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Proposition 24.24. The annulus, the Möbius band, the torus and the Klein bottle are, up to homeomorphism, the only connected, compact 2-dimensional topological manifolds that can be written as a mapping torus \( \operatorname{Tor}\left( {X, f}\right) \) of a connected 1-dimensional topological manifold \( X \) . | Proof. By the examples (1), (2) on page 713 and by Lemma 24.21 we know that there exist the following types of homeomorphisms:\n\n\[ \begin{aligned} \text{ annulus }{S}^{1} \times \left\lbrack {0,1}\right\rbrack & \cong \operatorname{Tor}\left( {\left\lbrack {0,1}\right\rbrack ,\mathrm{{id}}}\right) \\ \text{ Möbius ba... | Yes |
Let \( X \) be a topological space and let \( f : X \rightarrow X \) be a homeomorphism. Then the map\n\n\[ \nX \times \mathbb{R} \rightarrow \left( {X \times \mathbb{R}}\right) /{\mathbb{Z}}_{f} = \operatorname{Tor}\left( {X, f}\right) \n\]\n\n\[ \n\left( {x, t}\right) \mapsto \left\lbrack \left( {x, t}\right) \right\... | Proof. Let \( X \) be a topological space and let \( f : X \rightarrow X \) be a homeomorphism. The first statement is an immediate consequence of Lemma 24.25 and Proposition 16.10.\n\nNow let \( k \in \mathbb{N} \). We consider the following commutative diagram of maps:\n\n\n\nThus we have obtained an isomorphism \( \psi : {\pi }_{1}\left( {\operatorname{Tor}\left( {X, f}\right) }\right) \overset{ \cong }{ \rightarrow }{\pi }_{1}\left( X\right) \rtimes \mathbb{Z} \) . We... | No |
Proposition 25.1. The direct limit of any direct system exists in the following categories:\n\n\( \left( 0\right) \) the category of sets,\n\n(1) the category Top of topological spaces,\n\n(2) the category \( \mathcal{G}r \) of groups,\n\n(3) the category AbGr of abelian groups,\n\n(4) the category Ring of rings,\n\n(5... | Sketch of PROOF. Let \( \left( {I, \leq }\right) \) be a preordered set and let \( \mathcal{C} \) be one of the six given categories. Furthermore let \( \left( {{\left\{ {X}_{i}\right\} }_{i \in I},{\left\{ {f}_{ij} : {X}_{i} \rightarrow {X}_{j}\right\} }_{i \leq j}}\right) \) be a direct system in \( \mathcal{C} \) .\... | No |
Lemma 25.2. Let \( {f}_{n} : {X}_{n} \rightarrow {X}_{n + 1} \) be a sequence of morphisms in any category. Suppose there exists an \( N \in \mathbb{N} \) such that all \( {f}_{n} \) for \( n \geq N \) are isomorphisms. Then \( \underline{\lim }{X}_{n} \) exists and it is naturally isomorphic to \( {X}_{N} \), more pre... | Proof. We will provide the proof in Exercise 25.2. | No |
Lemma 25.3. Let \( \left( {{\left\{ {X}_{i}\right\} }_{i \in {\mathbb{N}}_{0}},{\left\{ {f}_{ij}\right\} }_{i \leq j}}\right) \) be a direct system in the category of topological spaces. We assume that all the maps \( {f}_{ij} : {X}_{i} \rightarrow {X}_{j} \) are inclusion maps.\n\n(1) In this case one can in fact take... | Proof \( \left( *\right) \).\n\n(1) The argument is the same as in the proof of Proposition 25.1 (1).\n\n(2) Let \( i \in \mathbb{N} \) . The map \( {X}_{i} \rightarrow \mathop{\lim }\limits_{ \rightarrow }{X}_{i} \) is evidently injective and continuous. By Lemma 2.42 (1) it remains to show that the map is an open map... | Yes |
Lemma 25.4. (*) Let \( \\left( {I, \\leq }\\right) \) be a preordered set and let \( \\left( {{\\left\{ {X}_{i}\\right\} }_{i \\in I},{\\left\{ {f}_{ij} : {X}_{i} \\rightarrow {X}_{j}\\right\} }_{i \\leq j}}\\right) \) be a direct system in the category of topological spaces. We denote by \( {g}_{i} : {X}_{i} \\rightar... | SKETCH OF PROOF. This lemma follows easily from the explicit description of the direct limit in the category of topological spaces that we gave in Proposition 25.1 together with Lemma 5.18. | No |
Lemma 25.5. (*) Let \( \left( {I, \leq }\right) \) be a preordered set and let \( \left( {{\left\{ {X}_{i}\right\} }_{i \in I},{\left\{ {f}_{ij} : {X}_{i} \rightarrow {X}_{j}\right\} }_{i \leq j}}\right) \) be a direct system in the category of topological spaces. Let \( G \) be a group. Suppose that for each \( i \in ... | Sketch of PROOF. This lemma follows easily from the explicit description of the direct limit in the category of topological spaces that we gave in Proposition 25.1. | No |
Lemma 25.8. Let \( {X}_{i}, i \in \mathbb{N} \) be a sequence of topological spaces such that each \( {X}_{i} \) is open in \( {X}_{i + 1} \) . Then every compact subset of \( \underline{\lim }{X}_{i} \) is already contained in one of the \( {X}_{i} \) . | Proof of Lemma 25.8. Let \( {X}_{i}, i \in \mathbb{N} \) be a nested sequence of topological spaces such that each \( {X}_{i} \) is open in \( {X}_{i + 1} \) .\n\nClaim. For every \( i \in \mathbb{N} \) the subset \( {X}_{i} \) is open in \( X \) .\n\nBy the definition of the topology on \( X \) we have to show that fo... | Yes |
Lemma 25.9. Let \( \left( {X,{x}_{0}}\right) \) be a pointed topological space.\n\n(1) Given any \( g \in {\pi }_{1}\left( {X,{x}_{0}}\right) \) there exists a compact subset \( K \) of \( X \) with \( {x}_{0} \in K \) such that \( g \) lies in the image of the inclusion induced map \( {\pi }_{1}\left( {K,{x}_{0}}\righ... | (1) Let \( g \in {\pi }_{1}\left( {X,{x}_{0}}\right) \) . We represent \( g \) by a loop \( \gamma : \left\lbrack {0,1}\right\rbrack \rightarrow X \) . By Lemma 2.40 the image \( \gamma \left( \left\lbrack {0,1}\right\rbrack \right) \) is compact. Thus \( K \mathrel{\text{:=}} \gamma \left( \left\lbrack {0,1}\right\rbr... | Yes |
Lemma 25.11. The fundamental group of the surface of infinite genus is the free group on infinitely many generators. | Proof. We continue with the notation from Lemma 25.10. By Lemma 25.10 (2) it suffices to show that the fundamental group of \( \underline{\lim }{X}_{i} \) is a free group on infinitely many generators. By Lemma 23.11 we have 451\n\n\[ \n{\pi }_{1}\left( F\right) = \left\langle {a, b, c,{c}^{\prime } \mid c = \left\lbra... | No |
(1) There exists an action of \( \mathbb{Z} \) on \( {\sum }_{\infty } \) that is discrete and continuous and such that the quotient \( {\sum }_{\infty }/\mathbb{Z} \) is homeomorphic to \( {\sum }_{g} \) . | Sketch of Proof.\n\n\( {}^{452} \) Indeed, we write \( X = \left\{ {{x}_{1},{y}_{1},\ldots ,{x}_{m},{y}_{m},{a}_{1},{b}_{1},\ldots ,{a}_{m},{b}_{m}}\right\}, u = \mathop{\prod }\limits_{{j = 1}}^{m}\left\lbrack {{a}_{j},{b}_{j}}\right\rbrack, v = \mathop{\prod }\limits_{{j = 1}}^{m}\left\lbrack {{x}_{j},{y}_{j}}\right\... | No |
Proposition 25.13. Let \( g \geq 2 \) . We consider the automorphism\n\n\[ \n\varphi : \underset{\text{free group of infinite rank }}{\underbrace{\left\langle {\left\{ {x}_{1, i},{y}_{1, i},\ldots ,{x}_{g - 1, i},{y}_{g - 1, i}\right\} }_{i \in \mathbb{Z}}\right\rangle }} \rightarrow \underset{\text{same free group of ... | Sketch of PROOF. We consider the surface \( {\sum }_{\infty } \) of infinite genus together with the \( \mathbb{Z} \) - action that is provided by Lemma 25.12 (1). By Proposition 16.9 (2) we know that the projection \( p : {\sum }_{\infty } \rightarrow {\sum }_{\infty }/\mathbb{Z} = {\sum }_{g} \) is a covering.\n\nWe ... | Yes |
Proposition 25.14. Let \( g \geq 2 \) . Every abelian subgroup of \( {\pi }_{1}\left( {\sum }_{g}\right) \) is isomorphic to \( \mathbb{Z} \) . | Proof. This proposition is a pleasant application of Proposition 25.13 and Exercise 19.11. We will provide the details in Exercise 25.9. | No |
Proposition 25.15. The inverse limit of any inverse system exists in the following categories:\n\n(0) the category Set of sets,\n\n(1) the category Top of topological spaces,\n\n(2) the category \( \mathcal{G}r \) of groups,\n\n(3) the category TopGr of topological groups,\n\n(4) the category AbGr of abelian groups,\n\... | Proof. Let \( \left( {I, \leq }\right) \) be a preordered set and let \( \mathcal{C} \) be one of the seven given categories. Furthermore let \( \left( {{\left\{ {X}_{i}\right\} }_{i \in I},{\left\{ {f}_{ji} : {X}_{j} \rightarrow {X}_{i}\right\} }_{i \leq j}}\right) \) be an inverse system in the category \( \mathcal{C... | No |
Lemma 25.16. Let \( p : X \rightarrow B \) be a map of topological spaces, let \( C \) be a topological space and let \( f : C \rightarrow B \) be a map.\n\n(1) In the category of topological spaces the pullback of the pullback system  \). \n\n(1) This statement follows quite easily from the definitions or alternatively from the proof of Proposition 25.15. | No |
Lemma 25.17. Let \( \left( {I, \leq }\right) \) be a preordered set and let \( \left( {{\left\{ {X}_{i}\right\} }_{i \in I},{\left\{ {f}_{ji} : {X}_{j} \rightarrow {X}_{i}\right\} }_{i \leq j}}\right) \) be an inverse system of topological spaces.\n\n(1) If all the \( {X}_{i} \) ’s are Hausdorff, then the inverse limit... | Proof. Let \( \left( {I, \leq }\right) \) be a preordered set and let \( \left( {{\left\{ {X}_{i}\right\} }_{i \in I},{\left\{ {f}_{ji} : {X}_{j} \rightarrow {X}_{i}\right\} }_{i \leq j}}\right) \) be an inverse system of topological spaces.\n\n(1) We suppose that all the \( {X}_{i} \) are Hausdorff. Given \( i \in I \... | No |
Proposition 25.18. Let \( \left( {I, \leq }\right) \) be a preordered set and let \( \left( {{\left\{ {G}_{i}\right\} }_{i \in I},{\left\{ {f}_{ji} : {G}_{j} \rightarrow {G}_{i}\right\} }_{i \leq j}}\right) \) be an inverse system of finite groups. Then the inverse limit \( \lim {G}_{i} \) exists in the category of top... | Proof. It follows from Proposition 25.15 that the inverse limit \( \mathop{\lim }\limits_{ \leftarrow }{G}_{i} \) exists in the category of topological groups. Since all the \( {G}_{i} \) ’s are finite, in particular compact, it follows from Lemma 25.17 (2) that the inverse limit \( \mathop{\lim }\limits_{ \leftarrow }... | Yes |
(1) Let \( \pi = \Gamma \) . Furthermore let \( \alpha : \pi \rightarrow \pi \) and \( \beta : \pi \rightarrow \pi \) be isomorphisms. We write\n\n\( \varphi = \beta \circ {\alpha }^{-1} : \pi \rightarrow \pi \) . In this setting the map\n\n\[ \Phi : \pi { \rtimes }_{\varphi }\mathbb{Z} \rightarrow \left\langle {\pi, t... | (1) We first verify that the given map is indeed a homomorphism. Let \( \left( {g, m}\right) \) and \( \left( {h, n}\right) \) be two elements in \( \pi { \rtimes }_{\varphi }\mathbb{Z} \) . First we assume that \( m \geq 0 \) . Then the following equalities hold in \( \left\langle {\pi, t \mid \alpha \left( \Gamma \ri... | Yes |
Proposition 26.2. (*) Let \( \pi \) and \( \Gamma \) be two groups and let \( \alpha ,\beta : \Gamma \rightarrow \pi \) be two homomorphisms. If \( \alpha \) and \( \beta \) are monomorphisms, then the obvious inclusion map\n\n\[ \pi \; \rightarrow \;\langle \pi, t \mid \alpha \left( \Gamma \right) = {t\beta }\left( \G... | Proof. The proposition is proved in [Ser80, p. 45]. Alternatively the proposition follows from the Normal Form Theorem for HNN-extensions which is formulated and proved in [LS77, Theorem IV.2.1] or alternatively in [CgRR08, Satz 6.3] and [Sti93, Chapter 9.2]. | No |
Lemma 26.4. With the above notation there exists a well-defined map\n\n\\[ \n\\left( {\\ldots {\\pi }_{-2}\\underset{{\\Gamma }_{-1}}{ * }{\\pi }_{-1}\\underset{{\\Gamma }_{0}}{ * }{\\pi }_{0}\\underset{{\\Gamma }_{1}}{ * }{\\pi }_{1}\\underset{{\\Gamma }_{2}}{ * }{\\pi }_{2}\\ldots }\\right) { \\rtimes }_{\\Xi }\\math... | SKETCH OF PROOF OF LEMMA 26.4. It follows easily from the universal property of the direct limit and of the amalgamated product that there exists a well-defined map\n\n\\[ \n\\ldots {\\pi }_{-2}\\underset{{\\Gamma }_{-1}}{ * }{\\pi }_{-1}\\underset{{\\Gamma }_{0}}{ * }{\\pi }_{0}\\underset{{\\Gamma }_{1}}{ * }{\\pi }_{... | No |
Sketch of the proof of Theorem [26.3] (2) (*). | The proof of Theorem [26.3] (2) is basically the same as the proof of Theorem 26.3 (1), we just need to replace the Seifert-van Kampen Theorem 22.1 by the Seifert-van Kampen Theorem 22.2 for smooth manifolds. We leave it to the reader to fill in the details and to make the necessary changes to the proof. | No |
Let \( X \) be a topological space. We suppose that it can be written as a union \( X = Y \cup Z \) such that the following conditions are satisfied:\n\n(1) \( Y \) and \( Z \) are open subsets,\n\n(2) \( Y \) is path-connected,\n\n(3) \( Z \) is simply connected,\n\n(4) \( Y \cap Z \) consists of two simply connected ... | Proof. Let \( {Z}^{\prime } \) be a disjoint copy of \( Z \) (e.g. we could set \( {Z}^{\prime } = Z \times \{ 0\} \) ). We consider \( W = \left( {Y \sqcup {Z}^{\prime }}\right) /a \sim {a}^{\prime } \) where we identify each point \( a \in A \) with the corresponding point in \( {Z}^{\prime } \) . We denote by \( {B}... | Yes |
Lemma 26.6. (*) Let \( n \in \mathbb{N} \) and let \( M \) be the 2-sphere with \( n \) open disks removed. We enumerate the boundary components by \( {C}_{1},\ldots ,{C}_{n} \) . For \( i = 1,\ldots, n \) we pick a path \( {\beta }_{i} : \left\lbrack {0,1}\right\rbrack \rightarrow {C}_{i} \) that goes once around \( {... | Sketch of A PROOF. Without loss of generality we can assume that \( j = n \) . We consider \( N \mathrel{\text{:=}} M \smallsetminus {\alpha }_{n}\left( \left\lbrack {0,1}\right\rbrack \right) \) . This is an open subset of the smooth manifold \( M \), thus \( N \) itself is a smooth manifold. Furthermore by Exercise 1... | No |
Lemma 27.1. Let \( m, n \in {\mathbb{N}}_{0} \) . We consider the following two maps:\n\n\[ \Phi : {S}^{m - 1} \times {\bar{B}}^{n} \rightarrow A \mathrel{\text{:=}} \left\{ {\left( {z, w}\right) \in {\mathbb{R}}^{m} \times {\mathbb{R}}^{n} = {\mathbb{R}}^{m + n} \mid \parallel w{\parallel }^{2} + \parallel z{\parallel... | (2) Since the orientations are a little more interesting in the second case let us treat the second case in detail. We consider the map\n\n\[ \widetilde{\Psi } : B = \left\{ {\left( {z, w}\right) \in {\mathbb{R}}^{m} \times {\mathbb{R}}^{n} = {\mathbb{R}}^{m + n}\left| {\;\parallel w{\parallel }^{2} + \parallel z{\para... | No |
Lemma 27.2. (*) We denote by \( \Omega : {S}^{3} \smallsetminus \{ \left( {0,1}\right) \} \rightarrow {\mathbb{R}}^{3} \) the stereographic projection as defined in Lemma 2.44. We consider the two smooth embeddings \[ \left( {\left( {x, y}\right) ,{e}^{\mathrm{i}\varphi }}\right) \mapsto \left( \begin{matrix} \cos \var... | Sketch of PROOF (*). By the Isotopy Extension Theorem 8.27 it suffices to show that there exists a smooth isotopy rel \( \{ \left( {0,0}\right) \} \times {S}^{1} \) from \( \Theta \) to \( \Omega \circ \Xi \) . We write \( H = \left\{ {\left( {x,0, z}\right) \in {\mathbb{R}}^{3} \mid x > 0}\right\} \) . Let us first co... | No |
Proposition 27.4. If \( U \) is the trivial knot, then \( {S}^{3} \smallsetminus U \) is diffeomorphic to \( {S}^{1} \times \mathbb{C} \), in particular \( {\pi }_{1}\left( {{S}^{3} \smallsetminus U}\right) \cong \mathbb{Z} \) . | Proof. Note that\n\n\[ \n{S}^{3} \smallsetminus U = \left\{ {\left( {z, w}\right) \in {\mathbb{C}}^{2} \mid {\left| z\right| }^{2} + {\left| w\right| }^{2} = 1}\right\} \smallsetminus \{ \left( {z,0}\right) \mid \left| z\right| = 1\}\n\]\n\n\[ \n= \left\{ {\left( {z, w}\right) \in {\mathbb{C}}^{2}\left| \right| z\left|... | Yes |
(1) Let \( p, q \in \mathbb{N} \) be coprime. Then \( {\pi }_{1}\left( {{S}^{3} \smallsetminus T\left( {p, q}\right) }\right) \cong \left\langle {x, y \mid {x}^{p} = {y}^{q}}\right\rangle \) . | Proof. By definition of the trefoil Statement (2) is just a special case of Statement (1). Thus we only need to prove Statement (1). Let \( p, q \in \mathbb{N} \) be coprime. By Lemma 27.1 there exists a diffeomorphism\n\n\[ \Phi : {S}^{3}\overset{ \cong }{ \rightarrow }\underset{ = : A}{\underbrace{\left( {S}^{1} \tim... | Yes |
(1) The group \( \left\langle {x, y \mid {x}^{2} = {y}^{3}}\right\rangle \) admits an epimorphism onto the permutation group \( {S}_{3} \) , in particular the group \( \left\langle {x, y \mid {x}^{2} = {y}^{3}}\right\rangle \) is not isomorphic to \( \mathbb{Z} \) . | (1) It follows from Lemma 21.4 and a straightforward calculation that there exists a unique homomorphism\n\n\( \Phi : \left\langle {x, y \mid {x}^{2} = {y}^{3}}\right\rangle \rightarrow {S}_{3} = \) permutation group on three elements\n\nwith\n\n\[ \Phi \left( x\right) = \sigma \mathrel{\text{:=}} \left( {12}\right) = ... | Yes |
(1) Let \( K \subset {S}^{3} \) be a knot. The reflections in any two hyperplanes of \( {\mathbb{R}}^{4} \) give rise to smoothly isotopic knots. | SKETCH OF PROOF. (1) We will prove this statement in Exercise 27.1, making use of Lemma 2.65 and Exercise 18.7 | No |
Proposition 27.11. Let \( M \) be a closed oriented \( n \) -dimensional smooth manifold. Furthermore let \( K \subset M \) be a closed oriented \( k \) -dimensional submanifold.\n\n(1) There exists a \( K \) -admissible map.\n\n(2) If \( M \) is connected, then for any \( K \) -admissible maps \( \varphi : {\bar{B}}_{... | Sketch of Proof. The first statement follows easily from the definition of a submanifold. The second statement is an analogue of Theorem 8.36. Unfortunately it seems like nobody ever bothered to write down a proof for Statement (2). Thus we muster all the authority we have and we claim that this statement can be proved... | No |
Let \( M \) be a closed oriented \( n \) -dimensional smooth manifold and let \( K \) be a closed oriented \( k \) -dimensional submanifold of \( M \) . Furthermore let \( N \) be a closed oriented \( n \) -dimensional smooth manifold and let \( L \) be a closed oriented \( k \) -dimensional sub-manifold of \( N \) .\n... | SKETCH OF PROOF.\n\n(1) We leave it to the reader to verify this statement. Note that here one needs to make use of the fact that the (anti-) admissible maps give us control on \( {\bar{B}}_{2}^{n} \) and not just on \( {\overline{B}}^{n} \) . | No |
Proposition 27.13. On the set of smooth isotopy classes of oriented knots the connected sum operation is well-defined and commutative. | Sketch of PROOF. First note that it follows almost immediately from the Isotopy Extension Theorem 8.27 together with Proposition 27.12 that the connected sum operation is indeed well-defined \( {}^{495} \) A slight variation on the proof of Proposition 8.35 (5) shows that the connected sum of knots is commutative. | No |
Proposition 27.14. Let \( K \) and \( L \) be two oriented knots in \( {S}^{3} \) . We pick meridians \( {\mu }_{K} \) and \( {\mu }_{L} \) and base points \( {x}_{0} \in {\mu }_{K} \) and \( {y}_{0} \in {\mu }_{L} \) . There exists an isomorphism\n\n\[ \n{\pi }_{1}\left( {{S}^{3} \smallsetminus \left( {K\# L}\right) }... | Sketch of PROOF. To preserve our sanity we only provide a sketch of the proof. We pick a \( K \) -admissible map \( \varphi : {\bar{B}}_{2}^{3} \rightarrow {S}^{3} \) and we pick an \( L \) -anti-admissible map \( \psi : {\bar{B}}_{2}^{3} \rightarrow {S}^{3} \) . We write \( {D}_{ + }^{3} \mathrel{\text{:=}} {S}^{3} \s... | No |
Theorem 27.16. (Whitten Theorem) Let \( K \) and \( L \) be two prime knots in \( {\mathbb{R}}^{3} \subset {\mathbb{R}}^{3} \cup \) \( \{ \infty \} = {S}^{3} \) . If \( {\pi }_{1}\left( {{S}^{3} \smallsetminus K}\right) \) is isomorphic to \( {\pi }_{1}\left( {{S}^{3} \smallsetminus L}\right) \), then \( K \) is smooth... | Proof. Let \( K \) and \( L \) be two prime knots such that the fundamental groups \( {\pi }_{1}\left( {{S}^{3} \smallsetminus K}\right) \) and \( {\pi }_{1}\left( {{S}^{3} \smallsetminus L}\right) \) are isomorphic. Wilbur Whitten [Whn87, Corollary 2.1], building on the Gordon-Luecke Theorem 8.46, showed that there ex... | Yes |
Proposition 27.19. Given any knot \( K \subset {S}^{3} \) there exists a knot diagram such that \( K \) is smoothly isotopic to the knot associated to the knot diagram. | The proof of Proposition 27.19 mostly rests on the next lemma. The formulation of this lemma requires the following notation.\n\nNotation. Given \( v \in {S}^{2} \) we denote by\n\n\[ \n{\pi }_{v} : {\mathbb{R}}^{3} \rightarrow {v}^{ \bot } \mathrel{\text{:=}} \left\{ {w \in {\mathbb{R}}^{3}\mid \langle v, w\rangle = 0... | No |
Lemma 27.20. Let \( K \subset {\mathbb{R}}^{3} \) be a knot. We pick a diffeomorphism \( f : {S}^{1} \rightarrow K \) . We consider the following three properties of a vector \( v \in {S}^{2} \) :\n\n(a) the map \( {\pi }_{v} \circ f : {S}^{1} \rightarrow {v}^{ \bot } = \left\{ {w \in {\mathbb{R}}^{3}\mid \langle v, w\... | Proof. Let \( K \subset {\mathbb{R}}^{3} \) be a knot. We pick a diffeomorphism \( f : {S}^{1} \rightarrow K \) . We say that a subset of a smooth manifold \( W \) is large if it is open and if it has full measure in \( W \) . It follows from Proposition 6.62 (5b) that the intersection of two large subsets is again lar... | Yes |
Theorem 27.21. Two knot diagrams give rise to smoothly isotopic knots if and only if the two diagrams are related by a finite sequence of smooth isotopies of \( {\mathbb{R}}^{2} \) and Reidemeister moves. The three Reidemeister moves are illustrated in Figure 513. | Proof. The theorem, not surprisingly, goes back to work of Kurt Reidemeister Rei32,  The statement of the theorem is well known in our setting, where knots are defined as submanifolds of \( {S}^{3} \) and it appears ... | No |
Given any knot \( K \) the abelianization of the group \( {\pi }_{1}\left( {{S}^{3} \smallsetminus K}\right) \) is isomorphic to \( \mathbb{Z} \) | We will prove the corollary in Exercise 27.5. Later, on page 1318, once we have introduced homology groups we will give a different, arguably more conceptual proof of the lemma. | No |
Lemma 27.26. Let \( J \) be the figure-8 knot. There exists an isomorphism\n\n\[{\pi }_{1}\left( {{S}^{3} \smallsetminus J}\right) \cong \left\langle {{x}_{1},{x}_{2},{x}_{3},{x}_{4} \mid {x}_{3}{x}_{2}{x}_{3}^{-1}{x}_{1}^{-1},{x}_{4}^{-1}{x}_{3}{x}_{4}{x}_{2}^{-1},{x}_{1}{x}_{4}{x}_{1}^{-1}{x}_{3}^{-1}}\right\rangle . | Proof. We leave it to the reader to use the knot diagram shown in Figure 521 to prove the lemma. | No |
Lemma 27.30. If \( \pi \) is a group whose abelianization is isomorphic to \( \mathbb{Z} \), then given any \( n \in \mathbb{N} \) there exists a unique subgroup that is cocyclic of order \( n \) . It is given by the kernel of any epimorphism \( \pi \rightarrow {\mathbb{Z}}_{n} \) . | Proof (*). Let \( \pi \) be a group whose abelianization is isomorphic to \( \mathbb{Z} \) and let \( n \in \mathbb{N} \) . We denote by \( \psi : \pi \rightarrow {\pi }_{\mathrm{{ab}}} \) the obvious projection and we pick an isomorphism \( \varphi : {\pi }_{\mathrm{{ab}}}\overset{ \cong }{ \rightarrow }\mathbb{Z} \) ... | Yes |
Proposition 27.32. Let \( \pi = \left\langle {{x}_{1},\ldots ,{x}_{m}, t \mid {r}_{1},\ldots ,{r}_{l}}\right\rangle \) be a group and let \( \phi : \pi \rightarrow {\mathbb{Z}}_{n} \) be an epimorphism such that \( \phi \left( {x}_{1}\right) = \cdots = \phi \left( {x}_{m}\right) = 0 \) and \( \phi \left( t\right) = 1 \... | Proof. As we had mentioned above, the proposition follows fairly easily from the more general Reidemeister-Schreier process that is explained in [LS77, Chapter II.4], [MKS76, Chapter 2.3] or [Bog08, Chapter 2.9]. | No |
Proposition 27.34. If \( \pi \) is a finitely presented group and \( \phi : \pi \rightarrow G \) is an epimorphism onto a finite group, then \( \ker \left( {\phi : \pi \rightarrow G}\right) \) is also finitely presented. | The proof of the proposition is given in most books on combinatorial group theory, see e.g. [LS77, Chapter II.4], [MKS76, Chapter 2.3] or [Bog08, Chapter 2.9]. In fact the proof is algorithmic, in particular it also gives an affirmative answer to the second question. The algorithm is called the Reidemeister-Schreier pr... | No |
Theorem 27.35. (Haefliger's (Un-)knotting Theorem)\n\n(1) Let \( n, k \in \mathbb{N} \) . If \( {2k} > 3\left( {n + 1}\right) \), then every \( n \) -dimensional knot in \( {S}^{k} \) is trivial.\n\n(2) Given any \( m \in \mathbb{N} \) there exists a \( \left( {{4m} - 1}\right) \) -dimensional knot in \( {S}^{6m} \) th... | Proof. The first statement was proved by André Haefliger [Hae61a, p. 47] in 1961. (An alternative proof is given in [Wall16, p. 192].) Shortly afterwards André Haefliger Hae62b also proved the second statement. | Yes |
Let \( n \in \mathbb{N} \) . (1) Let \( K \) and \( L \) be two \( n \) -dimensional knots in \( {S}^{n + 2} \) . If \( K \) and \( L \) are smoothly isotopic, then \( {\pi }_{1}\left( {{S}^{n + 2} \smallsetminus K}\right) \cong {\pi }_{1}\left( {{S}^{n + 2} \smallsetminus L}\right) \) . | (1) The proof of this statement is verbatim the same as the proof of Proposition 8.43. | No |
Proposition 27.37. Let \( K \subset {S}^{n + 2} \) be an \( n \) -dimensional knot.\n\n(1) For the 0-twist spin there exists an isomorphism \( {\pi }_{1}\left( {{S}^{n + 3} \smallsetminus {S}_{0}\left( K\right) }\right) \cong {\pi }_{1}\left( {{S}^{n + 2} \smallsetminus K}\right) \) . | SKETCH OF PROOF.\n\n(1) We have the following isomorphisms:\n\nby the Seifert-van Kampen Theorem 22.2, we also use that \( {\Phi }_{0}\left( {{S}^{1} \times {J}_{ + }}\right) = {S}^{1} \times {J}_{ + }\)\n\n\[ \n{\pi }_{1}\left( {{S}^{n + 3} \smallsetminus {S}_{0}\left( K\right) }\right) \overset{ \downarrow }{ \cong }... | No |
Corollary 27.38. Given any \( n \in \mathbb{N} \) there exists an \( n \) -dimensional knot in \( {S}^{n + 2} \) that is non-trivial. | Proof. Let \( K \subset {S}^{3} \) be the trefoil and let \( n \in \mathbb{N} \) . We see that\n\n\[ \n{\pi }_{1}\left( {{S}^{n + 2} \smallsetminus \left( {n - 1}\right) \text{-st iterated 0-twist spin of }K}\right) \;\widetilde{ \cong }\;{\pi }_{1}\left( {{S}^{3} \smallsetminus K}\right) \;\underset{ \uparrow }{\widet... | Yes |
Theorem 28.2. The answer to all four questions raised in Question 28.1 is no. | (1) Adyan Ady55 and Michael Rabin Rabi58 showed in 1955 that the answer to the first question is no. Alternatively see [MillC92, Theorem 3.3] for a proof.\n\n(2) An algorithm that performs (2) would also give an algorithm for determining whether or not a given presentation is isomorphic to the trivial group with the em... | Yes |
Theorem 28.4. Let \( n \geq 4 \) .\n\n(1) Then there is no algorithm that can decide whether or not two closed orientable \( n \) -dimensional smooth manifolds are homeomorphic.\n\n(2) Then there is no algorithm that can decide whether or not two closed orientable \( n \) -dimensional smooth manifolds are diffeomorphic... | Proof. The first statement was first proved by Markov [Mark58]. We refer to [BHP68, Theorem 1] and Hak73 for a detailed discussion of the proofs of both statements. | Yes |
Theorem 28.5. There exists an algorithm that can decide whether or not two closed orientable connected 3-manifolds are diffeomorphic. | In this case the algorithm is considerably more complicated than in the 1-dimensional and the 2-dimensional case. The proof that such an algorithm exists is due to the work of  The proof of the theorem was completed ... | No |
Lemma 29.3. Let \( p : \\left( {X,{x}_{0}}\\right) \\rightarrow \\left( {B,{b}_{0}}\\right) \) be a covering of pointed spaces, let \( \\gamma \) be a path in \( B \) with starting point \( {b}_{0} \) and let \( \\delta \) be another path in \( B \) with \( \\delta \\left( 0\\right) = \\gamma \\left( 1\\right) \) . We ... | Proof. Let \( p : \\left( {X,{x}_{0}}\\right) \\rightarrow \\left( {B,{b}_{0}}\\right) \) be a covering of pointed spaces, let \( \\gamma \) be a path in \( B \) with starting point \( {b}_{0} \) and let \( \\delta \) be another path in \( B \) with \( \\delta \\left( 0\\right) = \\gamma \\left( 1\\right) \) . We denot... | Yes |
Lemma 29.7. Let \( Y \) be a path-connected topological space and let \( {y}_{0} \in Y \). Furthermore let \( \Gamma \subset {\pi }_{1}\left( {Y,{y}_{0}}\right) \) be a subgroup.\n\n(i) We equip \( {Y}^{\left\lbrack 0,1\right\rbrack } = \) set of all maps \( \left\lbrack {0,1\right\rbrack \rightarrow Y \) with the comp... | Proof. Let \( Y \) be a topological space that is path-connected, locally path-connected and semi-locally simply connected, let \( {y}_{0} \in Y \) and let \( \Gamma \subset {\pi }_{1}\left( {Y,{y}_{0}}\right) \) be a subgroup. We write \( X \mathrel{\text{:=}} {X}_{{y}_{0},\Gamma } \) and we equip \( X \mathrel{\text{... | No |
Proposition 29.8. Let \( B \) be a topological space, let \( {b}_{0} \in B \) and let \( \Gamma \subset {\pi }_{1}\left( {B,{b}_{0}}\right) \) be a subgroup.\n\n(1) Let \( p : \left( {X,{x}_{0}}\right) \rightarrow \left( {B,{b}_{0}}\right) \) be a path-connected and locally path-connected covering such that \( {p}_{ * ... | Proof. Let \( B \) be a topological space and let \( {b}_{0} \in B \) .\n\n(1) Let \( p : \left( {X,{x}_{0}}\right) \rightarrow \left( {B,{b}_{0}}\right) \) and \( q : \left( {Y,{y}_{0}}\right) \rightarrow \left( {B,{b}_{0}}\right) \) be two path-connected coverings such that \( {p}_{ * }\left( {{\pi }_{1}\left( {X,{x}... | No |
Let \( X \) be a topological space that is path-connected, locally path-connected and semi-locally simply connected. Furthermore let \( {x}_{0} \in X \) . Then the following holds:\n\n(1) There exists, up to equivalence, a unique path-connected covering\n\n\[ p : \left( {\widetilde{X},{\widetilde{x}}_{0}}\right) \right... | Proof. Let \( X \) be a topological space that is path-connected, locally path-connected and semi-locally simply connected and let \( {x}_{0} \in X \).\n\n(1) We apply Proposition 29.8 to the trivial subgroup of \( {\pi }_{1}\left( {X,{x}_{0}}\right) \) and we obtain, up to equivalence, a unique path-connected covering... | Yes |
Lemma 29.11. Let \( X \) be a topological space that is path-connected, locally path-connected and semi-locally simply connected with universal covering \( p : \widetilde{X} \rightarrow X \). Given any map \( f : X \rightarrow X \), given any \( {x}_{0} \in X \), given any \( {\widetilde{x}}_{0} \in \widetilde{X} \) an... | Proof. We consider the map \( f \circ g \mathrel{\text{:=}} p : \widetilde{X} \rightarrow X \). Since \( {\widetilde{X}}_{0} \) is the universal covering, since \( {\pi }_{1}\left( {\widetilde{X},{\widetilde{x}}_{0}}\right) \) and since \( X \) is locally path-connected and semi-locally simply connected we obtain from ... | No |
Proposition 30.1. Let \( f : {S}^{1} \rightarrow {S}^{1} \) be a diffeomorphism.\n\n(1) If \( f \) is orientation-preserving, then \( f \) is diffeotopic to the identity, otherwise \( f \) is diffeotopic to the reflection \( r : {S}^{1} \rightarrow {S}^{1} \) given by \( \left( {x, y}\right) \mapsto {\left( x, - y\righ... | Proof.\n\n(1) As usual we identify \( {S}^{1} \) with \( \mathbb{R}/\mathbb{Z} \) . Thus let \( f : \mathbb{R}/\mathbb{Z} \rightarrow \mathbb{R}/\mathbb{Z} \) be a diffeomorphism. First we consider the case that \( f \) is orientation-preserving. We denote by \( p : \mathbb{R} \rightarrow \mathbb{R}/\mathbb{Z} \) the u... | Yes |
Corollary 30.3. Let \( M \) be a 2-dimensional smooth manifold (possibly disconnected) and let \( A \) and \( B \) be two distinct boundary components of \( M \) . Suppose \( A \) and \( B \) are equipped with orientations. Given any two orientation-preserving diffeomorphisms \( f, g : A \rightarrow B \) the resulting ... | Proof. It follows from Proposition 6.27 and Theorem 7.5 that \( A \) and \( B \) are both diffeomorphic to \( {S}^{1} \) . The corollary is now an immediate consequence of Lemma 8.16 and Corollary 30.2 | No |
Proposition 30.4. (*) Let \( U \subset \mathbb{C} \smallsetminus \{ 0\} \) be an open path-connected subset. The following statement holds.\nthe complex logarithm for some \( {x}_{0} \in U \) the inclusion induced map\nexists on \( U \) \( {\pi }_{1}\left( {U,{x}_{0}}\right) \rightarrow {\pi }_{1}\left( {\mathbb{C}\sma... | Proof (*). Let \( U \subset \mathbb{C} \smallsetminus \{ 0\} \) be an open path-connected subset and let \( {x}_{0} \in U \) . We consider the following rather simple minded diagram\n\n\n\nWe make the following obser... | Yes |
Proposition 30.6. Let \( \left( {B,{b}_{0}}\right) \) be a pointed topological space that is path-connected, locally path-connected and semi-locally simply connected. Let \( \varphi : {\pi }_{1}\left( {B,{b}_{0}}\right) \rightarrow G \) be a group epimorphism and let \( p : \left( {X,{x}_{0}}\right) \rightarrow \left( ... | Proof. To simplify the notation a little bit we assume that \( {a}_{0} = {b}_{0} \), in particular we ignore the role of \( \gamma \) . Throughout the proof we refer to Figure 546 for an illustration. (1) Let \( W \) be a path-component of \( {p}^{-1}\left( A\right) \) . We pick a point \( {w}_{0} \in W \) with \( p\le... | No |
Proposition 31.1. Let \( G = \left( {V, E, i, t}\right) \) be a connected abstract graph and let \( p : \widetilde{X} \rightarrow \left| G\right| \) be a covering of the topological realization \( X \mathrel{\text{:=}} \left| G\right| \) of \( G \) . Then there exists a natura \( {}^{532} \) abstract graph \( \widetild... | Proof. Let \( G = \left( {V, E, i, t}\right) \) be a connected abstract graph. Recall that on page 222 we defined the topological realization of \( G \) as follows:\n\n\( \left| G\right| \mathrel{\text{:=}} \left( {V \sqcup \left( {E \times \left\lbrack {0,1}\right\rbrack }\right) }\right) / \sim \; \) where for \( e \... | No |
Lemma 31.3. Let \( H \) be a finite connected graph and let \( G \) be a connected non-empty subgraph of \( H \) . Let \( v \) be a vertex of \( G \) . We denote by \( i : \left| G\right| \rightarrow \left| H\right| \) the inclusion. We set \( s \mathrel{\text{:=}} \chi \left( H\right) - \chi \left( G\right) \) . Then ... | Proof of Lemma 31.3. Let \( H = \left( {W, F, i, t}\right) \) be a finite connected graph. Furthermore let \( G = \left( {V, E, i, t}\right) \) be a connected non-empty subgraph of \( H \) . By Proposition 4.8 (1) there exists a spanning tree \( S \) for \( G \) . By a rather elementary argument, which we outsourced to... | No |
Proposition 31.4. Let \( F \) be a free group of countable rank.\n\n(1) Every subgroup of \( F \) is again a free group.\n\n(2) If \( F \) is finitely generated and if \( G \) is a finite-index subgroup of \( F \), then\n\n\[ \operatorname{rank}\left( G\right) - 1 = \left\lbrack {F : G}\right\rbrack \cdot \left( {\oper... | Proof. Let \( F = \langle E\rangle \) be a free group on a countable generating set \( E \) and let \( G \) be a subgroup of \( F \) . Note that it follows easily from Lemma 1.7 that \( G \) is a countable set. Let \( \Gamma = \left( {\left\{ {x}_{0}\right\}, E, i, t}\right) \) be the abstract graph with one vertex and... | No |
Proposition 31.6. Let \( M \) be the surface of genus \( g \) and let \( \Gamma \subset {\pi }_{1}\left( M\right) \) be a subgroup.\n\n(1) If the index \( \left\lbrack {\pi : \Gamma }\right\rbrack \) is finite, then \( \Gamma \) is the fundamental group of the surface of genus \( \left\lbrack {\pi : \Gamma }\right\rbra... | Proof. Let \( g \in {\mathbb{N}}_{ \geq 2} \) and let \( \sum \) be the surface of genus \( g \) . We pick a base point \( {x}_{0} \in \sum \) . Let \( \Gamma \subset {\pi }_{1}\left( {\sum ,{x}_{0}}\right) \) be a subgroup. We obtain from Proposition 36.10 (6) and Proposition 29.5 that there exists a path-connected co... | No |
Proposition 31.8. Let \( M \) be a surface of genus \( \geq 2 \) . If \( \Gamma \subset {\pi }_{1}\left( M\right) \) is a normal non-trivial subgroup of infinite-index, then \( {\pi }_{1}\left( M\right) \) is a free group of infinite rank. | Proof. Let \( M \) be a surface of genus \( \geq 2 \) and let \( \Gamma \subset {\pi }_{1}\left( M\right) \) be a normal non-trivial subgroup of infinite-index. By Proposition 31.6 it remains to show that \( \Gamma \) is infinitely generated. A proof for this statement is given in Cat03, Lemma 3.4, alternatively one ca... | No |
Proposition 31.11. Let \( \pi \) be a finitely generated group. If \( \pi \) is residually finite, then it is Hopfian. | Proof. Let \( \pi \) be a finitely generated group that is residually finite. Let \( \varphi : \pi \rightarrow \pi \) be an epimorphism. Now we assume that \( \varphi \) is not a monomorphism. We pick a non-trivial element \( g \neq e \in \ker \left( \varphi \right) \) . Since \( \pi \) is residually finite there exist... | Yes |
Lemma 31.13. Let \( G \) be a group and let \( H \) be a subgroup.\n\n(1) The normal core of \( H \) is a normal subgroup of \( G \) . | (1) Let \( k \in G \) . We have\n\n\[ k{H}^{c}{k}^{-1} = \mathop{\bigcap }\limits_{{g \in G}}k \cdot {gH}{g}^{-1} \cdot {k}^{-1} = \mathop{\bigcap }\limits_{{g \in G}}\left( {kg}\right) H{\left( kg\right) }^{-1} = \mathop{\bigcap }\limits_{{g \in G}}{gH}{g}^{-1} = {H}^{c}. \] | Yes |
Lemma 31.14. Let \( \left( {X,{x}_{0}}\right) \) be a pointed topological space that is path-connected, locally path-connected and semi-locally simply connected. The following two statements are equivalent:\n\n(1) The group \( {\pi }_{1}\left( {X,{x}_{0}}\right) \) is residually finite.\n\n(2) Given any loop \( \gamma ... | Proof. Let \( \left( {X,{x}_{0}}\right) \) be a topological space that is path-connected, locally path-connected and semi-locally simply connected. First we prove the \( \left( 1\right) \Rightarrow \left( 2\right) \) -implication. Thus we assume that the group \( {\pi }_{1}\left( {X,{x}_{0}}\right) \) is residually fin... | Yes |
Proposition 31.15. Every free group is residually finite. | First Proof of Proposition 3.1.15. Let \( X \) be a set and let \( g \in \langle X\rangle \) be a non-trivial element in the free group generated by the set \( X \) . We can and will write \( g \) as a product \( g = {y}_{1}^{{\epsilon }_{1}}\cdots \cdots {y}_{k}^{{\epsilon }_{k}} \) with \( {y}_{1},\ldots ,{y}_{k} \in... | Yes |
Corollary 31.16. Every finitely generated free group is Hopfian. | Proof. This statement follows immediately from Proposition 31.15 together with Proposition 31.11. An alternative proof of the corollary is given in [MKS76, p. 109]. | No |
Proposition 31.18. Let \( X = \left\{ {{x}_{1},\ldots ,{x}_{k}}\right\} \) be a finite set with \( k \) elements. If \( r \) is a non-trivial element in the free group \( \left\langle {{x}_{1},\ldots ,{x}_{k}}\right\rangle \), then the group \( \left\langle {{x}_{1},\ldots ,{x}_{k} \mid r}\right\rangle \) is not isomor... | Proof. Let \( X \) is a finite set and let \( r \) be an element in \( \langle X\rangle \) . Suppose there exists an isomorphism \( \varphi : \langle X \mid r\rangle \rightarrow \langle X\rangle \) . We denote by \( \psi : \langle X\rangle \rightarrow \langle X \mid r\rangle \) the obvious epimorphism. Then \( \varphi ... | Yes |
Proposition 31.19. Let \( G \) be a finitely generated group. If \( G \) is linear over \( \mathbb{C} \), then \( G \) is residually finite. | SKETCH OF PROOF. We only provide a full proof of the following easier claim which is the content of Exercise 19.14 (a).\n\nClaim. The group \( \mathrm{{GL}}\left( {n,\mathbb{Z}}\right) \) is residually finite.\n\nLet \( A = \left( {a}_{ij}\right) \in \mathrm{{GL}}\left( {n,\mathbb{Z}}\right) \) be a matrix with \( A \n... | No |
Theorem 31.21.\n\n(1) Thompson’s group \( F \) and Thompson’s group \( T \) are both finitely presented.\n\n(2) The commutator subgroup of Thompson’s group \( F \) is simple, in particular Thompson’s group \( F \) is not residually finite.\n\n(3) Thompson’s group \( T \) is simple, in particular it is not residually fi... | Proof.\n\n(1) The statements are proved in [CFP96] and Geo08, Chapters 9.2].\n\n(2) By CFP96 and Geo08, Chapters 9.2 the commutator subgroup of Thompson's group \( F \) is simple. One can easily verify that Thompson’s group \( F \) is non-abelian and torsion-free. It now follows from an elementary argument that it is n... | No |
Proposition 32.1. The map\n\n\[ \n\\mathrm{{SL}}\\left( {2,\\mathbb{R}}\\right) \\rightarrow {\\operatorname{Isom}}^{ + }\\left( \\mathbb{H}\\right) \n\]\n\n\[ \n\\left( \\begin{array}{ll} a & b \\\\ c & d \\end{array}\\right) \\mapsto \\left( \\begin{matrix} \\mathbb{H} & \\rightarrow & \\mathbb{H} \\\\ z & \\mapsto &... | SKETCH OF PROOF. One can easily calculate by hand that each Möbius transformation is an orientation-preserving isometry of \( \\mathbb{H} \) . Furthermore, it is an amusing calculation to verify that the given map is indeed a group homomorphism. It is elementary to see that the map is a monomorphism. The only bit of th... | No |
Proposition 32.2. Every Möbius transformation of \( \mathbb{H} \) is the composition of scalar multiplications, horizontal translations and inversions. | Proof. Let \( a, b, c, d \in \mathbb{R} \) with \( {ad} - {bc} = 1 \) and denote by\n\n\[ \Phi \left( z\right) = \frac{{az} + b}{{cz} + d} \]\n\nthe corresponding Möbius transformation. If \( c = 0 \), then\n\n\[ \Phi \left( z\right) = \frac{a}{d}z + \frac{b}{d} \]\n\nis the composition of the scalar multiplication \( ... | Yes |
Proposition 32.3. The maps\n\n\[ \begin{matrix} \mathfrak{p} : \mathbb{H} & \rightarrow & \mathbb{D} & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & &... | Proof. All five statements follow from a straightforward calculation. The statements can also be deduced, more or less directly, from any book that deals with Möbius transformations, see e.g. Schwe79, Chapter 6] and [And05, Chapter 2]. | No |
(1) Compositions of Möbius transformations are again Möbius transformations and the inverse of a Möbius transformation is again a Möbius transformation. | Proof. By Proposition 32.3 it suffices to prove the statements for \( \mathbb{D} \) or for \( \mathbb{H} \), whichever is more convenient.\n\n(1),(2) The first two statements follow from a straightforward calculation, see e.g. Schwe79, p. \( {43}\rbrack \) . | No |
Lemma 32.5. Let \( \{ P, Q, R\} \) and let \( \{ S, T, U\} \) be two sets of three distinct points on the boundary \( \partial \overline{\mathbb{H}} = \mathbb{R} \cup \{ \infty \} \) . Then there exists a Möbius transformation \( \psi \) of \( \mathbb{H} \) with \( \psi \left( P\right) = S \) and \( \psi \left( {\{ Q, ... | Proof. Let \( P, Q, R \) be three distinct points on \( \partial \overline{\mathbb{H}} = \mathbb{R} \cup \{ \infty \} \) . It suffices to show that there exists a Möbius transformation \( \psi \) with \( \psi \left( P\right) = \infty \) and with \( \psi \left( {\{ Q, R\} }\right) = \{ 0,1\} \) . We proceed as follows:\... | Yes |
Lemma 32.8. Let \( r, s \) be two rays emanating from \( P \in \mathbb{D} \) and let \( {r}^{\prime },{s}^{\prime } \) be two rays emanating from \( {P}^{\prime } \in \mathbb{D} \) . Then the following holds:\n\nthere is a Möbius transformation \( \phi \Leftrightarrow \) angle between \( r, s = \) angle between \( {r}^... | Proof. By Lemma 32.7 Möbius transformations preserve angles and by Proposition 32.4 Möbius transformations act transitively on \( \mathbb{D} \) . Therefore we can without loss of generality assume that \( P = {P}^{\prime } = 0 \) . The lemma follows from the observation, made on page 876, that Möbius transformations pr... | Yes |
Lemma 32.10. Let \( \\left( {M, g}\\right) \) and \( \\left( {N, h}\\right) \) be two connected Riemannian manifolds. Furthermore let \( f : \\left( {M, g}\\right) \\rightarrow \\left( {N, h}\\right) \) be a local isometry. Then the following hold:\n\n(1) For any piecewise smooth path \( \\gamma : \\left\\lbrack {a, b}... | Proof. Let \( f : \\left( {M, g}\\right) \\rightarrow \\left( {N, h}\\right) \) be a local isometry. We start out with the following claim. Claim.\n\n(1) For a smooth path \( \\gamma : \\left\\lbrack {c, d}\\right\\rbrack \\rightarrow M \) and \( t \\in \\left( {c, d}\\right) \) we have \( {\\begin{Vmatrix}{\\left( f \... | Yes |
Lemma 32.12. Let \( \\left( {P, Q}\\right) \) and \( \\left( {S, T}\\right) \) be two pairs of distinct points on \( \\mathbb{D} \) such that \( d\\left( {P, Q}\\right) = d\\left( {S, T}\\right) \) . Then there exists a unique Möbius transformation \( \\phi \) with \( \\phi \\left( P\\right) = S \) and \( \\phi \\left(... | Proof of Lemma 32.12. It follows from Proposition 32.4 (3) that, without loss of generality, we can assume that \( P = 0 \) and \( S = 0 \) . We write \( \\bar{Q} = r{e}^{\\mathrm{i}\\varphi } \) and \( T = s{e}^{\\mathrm{i}\\psi } \) . It is a direct consequence of Proposition 32.11 (1) and our hypothesis that\n\n\[ d... | Yes |
Lemma 32.13. Let \( P \in \mathbb{D} \) and \( r > 0 \) . Let \( S, T \in {B}_{r}\left( P\right) \) . Then the image of the unique geodesic in \( \mathbb{D} \) from \( S \) to \( T \) lies in \( {B}_{r}\left( P\right) \) . | Proof. As usual, this time by Propositions 32.4, 32.6 and 32.11, it suffices to prove the lemma for \( P = 0 \) . In this special case the reader will have no troubles providing the elementary argument. We also refer to Figure 565 for an illustration. | No |
Proposition 32.14. Every compact metric space is complete. | Proof. Let \( \left( {X, d}\right) \) be a compact metric space and let \( {\left( {a}_{n}\right) }_{n \in \mathbb{N}} \) be a Cauchy sequence in \( \left( {X, d}\right) \) . We want to show that the sequence converges, i.e. we want to show that\n\n\[ \mathop{\exists }\limits_{{x \in X}}\mathop{\forall }\limits_{{\epsi... | Yes |
Lemma 32.15. The Riemannian manifolds \( \mathbb{D} \) and \( \mathbb{H} \) are complete. | Proof. By Proposition 32.3 the two Riemannian manifolds \( \mathbb{D} \) and \( \mathbb{H} \) are isometric. Therefore it suffices to show that \( \mathbb{D} \) is complete. We denote by \( d = {d}_{\left( \mathbb{D}, g\right) } \) the metric on \( \mathbb{D} \) corresponding to the Riemannian structure. Let \( {\left\... | Yes |
Lemma 33.1. Let \( M \) be a surface and let \( {\left\{ {\Phi }_{i} : {U}_{i} \rightarrow {V}_{i}\right\} }_{i \in I} \) be a Möbius structure for \( M \). Then the following hold:\n\n(1) The charts form a holomorphic atlas for \( M \), in particular \( M \) is a complex 1-dimensional smooth manifold.\n\n(2) (a) The s... | Proof. Let \( M \) be a surface and let \( {\left\{ {\Phi }_{i} : {U}_{i} \rightarrow {V}_{i}\right\} }_{i \in I} \) be a Möbius structure for \( M \).\n\n(1) The first statement is an immediate consequence of the definitions and of the fact,\n\nproved in Proposition 32.4 (2), that Möbius transformations are biholomorp... | Yes |
(1) We have \( {\Phi }_{1}\left( {Q}_{2}\right) = {Q}_{4} \) and \( {\Phi }_{1}\left( {Q}_{3}\right) = {Q}_{3} \) and we also have \( {\Phi }_{2}\left( {Q}_{2}\right) = {Q}_{4} \) and \( {\Phi }_{1}\left( {Q}_{1}\right) = {Q}_{1} \)\n(2) The restriction of \( {\Phi }_{1} \) to the hyperbolic line with endpoints \( {Q}_... | Proof. The first statement follows immediately from plugging in the points. A straightforward, albeit slightly painful calculation shows that for any \( \alpha \in \mathbb{R} \) we have\n\n\[ \n{\Phi }_{1}\left( {-1 + \mathrm{i} + {e}^{-\mathrm{i}\alpha }}\right) = - 1 - \mathrm{i} + {e}^{\mathrm{i}\alpha }.\n\]\n\nThi... | No |
Proposition 33.6. There exists a biholomorphism \( {H}_{4}/ \sim \rightarrow \mathbb{C} \smallsetminus \{ 0,1\} \) . | Proof of Proposition 33.6, We denote by \( \Delta \) the open subset of \( \mathbb{D} \) bounded by the three hyperbolic lines with endpoints \( - 1,1 \) and \( i \) . (We refer to Figure 576 for an illustration of \( \Delta \) .) It follows from applying the Riemann Mapping Theorem 12.11 together with Carathéodory’s T... | Yes |
Theorem 33.7. Let \( g \in {\mathbb{N}}_{0} \) and let \( n \in {\mathbb{N}}_{0} \) . If \( {2g} + n - 2 > 0 \), then \( {F}_{g, n} \) admits a hyperbolic structure. | Sketch of A PROOF. Let \( g \in {\mathbb{N}}_{0} \) and let \( n \in {\mathbb{N}}_{0} \) . If \( {2g} + n - 2 > 0 \) then variations on the argument of Propositions 33.2 and 33.5 show that \( {F}_{g, n} \) admits a hyperbolic metric. In Figure 577 we sketch the constructions needed to show that the once-punctured and t... | No |
(1) If \( U \subset \mathbb{D} \) is a connected subset and if \( f : U \rightarrow \mathbb{D} \) is a Möbius map, then \( f \) is given by a Möbius transformation. | (1) Let \( U \subset \mathbb{D} \) be a connected subset and let \( f : U \rightarrow \mathbb{D} \) be a Möbius map. It follows from Proposition 32.4 (4) that given \( P \in U \) there exists a unique Möbius transformation \( {\phi }_{P} \) such that \( f = {\phi }_{P} \) in an open neighborhood of \( P \) . We conside... | Yes |
Lemma 33.11. Let \( U \) be an open connected subset of \( \mathbb{D} \) and let \( M \) be a \( M\ddot{o} \) bius manifold. Furthermore let \( \Phi ,\Psi : U \rightarrow M \) be two Möbius maps \( {}^{553} \) Suppose one of the following two conditions is satisfied:\n\n(1) There exists an open non-empty subset of \( U... | Proof. Let \( U \) be an open connected subset of \( \mathbb{D} \), let \( M \) be a Möbius manifold and let \( \Phi ,\Psi : U \rightarrow M \) be two Möbius maps. We consider one more statement:\n\n(3) There exists a point \( Q \in U \) such that \( \Phi \left( Q\right) = \Psi \left( Q\right) \) and such that \( \math... | Yes |
Lemma 33.13. Let \( \\left( {X, g}\\right) \) and \( \\left( {Y, h}\\right) \) be Riemannian manifolds. Let \( Q \\in X \) and \( r > 0 \) . Let \( f : {B}_{3r}^{X}\\left( Q\\right) \\rightarrow \\left( {Y, h}\\right) \) be a local isometry and let \( S, T \\in {B}_{r}^{X}\\left( Q\\right) \) . Then we have\n\n\[ \n{d}... | Proof. Let \( S, T \\in {B}_{r}^{X}\\left( Q\\right) \) . Furthermore let \( \\epsilon > 0 \) . By the triangle inequality we have \( {d}_{X}\\left( {S, T}\\right) < {2r} \) . In particular there exists a piecewise smooth path \( \\gamma \) in \( X \) from \( S \) to \( T \) such that \( {\\ell }_{X}\\left( \\gamma \\r... | Yes |
Lemma 33.14. Let \( M \) be a Möbius manifold. Let \( Q \in M \) and let \( r > 0 \) such that \( {B}_{r}^{M}\left( Q\right) \) is small. Let \( \Phi : \mathbb{D} \rightarrow M \) be a Möbius map such that \( {B}_{r}^{M}\left( Q\right) \subset \Phi \left( \mathbb{D}\right) \) . Then the following hold:\n\n(1) For any \... | Proof. Let \( M \) be a Möbius manifold. Let \( Q \in M \) and let \( r > 0 \) such that \( {B}_{r}^{M}\left( Q\right) \) is small. We pick a Möbius isomorphism \( \Omega : {B}_{r}^{\mathbb{D}}\left( 0\right) \rightarrow {B}_{r}^{M}\left( Q\right) \) as in the definition of \ | No |
Lemma 33.16. The map \( p : \mathbb{D} \rightarrow M \) is a Möbius map. | Proof. By Lemma 33.10 (2) it suffices to show that \( p \) is locally a Möbius map. This means that it suffices to show that every point \( P \in \mathbb{D} \) admits an open neighborhood \( W \) such that the restriction of \( p \) to \( W \) is a Möbius map. Thus let \( P \in \mathbb{D} \) . By the above claim there ... | Yes |
Lemma 33.17. The map \( p : \mathbb{D} \rightarrow M = {H}_{8}/ \sim \) is a covering map. | Proof. First we show that the map \( p : \mathbb{D} \rightarrow M = {H}_{8}/ \sim \) is surjective. Recall that by construction \( p \) is the identity on \( {H}_{8} \) . Since \( p \) is continuous \( {}^{557} \) it follows that the restriction of \( p \) to \( {H}_{8} \rightarrow M = {H}_{8}/ \sim \) is surjective. \... | Yes |
Proposition 33.18. Let \( {H}_{4}/ \sim \) be the three-punctured sphere with the Möbius structure constructed in Proposition 33.5. Then there exists a covering map \( p : \mathbb{D} \rightarrow {H}_{4}/ \sim \) with the following properties:\n\n(1) \( p \) is a Möbius map,\n\n(2) \( p \) is a local isometry,\n\n(3) \(... | Proof. The proof of Proposition 33.18 is almost the same as the proof of Theorem 33.9. In the proof of Theorem 33.9 we used that \( {H}_{8}/ \sim \) is complete. In the present case it follows from Proposition 33.5 (3) that \( {H}_{4}/ \sim \) is complete. The construction of the Möbius map \( \Phi : \mathbb{D} \righta... | Yes |
Theorem 33.19. (Picard’s Theorem) Let \( f : \mathbb{C} \rightarrow \mathbb{C} \) be a non-constant holomorphic function. Then there exists at most one \( z \in \mathbb{C} \) which does not lie in the image of \( f \) . | Proof. Let \( f : \mathbb{C} \rightarrow \mathbb{C} \) be a holomorphic function such that there exist two different complex numbers \( a, b \) that do not lie in the image of \( f \) . We need to show that \( f \) is constant. We consider the biholomorphism\n\n\[ \alpha : \mathbb{C} \smallsetminus \{ a, b\} \rightarro... | Yes |
Proposition 33.20. Let \( l, m, n \in \mathbb{N} \) with \( \frac{1}{l} + \frac{1}{m} + \frac{1}{n} < 1 \). (1) There exists a hyperbolic triangle \( {\Delta }_{ABC} \) with interior angles \( \frac{\pi }{l},\frac{\pi }{m} \) and \( \frac{\pi }{n} \) at the vertices \( A, B \) and \( C \) . | Proof. (1) This statement is shown in [Rat19, Theorem 3.5.6]. In fact in that reference it is also shown that the triangle is unique up to an isometry. | Yes |
Lemma 33.21. Given any \( l, m, n \in \mathbb{N} \) the map\n\n\[ \left\langle {a, b \mid {a}^{l},{b}^{m},{\left( ab\right) }^{n}}\right\rangle \rightarrow T\left( {l, m, n}\right) \]\n\n\[ a \mapsto {xy} \]\n\n\[ b \mapsto {yz} \]\nis an isomorphism. | Proof. We will prove Lemma 33.21 in Exercise 33.2 | No |
Corollary 33.22. Let \( l, m, n \in \mathbb{N} \) . If \( \frac{1}{l} + \frac{1}{m} + \frac{1}{n} < 1 \), then the triangle group \( {T}^{ * }\left( {l, m, n}\right) \) and the von Dyck group \( T\left( {l, m, n}\right) \) are infinite. | Proof. Let \( l, m, n \in \mathbb{N} \) with \( \frac{1}{l} + \frac{1}{m} + \frac{1}{n} < 1 \) . We denote by \( G \) the group introduced in Proposition 33.20. Note that it follows from Proposition 33.20 (2), the fact that \( {\Delta }_{ABC} \) is compact and the fact that \( \mathbb{D} \) is non-compact that \( G \) ... | Yes |
Lemma 34.1. Let \( p : \left( {X,{x}_{0}}\right) \rightarrow \left( {B,{b}_{0}}\right) \) be a covering of pointed path-connected topological spaces and let \( {d}_{1} \) and \( {d}_{2} \) be two deck transformations of \( p \) . Then the following holds\n\n\[ \n{d}_{1} = {d}_{2} \Leftrightarrow {d}_{1}\left( {x}_{0}\r... | Proof. Evidently we only need to show the \ | No |
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