In mathematics, the Seifert–van Kampen theoremof algebraic topology (named after Herbert Seifert and Egbert van Kampen), sometimes just called van Kampen's theorem, expresses the structure of the fundamental group of a topological space X in terms of the fundamental groups of two open, path-connected subspaces that cover X.
在數學中，代數拓撲的塞夫特-範卡姆彭定理(以赫伯特·塞夫特和埃格伯特·范卡姆彭命名)，有時也被稱為範卡姆彭定理，用覆蓋X的兩個開放的、路徑相連的子空間的基本組來表示拓撲空間X的基本組的結構。
It can therefore be used for computations of the fundamental group of spaces that are constructed out of simpler ones.
因此，它可以用來計算由簡單空間構造而成的基本空間群。
<p data-bjh-box="template_header" contenteditable="false" data-bjh-params="{"templateId":"1","color":"000000","index":"01","content":"Van Kampen" s="" theorem="" for="" fundamental="" groups="" 基本群的範坎彭定理"}'="" data-diagnose-id="7b5527c43d961bef0df14ca4f7758fab">01Van Kampen's theorem for fundamental groups 基本群的範坎彭定理Let X be a topological space which is the union of two open and path connected subspaces U1, U2. Suppose U1 ∩ U2 is path connected and nonempty, and let x0 be a point in U1 ∩ U2 that will be used as the base of all fundamental groups. The inclusion maps of U1 and U2 into X induce group homomorphisms J1: π1（U1，x0）→π1（X，x0） and J2: π1（U2，x0）→π1（X，x0）. Then X is path connected and j1 and j2 form a commutative pushout diagram:
設X為一個拓撲空間，它是兩個開連通子空間U1, U2的併集。假設U1∩U2是連通的非空路徑，假設x0是U1∩U2中的一個點，是所有基類的基礎。包含地圖U1和U2到X誘導組同態j1:π1 (U1, x0)→π1 (X, x0)和j2:π1 (U2, x0)→π1 (X, x0)。則X為路徑連接，j1和j2構成交換推出圖:
the natural morphism kis an isomorphism, that is, the fundamental group of X is the free product of the fundamental groups of U1 and U2 with amalgamation of π1(U1∩U2,x0).
自然態射k是一個同構,即X的基本組織是免費產品的基本組U1和U2的融合π1 (U1∩U2, x0)。
Usually the morphisms induced by inclusion in this theorem are not themselves injective, and the more precise version of the statement is in terms of pushouts of groups.
通常這個定理中包含的詞形本身不是單射的，而更精確的說法是關於群的推出。
<p data-bjh-box="template_header" contenteditable="false" data-bjh-params="{"templateId":"1","color":"000000","index":"02","content":"Van Kampen" s="" theorem="" for="" fundamental="" groupoids="" 基本群胚的範坎彭定理"}'="" data-diagnose-id="9ad744730b280e2e46688fd400d93d48">02Van Kampen's theorem for fundamental groupoids 基本群胚的範坎彭定理Unfortunately, the theorem as given above does not compute the fundamental group of the circle, which is the most important basic example in algebraic topology. The reason is that the circle cannot be realised as the union of two open sets with connected intersection. This problem can be resolved by working with the fundamental groupoid π1(X,A) on a set A of base points, chosen according to the geometry of the situation. Thus for the circle, one uses two base points.
不幸的是，上面給出的定理沒有計算圓的基本群，而圓是代數拓撲中最重要的基本例子。其原因是圓不能實現為兩個開口集的併集與相交連通。這個問題可以通過使用在一組基點集A上的廣群π1(X,A), 根據情況的幾何形狀進行選擇來解決。因此，對於圓，使用兩個基點。
This groupoid consists of homotopy classes relative to the end points of paths in Xjoining points of A ∩ X. In particular, if X is a contractible space, and A consists of two distinct points of X, then π1(X,A) is easily seen to be isomorphic to the groupoid often written L with two vertices and exactly one morphism between any two vertices. This groupoid plays a role in the theory of groupoids analogous to that of the group of integers in the theory of groups. The groupoid L also allows for groupoids a notion of homotopy: it is a unit interval object in the category of groupoids.
這個廣群由同倫類組成，同倫類相對於A∩ X的連接點中的路徑的端點。特別是，如果X是一個可收縮的空間，而A由X的兩個不同的點組成，那麼很容易看到π1(X,A)與廣群同構，通常寫成L有兩個頂點並且任意兩個頂點之間只有一個態射。這個群胚在群胚理論中所起的作用類似於群論中整數群的作用。廣群 L還允許廣群有一個同倫的概念:它是廣群類別中的一個單位間隔對象。
The category of groupoids admits all colimits, and in particular all pushouts.
廣群的類別允許所有的共極限，特別是所有的推出。
Theorem.Let the topological space X be covered by the interiors of two subspaces X1, X2 and let A be a set which meets each path component of X1, X2 and X0 = X1 ∩ X2. Then A meets each path component of X and the diagram P of morphisms induced by inclusion
定理。令拓撲空間X被兩個子空間X1、X2的內部所覆蓋，令A為滿足X1、X2的各路徑分量的集合，X0 = X1∩X2。則A滿足X的各路徑分量和包含誘導的形態圖P
is a pushout diagram in the category of groupoids.
是廣群類別中的推出圖。
This theorem gives the transition from topology to algebra, in determining completely the fundamental groupoid π1(X,A); one then has to use algebra and combinatorics to determine a fundamental group at some basepoint.
這個定理給出了從代數拓撲,在決定完全基本廣群π1(X,A);然後必須使用代數和組合學來確定某個基的基群。
One interpretation of the theorem is that it computes homotopy 1-types. To see its utility, one can easily find cases where Xis connected but is the union of the interiors of two subspaces, each with say 402 path components and whose intersection has say 1004 path components. The interpretation of this theorem as a calculational tool for "fundamental groups" needs some development of 'combinatorial groupoid theory'. This theorem implies the calculation of the fundamental group of the circle as the group of integers, since the group of integers is obtained from the groupoid L by identifying, in the category of groupoids, its two vertices.
該定理的一種解釋是，它計算同倫1-類型。要查看它的用途，可以很容易地找到X是連接的情況，但兩個子空間的內部的結合，每個有402個路徑分量，其交點有1004個路徑分量。將該定理解釋為「基本群」的計算工具需要「組合群理論」的一些發展。這個定理意味著將圓的基本組作為整數組來計算，因為整數組是從廣群L中通過在廣群類別中標識它的兩個頂點而得到的。
There is a version of the last theorem when Xis covered by the union of the interiors of a family {Uλ：λ∈Λ} of subsets.
有一個版本的最後一個定理,當X是被一個家族{Uλ:λ∈Λ}集合內部的子集覆蓋時。
The conclusion is that if Ameets each path component of all 1,2,3-fold intersections of the sets Uλ, then A meets all path components of X and the diagram
結論是,如果一個滿足所有的每個路徑組件1,2,3-重集Uλ的交集,那麼A滿足所有的路徑組件，
of morphisms induced by inclusions is a coequaliser in the category of groupoids.
由內含物引起的上述態射圖表是群胚類中的一種均衡現象。
people still obstinately persist, when calculating with fundamental groups, in fixing a single base point, instead of cleverly choosing a whole packet of points which is invariant under the symmetries of the situation, which thus get lost on the way. In certain situations (such as descent theorems for fundamental groups à la van Kampen) it is much more elegant, even indispensable for understanding something, to work with fundamental groupoids with respect to a suitable packet of base points.—Alexander Grothendieck人們在用基群進行計算，在固定一個基點時，仍然固執地堅持，而不是聰明地選擇整包的點，這些點在情況的對稱性下是不變的，從而在途中丟失。在某些情況下(例如van Kampen提出的關於基本群的下降定理)，處理關於合適的一組基點的基本群胚要優雅得多，甚至對於理解某些東西是必不可少的。—亞歷山大·格羅滕迪克
03Equivalent formulations 等價公式
In the language of combinatorial group theory, if Xis a topological space; U and V are open, path connected subspaces of X; U∩V is nonempty and path-connected; and ω∈U∩V; then π1(X,ω) is the free product with amalgamation of π1(U,ω) and π1(V,ω), with respect to the (not necessarily injective) homomorphisms I: π1(U∩V,ω)→π1(U,ω) and J:.π1(U∩V,ω)→π1(V,ω). Given group presentations:
在組合群理論的語言中，如果X是一個拓撲空間;U和V是開集，X路徑連通的子空間;U∩V非空，路徑連接; ω∈U∩V; 那麼π1 (X,ω)是π1 (U,ω)和π1 (V,ω)的自由乘積,關於(不一定是單射)同態I: π1(U∩V,ω)→π1(U,ω) 和 J:.π1(U∩V,ω)→π1(V,ω)。給出群表示：
the amalgamation can be presented as
合併可以表示為
π1(X,ω)=〈u,...,uk,v,...,vm|α,...αl,β,...βn,I(ω)(1/J(ω)),...,I(ωp)(1/J(ωp))〉.
In category theory, π1(X,ω) is the pushout, in the category of groups, of the diagram:
在範疇論中,π1 (X,ω)是群論中下圖的推出:
π1(U,ω)←π1(U∩V,ω)→π1(V,ω).
04Generalizations 一般化
As explained above, this theorem was extended by Ronald Brown to the non-connected case by using the fundamental groupoid π1(X,A) on a set A of base points. The theorem for arbitrary covers, with the restriction that A meets all threefold intersections of the sets of the cover, is given in the paper by Brown and Abdul Razak Salleh.The theorem and proof for the fundamental group, but using some groupoid methods, are also given in J. Peter May's book. The version that allows more than two overlapping sets but with A a singleton is also given in Allen Hatcher's book.如上所述,這個定理是延長羅納德·布朗非連通案例通過使用基點集A上的基本廣群π1(X,A)。任意復蓋定理，在滿足復蓋集合的所有三重交點的條件下，由布朗和阿卜杜勒·拉扎克·薩雷給出。基本群的定理和證明，但是使用了一些廣群方法，也在J. Peter May的書中給出。在Allen Hatcher的書中也給出了一個單例A允許兩個以上重疊的集合的版本。Applications of the fundamental groupoid on a set of base points to the Jordan curve theorem, covering spaces, and orbit spaces are given in Ronald Brown's book. In the case of orbit spaces, it is convenient to take A to include all the fixed points of the action. An example here is the conjugation action on the circle.
在羅納德·布朗的書中給出了基本群點在一組基點上的應用，包括空間和軌道空間。在軌道空間的情況下，取A來包含作用的所有不動點是方便的。這裡的一個例子是對圓的共軛作用。
References to higher-dimensional versions of the theorem which yield some information on homotopy types are given in an article on higher-dimensional group theories and groupoids. Thus a 2-dimensional van Kampen theorem which computes nonabelian second relative homotopy groups was given by Ronald Brown and Philip J. Higgins. A full account and extensions to all dimensions are given by Brown, Higgins, and Rafael Sivera, while an extension to n-cubes of spaces is given by Ronald Brown and Jean-Louis Loday.
在一篇關於高維群理論和群胚的文章中給出了該定理的高維版本的參考文獻，這些版本提供了關於同倫類型的一些信息。由此，羅納德·布朗和菲利普·j·希金斯給出了計算非阿貝爾第二相對同倫群的二維範坎彭定理。Brown、Higgins和Rafael Sivera給出了對所有維度的完整描述和擴展，而Ronald Brown和Jean-Louis Loday給出了對n-cubes空間的擴展。
Fundamental groups also appear in algebraic geometry and are the main topic of Alexander Grothendieck's first Séminaire de géométrie algébrique (SGA1). A version of van Kampen's theorem appears there, and is proved along quite different lines than in algebraic topology, namely by descent theory. A similar proof works in algebraic topology.
基本群也出現在代數幾何中，並且是Alexander Grothendieck的第一個幾何代數研討(SGA1)的主要主題。範坎本定理的一個版本出現在那裡，並沿著與代數拓撲不同的路線被證明，即通過下降理論。代數拓撲中也有類似的證明。