Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Lemma 8.4. (*) Let \( M \) and \( N \) be smooth manifolds and let \( F : M \times \left\lbrack {0,1}\right\rbrack \rightarrow N \) be a smooth homotopy. There exists an \( \epsilon \in \left( {0,\frac{1}{2}}\right) \) and a smooth homotopy \( G : M \times \left\lbrack {0,1}\right\rbrack \rightarrow N \) with \( {G}_{0... | Proof. Using Lemma 6.13 one can easily find a smooth map \( \varphi : \left\lbrack {0,1}\right\rbrack \rightarrow \left\lbrack {0,1}\right\rbrack \) such that \( \varphi \left( 0\right) = 0,\varphi \left( 1\right) = 1,\varphi \left( t\right) = 0 \) for all \( t \in \left\lbrack {0,\epsilon }\right\rbrack \) and \( \var... | Yes |
Lemma 8.5. Let \( n \in \mathbb{N} \). (1) If \( \gamma : \left\lbrack {0,1}\right\rbrack \rightarrow \mathrm{O}\left( n\right) \) is a smooth path, then the map \[ F : {\bar{B}}^{n} \times \left\lbrack {0,1}\right\rbrack \rightarrow {\bar{B}}^{n} \] \[ \left( {x, t}\right) \mapsto \gamma \left( t\right) \cdot x \] is ... | Proof. (1) The first statement is basically obvious. | No |
Lemma 8.8. (*) Let \( n \in {\mathbb{N}}_{0} \) and let \( k \in \{ 1,\ldots, n\} \) . We write\n\n\( \mathrm{{GL}}\left( {n, k}\right) = \) set of \( k \) -tuples of linearly independent vectors of \( {\mathbb{R}}^{n} \)\n\n\( \mathrm{O}\left( {n, k}\right) = \) set of \( k \) -tuples of orthonormal vectors of \( {\ma... | Sketch of Proof Proof (*).\n\n(1) One easily verifies that each of the \( {2k} \) steps of the Gram-Schmidt process is realized by a smooth deformation retraction that satisfies \( \left( *\right) \) . The proof of Lemma 8.7 shows that we can combine these smooth deformation retractions to obtain the desired smooth def... | Yes |
Lemma 8.10. Let \( N \) be a smooth manifold. The property of being (properly) smoothly isotopic is an equivalence relation on the set of (proper) submanifolds of (oriented) sub-manifolds of \( M \) . | Proof. It follows from Proposition 8.1 (1) that being (properly) smoothly isotopic is a symmetric relation. It now follows as before from Lemma 8.7 that being (properly) smoothly isotopic is an equivalence relation. | No |
Lemma 8.11. Let \( M \) be a smooth manifold and let \( f : \left\lbrack {0,1}\right\rbrack \times \partial M \rightarrow M \) be a collar. For any \( t \in (0,1\rbrack \) the set \( \lbrack 0, t) \times \partial M \) is an open subset of \( M \), in particular it is in fact a neighborhood of \( \partial M \) in the se... | Proof. The proof of this lemma is quite similar to the proof of Proposition 8.2, i.e. the proof is a reasonably straightforward consequence of Theorem 6.19. Once again we leave it to the reader to fill in the details. | No |
Proposition 8.13. (*) Let \( M \) be a smooth manifold and let \( {f}_{0},{f}_{1} : \left\lbrack {0,1}\right\rbrack \times \partial M \rightarrow M \) be two smooth embeddings that are the identity map on \( \partial M \times \{ 0\} = \partial M \) . If \( \partial M \) is compact \( {}^{142} \) , then there exists a d... | Proof. The proposition is proved in [Wall16, Proposition 2.5.7]. A very similar statement is also proved in [BJ82, Theorem 13.7]. | No |
Corollary 8.14. (*) Let \( M \) be an \( n \) -dimensional smooth manifold. Let \( \left\lbrack {0,1}\right\rbrack \times \partial M \) be a collar neighborhood.\n\n(1) The subset \( W \mathrel{\text{:=}} M \smallsetminus \left( {\lbrack 0,1}\right) \times \partial M) \) is an \( n \) -dimensional smooth manifold with ... | Proof.\n\n(1) This statement follows fairly easily from Proposition 6.27. We leave it to the reader to fill in the details.\n\n(2) We write \( W \mathrel{\text{:=}} M \smallsetminus \left( {\lbrack 0,1}\right) \times \partial M) \) and we set \( {M}^{\prime } \mathrel{\text{:=}} M \smallsetminus \partial M \) . We cons... | No |
Proposition 8.15. Let \( M \) be an \( n \) -dimensional smooth manifold (we do not assume that \( M \) is connected). Let \( A \) and \( B \) be disjoint unions of boundary components of \( M \) . Furthermore let \( f : A \rightarrow B \) be a diffeomorphism. \( {}^{144} \n\n(1) (a) The topological space\n\n\[ \nM/a \... | (b) Any choice of a collar \( {}^{145}\varphi : \left\lbrack {0,1}\right\rbrack \times \partial M \rightarrow M \) for \( \partial M \) equips the topological manifold \( M/a \sim f\left( a\right) \) canonically \( {}^{146} \) with the structure of an n-dimensional smooth manifold such that \( M \smallsetminus \left( {... | No |
Lemma 8.16. Let \( M \) be an \( n \) -dimensional smooth manifold (we do not assume that \( M \) is connected) with compact boundary. Let \( A \) and \( B \) be two distinct compact boundary components of \( M \) . Furthermore let \( {f}_{0} \) and \( {f}_{1} \) be two diffeomorphisms from \( A \) to \( B \) . If \( {... | Sketch of PROOF. Since \( {f}_{0} \) and \( {f}_{1} \) are diffeotopic and since they are diffeomorphisms there exists a diffeotopy \( F : A \times \left\lbrack {0,1}\right\rbrack \rightarrow A \) from \( {f}_{1} \circ {f}_{0}^{-1} \) to \( {\mathrm{{id}}}_{A} \) . With the same approach as in the proof of Lemma 8.7 we... | Yes |
Proposition 8.17. Let \( M \) be an orientable n-dimensional smooth manifold and let \( F \subset M \) be a compact orientable proper submanifold of codimension one. | SKETCH OF PROOF.\n\n---\n\n\( {}^{150} \) Note that here \ | No |
Proposition 8.18. Let \( M \) be an orientable n-dimensional smooth manifold and let \( A \) and \( B \) be disjoint unions of boundary components of \( M \) . Furthermore let \( f : A \rightarrow B \) be an orientation-preserving diffeomorphism. Let \( W = M/a \sim f\left( a\right) \) be the smooth manifold that is ob... | Proof. We leave it to the reader to verify that the obvious maps are indeed diffeomorphisms. | No |
Let \( M \) be an \( n \) -dimensional smooth manifold (we do not assume that \( M \) is connected). Let \( A \) and \( B \) be two distinct boundary components of \( M \). Furthermore let \( f : A \rightarrow B \) be a diffeomorphism. Let \( N \) be a proper submanifold of \( M \) such that the map \( f \) restricts t... | The proof follows easily from the definition of the smooth structure on \( M/a \sim \) \( f\left( a\right) \) as provided in the proof of Proposition 8.15 (1). We will not bore the reader with unnecessary details. | No |
Theorem 8.20. (Collar Neighborhood Theorem) Let \( M \) be a smooth manifold. If \( N \) is a proper submanifold of \( M \), then \( M \) admits a collar neighborhood \( \left\lbrack {0,1}\right\rbrack \times \partial M \) such that \( N \) is a product with respect to \( \left\lbrack {0,1}\right\rbrack \times \partial... | ## Proof. This statement is precisely [Wall16, Proposition 1.5.6 (ii)]. | No |
Proposition 8.21. (*) Let \( M \) be a smooth manifold, let \( N \) be a compact proper subman-ifold of \( M \) and let \( \left\lbrack {0,1}\right\rbrack \times \partial M \) be a collar neighborhood for \( M \) . There exists a diffeotopy \( F \) of \( M \) rel \( \partial M \) and an \( \epsilon > 0 \) such that \( ... | Sketch of PROOF (*). If \( \partial M \) is compact, then we can take \( \epsilon = 1 \) and the statement follows immediately from the above Collar Neighborhood Theorem 8.20 and the uniqueness statement given by Proposition 8.13.\n\nIt \( \partial M \) is non-compact (and later in Proposition 8.22 we will be intereste... | No |
Proposition 8.22. Let \( M \) and \( N \) be two n-dimensional smooth manifolds with boundary components \( A \subset M \) and \( B \subset N \) and let \( f : A \rightarrow B \) be a diffeomorphism. Suppose we have chosen collar neighborhoods for \( \partial M \) and \( \partial N \) such that \( M{ \cup }_{f}N \) com... | Proof. The proposition can be deduced quite easily from Propositions 8.19 and 8.21. We leave it to the reader to fill in the details. | No |
Lemma 8.23. Let \( W \) be a smooth manifold with compact boundary and let \( X \) be a compact submanifold of \( \partial W \) . The push-in \( Y \) of \( X \) is a compact proper submanifold of \( W \) whose boundary equals \( \partial Y = \partial X \) . | Proof. We continue with the notation from the definition of the push-in. First we claim that \( f \) is smooth. This can be seen as follows: It follows easily from the choice of \( \varphi \) that the map \( f \) is evidently smooth on \( \lbrack 0,1) \times \partial X \) and \( X \smallsetminus \left( {\left\lbrack {0... | Yes |
Proposition 8.25. Let \( N \) be a closed \( k \) -dimensional smooth submanifold of \( {\mathbb{R}}^{n} \) . Given \( P \in N \) we write\n\n\[ \n{\left( {\mathrm{V}}_{P}N\right) }^{ \bot } = \left\{ {w \in {\mathbb{R}}^{n}\mid \langle v, w\rangle = 0\text{ for all }v \in {\mathrm{V}}_{P}N}\right\} .\n\]\n\nThere exis... | Sketch of A PROOF OF Proposition 8.25. Let \( N \) be a closed \( k \) -dimensional submani-fold of \( {\mathbb{R}}^{n} \) . We consider\n\n\[ \n{WN} \mathrel{\text{:=}} \left\{ {\left( {w, P}\right) \in {\mathbb{R}}^{n} \times {\mathbb{R}}^{n} \mid P \in N\text{ and }w \in {\left( {\mathrm{V}}_{P}N\right) }^{ \bot }}\... | Yes |
Proposition 8.26. (Uniqueness of tubular maps and tubular neighborhoods) Let \( M \) be an \( m \) -dimensional smooth manifold and let \( N \) be a closed \( n \) -dimensional submanifold. Furthermore let \( F, G : {\bar{B}}^{m - n} \times N \rightarrow M \) be two tubular maps for \( N \) . Then there exists a diffeo... | Proof. A proof for the proposition is given in [Wall16, Theorem 2.5.5 and Theorem 2.5.8]. We refer to [BJ82, Theorem 12.13], [Kos93, Corollary III.3.2] and [Lan01, Theorem IV.6.2] for alternative approaches, which, with a little bit of mental gymnastics, can also be used to prove the proposition. | No |
Theorem 8.27. (Isotopy Extension Theorem) Let \( M \) be a smooth manifold and let \( N \) be a compact smooth manifold.\n\n(1) Let \( f : N \rightarrow M \smallsetminus \partial M \) be a smooth embedding. If \( F : N \times \left\lbrack {0,1}\right\rbrack \rightarrow M \smallsetminus \partial M \) is a smooth isotopy... | Proof of the Isotopy Extension Theorem 8.27. The theorem follows almost immediately from the statements and the proofs of Theorem 2.4.2 and Theorem 2.4.6 of Wall16. Note that the results of Wall16 do not explicitly say that we can assume that \( {G}_{0} = \mathrm{{id}} \), but a quick look at the proof of [Wall16, Theo... | No |
Corollary 8.28. Let \( M \) be a smooth manifold without boundary and let \( N \) be a compact smooth manifold. Let \( f : N \rightarrow M \) and \( g : N \rightarrow M \) be two smooth embeddings. If there exists a smooth isotopy \( F : N \times \left\lbrack {0,1}\right\rbrack \rightarrow M \) from \( f \) to \( g \),... | Proof. First note that by the Isotopy Extension Theorem 8.27 there exists a diffeotopy \( G : M \times \left\lbrack {0,1}\right\rbrack \rightarrow M \) with \( {G}_{0} = \) id such that \( {G}_{1}\left( {f\left( N\right) }\right) = \overline{g\left( N\right) } \) . But this implies that \( {G}_{1} : M \rightarrow M \) ... | Yes |
Proposition 8.31. (*) Let \( M \) be a connected non-orientable \( n \) -dimensional smooth manifold, let \( {P}_{1},\ldots ,{P}_{m} \) be points in \( M \smallsetminus \partial M \) and let \( I \subset \{ 1,\ldots, m\} \) be a subset. There exists a diffeotopy rel a neighborhood of \( \partial M \) from the identity ... | Proof. We leave it to the reader to use the paths from Lemma 17.5 to modify the proof of Proposition 8.29 to obtain a proof of the proposition. | No |
Theorem 8.32. (Extension Theorem)(*) Let \( f : N \rightarrow M \) be a smooth embedding of a smooth manifold with boundary into a smooth manifold \( M \) such that \( f\left( N\right) \subset M \smallsetminus \partial M \) . Then there exists a smooth embedding \( {}^{161} \) | \[ f : \underset{\text{smooth manifold by Proposition 8.15 }}{\underbrace{\left( {N \sqcup \left( {\partial N \times \left\lbrack {0,1}\right\rbrack }\right) }\right) /{}_{\partial N = \partial N \times 0}}} \rightarrow M \] which restricts to the original \( f \) on \( N \) and such that the image lies in \( M \smalls... | No |
(1) Let \( g \in \mathbb{N} \) . The iterated connected sum of \( g \) tori is diffeomorphic to the surface of genus \( g \), as defined on page 206. | Sketch of PROOF. (1) We prove the statement for \( g = 2 \) . More precisely, in Figure 222 to the left we show two copies of the torus and two embeddings of \( {\bar{B}}^{2} \), one is orientation-preserving and the other one is orientation-reversing. Furthermore we indicate a map from the corresponding connected sum ... | No |
Lemma 8.34. The connected sum of an \( n \) -dimensional smooth manifold \( M \) with \( {S}^{n} \) is diffeomorphic to \( M \) . | Proof. Indeed, this can be seen as follows: in the construction of the connected sum \( M\# {S}^{n} \) we consider the ball \( {S}_{ \geq 0}^{n} \) in \( {S}^{n} \) . Since \( {S}^{n} \smallsetminus {S}_{ > 0}^{n} = {S}_{ \leq 0}^{n} \) is again a closed ball it is very believable that \( M\# {S}^{n} \) is again diffeo... | No |
Proposition 8.35. Let \( n \in \mathbb{N} \) . Let \( M \) and \( {M}^{\prime } \) be two connected \( n \) -dimensional smooth manifolds. (If both \( M \) and \( {M}^{\prime } \) are orientable, then we demand that \( M \) and \( {M}^{\prime } \) are oriented.) Let \( \varphi : {\bar{B}}^{n} \rightarrow M \smallsetmin... | (1) The resulting connected sum \( M\# {M}^{\prime } \) is a \( n \) -dimensional topological manifold that admits a canonical smooth structure which has the property that \( M \smallsetminus \varphi \left( {B}^{n}\right) \) and \( {M}^{\prime } \smallsetminus {\varphi }^{\prime }\left( {B}^{n}\right) \) are submanifol... | Yes |
Theorem 8.36. Let \( M \) be an \( n \) -dimensional smooth manifold. (If \( M \) is orientable, then we pick an orientation for \( M \) .) In the following let \( {\varphi }_{1},\ldots ,{\varphi }_{m} : {\bar{B}}^{n} \rightarrow M \smallsetminus \partial M \) and \( {\psi }_{1},\ldots ,{\psi }_{m} : {\bar{B}}^{n} \rig... | Proof of Theorem 8.36. Before we start with the actual proof we want to point out that on several occasions we implicitly use Lemma 8.6 which allows us to \ | No |
Lemma 8.37. (*) Let \( M \) be a non-empty \( n \) -dimensional smooth manifold \( M \). The connected sum \( M\# {S}^{n} \) is diffeomorphic to \( M \). | Proof (*). First we pick an orientation-preserving embedding \( \varphi : {\bar{B}}^{n} \rightarrow M \smallsetminus \partial \underline{M} \). Next let \( {g}_{ - } : {\bar{B}}^{n} \cong {S}_{ \leq 0}^{n} \subset {S}^{n} \) and \( {g}_{ + } : {\bar{B}}^{n} \cong {S}_{ \geq 0}^{n} \subset {S}^{n} \) be the two embeddin... | Yes |
Lemma 8.39. Let \( n \geq 1 \) .\n\n(1) Given two connected non-empty \( n \) -dimensional smooth manifolds \( X \) and \( Y \) the connected sums \( X\# Y \) and \( Y\# X \) are diffeomorphic.\n\n(2) Given three connected non-empty \( n \) -dimensional smooth manifolds \( X, Y \) and \( Z \) the connected sums \( \lef... | Proof. The first statement is just an immediate consequence of the definitions, whereas the second statement is a fairly straightforward consequence of Theorem 8.36. We leave it to the reader to fill in the details. | No |
Proposition 8.43. Let \( K \) and \( J \) be two knots. If they are smoothly isotopic, then the following two statements hold:\n\n(1) There exists an orientation-preserving diffeomorphism \( \Phi : {S}^{3} \rightarrow {S}^{3} \) with \( \Phi \left( K\right) = J \) . If \( K \) and \( J \) are oriented, then \( \Phi \) ... | Proof. Let\n\n\[ F : {S}^{1} \times \left\lbrack {0,1}\right\rbrack \rightarrow {S}^{3} \]\n\n\[ \left( {z, t}\right) \mapsto F\left( {z, t}\right) \]\n\nbe a smooth isotopy from a knot \( K \) to a knot \( J \) . By the Isotopy Extension Theorem 8.27 we can extend the smooth isotopy \( F \) to a diffeotopy of \( {S}^{... | Yes |
Corollary 8.45. Let \( K \) and \( J \) be two knots. If there exists an orientation-preserving diffeomorphism \( \Phi : {S}^{3} \rightarrow {S}^{3} \) with \( \Phi \left( K\right) = J \) | Proof. Let \( \Phi : {S}^{3} \rightarrow {S}^{3} \) be an orientation-preserving diffeomorphism with \( \Phi \left( K\right) = J \) . By the Cerf Theorem 8.44 there exists a diffeotopy \( F : {S}^{3} \times \left\lbrack {0,1}\right\rbrack \rightarrow {S}^{3} \) with \( {F}_{0} = \mathrm{{id}} \) and \( {F}_{1} = \Phi \... | Yes |
Theorem 8.46. (Gordon-Luecke) Let \( K \) and \( J \) be two knots in \( {S}^{3} \) . If there exists an orientation-preserving diffeomorphism between the knot complements \( {S}^{3} \smallsetminus K \) and \( {S}^{3} \smallsetminus J \) , then \( K \) and \( J \) are smoothly isotopic. | Proof. This theorem is significantly more difficult to prove than Proposition 8.43. In fact a proof was given only in the 1980s by Cameron Gordon and John Luecke GL89, Theorem 1]. The proof builds on the work of William Thurston Thu82 for which he got the fields medal in 1982. | No |
Lemma 8.48. Let \( n \in \mathbb{N} \) and let \( \varphi : {\bar{B}}^{n} \rightarrow {\mathbb{R}}^{n} \) be an embedding with \( \varphi \left( 0\right) = 0 \) .\n\n(1) If \( \varphi \) is orientation-preserving, then \( \varphi \) is smoothly isotopic, rel 0, to the identity.\n\n(2) If \( \varphi \) is orientation-re... | Proof of Lemma 8.48. This lemma is a special case of Proposition 8.26. But it is more fun to prove the lemma \ | No |
Proposition 8.50. Let \( K \) be an oriented knot in \( {S}^{3} \) . Let \( \mu \subset {S}^{3} \smallsetminus K \) be a closed oriented curve. Suppose there exists a smooth embedding \( \varphi : {\bar{B}}^{2} \subset {S}^{3} \) with the following two properties:\n\n(1) the \( {\left. \varphi \right| }_{{S}^{1}} \) de... | Sketch of Proof. By the Tubular Neighborhood Theorem 8.24 there exists an orientation-preserving smooth embedding \( F : {\bar{B}}^{2} \times K \rightarrow {S}^{3} \) such that \( F\left( \overline{0, P}\right) = P \) for all \( P \in K \) . For\n\n---\n\n\( {}^{171} \) It follows immediately from Proposition 8.1 that ... | No |
Lemma 8.51. Given any knot \( K \subset {S}^{3} \) the complement \( {S}^{3} \smallsetminus K \) is path-connected. | Proof. We will provide the proof in Exercise 8.28, | No |
Corollary 9.2. Every topological manifold \( M \) is metrizable. | Proof. Let \( M \) be a topological manifold. By Proposition 9.1 (2) there exists an \( n \in \mathbb{N} \) and a map \( \varphi : M \rightarrow {\mathbb{R}}^{n} \) which is an embedding. In other words, \( M \) is homeomorphic to a subset of \( {\mathbb{R}}^{n} \) . This means that the usual Euclidean metric \( d \) d... | Yes |
Theorem 9.3. (Whitney Approximation Theorem) Let \( M \) be smooth manifold (with or without boundary) and let \( N \) be a smooth manifold (with or without boundary).\n\n(1) Let \( f : M \rightarrow N \) be a map that is smooth on a (possibly empty) closed subset \( A{1}^{174} \) There exists a homotopy \( F : M \time... | (1) The first statement, without Condition (d), is the content of [Lee02, Theorem 6.26] and Lee02, Theorem 9.27. The full statement including Condition (d) follows by almost the same argument as in the proof of [Lee02, Theorem 6.26] and [Lee02, Theorem 9.27], one just needs to make full use of [Lee02, Theorem 6.21]. We... | Yes |
Corollary 9.4. Let \( n \in \mathbb{N} \) . We pick a base point \( * \) on \( {S}^{n} \) . Furthermore let \( M \) be a smooth manifold without boundary. We pick a base point \( {x}_{0} \) . The obvious map \{all smooth maps \( f : \left( {{S}^{n}, * }\right) \rightarrow \left( {M,{x}_{0}}\right) \) up to smooth homot... | Proof. Note that it follows immediately from the definition on page 278 that the restriction of any map \( {S}^{n} \rightarrow M \) to any one-point subset \( \{ * \} \) is smooth. Thus the corollary follows immediately from the Whitney Approximation Theorem 9.3, applied to \( A = \{ * \} \) . | No |
Proposition 9.5. (*) Let \( M \) be smooth manifold (with or without boundary) and let \( N \) be a smooth manifold without boundary. Furthermore let \( f : M \rightarrow N \) be a map. Finally let \( A \) and \( X \) be disjoint closed subsets such that the restriction of \( f \) to \( A \) is smooth and let \( U \sub... | Proof. First we consider the case that \( U = N \) . By Corollary 9.2 we can pick a metric \( c \) on \( M \) such that the topology from the metric \( c \) agrees with the given topology on \( M \) . The same way we can pick a metric \( d \) on \( N \) . We consider the open subset \( W \mathrel{\text{:=}} M \smallset... | Yes |
Proposition 9.6. Let \( M \) be a smooth manifold and let \( f : M \rightarrow X \) be a map to a Hausdorff space \( X \) . Let \( \varphi : {\bar{B}}^{n} \rightarrow X \) be an injective map such that \( \varphi \left( {B}^{n}\right) \) is an open subset of \( X \) . Given any \( r \in \left( {0,1}\right) \) there exi... | Proof. Since \( {\bar{B}}^{n} \) is compact, since \( X \) is Hausdorff and since \( \varphi : {\bar{B}}^{n} \rightarrow X \) is injective we know by Proposition 2.43 (2) that \( \varphi \) is a closed embedding. Thus we might as well identify \( {\bar{B}}^{n} \) with its image under the map \( \varphi : {\bar{B}}^{n} ... | Yes |
Proposition 9.7. Let \( M \) be a smooth manifold and let \( f : M \rightarrow X \) be a map to a Hausdorff space \( X \) . Let \( n \in \mathbb{N} \) and let \( * \in {S}^{n} \) be a point. Finally let \( \varphi : {S}^{n} \rightarrow X \) be an injective map such that \( \varphi \left( {{S}^{n}\smallsetminus \{ * \} ... | Proof. We leave it to the reader to modify the proof of Proposition 9.6 to obtain the desired result. As the reader will notice, in this case there is no need to work with some \( r \in \left( {0,1}\right) \) . In particular the conclusion we get is slightly more satisfactory. | No |
Lemma 9.8. Let \( M \) be an orientable n-dimensional smooth manifold, let \( f : M \rightarrow X \) be a map to some Hausdorff space \( X \), let \( \varphi : {B}^{k} \rightarrow X \) be a map and let \( x \in \varphi \left( {B}^{k}\right) \) be a point such that the following conditions are satisfied:\n\n(1) the imag... | Proof \( \left( *\right) \) . Note that \( U \mathrel{\text{:=}} {f}^{-1}\left( {\varphi \left( {B}^{k}\right) }\right) \) is an open subset of \( M \), in particular, as discussed in Lemma 6.22, it is a smooth manifold in its own right. It follows from this observation and (3) that we can apply the Regular Value Theor... | Yes |
Lemma 9.9. Let \( M \) be a smooth manifold and let \( Y \) be a proper submanifold of \( M \) of codimension l. Let \( X \) be some other smooth manifold and let \( f : X \rightarrow M \) be a smooth map. If \( f \) intersects \( Y \) transversely, then the following statements hold:\n\n(1) The preimage \( {f}^{-1}\le... | Proof. The proof of this lemma is not overly hard. For space reasons we skip the proof and instead we refer to [Lee02, Theorem 6.30] for a proof of (1). We leave the task of verifying the remaining statements to the reader. | No |
Theorem 9.10. (Transversality Theorem) Let \( M \) be a smooth manifold and let \( Y \) be a proper submanifold of \( M \). (1) If \( X \) is a smooth manifold and if \( f : X \rightarrow M \) is a smooth map, then there exists a smooth homotopy from \( f \) to a smooth map that is transverse to \( Y \). | Proof. (1) The first part of this statement is proved in [Lee02, Theorem 6.36]. A slightly less general statement is proved in GP74, p. 70]. | No |
Theorem 9.12. Let \( M \) be a compact smooth manifold and let \( N \) be a smooth manifold. If \( \dim \left( M\right) \geq 2 \cdot \dim \left( N\right) \), then any map \( f : M \rightarrow N \) is homotopic to an immersion. | Proof. This theorem follows from GG73, Theorem II.5.6 and the argument in [Wall16, Proof of Proposition 4.4.4]. | No |
Theorem 9.13. (Self-Transversality Theorem) Let \( M \) be a smooth manifold, let \( X \) be a smooth manifold, let \( \varphi : \left( {X,\partial X}\right) \rightarrow \left( {M,\partial M}\right) \) be an immersion and let \( U \) be a neighborhood of \( \varphi \left( X\right) \) . (1) There exists a smooth isotopy... | Proof. (1) This statement is a straightforward consequence of [Wall16, Proposition 4.6.6]. (2) With a little bit of an effort one can show that this statement follows from (1) together with [Adac93, Theorem II.6.2.8]. | No |
Corollary 9.14. Let \( M \) be a connected \( n \) -dimensional smooth manifold and let \( Y \) be a proper \( k \) -dimensional smooth submanifold of \( M \) . If \( k \leq n - 2 \), then \( M \smallsetminus Y \) is path-connected. | Proof. We leave the basically trivial case \( n = 2 \) to the reader. Furthermore, to simplify the discussion we assume that \( \partial M = \varnothing \) . We leave it to the reader to modify the argument below to deal with the case that \( \partial M \neq \varnothing \) .\n\nThus in the following we assume that \( n... | No |
Lemma 10.1. Let \( K \) be a compact subset of a topological manifold \( M \) . If \( N \) is a regular neighborhood of \( K \), then the following statements hold:\n\n(1) \( M \smallsetminus \overset{ \circ }{N} \) is a deformation retract of \( M \smallsetminus K \) .\n\n(2) The inclusion \( M \smallsetminus \widehat... | Proof.\n\n(1) By definition of a regular neighborhood there exists a deformation retraction \( r \) from \( N \smallsetminus K \) to \( {\partial }_{0}N \) . We extend this deformation retraction to all of \( M \smallsetminus K \) in the obvious way. More precisely, we consider the map\n\n\[ f : \left( {M \smallsetminu... | Yes |
Lemma 10.2. Let \( M \) be a compact topological manifold and let \( K \subset M \) be a compact subset. If \( K \) admits a regular neighborhood, then the following two statements hold:\n\n(1) the fundamental group \( {\pi }_{1}\left( {M \smallsetminus K}\right) \) is finitely presented, and\n\n(2) for each \( k \in {... | Proof. Note that by Lemma 10.1 the topological space \( M \smallsetminus K \) is homotopy equivalent to \( M \smallsetminus N \) . Since \( M \) is compact it follows from Lemma 89.1 (3) that \( M \smallsetminus N \) is also compact. Furthermore, by Lemma 89.1 (4) we know that \( M \smallsetminus N \) is an \( n \) -di... | No |
Theorem 10.3. (Regular Neighborhood Theorem) Let \( M \) be a smooth manifold.\n\nEvery compact proper submanifold of \( M \) admits a regular neighborhood. | An outline of the proof will be sketched in the next section. | No |
(1) Let \( N \) be an oriented \( n \) -dimensional smooth manifold and let \( k \in \mathbb{N} \) . The map\n\n\[ \n\\{ \\text{ maps }N \\rightarrow \\mathrm{{SO}}\\left( k\\right) \\} /\\text{ homotopy } \\rightarrow \\left\\{ \\begin{matrix} \\text{ orientation-preserving } \\\\ \\text{ isomorphisms of }{\\bar{B}}^{... | (1) This statement follows fairly easily from the definitions. We leave the details to the reader. | No |
Theorem 10.5. (General Tubular Neighborhood Theorem) Let \( M \) be a smooth manifold and let \( K \) be a proper submanifold. If \( K \) is compact, then \( K \) admits a tubular neighborhood. | Proof. The sketch for the existence of a tubular neighborhood that we provided for the previous Tubular Neighborhood Theorem 8.24 is also valid in this more general context. As before a full proof is provided in [Wal116, Theorem 2.3.3]. Closely related and similar results are also proved in [Kos93, Section III.2], [Bre... | No |
Proposition 10.6. (*) Let \( M \) be an orientable smooth manifold and let \( N \) be an orientable proper submanifold. Suppose one of the following holds:\n\n(1) \( N \) is one-dimensional,\n\n(2) \( N \) is of codimension one,\n\n(3) \( N \) is 2-dimensional and every component of \( N \) has non-empty boundary.\n\nT... | Proof. By the General Tubular Neighborhood Theorem 10.5 it remains to show that a tubular neighborhood is already trivial. For (1) and (2) this gets taken care of by the same references as in the Tubular Neighborhood Theorem 8.24. It remains to deal with (3). Note that with our hypothesis the topological space \( N \) ... | No |
Proposition 10.7. (Uniqueness of tubular neighborhoods) Let \( M \) be an \( n \) -dimensional smooth manifold and let \( K \) be a compact proper \( k \) -dimensional submanifold. Furthermore let \( p : U \rightarrow K \) and \( q : V \rightarrow K \) be two tubular neighborhoods of \( K \) . Then there exists a diffe... | Proof. As in the case of Proposition 8.26 a proof is provided in [Wall16, Chapter 2.5]. | No |
Lemma 10.8. Let \( M \) be an orientable smooth manifold and let \( K \) be a proper submanifold. If \( K \) is non-orientable, then the tubular neighborhood is non-trivial. | Sketch of Proof. We prove the converse, if the \( K \) admits a trivial tubular neighborhood, then \( K \) is orientable. So suppose that \( K \) is a proper \( k \) -dimensional submanifold of an orientable \( m \) -dimensional smooth manifold \( M \) and suppose that \( K \) admits a trivial neighborhood \( {\bar{B}}... | Yes |
Proposition 10.9. Let \( M \) be an oriented \( {}^{194} \) m-dimensional smooth manifold and let \( K \) be a compact oriented proper submanifold.\n\n(1) If \( K \) is of dimension one and \( m \geq 2 \), then there exists a trivial tubular neighborhood \( N\left( K\right) = {\bar{B}}^{m - 1} \times K \) such that the... | Proof. In both cases it follows from the General Tubular Neighborhood Theorem 10.5 together with Hau14, Proposition 9.2.3 and [MiS74, Axiom 3] that \( K \) admits a trivial neighborhood \( {}^{195} \) The statement regarding the orientations can be achieved by possibly using an orientation-reversing diffeomorphism of \... | No |
Lemma 10.12. (*) Let \( M \) be a smooth manifold and let \( X \) and \( Y \) be proper submanifolds of \( M \). If the submanifolds \( X \) and \( Y \) are compact and if they intersect transversally, then there exist tubular neighborhoods \( p : U \rightarrow X \) and \( q : V \rightarrow Y \) such that \n\n\[ \nU \c... | Proof. I could not find a reference for the third statement in any of the standard books on differential topology [Lee02, Wall16, Kos93, GP74]. But the standard proof for the existence of tubular neighborhoods can easily be adapted to prove the statement. The restriction to compact \( X \) and \( Y \) is almost certain... | No |
Proposition 10.13. Every finite spatial graph \( G \) in \( {\mathbb{R}}^{3} \) admits a regular neighborhood. | Proof. This proposition is proved in FH18, FH19. Note that in FH18, FH19 it is also shown that there is a type of regular neighborhood which is unique in an appropriate sense. | No |
Theorem 11.1. (Regular Value Theorem) Let \( M \) be an \( m \) -dimensional smooth manifold, let \( N \) be an \( n \) -dimensional smooth manifold without boundary, let \( f : M \rightarrow N \) be a smooth map and let \( s \in N \) be a regular value of \( f \) .\n\n(1) The preimage \( X \mathrel{\text{:=}} {f}^{-1}... | Sketch of PROOF. As mentioned above, the first three statements of the theorem form precisely the content of the original Regular Value Theorem 6.53 So it remains to deal with the final statement. This can be deduced quite easily from the Submersion Theorem 6.56 We leave it to the reader to fill in the details. | No |
Theorem 11.2. (Regular Value Theorem for \( N = \mathbb{R} \) ) Let \( M \) be an \( n \) -dimensional smooth manifold, let \( f : M \rightarrow \mathbb{R} \) be a smooth map and let \( s \in \mathbb{R} \) be regular value of \( f \) . Then beyond statements (1), (2), (3) of the Regular Value Theorem 11.1 the following... | SKETCH OF PROOF.\n\n(4) Note that given any neighborhood \( U \) of \( s \in \mathbb{R} \) there exists an \( \epsilon > 0 \) such that \( \left\lbrack {s - \epsilon, s + \epsilon }\right\rbrack \subset U \) . The statement now follows immediately from the Regular Value Theorem 11.1 (4) applied to the map \( \varphi : ... | No |
Theorem 11.3. (Regular Value Theorem for \( N = {S}^{1} \) ) Let \( M \) be an \( n \) -dimensional smooth manifold, let \( f : M \rightarrow {S}^{1} \) be a smooth map and let \( z = {e}^{{2\pi }\mathrm{i}t} \in {S}^{1} \) be a regular value. Then beyond statements (1), (2), (3) of the Regular Value Theorem 11.1 the f... | Proof. As in the case of Theorem 11.2 (4) this statement can be deduced easily from the Regular Value Theorem 11.1 (4). | No |
Proposition 11.4. Let \( W \) be an \( n \) -dimensional smooth manifold. There exists a sequence \( {X}_{1},{X}_{2},\ldots \) of n-dimensional submanifolds with corner of \( W \) with the following three properties:\n\n(1) The sequence is nested, i.e. for each \( i \in \mathbb{N} \) we have \( {X}_{i} \subset {X}_{i +... | Proof. The proof of Proposition 11.4 is almost identical to the proof of Proposition 6.64, for the most part we just need to replace the Exercise 6.31 by Regular Value Theorem 11.2. In particular we again pick a suitable smooth function \( f : W \rightarrow \mathbb{R} \), an increasing sequence of regular values \( {\m... | No |
Lemma 11.5. (*) Let \( M \) be a smooth manifold.\n\n(1) Let \( K \) be a compact subset of \( M \smallsetminus \partial M \) and let \( U \) be a neighborhood of \( K \) in \( M \) . There exists a submanifold \( X \) such that \( K \subset \overset{ \circ }{X} \subset X \subset U \) . | Proof (*). Let \( M \) be a smooth manifold. Recall that by Proposition 6.27 we know that \( \partial M \) is a closed subset of \( M \) .\n\n(1) Let \( K \) be a compact subset of \( M \smallsetminus \partial M \) . Basically by definition of a neighborhood we might as well assume that \( U \) is an open subset of \( ... | Yes |
Lemma 11.7. Let \( M \) be an \( m \) -dimensional submanifold of \( {\mathbb{R}}^{n} \). If \( n > {2m} + 1 \), then there exists a non-zero vector \( v \in {\mathbb{R}}^{n} \) which does not lie in any tangent space \( {}^{201}{\Gamma }_{P}M \) and that is not secant to \( M \), in other words, for any two distinct p... | Proof of Lemma 11.7. Let \( M \) be an \( m \) -dimensional submanifold of \( {\mathbb{R}}^{n} \). We suppose that \( n > {2m} + 1 \). We consider the maps\n\n\[ f : \left\{ {\left( {P, w}\right) \mid P \in M, w \in {\mathrm{T}}_{P}M\smallsetminus \{ 0\} }\right\} \rightarrow {\mathbb{{RP}}}^{n - 1} \]\n\n\[ \left( {P,... | Yes |
Proposition 11.10. Let \( M \) be a closed smooth manifold. Suppose we are given two smooth embeddings \( {f}_{0},{f}_{1} : M \rightarrow {\mathbb{R}}^{k} \). There exists a smooth isotopy \( F : M \times \left\lbrack {0,1}\right\rbrack \rightarrow {\mathbb{R}}^{2k} \) such that for \( i = 0,1 \) the following diagram ... | Proof. By Lemma 6.13 there exists a smooth function \( \nu : \left\lbrack {0,1}\right\rbrack \rightarrow \left\lbrack {0,1}\right\rbrack \) with \( \nu \left( t\right) = 1 \) for \( t \in \left\lbrack {0,\frac{1}{4}}\right\rbrack \) and \( \nu \left( t\right) = \overline{0\text{for}}t \in \left\lbrack {\frac{3}{4},1}\r... | Yes |
Proposition 11.11. Let \( M \) be a closed \( n \) -dimensional smooth manifold.\n\n(1) Given any two smooth embeddings \( {f}_{0},{f}_{1} : M \rightarrow {\mathbb{R}}^{{2n} + 2} \) there exists a smooth isotopy \( F : M \times \left\lbrack {0,1}\right\rbrack \rightarrow {\mathbb{R}}^{{2n} + 2} \) from \( {f}_{0} \) to... | Proof of Proposition 11.11 (1). Let \( M \) be a closed \( n \) -dimensional smooth manifold. Suppose we are given two smooth embeddings \( {f}_{0},{f}_{1} : M \rightarrow {\mathbb{R}}^{{2n} + 2} \) . By Proposition 11.10 there exists an \( N \geq {2n} + 2 \) and a smooth isotopy \( F : M \times \left\lbrack {0,1}\righ... | Yes |
Proposition 11.12. (*) Let \( M \) be a compact smooth manifold and let \( Y \) be a (possibly empty) union of boundary components of \( M \) . Suppose we are given two proper smooth embeddings \( {f}_{0},{f}_{1} : M \rightarrow {H}_{k} \) .\n\n(1) If \( {\left. {f}_{0}\right| }_{Y} = {\left. {f}_{1}\right| }_{Y} \), t... | Proof \( \left( *\right) \) .\n\n(1) We suppose that \( {\left. {f}_{0}\right| }_{Y} = {\left. {f}_{1}\right| }_{Y} \) . As in the proof of Proposition 11.10 we pick a smooth function \( \nu : \left\lbrack {0,1}\right\rbrack \rightarrow \left\lbrack {0,1}\right\rbrack \) with \( \nu \left( t\right) = 1 \) for \( t \in ... | Yes |
Theorem 11.14. (Whitney Embedding Theorem) Every n-dimensional smooth manifold admits a smooth embedding into \( {\mathbb{R}}^{2n} \) . | Proof. This result was first proved by Hassler Whitney [Why44a, p. 236] in 1944. For closed smooth manifolds more recent accounts of the proof can be found in Adac93, Theorem II.2.11] or [Pra07, Chapter 6.3]. | No |
Let \( M \) be a closed \( n \) -dimensional smooth manifold. If \( m \geq {2n} + 1 \) and \( m \geq n + 3 \) , then any two smooth embeddings \( {f}_{0},{f}_{1} : M \rightarrow {\mathbb{R}}^{m} \) are smoothly isotopic. | Both statements follow from [Hud72, Theorem 4]. For connected smooth manifolds the first statement also follows from [Hae61a, p. 47] or alternatively from [Wall16, Theorem 6.4.11]. | No |
Proposition 11.16. Every n-dimensional smooth manifold admits an immersion into \( {\mathbb{R}}^{2n} \) . | ## Proof. We will prove Proposition 11.16 in Exercise 11.8, | No |
Theorem 11.17. (Whitney Immersion Theorem) Every smooth manifold of dimension \( n \geq 2 \) admits an immersion into \( {\mathbb{R}}^{{2n} - 1} \) . | Proof. This theorem is proved in [Why44b, p. 247] and [Wall16, Theorem 6.3.6]. | No |
Lemma 12.5. Let \( n \in \mathbb{N} \) . On page 194 we introduced the complex projective space \[ {\mathbb{{CP}}}^{n} = \left( {{\mathbb{C}}^{n + 1}\smallsetminus \{ 0\} }\right) /\left( {\mathbb{C}\smallsetminus \{ 0\} }\right) . \] For \( i = 0,\ldots, n \) we consider the map \[ {\Phi }_{i} : \left\{ {\left\lbrack ... | Proof. In Proposition 3.40 (see also Exercise 3.39) we saw that each \( {\mathbb{{CP}}}^{n} \) is compact and Hausdorff. As we will show below, \( {\mathbb{{CP}}}^{n} \) admits a finite atlas, thus it follows from Lemma 6.4 that \( {\mathbb{{CP}}}^{n} \) is second-countable. For the remaining statements it is easy to o... | Yes |
(1) Let \( V \) be an \( n \)-dimensional complex vector space.\n\n(a) \( V \) admits a set of totally real vectors.\n\n(b) If \( \left\{ {{v}_{1},\ldots ,{v}_{n}}\right\} \) and \( \left\{ {{w}_{1},\ldots ,{w}_{n}}\right\} \) are two sets of totally real vectors for a complex vector space \( V \), then the bases \( \l... | (1) (a) This statement follows immediately from the observation that \( V \) is isomorphic, as a complex vector space, to \( {\mathbb{C}}^{n} \), and for \( {\mathbb{C}}^{n} \) a totally real basis is given by vectors \( {e}_{i} = \left( {0,\ldots ,1,\ldots ,0}\right), i = 1,\ldots, n \).\n\n(b) First we consider the c... | No |
(1) Let \( M \) be an \( n \) -dimensional complex manifold.\n\n(a) Given any \( P \in M \) we can use charts from the holomorphic atlas to equip \( {\mathrm{T}}_{P}M \) with the structure of an \( n \) -dimensional vector space.\n\n(b) If we equip each complex vector space \( {\mathrm{T}}_{P}M \) with the correspondin... | (1) (a) Let \( M \) be an \( n \) -dimensional complex manifold and let \( P \in M \) . We pick a chart \( \Phi : U \rightarrow V \) from the holomorphic atlas around \( P \) . We get an induced isomorphism \( \mathrm{D}{\Phi }_{P} : {\mathrm{T}}_{P}M \rightarrow {\mathrm{T}}_{P}V = {\mathbb{C}}^{n} \) . We use this is... | No |
Corollary 12.8. The Klein bottle and the real projective plane \( {\mathbb{{RP}}}^{2} \) do not admit the structure of a complex manifold. | Proof. In Lemma 6.48 we showed that neither the Klein bottle nor the real projective plane \( {\mathbb{{RP}}}^{2} \) are orientable. This is in contrast to the fact, shown in Proposition 12.7, that complex manifolds are orientable. | Yes |
Theorem 12.9. (Regular Value Theorem for Complex Manifolds) Let \( M \) be an \( m \) -dimensional complex manifold and let \( N \) be an \( n \) -dimensional complex manifold. Furthermore \( f : M \rightarrow N \) be a holomorphic map and let \( s \in N \) be a regular value of \( f \) .\n\n(1) The preimage \( X \math... | Proof. The proof is almost identical to the proof of the \ | No |
Lemma 14.3. (*) Let \( X \) be a topological space and let \( a < b < c \) be real numbers. Let \( f : X \times \left\lbrack {a, c}\right\rbrack \rightarrow Y \) be a map to some topological space \( Y \) . If the restrictions of \( f \) to \( X \times \left\lbrack {a, b}\right\rbrack \) and to \( X \times \left\lbrack... | Proof of Lemma 14.3. This statement is an immediate consequence of Lemma 2.35 (2) and the observation, see Lemma 3.9, that \( X \times \left\lbrack {a, b}\right\rbrack \) and \( X \times \left\lbrack {b, c}\right\rbrack \) are in fact closed subsets of \( X \times \left\lbrack {a, c}\right\rbrack \) . | Yes |
Lemma 14.4. Let \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow X \) be a path in a topological space. If a path \( \delta : \left\lbrack {a, b}\right\rbrack \rightarrow X \) is obtained from \( \gamma \) by a reparametrization, then \( \gamma \) and \( \delta \) are path-homotopic. | Proof. Let \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow X \) be a path in a topological space and let \( \varphi : \left\lbrack {a, b}\right\rbrack \rightarrow \left\lbrack {a, b}\right\rbrack \) be a map with \( \varphi \left( a\right) = a,\varphi \left( b\right) = b \) . Then a path-homotopy between \( \g... | Yes |
Lemma 14.5. Let \( \\alpha ,{\\alpha }^{\\prime } : \\left\\lbrack {a, b}\\right\\rbrack \\rightarrow X \) and \( \\beta ,{\\beta }^{\\prime } : \\left\\lbrack {c, d}\\right\\rbrack \\rightarrow X \) be two pairs of paths in a topological space \( X \) such that \( \\alpha \\left( b\\right) = {\\alpha }^{\\prime }\\lef... | Proof. We leave it to the reader to provide the fairly elementary proof. | No |
Proposition 14.6. Let \( \alpha ,\beta ,\gamma : \left\lbrack {0,1}\right\rbrack \rightarrow X \) be three paths in a topological space \( X \) . Then the following hold:\n\n(1) If \( \alpha \left( 1\right) = \beta \left( 0\right) \) and \( \beta \left( 1\right) = \gamma \left( 0\right) \), then \( {}^{218} \n\n\[ \lef... | Proof (*). Let \( \alpha ,\beta ,\gamma : \left\lbrack {0,1}\right\rbrack \rightarrow X \) be three paths in a topological space \( X \) .\n\n(1) Suppose that \( \alpha \left( 1\right) = \beta \left( 0\right) \) and \( \beta \left( 1\right) = \gamma \left( 0\right) \) . We need to show that\n\n\[ \alpha * \left( {\beta... | No |
Proposition 14.7. Let \( X \) be a topological space and \( {x}_{0} \in X \) . The set\n\n\[ \n{\pi }_{1}\left( {X,{x}_{0}}\right) \mathrel{\text{:=}} \left\{ \right. \text{path-homotopy classes of loops in}\left. \left( {X,{x}_{0}}\right) \right\} \n\]\n\ntogether with the product map \( \left\lbrack \alpha \right\rbr... | Proof. It is clear that the product of any two loops in \( \left( {X,{x}_{0}}\right) \) is defined. According to Lemma 14.5 the product of paths descends to a well-defined map\n\n\[ \n{\pi }_{1}\left( {X,{x}_{0}}\right) \times {\pi }_{1}\left( {X,{x}_{0}}\right) \rightarrow {\pi }_{1}\left( {X,{x}_{0}}\right) \n\]\n\n\... | Yes |
Theorem 14.8. Let \( X \) be a proper subse \( {}^{221} \) of \( \mathbb{C} \), let \( {x}_{0} \in X \) be a point and let \( w \in \mathbb{C} \smallsetminus X \) be a point in the complement of \( X \) . The map\n\n\[ \n{\pi }_{1}\left( {X,{x}_{0}}\right) \rightarrow \mathbb{Z} \n\]\n\n\[ \n\left\lbrack \gamma \right\... | Proof. A vigilant reader might first have noticed that the statement of Theorem 14.8 contains a slight cheat. We use the path integral over a continuous path \( \gamma \), whereas so far on page 458 we gave only the definition for the integral over a smooth path. The more general definition of a path integral is given ... | No |
Corollary 14.9. There exists an epimorphism \( {\pi }_{1}\left( {{S}^{1},1}\right) \rightarrow \mathbb{Z} \) . | Proof. The corollary follows from Theorem 14.8 and the calculation on page 458. | No |
Proposition 14.11. Let \( X \) be a topological space, let \( {x}_{0} \) and \( {x}_{1} \) be two points in \( X \) and let \( p : \left\lbrack {0,1}\right\rbrack \rightarrow X \) be a path from \( {x}_{0} \) to \( {x}_{1} \) . Then the following five statements hold:\n\n(1) The map\n\n\[ \n{p}_{ * } : {\pi }_{1}\left(... | Proof (*). Let \( X \) be a topological space and let \( {x}_{0} \) and \( {x}_{1} \) be two points in \( X \) that are connected via a path \( p : \left\lbrack {0,1}\right\rbrack \rightarrow X \) from \( {x}_{0} \) to \( {x}_{1} \) .\n\n(1) Let \( \gamma \) be a loop in \( \left( {X,{x}_{1}}\right) \) . Then \( p * \g... | Yes |
Lemma 14.15. For any \( P \in {S}^{n} \) the topological space \( {S}^{n} \smallsetminus \{ P\} \) is simply connected. | Proof. In Lemma 2.44 we showed that there exists a homeomorphism \( f : {S}^{n} \rightarrow {\mathbb{R}}^{n} \cup \{ \infty \} \) . We set \( Q \mathrel{\text{:=}} {f}^{-1}\left( \infty \right) \) . Thus \( f \) restricts to homeomorphism \( f : {S}^{n} \smallsetminus \{ Q\} \rightarrow {\mathbb{R}}^{n} \) . Now let \(... | Yes |
Lemma 15.2. The following two statements hold:\n\n(1) Let \( G = \\left( {V, E, i, t}\\right) \) and \( {G}^{\\prime } = \\left( {{V}^{\\prime },{E}^{\\prime },{i}^{\\prime },{t}^{\\prime }}\\right) \) be two abstract graphs. Furthermore let \( f \\mathrel{\\text{:=}} \\left( {\\alpha : V \\rightarrow {V}^{\\prime },\\... | Proof.\n\n(1) It follows almost immediately from the definitions that the map is well-defined. We leave it to the reader to verify, e.g. using Lemma 3.28, that the map is continuous. | No |
Lemma 15.3. Let \( \mathcal{D} \) be a category and let Set be the category of sets. Furthermore let \( W \in \mathrm{{Ob}}\left( \mathcal{D}\right) \) . The map \[ F : \mathrm{{Ob}}\left( \mathcal{D}\right) \rightarrow \mathrm{{Ob}}\left( {\mathcal{S}{et}}\right) \] \[ X \mapsto {\operatorname{Mor}}_{\mathcal{D}}\left... | Proof. The statement follows easily from the axioms of a category. | No |
Lemma 15.4. Let \( \mathcal{D} \) be a category and let Set be the category of sets. Furthermore let \( W \in \mathrm{{Ob}}\left( \mathcal{D}\right) \) . The map\n\n\[ F : \mathrm{{Ob}}\left( \mathcal{D}\right) \rightarrow \mathrm{{Ob}}\left( {\mathcal{S}{et}}\right) \]\n\n\[ X \mapsto {\operatorname{Mor}}_{\mathcal{D}... | Proof. Again the statement follows easily from the axioms of a category. | No |
Lemma 15.7. The circle \( {S}^{1} \) is not a retract of \( {\bar{B}}^{2} \) . | Proof. We denote by \( i : {S}^{1} \rightarrow {\bar{B}}^{2} \) the inclusion map. Suppose there exists a retraction \( r : {\bar{B}}^{2} \rightarrow {S}^{1} \) . By definition this means that \( r \circ i = {\operatorname{id}}_{{S}^{1}} \) . We obtain the following commutative diagram of maps between topological space... | Yes |
Lemma 15.10. (*) Let \( {E}_{8} \) be the regular octagon from the definition of the surface of genus 2, see page 205 for details, and let \( \sum = {E}_{8}/ \sim \) be the surface of genus 2. Let \( P \) and \( Q \) be two adjacent vertices of \( {E}_{8} \) . Then the following hold:\n\n(1) The map\n\n\[ \n\gamma : {S... | Proof \( \left( *\right) \) .\n\n(1) It is clear that \( \gamma \) is continuous and injective. By Proposition 6.8 we know that \( \sum = {E}_{8}/ \sim \) is Hausdorff. It follows from Proposition 2.43 (2) that \( \gamma \) is a closed embedding. | Yes |
Proposition 15.11. Let \( \varphi : X \rightarrow Y \) be a map between topological spaces and let \( {x}_{0} \in X \) .\n\n(1) If \( \gamma : \left\lbrack {0,1}\right\rbrack \rightarrow X \) is a path from \( {x}_{0} \) to some point \( {x}_{1} \), then the following diagram commutes \( {}^{238} \n\n\[ \n\begin{matrix... | (1) This statement follows immediately from the definitions. | Yes |
Lemma 16.3. Let \( p : X \rightarrow B \) be a covering of topological spaces. Then the following hold:\n\n(1) The map \( p : X \rightarrow B \) is open. | Proof \( \left( *\right) \) . Let \( p : X \rightarrow B \) be a covering of topological spaces.\n\n(1) Let \( U \) be an open subset of \( X \) . We need to show that \( p\left( U\right) \) is open. It suffices to show that for each \( P \in p\left( U\right) \) there exists an open neighborhood \( V \) such that \( P ... | Yes |
Lemma 16.5. Let \( G \) be a group which acts continuously on a topological space \( X \) .\n\n(1) If \( G \) is finite, then the action is proper.\n\n(2) If \( G \) acts discretely, then it also acts freely.\n\n(3) If \( X \) is a Hausdorff space and if the action by \( G \) is free and proper, then the action is also... | Proof. The first two statements follow immediately from the definitions. The third statement is the content of Lemma 6.33 | No |
Lemma 16.6. Let \( p \in \mathbb{N} \) and furthermore let \( q, r \in \mathbb{Z} \) with \( \gcd \left( {p, q}\right) = \gcd \left( {p, r}\right) = 1 \) . If \( q \equiv \pm {r}^{\pm 1}{\;\operatorname{mod}\;p}.{}^{256} \) then \( L\left( {p, q}\right) \) and \( L\left( {p, r}\right) \) are diffeomorphic \( {}^{257} \... | Proof. Let \( p \in \mathbb{N} \) and let \( q, r \in \mathbb{Z} \) with \( \gcd \left( {p, q}\right) = \gcd \left( {p, r}\right) = 1 \) . We suppose that \( q \equiv \pm {r}^{\pm 1}{\;\operatorname{mod}\;p} \) . We define a map\n\n\[ L\left( {p, q}\right) \rightarrow L\left( {p, r}\right) \]\n\nas follows:\n\nCase 1: ... | Yes |
Proposition 16.9. Let \( X \) be a topological space together with a discrete and continuous action by a group \( G \) . Then the canonical projection \( p : X \rightarrow X/G \) is a covering. | Proof. Let \( G \) be a group which acts discretely and continuously on a topological space \( X \) . We want to show that the projection map \( p : X \rightarrow X/G \) is a covering. Let \( \left\lbrack x\right\rbrack \in X/G \) . Since the group acts discretely there exists an open neighborhood \( V \) of \( x \in X... | Yes |
Proposition 16.10. Let \( X \) be a topological space together with a discrete and continuous action by a group \( G \) . Furthermore let \( H \) be a subgroup of \( G \) . Then the canonical projection\n\n\[ p : X/H \rightarrow X/G \]\n\n\[ \left\lbrack x\right\rbrack \mapsto \left\lbrack x\right\rbrack \]\n\nis a cov... | For \( H \) the trivial group we recover the statement of Proposition 16.9. In fact the proof of Proposition 16.10 is almost identical to the proof of Proposition 16.9. We leave the details to the reader. | No |
Proposition 16.12. Let \( p : X \rightarrow B \) be a covering, let \( B \) be a topological space, let \( Y \) be a topological manifold \( {}^{262} \) and let \( f : Y \times \left\lbrack {0,1}\right\rbrack \rightarrow B \) be a map. Furthermore let \( \widetilde{f} : Y \times 0 \rightarrow X \) be a lift of \( {\lef... | Proof (*). Let \( p : X \rightarrow B \) be a covering, let \( f : Y \times \left\lbrack {0,1}\right\rbrack \rightarrow B \) be a continuous map and let \( \widetilde{f} : Y \times 0 \rightarrow X \) be a continuous lift of \( {\left. f\right| }_{Y \times 0} \) . (Here and throughout this proof we do not suppose that a... | No |
Corollary 16.13. Let \( p : X \rightarrow B \) be a covering and let \( f, g : \left\lbrack {0,1}\right\rbrack \rightarrow B \) be two paths. Let \( \widetilde{f},\widetilde{g} : \left\lbrack {0,1}\right\rbrack \rightarrow X \) be two lifts with the same starting point. If \( f \) and \( g \) are path-homotopic, then t... | Proof \( \left( *\right) \) . We write \( P \mathrel{\text{:=}} f\left( 0\right) = g\left( 0\right) \) and \( Q \mathrel{\text{:=}} f\left( 1\right) = g\left( 1\right) \) . We denote by \( \widetilde{P} \mathrel{\text{:=}} \widetilde{f}\left( 0\right) = \) \( \widetilde{g}\left( 0\right) \) the common starting point of... | Yes |
For any covering \( p : \left( {X,{x}_{0}}\right) \rightarrow \left( {B,{b}_{0}}\right) \) of pointed topological spaces the induced map \( {p}_{ * } : {\pi }_{1}\left( {X,{x}_{0}}\right) \rightarrow {\pi }_{1}\left( {B,{b}_{0}}\right) \) is a monomorphism. | We have \( {}^{268} \)\n\n\[ \left\lbrack f\right\rbrack \in \ker \left( {{\pi }_{1}\left( {X,{x}_{0}}\right) \overset{{p}_{ * }}{ \rightarrow }{\pi }_{1}\left( {B,{b}_{0}}\right) }\right) \Rightarrow \left\lbrack {p \circ f}\right\rbrack = e \in {\pi }_{1}\left( {B,{b}_{0}}\right) \Leftrightarrow p \circ f \simeq {e}_... | Yes |
Lemma 16.15. Let \( p : \left( {X,{x}_{0}}\right) \rightarrow \left( {B,{b}_{0}}\right) \) be a covering of pointed topological spaces.\n\n(1) Let \( f \) be a loop in \( \left( {B,{b}_{0}}\right) \) . We denote by \( \widetilde{f} \) the lift of \( f \) to the starting point \( {x}_{0} \) . Then the following holds:\n... | (1) We prove the \ | No |
Theorem 16.16. Let \( X \) be a topological space and let \( G \) be a group which acts continuously and discretely on \( X \) . We choose an \( x \in X \) . We denote by \( p : X \rightarrow X/G \) the canonical projection map.\n\n(1) If \( X \) is simply connected, then the map\n\n\[ G \rightarrow {\pi }_{1}\left( {X... | Proof of Theorem 16.16 (*). To simplify the discussion we only provide the proof of Theorem 16.16 (1). The proof of Theorem 16.16 (2) is almost the same. We leave it to the reader to make the necessary modifications.\n\nLet \( X \) | No |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.