Seifert-van Kampten Theorem

11 Seifert-van Kampten Theorem

11.1 Direct sum of abelian groups

11.1.1 G_{\alpha} generates G ( an indexed family {G_{\alpha}}{\alpha \in J} of subgroups of an abelian group G) : Every element x of G can be written as a finite sum of elements of the groups G{\alpha}

11.1.1.1 G is the sum of the groups G_{\alpha}. G = \sum_{\alpha \in J} G_{\alpha}

11.1.1.2 Direct sum G of the groups G_{\alpha} : groups G_{\alpha} generate G, and for each x \in G the expression x = \sum x_{\alpha} for x is unique.

11.1.1.2.1 There is only one J-tuple (x_{\alpha})(\alpha \in J) with x{\alpha} = 0 for all but finitely many \alpha s.t. x = \sum x_{\alpha}.

11.1.1.3 G = \bigoplus_{\alpha \in J} G_{\alpha} or in the finite case, G = G_{1} \oplus \cdots \oplus G_{n}.

11.1.2 (Extension condition for direct sums) (Lem 67.1) Let G be an abelian group; let {G_{\alpha} } be a family of subgroups of G. If G is the direct sum of the groups G_{\alpha}, then G satisfies the following condition:

11.1.2.1 Given any abelian group H and any family of homomorphisms h_{\alpha} : G_{\alpha} -> H, there exists a homomorphism h : G -> H whose restriction to G-{\alpha} equals h_{\alpha}, for each \alpha.

11.1.2.1.1 h is unique.

11.1.2.2 Conversely, If the groups G_{\alpha} generate G and the extension condition holds, then G is the direct sum of the groups G_{\alpha}.

11.1.2.3 (Cor 67.2)Let G = G_{1} \oplus G_{2}. Suppose G_{1} is the direct sum of subgroups H_{\alpha} for \alpha \in J, and G_{2} is the direct sum of subgroups H_{\beta} for \beta \in K, where the index sets J and K are disjoint. Then G is the direct sum of the subgroups H_{\gamma}, for \gamma \in J \cup K.

11.1.2.4 (Cor 67.3) If G = G_{1} \oplus G__{2}, then G/G_{2} is isomorphic to G_{1}.

11.1.3 External direct sum G of the groups G_{\alpha} relative to the monomorphisms i_{\alpha}. ( an indexed family {G_{\alpha}}{\alpha \in J} of subgroups of an abelian group G) Let i{\alpha} : G_{\alpha} -> G is a family of monomorphisms, such that G is the direct sum of the groups i_{\alpha} (G_{\alpha}).

11.1.4 (Thm 67.4) Given a fimly of abelian groups {G_{\alpha}}{\alpha \in J}, there exists an abelian group G and a family of monomorphisms i{\alpha} : G_{\alpha} -> G s.t. G is the direct sum of the groups i_{\alpha} (G_{\alpha}).

11.1.5 (Lem 67.5) Let {G_{\alpha}}{\alpha \in J} be an indexed family of abelian groups; Let G be an abelian group; Let i{\alpha} : G_{\alpha} -> G be a family of homomorphisms. If each i_{\alpha} is a monomorphism and G is the direct sum of the groups i_{\alpha} (G_{\alpha})., then G satisfies the following extension condition:

11.1.5.1 Given any abelian group H and any family of homomorphisms h_{\alpha} : G_{\alpha} -> H, there exists a homomorphism h : G -> H whose restriction to G-{\alpha} equals h_{\alpha}, for each \alpha

11.1.5.1.1 h is unique.

11.1.5.2 Conversely, If the groups i_{\alpha} (G_{\alpha}) generate G and the extension condition holds, then each i_{\alpha} is a monomorphism, and G is the direct sum of the groups G_{\alpha}.

11.1.6 (Uniqueness of direct sums) (Thm 67.6) Let {G_{\alpha}}{\alpha \in J} be an indexed family of abelian groups; Let G and G’ be an abelian group; Let i{\alpha} : G_{\alpha} -> G and i’{\alpha} : G{\alpha} -> G’ be a family of monomorphisms, such that G is the direct sum of the groups i_{\alpha} (G_{\alpha}) and G’ is the direct sum of the groups i’{\alpha} (G{\alpha}).. Then there is a unique isomorphism \phi : G -> G’ s.t. \phi \bullet i_{\alpha} = i’_{\alpha} for each \alpha.

11.1.6.1 pf) (Lem 67.5)

11.1.7 Free abelian group G having the elements {a_{\alpha}} as a basis. ( an indexed family {a_{\alpha}} of elements of an abelian group G) : Let G_{\alpha} be the subgroup of G generated by a_{\alpha}. If the groups G_{\alpha} generates, we also say that elements a_{\alpha}} generate G. If each group G_{\alpha} is infinite cyclic, and if G is the direct sum of the groups G_{\alpha}.

11.1.8 (Lem 67.6) Let G be an abelian group; let {a_{\alpha}}{\alpha \in J} be a family of elements of G that generates G. Then G is a free abelian group with basis {a{\alpha}} iff for any abelian group H and any family {y_{\alpha}} of elements of H, there is a homomorphism h of G into H s.t. h(a_{\alpha}) = y_{\alpha} for each \alpha. In such case, h is unique.

11.1.8.1 pf) (Lem 67.1)

11.1.9 (Thm 67.8) If G is a free abelian group with basis {a_{1}, …., a_{n}}, then n is uniquely determined by G.

11.1.10 Rank of G ( a free abelian group with a finite basis) : the number of elements in a basis of G.

11.2 Free products of Groups

11.2.1 Let G be a group. Let {G_{\alpha}}{\alpha \in J} be a family of subgroups of G, and {G{\alpha}}_{\alpha \in J} generate G.

11.2.1.1 word of length n in the groups G_{\alpha} : a finite sequence (x_{1}, …, x_{n}) of elements of the groups G_{\alpha} s.t. x = x_{1} \cdots x_{n}.

11.2.1.1.1 Represents the element x of G.

11.2.1.2 Reduced word : a word representing x of the form (y_{1}, …, y_{m}) where no group G_{\alpha} contains both y_{i} and y_{i+1}, and where y_{i} \neq 1 for all i.

11.2.2 G is the free product of the groups G_{\alpha} ( an indexed family {G_{\alpha}}{\alpha \in J} of subgroups of an abelian group G that generates G) : Suppose that G{\alpha} \cap G_{\beta} consists of the identity element alone whenever \alpha \neq \beta. For each x \in G, there is only one reduced word in the groups G_{\alpha} that represents x.

11.2.2.1 G = \prod_{\alpha \in J}^{*} G_{\alpha} or in the finite case, G = G_{1} * \cdots * G_{n}.

11.2.3 (Lem 68.1) a family {G_{\alpha}} } of subgroups of a group G. If G is the free product of the groups G_{\alpha}, then G satisfies the following condition:

11.2.3.1 Given any group H and any family of homomorphisms h_{\alpha} : G_{\alpha} -> H, there exists a homomorphism h : G -> H whose restriction to G_{\alpha} equals h_{\alpha}, for each \alpha.

11.2.3.1.1 h is unique

11.2.4 External free product G of the groups G_{\alpha} relative to the monomorphisms i_{\alpha}. ( an indexed family {G_{\alpha}}{\alpha \in J} of groups, a group G) Let i{\alpha} : G_{\alpha} -> G is a family of monomorphisms, such that G is the free product of the groups i_{\alpha} (G_{\alpha}).

11.2.5 (Lem 68.2)Given a family {G_{\alpha}}{\alpha \in J} of groups, There exists a group G and a family of monomorphisms i{\alpha} : G_{\alpha} -> G s.t. G is the free product of the groups i_{alpha}(G_{\alpha}).

11.2.6 (Extension condition for ordinary free products) (Lem 68.3) Let {G_{\alpha}} be a family of groups; Let G be a group ; Let i_{\alpha} : G_{\alpha} -> G be a family of homomorphisms. If each i_{\alpha} is a monomorphism and G is the free product of the groups i_{\alpha}(G_{\alpha}, then G satisfies the following condition:

11.2.6.1 Given any group H and any family of homomorphisms h_{\alpha} : G_{\alpha} -> H, there exists a homomorphism h : G -> H whose restriction to G_{\alpha} equals h_{\alpha}, for each \alpha.

11.2.6.1.1 h is unique

11.2.7 (Uniqueness of free products)( Thm 68.4) Let {G_{\alpha}}{\alpha \in J} be a family of groups; Let G and G’ be groups; Let i{\alpha} : G_{\alpha} -> G and i’{\alpha} : G{\alpha} -> G’ be a family of monomorphisms, such that the families [ i_{\alpha} (G_{\alpha}) ] and [ i’{\alpha} (G{\alpha}) ] generate G and G’. If both G and G’ have the extension property stated in the preceding lemma, then there is a unique isomorphism \phi : G -> G’ s.t. \phi \bullet i_{\alpha} = i’_{\alpha} for all \alpha.

11.2.8 (Lem 68.5) Let {G_{\alpha}} be a family of groups; Let G be a group ; Let i_{\alpha} : G_{\alpha} -> G be a family of homomorphisms. If rhe extension condition of (Lem 68.3) holds, then each i_{\alpha} is a monomorphism and G is the free product of the groups i_{\alpha}(G_{\alpha}

11.2.8.1 (Cor 68.6) Let G = G_{1} * G_{2}. Suppose G_{1} is the free product of the subgroups H_{\alpha} for \alpha \in J, and G_{2} is the free product of subgroups H_{\beta} for \beta \in K. If the index sets J and K are disjoint, then G is the free product of the subgroups {H_{\gamma}}{\gamma \in J \cup K}.

11.2.9 (Thm 68.7) Let G = G_{1} * G_{2}. Let N_{i} be a normal subgroup of G_{i}, for i = 1,2. If N is the least normal subgroup of G that contains N_{1} and N_{2}, then G / N \cong (G_{1} / N_{1}) * (G_{2} / N_{2}).

11.2.9.1 (Cor 68.8) If N is the least normal subgroup of G_{1} * G_{2} that contains G_{1}, then (G_{1} * G_{2}) / N \cong G_{2}.

11.2.10 (Lem 68.9) Let S be a subset of the group G. If N is the least normal subgroup of G containing S, then N is generatied by all conjugates of elements of S.

11.3 Free groups

11.3.1 Let G be a group; let {a_{\alpha}} be a family of elements of G, for \alpha \in J. We say trhe elements {a_{\alpha}} generate G if every element of G can be written a s a product of powers of the elements a_{\alpha}.

11.3.1.1 If the family {a_{\alpha}} is finite, we say G is finitely generated.

11.3.2 Let {a_{\alpha}} be a family of elements of a group G. Suppose each a_{\alpha} generates an infinite cyclic subgroup G_{\alpha} of G. If G is a free product of the groups {G_{\alpha}}, then G is said to be a free group.

11.3.2.1 the family {a_{\alpha}} is called system of free generators for G.

11.3.3 (Lem 69.1) Let G be a group; let {a_{\alpha}}{\alpha \in J} be a family of elements of G. If G is a free group with system of free generators {a{\alpha}}, then G satisfies the following condition(*) : Furthermore, h is unique.

11.3.3.1 (*) Given any group H and any family {y_{\alpha}} of elements of H, there exists a homomorphism h : G -> H whose h(a_{\alpha}) = y_{\alpha}, for each \alpha.

11.3.3.2 If the extension condition (*) holds, then G is a free group with system of free generators {a_{\alpha}}.

11.3.4 (Thm 69.2) Let G = G_{1} * G_{2}, where G_{1} and G_{2} are free groups with {a_{\alpha}}{\alpha \in J} and {a{\alpha}}{\alpha \in K} as respective systems of free generators. If J and K are disjoint, then G is a free group with {a{\alpha}}_{\alpha \in J \cup K} as a system of free generators.

11.3.5 Let {a_{\alpha}}{\alpha \in J} be an arbitrary indexed family. Let G{\alpha} denote the set of all symbols of the form a_{\alpha}^{n} for n \in \mathbb{Z}. We make G_{\alpha} into a group by defining a_{\alpha}^{n} \cdot a_{\alpha}^{n} = a_{\alpha}^{n+m}. Then a_{\alpha}^{0} is the identity element of G_{\alpha}, and a_{\alpha}^{-n} is the inverse of a_{\alpha}^{n} The external free product of the groups {G_{\alpha}} is called the free group on the elements a_{\alpha}.

11.3.5.1 We denote a_{\alpha}^{1} simply by a_{\alpha}.

11.3.6 Let G be a group. If x , y \in G, we denote by [x,y] the element [x,y] = xyx^{-1}y^{-1} of G; it is called the commutator of x and y. The subgroup of G generated by the set of all commutators in G is called the commutator subgroup of G and denoted [G,G].

11.3.7 (Lemma 69.3) Given G, the subgroup [G,G] is a normal subgroup of G and the quotient group G/[G,G] is abelian, If h:G->H is any homomorphism from G to an abelian group H, then the kernel of h contained [G,G], so h induces a homomorphism k : G/[G,G] -> H.

11.3.8 (Thm 69.4) If G is a free group with free generators a_{\alpha}, then G/[G,G] is a free abelian group with basis [a_{\alpha}], where [a_{\alpha]} denotes the coset of a_{\alpha} in G/[G,G].

11.3.8.1 (Cor 69.5) If G is a free group with n free generators, then any system of free generators for G has n elements.

11.3.9 If G is a group, a presentation of G consists of a family {a_{\alpha}} of generators for G, along with a complete set {r_{\beta}} of relations for G, where each r_{\beta is an element of the free group on the set {a_{\alpha}}.

11.3.9.1 If the family {a_{\alpha}} is finite, the G is finitely generated. If both the families {a_{\alpha} } and {r_{\beta}} are finite, then G is said to be finitely presented, and these families form a finite presentation for G.

11.4 Seifert-van Kampen Theorem

11.4.1 (Seifert-van Kampen Theorem) (Thm 70.1) : Let X = U \cup V, where U and V are open in X; assume U, V, and U \cap V are path connected; let x_{0} \in U \cap V. Let H be a group, and let \phi_{1} : \pi_{1}(U,x_{0}) -> H and \phi_{2} : \pi_{1}(V,x_{0}) -> H be homomorphisms. Let i_{1}, i_{2}, j_{1}, j_{2} be the homomorphisms indicated in the following diagram, each induced by inclusion. If \phi_{1} \bullet i_{1} = \phi_{2} \bullet i_{2}, then there is a unique homomorphism \Phi : \phi_{1} : \pi_{1}(X,x_{0}) -> H s.t. \Phi \bullet j_{1} = \phi_{1} and \Phi \bullet j_{2} = \phi_{2}.

11.4.1.1 \begin {tikzpicture} \matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em] {& \pi_{1} (U,x_{0}) &\ \pi_{1} (U,x_{0})&\pi_{1}(U,x_{0}) & H \& \pi_{1} (U,x_{0})&\}; \path[-stealth] (m-1-2) edge node [right] {$ j_{1} $} (m-2-2) edge node[above] {$ \phi_{1} $} (m-2-3) (m-2-1) edge node [above] {$ i_{1} $} (m-1-2) edge (m-2-2) edge node [below] {$ i_{2} $} (m-3-2) (m-2-2) edge node [above] {$ \Phi $} (m-2-3) (m-3-2) edge node [right] {$ j_{2} $} (m-2-2) edge node[below] {$ \phi_{2} $} (m-2-3); \end {tikzpicture}

11.4.2 (Seifert-van Kampen theorem, classical version) (Thm 70.2) Assume the hypotheses of the preceding theorem. Let j : \pi_{1} (U,x_{0}) * \pi_{1}(V,x_{0}) -> \pi_{1} (X,x_{0}) be the homomorphism of the free product that extends the homomorphisms j_{1} and j_{2} induced by inclusion. Then j is surjective, and its kernel is the least normal subgroup N of the free product that contains all elements represented by words of the form (l_{1}(g)^{-1} i_{2}(g)), for g \in \pi_{1} (U \cap V ,x_{0}).

11.4.2.1 (Cor 70.3) Assume the hypotheses of the Seifert-van Kampen theorem. If U \cap V is simply connected, then there is an isomorphism k : pi (U,x_{0}) * \pi_{1}(V,x_{0}) -> \pi_{1} (X,x_{0})

11.4.2.2 (Cor 70.4) Assume the hypotheses of the Seifert-van Kampen theorem. If V is simply connected, then there is an isomorphism k : \pi_{1} (U,x_{0})/N -> \pi_{1} (X,x_{0}) where N is the least normal subgroup of \pi_{1} (U,x_{0}) containing the image of homomorphism i_{1} : \pi_{1} (U \cap V ,x_{0}) -> \pi_{1} (U,x_{0})

11.5 Fundamental group of a wedge of circles

11.5.1 Wedge of the circles S_{1} , …, S_{n} (a Hausdorff space X ) : X is the union of the subspaces S_{1} , …, S_{n} each of which is homeomorphic to the unit circle S^{1}. There is a point of X s.t. S_{i} \cap S_{j} = {p} whenever i \neq j.

11.5.2 (Thm 71.1) Let X be the wedge of the circles S_{1} , …, S_{n}; let p be the common point of these circles. then \pi_{1} (X,p) is a free group. If f_{i} is a loop in S_{i} that represents a generator of \pi_{1} (S_{i},p), then the loops f_{1} , …, f_{n} represent a system of free generators for \pi_{1} (X,p).

11.5.3 \Tau Coherent with X_{\alpha} ( a topology of a space X which is union of the subspaces X_{\alpha} for \alpha \in J) : a subset C of X is closed in X if C \cap X_{\alpha} is closed in X_{\alpha} for each \alpha.

11.5.3.1 equivalent condition is that a set be open in X if its intersection with each X_{\alpha} is open in X_{\alpha}.

11.5.4 wedge X of the circles S_{\alpha} ( a space X that is the union of the subspaces X_{\alpha} for \alpha \in J) : Let each of S_{\alpha} be homeomorphic to the unit circle. Assume there is a point p of X s.t. S_{\alpha} \cap S_{\beta} = {p} whenever \alpha \neq \beta. If the topology of X is coherent with the subspaces S_{\alpha}, then X is called the wedge of the circles S_{\alpha}.

11.5.5 (Lem 71.2) Let X be the wedge of the circles S_{\alpha}, for \alpha \in J. The X is normal.

11.5.5.1 Any compact subspace of X is contained in the union of finitely many circles S_{\alpha}.

11.5.6 (Thm 71.3) Let X be the wedge of the circles S_{\alpha}, for \alpha \in J; Let p be the common point of these circles. Then \pi_{1} (X,p) is a free group. If f_{\alpha} is a loop in S_{\alpha} representing a generator of \pi_{a} (S_{\alpha} , p), then the loops {f_{\alpha}} represent a system of free generators for \pi_{1} (X,p).

11.5.7 (Lem 71.4) Given an index set J, there exists a space X that is a wedge of circles S_{\alpha} for \alpha \in J.

11.6 Adjoining a Two-cell

11.6.1 (Thm 72.1)Let X be a Hausdorff space; let A be a closed path-connected subspace of X. Suppose that there is a continuous map h : B^{2} -> X that maps Int B^{2} bijectively onto X-A and maps S^{1} = Bd B^{2} into A. Let p \in S^{1} and let a = h(p); let k : (S^{1}, p) -> (A,a) be the map obtained by restricting h. Then the homomorphism i_{} : \pi_{1} (A,a) -> \pi_{1} (X,a) induced by inclusion is surjective, and its kernel is the least normal subgroup of \pi_{1} (A,a) containing the image of k_{} : \pi_{1} (S^{1},p) -> \pi_{1} (A,a)

11.6.1.1 The fundamental group of X is obtained from the fundamental group of A by killing off the class k_{*} [ f ] , where [f] generates \pi_{1} (S^{1},p).

11.7 Fundamental groups of the Torus and the Dunce cap

11.7.1 (Thm 73.1) The fundamental group of the torus has a presentation consisting of two generators \alpha, \beta and a single relation \alpha \beta \alpha^{-1} \beta^{-1} .

11.7.1.1 (Cor) The fundamental group of the torus is a free abelian group of rank 2.

11.7.2 n-fold dunce cap X : Let n be a positive integet with n > 1. Let r : S^{1} -> S^{1} be rotation through the angle \frac{2\pi}{n}, mapping the point (cos\theta, sin\theta) to the point (cos(\theta +\frac{2\pi}{n} ) , sin(\theta+\frac{2\pi}{n})). Form a quotient space X from the unit ball B^{2} by identifying each point x of S^{1} with the points r(x) , r^{2}(x), …, r^{n-1}(x).

11.7.2.1 X is a compact Hausdorff space.

11.7.3 (Lem 73.3) Let \pi : E -> X be a closed quotient map. If E is normal, then so is X.

11.7.4 (Thm 73.4) The fundamental group of the n-fold dunce cap is a cyclic group of order n.