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Corollary 7.26 (Topological Invariance of \( {\pi }_{1} \) ). Homeomorphic spaces have isomorphic fundamental groups. Specifically, if \( \varphi : X \rightarrow Y \) is a homeomorphism, then \( {\varphi }_{ * } : {\pi }_{1}\left( {X, p}\right) \rightarrow {\pi }_{1}\left( {Y,\varphi \left( p\right) }\right) \) is an i... | Proof. If \( \varphi \) is a homeomorphism, then \( {\left( {\varphi }^{-1}\right) }_{ * } \circ {\varphi }_{ * } = {\left( {\varphi }^{-1} \circ \varphi \right) }_{ * } = {\left( {\operatorname{Id}}_{X}\right) }_{ * } = \) \( {\operatorname{Id}}_{{\pi }_{1}\left( {X, p}\right) } \), and similarly \( {\varphi }_{ * } \... | Yes |
Proposition 7.28. Suppose \( A \) is a retract of \( X \) . If \( r : X \rightarrow A \) is any retraction, then for any \( p \in A,{\left( {\iota }_{A}\right) }_{ * } : {\pi }_{1}\left( {A, p}\right) \rightarrow {\pi }_{1}\left( {X, p}\right) \) is injective and \( {r}_{ * } : {\pi }_{1}\left( {X, p}\right) \rightarro... | Proof. Since \( r \circ {\iota }_{A} = {\operatorname{Id}}_{A} \), the composition \( {r}_{ * } \circ {\left( {\iota }_{A}\right) }_{ * } \) is the identity on \( {\pi }_{1}\left( {A, p}\right) \) , from which it follows that \( {\left( {\iota }_{A}\right) }_{ * } \) is injective and \( {r}_{ * } \) is surjective. | Yes |
Corollary 7.29. A retract of a simply connected space is simply connected. | Proof. If \( A \) is a retract of \( X \), the previous proposition shows that \( {\left( {\iota }_{A}\right) }_{ * } : {\pi }_{1}\left( {A, p}\right) \rightarrow \) \( {\pi }_{1}\left( {X, p}\right) \) is injective. Thus if \( {\pi }_{1}\left( {X, p}\right) \) is trivial, so is \( {\pi }_{1}\left( {A, p}\right) \) . | Yes |
For any \( n \geq 1 \), it is easy to check that the map \( r : {\mathbb{R}}^{n} \smallsetminus \{ 0\} \rightarrow {\mathbb{S}}^{n - 1} \) given by \( r\left( x\right) = x/\left| x\right| \) is a retraction. | Because \( {\mathbb{S}}^{1} \) is not simply connected, it follows from Corollary 7.29 that \( {\mathbb{R}}^{2} \smallsetminus \{ 0\} \) is not simply connected. Thus \( {\mathbb{R}}^{2} \smallsetminus \{ 0\} \) is not homeomorphic to \( {\mathbb{R}}^{2} \) . | No |
Proposition 7.34 (Fundamental Group of a Product). If \( {X}_{1},\ldots ,{X}_{n} \) are any topological spaces, the map \( P \) defined by (7.2) is an isomorphism. | Proof. First we show that \( P \) is surjective. Let \( \left\lbrack {f}_{i}\right\rbrack \in {\pi }_{1}\left( {{X}_{i},{x}_{i}}\right) \) be arbitrary for \( i = 1,\ldots, n \) . Define a loop \( f \) in the product space by \( f\left( s\right) = \left( {{f}_{1}\left( s\right) ,\ldots ,{f}_{n}\left( s\right) }\right) ... | Yes |
Proposition 7.37. For any \( n \geq 1,{\mathbb{S}}^{n - 1} \) is a strong deformation retract of \( {\mathbb{R}}^{n} \smallsetminus \{ 0\} \) and of \( {\overline{\mathbb{B}}}^{n} \smallsetminus \{ 0\} \) . | Proof. Define a homotopy \( H : \left( {{\mathbb{R}}^{n}\smallsetminus \{ 0\} }\right) \times I \rightarrow {\mathbb{R}}^{n} \smallsetminus \{ 0\} \) by\n\n\[ H\left( {x, t}\right) = \left( {1 - t}\right) x + t\frac{x}{\left| x\right| }.\]\n\nThis is just the straight-line homotopy from the identity map to the retracti... | Yes |
Let \( X \) be any space. If the identity map of \( X \) is homotopic to a constant map, we say that \( X \) is contractible. Other equivalent definitions are that any point of \( X \) is a deformation retract of \( X \), or \( X \) is homotopy equivalent to a one-point space (Exercise 7.42). Concretely, contractibilit... | In other words, the whole space \( X \) can be continuously shrunk to a point. Some simple examples of contractible spaces are convex subsets of \( {\mathbb{R}}^{n} \), and, more generally, any subset \( B \subseteq {\mathbb{R}}^{n} \) that is star-shaped, which means that there is some point \( {p}_{0} \in B \) such t... | Yes |
Lemma 7.45. Suppose \( \varphi ,\psi : X \rightarrow Y \) are continuous, and \( H : \varphi \simeq \psi \) is a homotopy. For any \( p \in X \), let \( h \) be the path in \( Y \) from \( \varphi \left( p\right) \) to \( \psi \left( p\right) \) defined by \( h\left( t\right) = H\left( {p, t}\right) \) , and let \( {\P... | Proof. Let \( f \) be any loop in \( X \) based at \( p \) . What we need to show is\n\n\[ \n{\psi }_{ * }\left\lbrack f\right\rbrack = {\Phi }_{h}\left( {{\varphi }_{ * }\left\lbrack f\right\rbrack }\right) \n\]\n\n\[ \n\Leftrightarrow \psi \circ f \sim \bar{h} \cdot \left( {\varphi \circ f}\right) \cdot h \n\]\n\n\[ ... | Yes |
Proposition 7.46. With notation as above, if \( f \) is a homotopy equivalence, then \( \widetilde{Y} \) and \( \widetilde{X} \) are deformation retracts of \( {Z}_{f} \) . Thus two spaces are homotopy equivalent if and only if they are both homeomorphic to deformation retracts of a single space. | Proof. For any \( \left( {x, s}\right) \in X \times I \), let \( \left\lbrack {x, s}\right\rbrack = q\left( {x, s}\right) \) denote its equivalence class in \( {Z}_{f} \) ; similarly, \( \left\lbrack y\right\rbrack = q\left( y\right) \) is the equivalence class of \( y \in Y \). First we show that \( \widetilde{Y} \) i... | Yes |
Example 7.47 (Categories). Here are some familiar examples of categories, which we describe by specifying their objects and morphisms. In each case, the source and target of a morphism are its domain and codomain, respectively; the composition law is given by composition of maps; and the identity morphism is the identi... | In each case, the verification of the axioms of a category is straightforward. The main point is to show that a composition of the appropriate structure-preserving maps again preserves the structure. Associativity is automatic because it holds for composition of maps. | No |
Theorem 7.51. For any categories \( \mathrm{C} \) and \( \mathrm{D} \), every (covariant or contravariant) functor from \( \mathrm{C} \) to \( \mathrm{D} \) takes isomorphisms in \( \mathrm{C} \) to isomorphisms in \( \mathrm{D} \) . | Proof. The proof is exactly the same as the proof for the fundamental group functor (Corollary 7.26). | No |
Theorem 7.54. If a product exists in any category, it is unique up to a unique isomorphism that respects the projections. More precisely, if \( \left( {P,\left( {\pi }_{\alpha }\right) }\right) \) and \( \left( {{P}^{\prime },\left( {\pi }_{\alpha }^{\prime }\right) }\right) \) are both products of the family \( \left(... | Proof. Given \( \left( {P,\left( {\pi }_{\alpha }\right) }\right) \) and \( \left( {{P}^{\prime },\left( {\pi }_{\alpha }^{\prime }\right) }\right) \) as in the statement of the theorem, the defining property of products guarantees the existence of unique morphisms \( f : P \rightarrow {P}^{\prime } \) and \( {f}^{\pri... | Yes |
Proposition 8.1. Each point \( z \in {\mathbb{S}}^{1} \) has a neighborhood \( U \) with the following property (see Fig. 8.1):\n\n\( {\varepsilon }^{-1}\left( U\right) \) is a countable union of disjoint open intervals\n\n\( {\widetilde{U}}_{n} \) with the property that \( \varepsilon \) restricts to a homeomor-\n\n(8... | Proof. This is just a straightforward computation from the definition of \( \varepsilon \) . We can cover \( {\mathbb{S}}^{1} \) by the four open subsets\n\n\[ \n{X}_{ + } = \{ \left( {x + {iy}}\right) : x > 0\} ,\;{Y}_{ + } = \{ \left( {x + {iy}}\right) : y > 0\} , \n\]\n\n(8.2)\n\n\[ \n{X}_{ - } = \{ \left( {x + {iy}... | Yes |
Corollary 8.2 (Local Section Property of the Circle). Let \( U \subseteq {\mathbb{S}}^{1} \) be any evenly covered open subset. For any \( z \in U \) and any \( r \) in the fiber of \( \varepsilon \) over \( z \), there is a local section \( \sigma \) of \( \varepsilon \) over \( U \) such that \( \sigma \left( z\right... | Proof. Given \( z \in U \) and \( r \in {\varepsilon }^{-1}\left( z\right) \), let \( \widetilde{U} \subseteq \mathbb{R} \) be the component of \( {\varepsilon }^{-1}\left( U\right) \) containing \( r \) . By definition of an evenly covered open subset, \( \varepsilon : \widetilde{U} \rightarrow U \) is a homeomorphism... | Yes |
Corollary 8.5 (Path Lifting Property of the Circle). Suppose \( f : I \rightarrow {\mathbb{S}}^{1} \) is any path, and \( {r}_{0} \in \mathbb{R} \) is any point in the fiber of \( \varepsilon \) over \( f\left( 0\right) \) . Then there exists a unique lift \( \widetilde{f} : I \rightarrow \mathbb{R} \) of \( f \) such ... | Proof. A path \( f \) can be viewed as a homotopy between two maps from a one-point space \( \{ * \} \) into \( {\mathbb{S}}^{1} \), namely \( * \mapsto f\left( 0\right) \) and \( * \mapsto f\left( 1\right) \) . Thus the existence and uniqueness of \( \widetilde{f} \) follow from the homotopy lifting property. To prove... | Yes |
Corollary 8.6 (Path Homotopy Criterion for the Circle). Suppose \( {f}_{0} \) and \( {f}_{1} \) are paths in \( {\mathbb{S}}^{1} \) with the same initial point and the same terminal point, and \( {\widetilde{f}}_{0},{\widetilde{f}}_{1} : I \rightarrow \) \( \mathbb{R} \) are lifts of \( {f}_{0} \) and \( {f}_{1} \) wit... | Proof. If \( {\widetilde{f}}_{0} \) and \( {\widetilde{f}}_{1} \) have the same terminal point, then they are path-homotopic by Exercise 7.14, because \( \mathbb{R} \) is simply connected. It follows that \( {f}_{0} = \varepsilon \circ {\widetilde{f}}_{0} \) and \( {f}_{1} = \varepsilon \circ {\widetilde{f}}_{1} \) are... | Yes |
Theorem 8.8 (Homotopy Classification of Loops in \( {\mathbb{S}}^{1} \) ). Two loops in \( {\mathbb{S}}^{1} \) based at the same point are path-homotopic if and only if they have the same winding number. | Proof. Suppose \( {f}_{0} \) and \( {f}_{1} \) are loops in \( {\mathbb{S}}^{1} \) based at the same point. By the path lifting property (Corollary 8.5), they have lifts \( {\widetilde{f}}_{0},{\widetilde{f}}_{1} : I \rightarrow \mathbb{R} \) starting at the same point, and by the path homotopy criterion (Corollary 8.6... | Yes |
Theorem 8.9 (Fundamental Group of the Circle). The group \( {\pi }_{1}\left( {{\mathbb{S}}^{1},1}\right) \) is an infinite cyclic group generated by \( \left\lbrack \omega \right\rbrack \) . | Proof. Define maps \( J : \mathbb{Z} \rightarrow {\pi }_{1}\left( {{\mathbb{S}}^{1},1}\right) \) and \( K : {\pi }_{1}\left( {{\mathbb{S}}^{1},1}\right) \rightarrow \mathbb{Z} \) by\n\n\[ J\left( n\right) = {\left\lbrack \omega \right\rbrack }^{n},\;K\left( \left\lbrack f\right\rbrack \right) = N\left( f\right) . \]\n\... | Yes |
Corollary 8.12 (Fundamental Groups of Tori). Let \( {\mathbb{T}}^{n} = {\mathbb{S}}^{1} \times \cdots \times {\mathbb{S}}^{1} \) be the \( n \) - dimensional torus with \( p = \left( {1,\ldots ,1}\right) \) as base point, and for each \( j = 1,\ldots, n \), let \( {\omega }_{j} \) denote the standard loop in the \( j \... | Proof. This is a direct consequence of Theorem 8.9 and Proposition 7.34. | No |
Lemma 8.14 (Another Characterization of the Degree). If \( \varphi : {\mathbb{S}}^{1} \rightarrow {\mathbb{S}}^{1} \) is continuous, the degree of \( \varphi \) is equal to the degree of the following group endomorphism:\n\n\[ \n{\left( {\rho }_{\varphi } \circ \varphi \right) }_{ * } : {\pi }_{1}\left( {{\mathbb{S}}^{... | Proof. Let \( \varphi \) be as in the statement of the lemma, and let \( n \) be the degree of \( \varphi \), which is the winding number of the loop \( \varphi \circ \omega \) . By Exercise 8.7, the winding number of \( {\rho }_{\varphi } \circ \varphi \circ \omega \) is also \( n \) . By Theorem 8.8, this implies tha... | Yes |
Theorem 8.17 (Homotopy Classification of Maps of the Circle). Two continuous maps from \( {\mathbb{S}}^{1} \) to itself are homotopic if and only if they have the same degree. | Proof. One direction was proved in Proposition 8.15. To prove the converse, suppose \( \varphi \) and \( \psi \) have the same degree. First consider the special case in which \( \varphi \left( 1\right) = \psi \left( 1\right) = 1 \) . Then the hypothesis means that \( \varphi \circ \omega \) and \( \psi \circ \omega \)... | Yes |
Theorem 8.18. Let \( \varphi : {\mathbb{S}}^{1} \rightarrow {\mathbb{S}}^{1} \) be continuous. If \( \deg \varphi \neq 0 \), then \( \varphi \) is surjective. | Proof. We prove the contrapositive. If \( \varphi \) is not surjective, then it actually maps into the subset \( {\mathbb{S}}^{1} \smallsetminus \{ c\} \) for some \( c \in {\mathbb{S}}^{1} \) . But \( {\mathbb{S}}^{1} \smallsetminus \{ c\} \) is homeomorphic to \( \mathbb{R} \) (by the 1-dimensional version of stereog... | Yes |
Theorem 8.19. Let \( \varphi : {\mathbb{S}}^{1} \rightarrow {\mathbb{S}}^{1} \) be continuous. If \( \deg \varphi \neq 1 \), then \( \varphi \) has a fixed point. | Proof. Again we prove the contrapositive. Assuming \( \varphi \) has no fixed point, it follows that for every \( z \in {\mathbb{S}}^{1} \), the line segment in \( \mathbb{C} \) from \( \varphi \left( z\right) \) to \( - z \) does not pass through the origin. Thus we can define a homotopy from \( \varphi \) to the anti... | Yes |
Proposition 9.1. Given an indexed family of groups \( {\left( {G}_{\alpha }\right) }_{\alpha \in A} \), their free product is a group under the multiplication operation induced by multiplication of words. | Proof. First we need to check that multiplication of words respects the equivalence relation. If \( {V}^{\prime } \) is obtained from \( V \) by an elementary reduction, then it is easy to see that \( {V}^{\prime }W \) is similarly obtained from \( {VW} \), as is \( W{V}^{\prime } \) from \( {WV} \) . If \( V \sim {V}^... | Yes |
Example 9.3. Let \( \mathbb{Z}/2 \) denote the group of integers modulo 2 . The free product \( \mathbb{Z}/2 * \mathbb{Z}/2 \) can be described as follows. If we let \( \beta \) and \( \gamma \) denote the nontrivial elements of the first and second copies of \( \mathbb{Z}/2 \), respectively, each element of \( \mathbb... | For example,\n\n\[ \left( {\beta \gamma \beta \gamma \beta }\right) \left( {\gamma \beta \gamma \beta }\right) = {\beta \gamma \beta \gamma \beta \gamma \beta \gamma \beta } \]\n\n\[ \left( {\gamma \beta \gamma \beta }\right) \left( {\beta \gamma \beta \gamma \beta }\right) = \beta \text{.} \]\n\nBecause these two prod... | Yes |
Theorem 9.5 (Characteristic Property of the Free Product). Let \( {\left( {G}_{\alpha }\right) }_{\alpha \in A} \) be an indexed family of groups. For any group \( H \) and any collection of homomorphisms \( {\varphi }_{\alpha } : {G}_{\alpha } \rightarrow H \), there exists a unique homomorphism \( \Phi : { * }_{\alph... | Proof. Suppose we are given a collection of homomorphisms \( {\varphi }_{\alpha } : {G}_{\alpha } \rightarrow H \) . The requirement that \( \Phi \circ {\iota }_{\alpha } = {\varphi }_{\alpha } \) implies that the desired homomorphism \( \Phi \) must satisfy\n\n\[ \Phi \left( g\right) = {\varphi }_{\alpha }\left( g\rig... | Yes |
Theorem 9.5 (Characteristic Property of the Free Product). Let \( {\left( {G}_{\alpha }\right) }_{\alpha \in A} \) be an indexed family of groups. For any group \( H \) and any collection of homomorphisms \( {\varphi }_{\alpha } : {G}_{\alpha } \rightarrow H \), there exists a unique homomorphism \( \Phi : { * }_{\alph... | Proof. Suppose we are given a collection of homomorphisms \( {\varphi }_{\alpha } : {G}_{\alpha } \rightarrow H \) . The requirement that \( \Phi \circ {\iota }_{\alpha } = {\varphi }_{\alpha } \) implies that the desired homomorphism \( \Phi \) must satisfy\n\n\[ \Phi \left( g\right) = {\varphi }_{\alpha }\left( g\rig... | Yes |
Corollary 9.6. The free product is the coproduct in the category of groups. | Proof. The characteristic property is exactly the defining property of the coproduct in a category. | No |
The free product is the unique group (up to isomorphism) satisfying the characteristic property. | Proof. Theorem 7.57 shows that coproducts in any category are unique up to isomorphism. | No |
Theorem 9.9 (Characteristic Property of the Free Group). Let \( S \) be a set. For any group \( H \) and any map \( \varphi : S \rightarrow H \), there exists a unique homomorphism \( \Phi : F\left( S\right) \rightarrow H \) extending \( \varphi \) : | Proof. This can be proved directly as in the proof of Theorem 9.5. Alternatively, recalling that the free group is defined as a free product, we can proceed as follows. There is a one-to-one correspondence between set functions \( \varphi : S \rightarrow H \) and collections of homomorphisms \( {\varphi }_{\sigma } : F... | No |
Theorem 9.9 (Characteristic Property of the Free Group). Let \( S \) be a set. For any group \( H \) and any map \( \varphi : S \rightarrow H \), there exists a unique homomorphism \( \Phi : F\left( S\right) \rightarrow H \) extending \( \varphi \). | Proof. This can be proved directly as in the proof of Theorem 9.5. Alternatively, recalling that the free group is defined as a free product, we can proceed as follows. There is a one-to-one correspondence between set functions \( \varphi : S \rightarrow H \) and collections of homomorphisms \( {\varphi }_{\sigma } : F... | No |
Proposition 9.12. A group \( G \) is free if and only if it has a generating set \( S \subseteq G \) such that every element \( g \in G \) other than the identity has a unique expression as a product of the form\n\n\[ g = {\sigma }_{1}^{{n}_{1}}\cdots {\sigma }_{k}^{{n}_{k}} \]\n\nwhere \( {\sigma }_{i} \in S,{n}_{i} \... | Proof. See Problem 9-3. | No |
Lemma 9.18. If an abelian group \( G \) has a finite basis, then every finite basis has the same number of elements. | Proof. Suppose \( G \) has a basis with \( n \) elements. Then \( G \cong {\mathbb{Z}}^{n} \) by Proposition 9.14(b), and the quotient group \( G/{2G} \) is easily seen to be isomorphic to \( {\left( \mathbb{Z}/2\right) }^{n} \) , which has exactly \( {2}^{n} \) elements. Since the order of \( G/{2G} \) is independent ... | Yes |
Proposition 9.19. Suppose \( G \) is a free abelian group of finite rank. Every subgroup of \( G \) is free abelian of rank less than or equal to that of \( G \) . | Proof. We may assume without loss of generality that \( G = {\mathbb{Z}}^{n} \) . We prove the proposition by induction on \( n \) . For \( n = 1 \), it follows from the fact that every subgroup of a cyclic group is cyclic.\n\nSuppose the result is true for subgroups of \( {\mathbb{Z}}^{n - 1} \), and let \( H \) be an... | Yes |
Proposition 9.21. Any abelian group that is finitely generated and torsion-free is free abelian of finite rank. | Proof. Suppose \( G \) is such a group. If \( S \subseteq G \) is a linearly independent subset, then the subgroup \( \langle S\rangle \subseteq G \) generated by \( S \) is easily seen to be free abelian with \( S \) as a basis.\n\nThe crux of the proof is the following claim: there exists a nonzero integer \( n \) an... | Yes |
Proposition 9.23. Suppose \( G \) and \( H \) are abelian groups and \( f : G \rightarrow H \) is a homomorphism. Then \( G \) is finitely generated if and only if both \( \operatorname{Im}f \) and \( \operatorname{Ker}f \) are finitely generated, in which case \( \operatorname{rank}G = \operatorname{rank}\left( {\oper... | Proof. Replacing \( H \) by the image of \( f \), we may as well assume that \( f : G \rightarrow H \) is surjective. Write \( K = \operatorname{Ker}f \subseteq G \). If \( G \) is finitely generated, then so is \( K \) by Corollary 9.20; and since \( f \) is surjective, it takes a set of generators for \( G \) to a se... | Yes |
Theorem 10.3 (Presentation of an Amalgamated Free Product). Let \( {f}_{1} : H \rightarrow \) \( {G}_{1} \) and \( {f}_{2} : H \rightarrow {G}_{2} \) be group homomorphisms. Suppose \( {G}_{1},{G}_{2} \), and \( H \) have the following finite presentations:\n\n\[ \n{G}_{1} \cong \left\langle {{\alpha }_{1},\ldots ,{\al... | Proof. This is an immediate consequence of Problems 9-4(b) and 9-5. | No |
Corollary 10.4 (First Special Case: Simply Connected Intersection). Assume the hypotheses of the Seifert-Van Kampen theorem, and suppose in addition that \( U \cap V \) is simply connected. Then \( \Phi \) is an isomorphism between \( {\pi }_{1}\left( {U, p}\right) * {\pi }_{1}\left( {V, p}\right) \) and \( {\pi }_{1}\... | \[ {\pi }_{1}\left( {U, p}\right) \cong \left\langle {{\alpha }_{1},\ldots ,{\alpha }_{m} \mid {\rho }_{1},\ldots ,{\rho }_{r}}\right\rangle ,\] \[ {\pi }_{1}\left( {V, p}\right) \cong \left\langle {{\beta }_{1},\ldots ,{\beta }_{n} \mid {\sigma }_{1},\ldots ,{\sigma }_{s}}\right\rangle \] then \( {\pi }_{1}\left( {X, ... | Yes |
Lemma 10.6. Suppose \( {p}_{i} \in {X}_{i} \) is a nondegenerate base point for \( i = 1,\ldots, n \) . Then \( * \) is a nondegenerate base point in \( {X}_{1} \vee \cdots \vee {X}_{n} \) . | Proof. For each \( i \), choose a neighborhood \( {W}_{i} \) of \( {p}_{i} \) that admits a strong deformation retraction \( {r}_{i} : {W}_{i} \rightarrow \left\{ {p}_{i}\right\} \), and let \( {H}_{i} : {W}_{i} \times I \rightarrow {W}_{i} \) be the associated homotopy from \( {\operatorname{Id}}_{{W}_{i}} \) to \( {\... | Yes |
Theorem 10.7. Let \( {X}_{1},\ldots ,{X}_{n} \) be spaces with nondegenerate base points \( {p}_{j} \in {X}_{j} \) . The map\n\n\[ \Phi : {\pi }_{1}\left( {{X}_{1},{p}_{1}}\right) * \cdots * {\pi }_{1}\left( {{X}_{n},{p}_{n}}\right) \rightarrow {\pi }_{1}\left( {{X}_{1} \vee \cdots \vee {X}_{n}, * }\right) \]\n\ninduce... | Proof. First consider the wedge sum of two spaces \( {X}_{1} \vee {X}_{2} \) . We would like to use Corollary 10.4 to the Seifert-Van Kampen theorem with \( U = {X}_{1}, V = {X}_{2} \) (considered as subspaces of the wedge sum), and \( U \cap V = \{ * \} \) . The trouble is that these spaces are not open in \( {X}_{1} ... | No |
The preceding proposition shows that the bouquet \( {\mathbb{S}}^{1} \vee \cdots \vee {\mathbb{S}}^{1} \) of \( n \) circles has a fundamental group isomorphic to \( \mathbb{Z} * \cdots * \mathbb{Z} \), which is a free group on \( n \) generators. | In fact, it shows more: since the isomorphism is induced by inclusion of each copy of \( {\mathbb{S}}^{1} \) into the bouquet, we can write explicit generators of this free group. If \( {\omega }_{i} \) denotes the standard loop in the \( i \) th copy of \( {\mathbb{S}}^{1} \), then the fundamental group of the bouquet... | Yes |
Proposition 10.11. Every finite connected graph contains a spanning tree. | Proof. Let \( \Gamma \) be a finite connected graph. If \( \Gamma = \varnothing \), then the empty subgraph is a spanning tree. Otherwise, we begin by showing that \( \Gamma \) contains a maximal tree, meaning a subgraph that is a tree and is not properly contained in any larger tree in \( \Gamma \) . To prove this, st... | Yes |
Proposition 10.13 (Attaching a Disk). Let \( X \) be a path-connected topological space, and let \( \widetilde{X} \) be the space obtained by attaching a closed 2-cell \( D \) to \( X \) along an attaching map \( \varphi : \partial D \rightarrow X \) . Let \( v \in \partial D,\widetilde{v} = \varphi \left( v\right) \in... | Proof. Let \( q : X \coprod D \rightarrow \widetilde{X} \) be the quotient map. As usual, we identify \( X \) with its image under \( q \), so we can consider \( X \) as a subspace of \( \widetilde{X} \) . First we set up some notation (see Fig. 10.7). Choose a point \( z \in \operatorname{Int}D \), set \( U = \operato... | Yes |
Proposition 10.14 (Attaching an \( n \) -cell). Let \( X \) be a path-connected topological space, and let \( \widetilde{X} \) be a space obtained by attaching an \( n \) -cell to \( X \), with \( n \geq 3 \) . Then inclusion \( X \hookrightarrow \widetilde{X} \) induces an isomorphism of fundamental groups. | Proof. We define open subsets \( \widetilde{U},\widetilde{V} \subseteq \widetilde{X} \) just as in the preceding proof. In this case, \( \widetilde{U} \cap \widetilde{V} \) is simply connected, because it is homeomorphic to \( {\mathbb{B}}^{n} \smallsetminus \{ 0\} \), and the result follows. | No |
Theorem 10.15 (Fundamental Group of a Finite CW Complex). Suppose \( X \) is a connected finite \( {CW} \) complex, and \( v \) is a point in the 1-skeleton of \( X \) that is contained in the closure of every 2-cell. Let \( {\beta }_{1},\ldots ,{\beta }_{n} \) be generators for the free group \( {\pi }_{1}\left( {{X}_... | Proof. This follows immediately by induction from the two preceding propositions, using the result of Exercise 5.19. | No |
Theorem 10.16 (Fundamental Groups and Polygonal Presentations). Let \( M\;{be} \) a topological space with a polygonal presentation \( \left\langle {{a}_{1},\ldots ,{a}_{n} \mid W}\right\rangle \) with one face, in which all vertices are identified to a single point. Then \( {\pi }_{1}\left( M\right) \) has the present... | Proof. As we observed in Chapter 6, a polygonal presentation determines a CW decomposition of \( M \) in a natural way. Under the assumption that all the vertices are identified to a single point, the 1 -skeleton \( {M}_{1} \) is a wedge sum of circles, one for each symbol in the presentation, and thus its fundamental ... | Yes |
Corollary 10.17 (Fundamental Groups of Compact Surfaces). The fundamental groups of compact connected surfaces have the following presentations:\n\n(a) \( {\pi }_{1}\left( {\mathbb{S}}^{2}\right) \cong \langle \varnothing \mid \varnothing \rangle \) (the trivial group).\n\n(b) \( {\pi }_{1}\left( {{\mathbb{T}}^{2}\# \c... | Proof. For \( {\mathbb{S}}^{2} \), this follows from Theorem 7.20. For all of the other surfaces, it follows from Theorem 10.16, using the standard presentations of Example 6.13, and noting that for each surface other than the sphere, the standard presentation identifies all of the vertices to one point, as you can eas... | Yes |
Theorem 10.22 (Classification of Compact Surfaces, Part II). Every nonempty, compact, connected 2-manifold is homeomorphic to exactly one of the surfaces \( {\mathbb{S}}^{2} \) , \( {\mathbb{T}}^{2}\widetilde{\# }\cdots \# {\mathbb{T}}^{2} \), or \( {\mathbb{P}}^{2}\# \cdots \# {\mathbb{P}}^{2} \) . | Proof. Theorem 6.15 showed that every nonempty, compact, connected surface is homeomorphic to one of the surfaces on the list, so we need only show that no two surfaces on the list are homeomorphic to each other. First note that the sphere cannot be homeomorphic to a connected sum of tori or projective planes, because ... | Yes |
Corollary 10.23. A connected sum of projective planes is not orientable. | Proof. By the argument in Chapter 6, if a manifold admits an oriented presentation, then it is homeomorphic to a sphere or a connected sum of tori. The preceding corollary showed that a connected sum of projective planes is not homeomorphic to any of these surfaces. | No |
Corollary 10.24. Orientability of a compact surface is a topological invariant. | Proof. Combining the results of Proposition 6.20 and Corollary 10.23, we can conclude that no surface that has an oriented presentation is homeomorphic to one that does not. | Yes |
Corollary 10.25. The Euler characteristic of a surface presentation is a topological invariant. | Proof. Suppose \( \mathcal{P} \) and \( \mathcal{Q} \) are polygonal surface presentations such that \( \left| \mathcal{P}\right| \approx \left| \mathcal{Q}\right| \) . Each of these presentations can be transformed into one of the standard ones by elementary transformations, and since the surfaces represented by diffe... | Yes |
The exponential quotient map \( \varepsilon : \mathbb{R} \rightarrow {\mathbb{S}}^{1} \) given by \( \varepsilon \left( x\right) = {e}^{2\pi ix} \) is a covering map; | this is the content of Proposition 8.1. | No |
The \( n \) th power map \( {p}_{n} : {\mathbb{S}}^{1} \rightarrow {\mathbb{S}}^{1} \) given by \( {p}_{n}\left( z\right) = {z}^{n} \) is also a covering map. | For each \( {z}_{0} \in {\mathbb{S}}^{1} \), the set \( U = {\mathbb{S}}^{1} \smallsetminus \left\{ {-{z}_{0}}\right\} \) has preimage equal to \( \left\{ {z \in {\mathbb{S}}^{1} : {z}^{n} \neq - {z}_{0}}\right\} \), which has \( n \) components, each of which is an open arc mapped homeomorphically by \( {p}_{n} \) ont... | Yes |
Lemma 11.10 (Existence of Local Sections). Let \( q : E \rightarrow X \) be a covering map. Given any evenly covered open subset \( U \subseteq X \), any \( x \in U \), and any \( {e}_{0} \) in the fiber over \( x \), there exists a local section \( \sigma : U \rightarrow E \) such that \( \sigma \left( x\right) = {e}_... | Proof. Let \( {\widetilde{U}}_{0} \) be the sheet of \( {q}^{-1}\left( U\right) \) containing \( {e}_{0} \) . Since the restriction of \( q \) to \( {\widetilde{U}}_{0} \) is a homeomorphism, we can just take \( \sigma = {\left( {\left. q\right| }_{{\widetilde{U}}_{0}}\right) }^{-1} \) . | Yes |
Proposition 11.11. For every covering map \( q : E \rightarrow X \), the cardinality of the fibers \( {q}^{-1}\left( x\right) \) is the same for all fibers. | Proof. Define an equivalence relation on \( X \) by saying that \( x \sim {x}^{\prime } \) if and only if \( {q}^{-1}\left( x\right) \) and \( {q}^{-1}\left( {x}^{\prime }\right) \) have the same cardinality. Suppose \( x \in X \), and let \( U \) be an evenly covered neighborhood of \( x \) . Then each sheet of \( {q}... | Yes |
Theorem 11.15 (Monodromy Theorem). Let \( q : E \rightarrow X \) be a covering map. Suppose \( f \) and \( g \) are paths in \( X \) with the same initial point and the same terminal point, and \( {\widetilde{f}}_{e},{\widetilde{g}}_{e} \) are their lifts with the same initial point \( e \in E \) .\n\n(a) \( {\widetild... | Proof. If \( {\widetilde{f}}_{e} \sim {\widetilde{g}}_{e} \), then \( f \sim g \) because composition with \( q \) preserves path homotopy. Conversely, suppose \( f \sim g \), and let \( H : I \times I \rightarrow X \) be a path homotopy between them. Then the homotopy lifting property implies that \( H \) lifts to a h... | Yes |
Theorem 11.16 (Injectivity Theorem). Let \( q : E \rightarrow X \) be a covering map. For any point \( e \in E \), the induced homomorphism \( {q}_{ * } : {\pi }_{1}\left( {E, e}\right) \rightarrow {\pi }_{1}\left( {X, q\left( e\right) }\right) \) is injective. | Proof. Suppose \( \left\lbrack f\right\rbrack \in {\pi }_{1}\left( {E, e}\right) \) is in the kernel of \( {q}_{ * } \) . This means that \( {q}_{ * }\left\lbrack f\right\rbrack = \left\lbrack {c}_{x}\right\rbrack \) , where \( x = q\left( e\right) \) . In other words, \( q \circ f \sim {c}_{x} \) in \( X \) . By the m... | Yes |
Theorem 11.18 (Lifting Criterion). Suppose \( q : E \rightarrow X \) is a covering map. Let \( Y \) be a connected and locally path-connected space, and let \( \varphi : Y \rightarrow X \) be a continuous map. Given any points \( {y}_{0} \in Y \) and \( {e}_{0} \in E \) such that \( q\left( {e}_{0}\right) = \varphi \le... | Proof. The necessity of the algebraic condition is easy to prove (and, in fact, does not require any connectivity assumptions about \( Y \) ). If \( \widetilde{\varphi } \) satisfies the conditions in the statement of the theorem, the following diagram commutes:\n\n. Suppose \( q : E \rightarrow X \) is a covering map and \( x \in X \) . There is a transitive right action of \( {\pi }_{1}\left( {X, x}\right) \) on the fiber \( {q}^{-1}\left( x\right) \), called the monodromy action, given by \( e \cdot \left\lbrack f\right\rbrack = {\widetilde{... | Proof. If \( e \) is any point in \( {q}^{-1}\left( x\right) \), the path lifting property shows that every loop \( f \) based at \( x \) has a unique lift to a path \( {\widetilde{f}}_{e} \) starting at \( e \) . The fact that \( f \) is a loop guarantees that \( {\widetilde{f}}_{e}\left( 1\right) \in {q}^{-1}\left( x... | Yes |
Proposition 11.23 (Isotropy Groups of Transitive \( G \) -sets). Suppose \( G \) is a group and \( S \) is a transitive right \( G \) -set.\n\n(a) For each \( s \in S \) and \( g \in G \), \n\n\[ \n{G}_{s \cdot g} = {g}^{-1}{G}_{s}g \n\] \n\n(b) The set \( \\left\\{ {{G}_{s} : s \in S}\\right\\} \) of all isotropy grou... | Proof. The proof of (a) is just a computation: for \( s \in S \) and \( g \in G \), \n\n\[ \n{G}_{s \cdot g} = \\left\\{ {{g}^{\\prime } \in G : \\left( {s \cdot g}\\right) \cdot {g}^{\\prime } = s \cdot g}\\right\\} \n\] \n\n\[ \n= \\left\\{ {{g}^{\\prime } \in G : s \cdot \\left( {g{g}^{\\prime }{g}^{-1}}\\right) = s... | Yes |
Proposition 11.24 (Properties of \( G \) -Equivariant Maps). Suppose \( G \) is a group, and \( {S}_{1},{S}_{2} \) are transitive right \( G \) -sets.\n\n(a) Any two \( G \) -equivariant maps from \( {S}_{1} \) to \( {S}_{2} \) that agree on one element of \( {S}_{1} \) are identical. | Proof. Suppose \( \varphi ,{\varphi }^{\prime } : {S}_{1} \rightarrow {S}_{2} \) are \( G \) -equivariant and \( \varphi \left( {s}_{1}\right) = {\varphi }^{\prime }\left( {s}_{1}\right) \) for some \( {s}_{1} \in \) \( {S}_{1} \) . Any \( s \in {S}_{1} \) can be written \( s = {s}_{1} \cdot g \) for some \( g \in G \)... | Yes |
Proposition 11.27 (Orbit Criterion for \( G \) -Automorphisms). Suppose \( S \) is a transitive right \( G \) -set. For any \( {s}_{1},{s}_{2} \in S \), there exists a (necessarily unique) \( \varphi \in \) \( {\operatorname{Aut}}_{G}\left( S\right) \) such that \( \varphi \left( {s}_{1}\right) = {s}_{2} \) if and only... | Proof. This is an immediate consequence of Proposition 11.26(a). | No |
Theorem 11.28 (Algebraic Characterization of \( G \) -Automorphism Groups). Let \( S \) be a transitive right \( G \) -set, and let \( {s}_{0} \) be any element of \( S \) . For each \( \gamma \in {N}_{G}\left( {G}_{{s}_{0}}\right) \) , there is a unique \( G \) -automorphism \( {\varphi }_{\gamma } \in {\operatorname{... | Proof. Suppose \( \gamma \in {N}_{G}\left( {G}_{{s}_{0}}\right) \) . Then \( {\gamma }^{-1} \in {N}_{G}\left( {G}_{{s}_{0}}\right) \) as well. Together with (11.1), this implies \( {G}_{{s}_{0}} = {\gamma }^{-1}{G}_{{s}_{0}}\gamma = {G}_{{s}_{0}} \cdot \gamma \) . Then Proposition 11.27 shows that there is a unique \( ... | Yes |
Theorem 11.29 (Isotropy Groups of the Monodromy Action). Suppose \( q \) : \( E \rightarrow \) \( X \) is a covering map and \( x \in X \) . For each \( e \in {q}^{-1}\left( x\right) \), the isotropy group of \( e \) under the monodromy action is \( {q}_{ * }{\pi }_{1}\left( {E, e}\right) \subseteq {\pi }_{1}\left( {X,... | Proof. Let \( e \in {q}^{-1}\left( x\right) \) be arbitrary, and suppose first that \( \left\lbrack f\right\rbrack \) is in the isotropy group of \( e \) . This means \( {\widetilde{f}}_{e}\left( 1\right) = e \cdot \left\lbrack f\right\rbrack = e \), which is to say that \( {\widetilde{f}}_{e} \) is a loop and thus rep... | Yes |
Corollary 11.30. Suppose \( q : E \rightarrow X \) is a covering map. The monodromy action is free on each fiber of \( q \) if and only if \( E \) is simply connected. | Proof. The action is free if and only if each isotropy group is trivial, which by Theorem 11.29 is equivalent to \( {q}_{ * }{\pi }_{1}\left( {E, e}\right) \) being the trivial group for each \( e \) in the fiber. Since \( {q}_{ * } \) is injective, this is true if and only if \( E \) is simply connected. | Yes |
Corollary 11.31. Suppose \( q : E \rightarrow X \) is a covering map and \( E \) is simply connected. Then each fiber of \( q \) has the same cardinality as the fundamental group of \( X \) . | Proof. By the previous corollary, the monodromy action is free. Choose a base point \( x \in X \) and a point \( e \) in the fiber over \( x \), and consider the map \( {\pi }_{1}\left( {X, x}\right) \rightarrow {q}^{-1}\left( x\right) \) given by \( \left\lbrack f\right\rbrack \mapsto e \cdot \left\lbrack f\right\rbra... | Yes |
Corollary 11.33 (Coverings of Simply Connected Spaces). If \( X \) is a simply connected space, every covering map \( q : E \rightarrow X \) is a homeomorphism. | Proof. The injectivity theorem shows that \( E \) is also simply connected. Then Corollary 11.31 shows that the cardinality of the fibers is 1, so \( q \) is injective. Thus it is a homeomorphism by Proposition 11.1(b). | Yes |
Theorem 11.34 (Conjugacy Theorem). Let \( q : E \rightarrow X \) be a covering map. For any \( x \in X \), as e varies over the fiber \( {q}^{-1}\left( x\right) \), the set of induced subgroups \( {q}_{ * }{\pi }_{1}\left( {E, e}\right) \) is exactly one conjugacy class in \( {\pi }_{1}\left( {X, x}\right) \) . | Proof. Given \( x \in X \), Theorem 11.29 shows that the set of subgroups \( {q}_{ * }{\pi }_{1}\left( {E, e}\right) \) as \( e \) varies over \( {q}^{-1}\left( x\right) \) is equal to the set of isotropy groups of points in \( {q}^{-1}\left( x\right) \) under the monodromy action. Then Proposition 11.23(b) shows that ... | Yes |
Proposition 11.35 (Characterizations of Normal Coverings). Suppose \( q : E \rightarrow \) \( X \) is a covering map. Then the following are equivalent:\n\n(a) The subgroup \( {q}_{ * }{\pi }_{1}\left( {E, e}\right) \) is normal for some \( e \in E \) (i.e., \( q \) is normal).\n\n(b) For some \( x \in X \), the subgro... | Proof. Because a subgroup is normal if and only if it is the sole member of its conjugacy class, the implications \( \left( \mathrm{d}\right) \Rightarrow \left( \mathrm{c}\right) \Rightarrow \left( \mathrm{b}\right) \Rightarrow \left( \mathrm{a}\right) \) are easy consequences of the conjugacy theorem. Thus we need onl... | Yes |
Proposition 11.36 (Properties of Covering Homomorphisms). Let \( {q}_{1} : {E}_{1} \rightarrow X \) and \( {q}_{2} : {E}_{2} \rightarrow X \) be coverings of the same space \( X \). (a) If two covering homomorphisms from \( {q}_{1} \) to \( {q}_{2} \) agree at one point of \( {E}_{1} \), then they are equal. | Proof. A covering homomorphism from \( {q}_{1} \) to \( {q}_{2} \) can also be viewed as a lift of \( {q}_{1} \):  (11.3) Thus (a) follows from the unique lifting property. | Yes |
Theorem 11.37 (Covering Homomorphism Criterion). Let \( {q}_{1} : {E}_{1} \rightarrow X \) and \( {q}_{2} : {E}_{2} \rightarrow X \) be two coverings of the same space \( X \), and suppose \( {e}_{1} \in {E}_{1} \) and \( {e}_{2} \in {E}_{2} \) are base points such that \( {q}_{1}\left( {e}_{1}\right) = {q}_{2}\left( {... | Proof. Because a covering homomorphism from \( {q}_{1} \) to \( {q}_{2} \) is a lift of \( {q}_{1} \) as in (11.3), both the necessity and the sufficiency of the subgroup condition follow from the lifting criterion (Theorem 11.18). | Yes |
Consider the following two coverings of \( {\mathbb{T}}^{2} \) : the first is \( {\varepsilon }^{2} : {\mathbb{R}}^{2} \rightarrow \) \( {\mathbb{T}}^{2} \), the covering of Example 11.5 (the product of two copies of \( \varepsilon : \mathbb{R} \rightarrow {\mathbb{S}}^{1} \) ); and the second is the map \( q : {\mathb... | It is easy to check that \( \varphi \left( {x, e}\right) = \left( {\varepsilon \left( x\right), e}\right) \) is such a homomorphism. | Yes |
Theorem 11.40 (Covering Isomorphism Criterion). Suppose \( {q}_{1} : {E}_{1} \rightarrow X \) and \( {q}_{2} : {E}_{2} \rightarrow X \) are two coverings of the same space \( X \). (a) Given \( {e}_{1} \in {E}_{1} \) and \( {e}_{2} \in {E}_{2} \) such that \( {q}_{1}\left( {e}_{1}\right) = {q}_{2}\left( {e}_{2}\right) ... | Proof. First we prove (a). Suppose there exists a covering isomorphism \( \varphi : {E}_{1} \rightarrow \) \( {E}_{2} \) such that \( \varphi \left( {e}_{1}\right) = {e}_{2} \), and let \( x = {q}_{1}\left( {e}_{1}\right) = {q}_{2}\left( {e}_{2}\right) \) . By Proposition 11.36(b), \( \varphi \) restricts to a \( {\pi ... | Yes |
Proposition 11.41 (Universality of Simply Connected Coverings). (a) Let \( q : E \rightarrow X \) be a covering map with \( E \) simply connected. If \( {q}^{\prime } : {E}^{\prime } \rightarrow X \) is any covering, there exists a covering map \( Q : E \rightarrow {E}^{\prime } \) such that the following diagram commu... | Proof. Since the trivial subgroup is contained in every other subgroup, part (a) follows from the covering homomorphism criterion and the fact that every covering homomorphism is a covering map. | No |
Theorem 11.43 (Existence of the Universal Covering Space). Every connected and locally simply connected topological space (in particular, every connected manifold) has a universal covering space. | Proof. To get an idea how to proceed, suppose for a moment that \( X \) does have a universal covering \( q : \widetilde{X} \rightarrow X \) . The key fact is that once we choose base points \( {\widetilde{x}}_{0} \in \widetilde{X} \) and \( {x}_{0} = q\left( {\widetilde{x}}_{0}\right) \in X \), the fiber \( {q}^{-1}\l... | Yes |
Proposition 12.1 (Properties of the Automorphism Group). Let \( q : E \rightarrow X \) be a covering map.\n\n(a) If two automorphisms of \( q \) agree at one point, they are identical.\n\n(b) Given \( x \in X \), each covering automorphism restricts to a \( {\pi }_{1}\left( {X, x}\right) \) -automorphism of the fiber \... | Proof. Parts (a) and (b) follow immediately from Proposition 11.36(a, b). To prove (c), let \( U \) be an evenly covered open subset, and let \( {U}_{\alpha } \) be a component of \( {q}^{-1}\left( U\right) \) . Since \( \varphi \left( {U}_{\alpha }\right) \) is a connected subset of \( {q}^{-1}\left( U\right) \), it m... | Yes |
For the covering \( \varepsilon : \mathbb{R} \rightarrow {\mathbb{S}}^{1} \), the integral translations \( x \mapsto x + k \) for \( k \in \mathbb{Z} \) are easily seen to be automorphisms. To see that every automorphism is of this form, let \( \varphi \in {\operatorname{Aut}}_{\varepsilon }\left( \mathbb{R}\right) \) ... | A similar argument shows that the automorphism group of \( {\varepsilon }^{n} : {\mathbb{R}}^{n} \rightarrow {\mathbb{T}}^{n} \) is isomorphic to \( {\mathbb{Z}}^{n} \) acting by \( \left( {{x}_{1},\ldots ,{x}_{n}}\right) \cdot \left( {{k}_{1},\ldots ,{k}_{n}}\right) = \left( {{x}_{1} + {k}_{1},\ldots ,{x}_{n} + {k}_{n... | Yes |
Theorem 12.4 (Orbit Criterion for Covering Automorphisms). Let \( q : E \rightarrow X \) be a covering map. If \( {e}_{1},{e}_{2} \in E \) are two points in the same fiber \( {q}^{-1}\left( x\right) \), there exists a covering automorphism taking \( {e}_{1} \) to \( {e}_{2} \) if and only if the induced subgroups \( {q... | Proof. This follows immediately from the covering isomorphism criterion (Theorem 11.40). | No |
Corollary 12.5 (Normal Coverings Have Transitive Automorphism Groups). If \( q : E \rightarrow X \) is a covering map, then \( {\operatorname{Aut}}_{q}\left( E\right) \) acts transitively on each fiber if and only if \( q \) is a normal covering. | Proof. Let \( q : E \rightarrow X \) be a covering map, and let \( x \) be an arbitrary point of \( X \) . By virtue of Proposition 11.35 and Theorem 12.4, we have the following equivalences:\n\n\[ \n{\operatorname{Aut}}_{q}\left( E\right) \text{acts transitively on}{q}^{-1}\left( x\right) \n\]\n\n\[ \n\Leftrightarrow ... | Yes |
Theorem 12.6. Suppose \( q : E \rightarrow X \) is a covering map and \( x \) is any point in \( X \) . The restriction map \( \varphi \mapsto {\left. \varphi \right| }_{{q}^{-1}\left( x\right) } \) is a group isomorphism between \( {\operatorname{Aut}}_{q}\left( E\right) \) and the group \( {\operatorname{Aut}}_{{\pi ... | Proof. Proposition 12.1(b) shows that each covering automorphism restricts to a \( {\pi }_{1}\left( {X, x}\right) \) -automorphism of \( {q}^{-1}\left( x\right) \) . Since \( {\left. \left( {\varphi }_{1} \circ {\varphi }_{2}\right) \right| }_{{q}^{-1}\left( x\right) } = {\left. {\left. {\varphi }_{1}\right| }_{{q}^{-1... | Yes |
Theorem 12.7 (Covering Automorphism Group Structure Theorem). Suppose \( q : E \rightarrow X \) is a covering map, \( e \in E \), and \( x = q\left( e\right) \) . Let \( G = {\pi }_{1}\left( {X, x}\right) \) and \( H = \) \( {q}_{ * }{\pi }_{1}\left( {E, e}\right) \subseteq {\pi }_{1}\left( {X, x}\right) \) . For each ... | Proof. We have two isomorphisms:\n\n\[ {N}_{G}\left( H\right) /H\overset{ \cong }{ \rightarrow }{\operatorname{Aut}}_{G}\left( {{q}^{-1}\left( x\right) }\right) \overset{ \cong }{ \rightarrow }{\operatorname{Aut}}_{q}\left( E\right) . \]\n\nThe first isomorphism is induced by the map of Theorem 11.28, which sends an el... | Yes |
Theorem 12.14 (Covering Space Quotient Theorem). Let \( E \) be a connected, locally path-connected space, and suppose we are given an effective action of a group \( \Gamma \) on \( E \) by homeomorphisms. Then the quotient map \( q : E \rightarrow E/\Gamma \) is a covering map if and only if the action is a covering s... | Proof. Assume first that \( q \) is a covering map. Then the action of each \( g \in \Gamma \) is an automorphism of the covering, because it is a homeomorphism satisfying \( q\left( {g \cdot e}\right) = \) \( q\left( e\right) \), so we can identify \( \Gamma \) with a subgroup of \( {\operatorname{Aut}}_{q}\left( E\ri... | No |
Proposition 12.15. Let \( \Gamma \) be a discrete subgroup of a connected and locally path-connected topological group \( G \) . Then the action of \( \Gamma \) on \( G \) by right translations is a covering space action, so the quotient map \( q : G \rightarrow G/\Gamma \) is a normal covering map. | Proof. Because \( \Gamma \) is discrete, there is a neighborhood \( V \) of 1 in \( G \) such that \( V \cap \) \( \Gamma = \{ 1\} \) . Consider the continuous map \( F : G \times G \rightarrow G \) given by \( F\left( {g, h}\right) = {g}^{-1}h \) . Since \( {F}^{-1}\left( V\right) \) is a neighborhood of \( \left( {1,... | Yes |
Corollary 12.16. Suppose \( G \) and \( H \) are connected and locally path-connected topological groups, and \( \varphi : G \rightarrow H \) is a surjective continuous homomorphism with discrete kernel. If \( \varphi \) is an open or closed map, then it is a normal covering map. | Proof. Let \( \Gamma = \operatorname{Ker}\varphi \) . By the preceding proposition, the quotient map \( q : G \rightarrow \) \( G/\Gamma \) is a normal covering map. The assumption that \( \varphi \) is either open or closed implies that it is a quotient map, and by the first isomorphism theorem the identifications mad... | Yes |
For any integers \( a, b, c, d \) such that \( {ad} - {bc} \neq 0 \), consider the map \( q : {\mathbb{T}}^{2} \rightarrow {\mathbb{T}}^{2} \) given by \( q\left( {z, w}\right) = \left( {{z}^{a}{w}^{b},{z}^{c}{w}^{d}}\right) \). This is easily seen to be a surjective continuous homomorphism, and it is a closed map by t... | Let \( A \) denote the invertible linear transformation of \( {\mathbb{R}}^{2} \) whose matrix is \( \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \). Then we have a commutative diagram\n\n\n\n(12.2)\n\n... | Yes |
Proposition 12.21 (Hausdorff Criterion for Orbit Spaces). Suppose \( E \) is a topological space and \( \Gamma \) is a group acting \( E \) by homeomorphisms. Then \( E/\Gamma \) is Hausdorff if and only if the action satisfies the following condition:\n\n\[ \text{if}e,{e}^{\prime } \in E\text{lie in different orbits, ... | Proof. Let \( q : E \rightarrow E/\Gamma \) denote the quotient map. If \( E/\Gamma \) is Hausdorff, then given \( e,{e}^{\prime } \) in different orbits, there are disjoint neighborhoods \( U \) of \( q\left( e\right) \) and \( {U}^{\prime } \) of \( q\left( {e}^{\prime }\right) \) , and then \( V = {q}^{-1}\left( U\r... | Yes |
Proposition 12.22. Every continuous action of a compact topological group on a Hausdorff space is proper. | Proof. Suppose \( G \) is a compact group acting continuously on a Hausdorff space \( E \), and let \( \Theta : G \times E \rightarrow E \times E \) be the map defined by (12.5). Given a compact set \( L \subseteq E \times E \), let \( K = {\pi }_{2}\left( L\right) \), where \( {\pi }_{2} : E \times E \rightarrow E \) ... | Yes |
Proposition 12.23. Suppose we are given a continuous action of a topological group \( G \) on a Hausdorff space \( E \). The action is proper if and only if for every compact subset \( K \subseteq E \), the set \( {G}_{K} = \{ g \in G : \left( {g \cdot K}\right) \cap K \neq \varnothing \} \) is compact. | Proof. Let \( \Theta : G \times E \rightarrow E \times E \) be the map defined by (12.5). Suppose first that \( \Theta \) is proper. Then for any compact set \( K \subseteq E \), we have\n\n\[ \n{G}_{K} = \{ g \in G : \text{ there exists }e \in K\text{ such that }g \cdot e \in K\} \n\]\n\n\[ \n= \{ g \in G\text{ : ther... | Yes |
Proposition 12.24. If a topological group \( G \) acts continuously and properly on a locally compact Hausdorff space \( E \), then the orbit space \( E/G \) is Hausdorff. | Proof. Let \( \mathcal{O} \subseteq E \times E \) be the orbit relation defined in Problem 3-22. By the result of that problem, the orbit space is Hausdorff if and only if \( \mathcal{O} \) is closed in \( E \times E \) . But \( \mathcal{O} \) is just the image of the map \( \Theta : G \times E \rightarrow E \times E \... | No |
Proposition 12.25. Suppose we are given a covering space action of a group \( \Gamma \) on a topological space \( E \), and \( E/\Gamma \) is Hausdorff. Then with the discrete topology, \( \Gamma \) acts properly on \( E \) . | Proof. For convenience, write \( X = E/\Gamma \), and let \( q : E \rightarrow X \) be the quotient map, which is a normal covering map by the covering space quotient theorem. It follows from Proposition 3.57 and Problem 3-22 that the orbit relation \( \mathcal{O} \) defined by (3.6) is closed in \( E \times E \) . Als... | Yes |
Theorem 12.26. Suppose \( E \) is a connected, locally path-connected, and locally compact Hausdorff space, and a discrete group \( \Gamma \) acts continuously, freely, and properly on \( E \) . Then the action is a covering space action, \( E/\Gamma \) is Hausdorff, and the quotient map \( q : E \rightarrow E/\Gamma \... | Proof. We need only show that the action is a covering space action, for then Proposition 12.24 shows that \( E/\Gamma \) is Hausdorff, and the covering space quotient theorem shows that \( q \) is a normal covering map.\n\nSuppose \( {e}_{0} \in E \) is arbitrary. Because \( E \) is locally compact, \( {e}_{0} \) has ... | Yes |
Theorem 12.29. Let \( M \) be a compact surface. The universal covering space of \( M \) is homeomorphic to\n\n(a) \( {\mathbb{S}}^{2} \) if \( M \approx {\mathbb{S}}^{2} \) or \( {\mathbb{P}}^{2} \) ,\n\n(b) \( {\mathbb{R}}^{2} \) if \( M \approx {\mathbb{T}}^{2} \) or \( {\mathbb{P}}^{2}\# {\mathbb{P}}^{2} \) ,\n\n(c... | Proof. Because \( {\mathbb{S}}^{2} \) is simply connected, it is its own universal covering space. It was shown in Example 11.42 that the universal covering space of \( {\mathbb{T}}^{2} \) is \( {\mathbb{R}}^{2} \), and that of \( {\mathbb{P}}^{2} \) is \( {\mathbb{S}}^{2} \) . If \( M \) is a connected sum of \( n \ge... | Yes |
Lemma 13.1. If \( c \) is a singular chain, then \( \partial \left( {\partial c}\right) = 0 \) . | Proof. Since each chain group \( {C}_{p}\left( X\right) \) is generated by singular simplices, it suffices to show this in the case in which \( c = \sigma \) is a singular \( p \) -simplex.\n\nFirst we note that the face maps satisfy the commutation relation\n\n\[ \n{F}_{i, p} \circ {F}_{j, p - 1} = {F}_{j, p} \circ {F... | Yes |
Proposition 13.2 (Functorial Properties of Homology). Let \( X \) , \( Y \) , and \( Z \) be topological spaces.\n\n(a) The homomorphism \( {\left( {\operatorname{Id}}_{X}\right) }_{ * } : {H}_{p}\left( X\right) \rightarrow {H}_{p}\left( X\right) \) induced by the identity map of \( X \) is the identity of \( {H}_{p}\l... | Proof. It is easy to check that both properties hold already for \( {f}_{\# } \) . | No |
Proposition 13.5. Let \( X \) be a space, let \( {\left\{ {X}_{\alpha }\right\} }_{\alpha \in A} \) be the set of path components of \( X \), and let \( {\iota }_{\alpha } : {X}_{\alpha } \hookrightarrow X \) be inclusion. Then for each \( p \geq 0 \) the map\n\n\[ \n{\bigoplus }_{\alpha \in A}{H}_{p}\left( {X}_{\alpha... | Proof. Since the image of any singular simplex must lie entirely in one path component, the chain maps \( {\left( {\iota }_{\alpha }\right) }_{\# } : {C}_{p}\left( {X}_{\alpha }\right) \rightarrow {C}_{p}\left( X\right) \) already induce isomorphisms\n\n\[ \n{\bigoplus }_{\alpha \in A}{C}_{p}\left( {X}_{\alpha }\right)... | Yes |
Proposition 13.6 (Zero-Dimensional Homology). For any topological space \( X \) , \( {H}_{0}\left( X\right) \) is a free abelian group with basis consisting of an arbitrary point in each path component. | Proof. It suffices to show that \( {H}_{0}\left( X\right) \) is the infinite cyclic group generated by the class of any point when \( X \) is path-connected, for then in the general case Proposition 13.5 guarantees that \( {H}_{0}\left( X\right) \) is the direct sum of infinite cyclic groups, one for each path componen... | Yes |
Proposition 13.7 (Homology of a Discrete Space). If \( X \) is a discrete space, then \( {H}_{0}\left( X\right) \) is a free abelian group with one generator for each point of \( X \), and \( {H}_{p}\left( X\right) = 0 \) for \( p > 0 \) . | Proof. The case \( p = 0 \) follows from the preceding proposition, so we concentrate on \( p > 0 \) . By Proposition 13.5, it suffices to show that \( {H}_{p}\left( *\right) = 0 \) when \( * \) is a one-point space. In that case, there is exactly one singular simplex in each dimension, namely the constant map \( {\sig... | Yes |
Theorem 13.8. If \( {f}_{0},{f}_{1} : X \rightarrow Y \) are homotopic maps, then for each \( p \geq 0 \) the induced homomorphisms \( {\left( {f}_{0}\right) }_{ * },{\left( {f}_{1}\right) }_{ * } : {H}_{p}\left( X\right) \rightarrow {H}_{p}\left( Y\right) \) are equal. | Proof of Theorem 13.8. We begin by considering the special case in which \( Y = \) \( X \times I \) and \( {f}_{i} = {\iota }_{i} \), where \( {\iota }_{0},{\iota }_{1} : X \rightarrow X \times I \) are the maps\n\n\[ \n{\iota }_{0}\left( x\right) = \left( {x,0}\right) ,\;{\iota }_{1}\left( x\right) = \left( {x,1}\righ... | Yes |
Corollary 13.9 (Homotopy Invariance of Singular Homology). Suppose \( f : X \rightarrow Y \) is a homotopy equivalence. Then for each \( p \geq 0, {f}_{ * } : {H}_{p}\left( X\right) \rightarrow {H}_{p}\left( Y\right) \) is an isomorphism. | Proof of Theorem 13.8. We begin by considering the special case in which \( Y = X \times I \) and \( {f}_{i} = {\iota }_{i} \), where \( {\iota }_{0},{\iota }_{1} : X \rightarrow X \times I \) are the maps \[ {\iota }_{0}\left( x\right) = \left( {x,0}\right) ,\;{\iota }_{1}\left( x\right) = \left( {x,1}\right) . \] (Se... | Yes |
Lemma 13.13. Suppose \( {f}_{0} \) and \( {f}_{1} \) are paths in \( X \), and \( {f}_{0} \sim {f}_{1} \) . Then, considered as a singular chain, \( {f}_{0} - {f}_{1} \) is a boundary. | Proof. We must show there is a singular 2-chain whose boundary is the 1-chain \( {f}_{0} - {f}_{1} \) . Let \( H : {f}_{0} \sim {f}_{1} \), and let \( b : I \times I \rightarrow {\Delta }_{2} \) be the map\n\n\[ b\left( {x, y}\right) = \left( {x - {xy},{xy}}\right) ,\]\n\nwhich maps the square onto the triangle by send... | Yes |
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