續映射。
同倫問題 對於給定的映射ƒ,g:x→Y,如何判斷ƒ與g是否同倫?如果ƒ與常值映射同倫,則稱ƒ為零倫的,記作ƒ埍0。如何判斷給定映射ƒ是否零倫是這個典型問題的特例。
同調群提供了處理這個問題的工具。對任意整數n≥0,如果ƒ埍g,則
![同倫論](/img/d/641/ml2ZuM3X4MTOzMDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL4MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![同倫論](/img/a/88b/ml2ZuM3X4MDM5MjNyMTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL4MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![同倫論](/img/a/88b/ml2ZuM3X4MDM5MjNyMTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL4MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
但應注意,即使對所有n,ƒ
![同倫論](/img/a/88b/ml2ZuM3X4MDM5MjNyMTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL4MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![同倫論](/img/a/88b/ml2ZuM3X4MDM5MjNyMTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL4MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![同倫論](/img/a/88b/ml2ZuM3X4MDM5MjNyMTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL4MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
擴張問題 設A嶅x,給定映射ƒ:A→Y能否擴張為x到Y的映射,即是否存在映射g:X→Y,使得
。
![同倫論](/img/f/094/ml2ZuM3X5QjNxMDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL5QzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![同倫論](/img/b/d6e/ml2ZuM3X3QjN2MDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL3QzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
一般說來,ƒ:A→Y不一定能擴張。例如,對n≥1,恆同映射
![同倫論](/img/1/f1e/ml2ZuM3X2cTN3MDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL2czLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![同倫論](/img/7/0c5/ml2ZuM3X2YDN4MDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL2YzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
很多重要的問題可以轉化為映射擴張問題,1912年由L.E.J.布勞威爾首先提出的布勞威爾不動點定理就是典型一例。設 n≥0,ƒ:Dn→Dn是n維單位實心球體的自映射。則Dn中存在一點 x使得ƒ(x)=x。n=0時結論顯然成立。設n>0,如果對任意x∈Dn,ƒ(x)≠x,則令g(x)是ƒ(x)到x的有向線段的延長線與Sn-1的交點,即得到映射
![同倫論](/img/6/7b6/ml2ZuM3X5QjMzQDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL5QzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![同倫論](/img/d/cce/ml2ZuM3X5EDN0QDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL5EzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
同倫問題實際上是擴張問題的一個特例。設ƒ,g:x→Y是映射,可定義映射G:x×0∪x×1→Y為
一般地稱同倫映射所共有的性質為同倫性質,對於很多空間偶(x,A)(例如x是單純復形,A是子復形)來說,ƒ:A→Y能否擴張成為x到Y的映射也是一個同倫性質。
提升問題 在研究流形上有沒有非零向量場時,需要考慮映射的提升問題,它與擴張問題相對偶。提法如下:設p:x→B與ƒ:Y→B是映射,是否存在映射g:Y→x,使得pg=ƒ:Y→B。如果存在這樣的映射g,則稱g為ƒ關於p的提升。是否存在g的問題就是提升問題。
又設p:x→B,愝:Y→x均為映射,ƒt:Y→B,0≤t≤1,是倫移,使得p愝=ƒ0。是否存在倫移
同倫分類問題 對於給定空間x與Y,如何由x與Y的已知的可計算的不變數去計算從x到Y的映射同倫類集合【x,Y】,這是代數拓撲學中經常碰到的問題,特別是同倫群的計算等。
如果x與Y滿足一定的條件,則【x,Y】形成一個群。對n≥1及任意道路連通空間Y,W.赫維茨定義了πn(Y)=【Sn,Y】。可以證明πn(Y)是一個群,而且π1(x)就是龐加萊所定義的基本群。當n≥2,πn(Y)是交換群。從而把πn(Y)稱為空間Y的n維同倫群,它也是同倫不變數。
近幾十年代數拓撲學的發展表明,同倫群起著十分重要的作用。和同調群不同的是,對一般單純復形來說,同調群可以計算,但如何計算同倫群卻是一個至今遠未解決的問題,即使對十分簡單的n維球面Sn,當m相當大時,至今仍沒有計算群πm(Sn)的辦法。因此,同倫群的計算一直是代數拓撲學的重要課題。
如果π1(x)=0,則稱空間x是單連通的。一般地,群π1(x)通過交換化所得的交換群恰是H1(x)。此外,赫維茨又研究了高維同倫群與同調群的關係。如果
![同倫論](/img/d/5bc/ml2ZuM3X1ETN3QDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL1EzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![同倫論](/img/a/c26/ml2ZuM3X5kDN4QDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL5kzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
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關於球面同倫群的研究概況,首先要提出的是H.弗勒登塔爾的結果,他證明了
。
50年代初,J.P.塞爾提出了研究同倫群的新方法,他利用纖維化的譜序列,取得了球面同倫群計算的突破性進展。
關於
![同倫論](/img/3/685/ml2ZuM3X1UTNxUDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL1UzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
擴張問題與同倫分類問題之間存在一定關係。這方面,S.艾倫伯格首先定義了阻礙上鏈與阻礙上同調類的概念。設K是任意單連通的單純復形,n是正整數,K
![同倫論](/img/a/303/ml2ZuM3XzcjNyUDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzLzczLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![同倫論](/img/a/303/ml2ZuM3XzcjNyUDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzLzczLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
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![同倫論](/img/f/7e2/ml2ZuM3X0MjN4UDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzL0MzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
,
![同倫論](/img/2/4bb/ml2ZuM3XzAzNzUDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzLzAzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![同倫論](/img/2/4bb/ml2ZuM3XzAzNzUDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzLzAzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![同倫論](/img/2/4bb/ml2ZuM3XzAzNzUDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzLzAzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![同倫論](/img/2/4bb/ml2ZuM3XzAzNzUDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzLzAzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![同倫論](/img/2/4bb/ml2ZuM3XzAzNzUDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzLzAzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
若π為群,n為正整數,當n>1時,還假定π為交換群。如果道路連通空間Y滿足條件
![同倫論](/img/5/a22/ml2ZuM3XyYzMzYDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzLyYzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
普通同調群滿足七個公理,滿足維數公理(m>dim X時,Hm(x)=0)以外的六個公理的函子稱為廣義同調論。現在已經出現了許多有意義的廣義同調論。例如,對研究向量叢有重要意義的K-同調論K*;對研究微分流形有重要意義的協邊同調群MU*;對研究球面同倫群有重要意義的BP同調群BP*。E.布朗在 20世紀60年代初就已經證明,只要廣義上同調函子還滿足一定的條件,則這個廣義上同調群就自然同構於空間到一個固定空間(或空間譜)的所有映射同倫類所成的群。例如K*(x)=【x,BU】,MU(x)=【x,MU】,其中BU表示U群的分類空間;MU表示BU 的托姆空間。上面的結果說明同調論的問題又可以轉化為同倫論的問題。代數拓撲學的這兩個主要分支就統一起來了。
倫型問題 M.M.波斯尼科夫利用阻礙上同調類引進了一組能確定許多空間倫型的同倫不變數。設 x為一個單連通胞腔復形,對任意正整數n;可以作胞腔復形xn與映射ƒn:x→xn,使得:①
![同倫論](/img/f/a84/ml2ZuM3XzEjM0YDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzLzEzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![同倫論](/img/6/398/ml2ZuM3XyIjM1YDM4QTNxgDM5ETMwADMwADMwADMwADMxAzL1EzLyIzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
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參考書目
廖山濤、劉旺金著:《同倫論基礎》,北京大學出版社,北京,1980。
E.H.Spanier, Algebraic Topology, McGraw-Hill,New York, 1966.
R.M.Switzer,Algebraic Topology-Homotopy and homology, Springer-Verlag, New York, 1975.
G.W.Whitehead,Elements of Homotopy Theory,Graduate Texts in Mathematics, Vol.61,Springer-Verlag,New York, 1978.