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Lemma 1.3.9. The subspace \( \left( {{B}^{n}\times \{ 0\} }\right) \cup \left( {{S}^{n - 1} \times I}\right) \) is a strong deformation retract of \( {B}^{n} \times I \) . | Proof. One standard proof involves \ | No |
Proposition 1.3.12. If \( Y \) is obtained from \( A \) by attaching \( n \) -cells, then \( (Y \times \) \( \{ 0\} ) \cup \left( {A \times I}\right) \) is a strong deformation retract of \( Y \times I \) . Hence any map \( \left( {Y\times \{ 0\} }\right) \cup \left( {A \times I}\right) \rightarrow Z \) extends to a ma... | Proof. By 1.3.9, there is a strong deformation retraction \( F : {B}^{n} \times I \times I \rightarrow \) \( {B}^{n} \times I \) of \( {B}^{n} \times I \) to \( \left( {{B}^{n}\times \{ 0\} }\right) \cup \left( {{S}^{n - 1} \times I}\right) \) . Let \( \mathcal{A} \) index the \( n \) -cells. Consider the diagram\n\n \) then, for some \( m,\{ z\} \) is in the image of \( {i}_{ * } : {H}_{n}^{\Delta }\left( {{X}^{m},{X}^{k}}\right) \rightarrow {H}_{n}^{\Delta }\left( {X,{X}^{k}}\right) \) . | Proof (of Theorem 2.3.5). Consider the following diagram:\n\n\n\nThe homomorphism \( \phi \) is not well defined but the relation \( {j}_{ * }^{\left( n\right) }{\left( {i}_{ * }^{\left( n\right) }\right) }^{-1} \) has... | No |
Proposition 2.3.8. The R-module \( {C}_{n}\left( {X;R}\right) \) is freely generated by the set \( \left\{ {{\bar{h}}_{\alpha * }\left( {\lambda }_{n}\right) \mid \alpha \in \mathcal{A}}\right\} . \) | Let \( {e}_{\beta }^{n - 1} \) be a cell of \( X \) . To understand the boundary homomorphism \( \partial : {C}_{n}\left( {X;R}\right) \rightarrow {C}_{n - 1}\left( {X;R}\right) \) we must understand the coefficient of \( {\bar{h}}_{\beta * }\left( {\lambda }_{n - 1}\right) \) in \( \partial {\bar{h}}_{\alpha * }\left(... | No |
Proposition 2.4.2. The map \( {f}_{1, d} \) has degree \( d \) . | Proof. This is an exercise in applying the definitions given in Sect. 2.2. Apply \( {f}_{1, d} \) to the singular 1-simplex \( \left( {1 - t}\right) {p}_{0} + t{p}_{1} \mapsto {e}^{2\pi it} \) . | No |
Proposition 2.4.3. For \( n \geq 1,{f}_{n,1} \) is the identity map of \( {S}^{n};{f}_{n, - 1} \) is the map \( \left( {{x}_{1},{x}_{2},{x}_{3},\ldots ,{x}_{n + 1}}\right) \mapsto \left( {{x}_{1}, - {x}_{2},{x}_{3},\ldots ,{x}_{n + 1}}\right) ;{f}_{n,0} \) is homotopic to a constant map. | Proof. The statements about \( {f}_{n,1} \) and \( {f}_{n, - 1} \) are obvious. For the statement about \( {f}_{n,0} \), one proves by induction on \( n \) that \( {P}_{n} \) does not lie in the image of \( {f}_{n,0} \), where \( {P}_{n} = \left( {-1,0,\ldots ,0}\right) \in {S}^{n} \subset {\mathbb{R}}^{n + 1} \) . Thi... | Yes |
Proposition 2.4.4. \( \deg \left( {f}_{n, d}\right) = d \) . | Proof. The case \( n = 1 \) is 2.4.2, so we may assume \( n \geq 2 \) . Let \( {e}_{ \pm }^{n} = {S}^{n} \cap {\mathbb{R}}_{ \pm }^{n + 1} \) . Then \( {S}^{n} = {e}_{ + }^{n} \cup {e}_{ - }^{n} \) and \( {S}^{n - 1} = {e}_{ + }^{n} \cap {e}_{ - }^{n} \) . The spaces \( {e}_{ \pm }^{n} \) are homeomorphic to \( {B}^{n}... | Yes |
Theorem 2.4.5. (Brouwer-Hopf Theorem) Two maps \( {S}^{n} \rightarrow {S}^{n} \) are homotopic iff they have the same degree. | In view of the previous propositions the only part of this theorem not yet \( {\text{proved}}^{5} \) is that for \( n \geq 1 \) every map \( f : {S}^{n} \rightarrow {S}^{n} \) of degree \( d \) is homotopic to \( {f}_{n, d} \) . We begin with the case \( n = 1 \) . | No |
Proposition 2.4.8. The number \( \widetilde{\omega }\left( 1\right) - \widetilde{\omega }\left( 0\right) \) is an integer and is independent of the choice of \( {t}_{0} \in {\exp }^{-1}\left( {f\left( 1\right) }\right) \) . | Proof. Let \( \widetilde{\omega }\left( 1\right) = {t}_{1}.\omega \left( 0\right) = \omega \left( 1\right) \), so \( {e}^{{2\pi i}{t}_{0}} = {e}^{{2\pi i}{t}_{1}} \), hence \( {t}_{0} - {t}_{1} \in \mathbb{Z} \) . For \( a \in \mathbb{R} \), let \( {T}_{a} : \mathbb{R} \rightarrow \mathbb{R} \) be the translation homeo... | Yes |
Proposition 2.4.9. Two maps \( f, g : {S}^{1} \rightarrow {S}^{1} \) are homotopic iff \( \delta \left( f\right) = \delta \left( g\right) \). | Proof. Let \( F : {S}^{1} \times I \rightarrow {S}^{1} \) be a homotopy. Define \( G : I \times I \rightarrow {S}^{1} \) by \( G\left( {t, s}\right) = F\left( {{e}^{2\pi it}, s}\right) \). Let \( \omega \left( t\right) = G\left( {0, t}\right) \) and let \( \widetilde{\omega } : I \rightarrow \mathbb{R} \) be such that ... | Yes |
Proposition 2.4.10. \( \delta \left( {f}_{1, d}\right) = d \) . | Proof. Let \( d \in \mathbb{Z} \subset \mathbb{R} \) . The map \( \widetilde{\omega } : I \rightarrow \mathbb{R}, t \mapsto {td} \) makes the following diagram commute:\n\n\n\nwhere \( \omega \) is defined by the diagr... | Yes |
Proposition 2.4.11. Let \( L \) be a subcomplex of \( K \) and let \( {K}^{\prime } \) be a subdivision of \( K \) . Then there is a subcomplex \( {L}^{\prime } \) of \( {K}^{\prime } \) which is a subdivision of \( L \) . | Proof. Let \( {L}^{\prime } \) consist of all cells \( e \) of \( {K}^{\prime } \) such that \( e \subset L \) . We show that \( {L}^{\prime } \) is a subcomplex and that \( {L}^{\prime } \) is a subdivision of \( L \) .\n\nLet \( e \) be a cell of \( {L}^{\prime } \) . By 1.2.13 there are only finitely many cells of \... | Yes |
Proposition 2.4.12. For each open cover \( \mathcal{U} \) of \( {I}^{n + 1} \) there exists \( {k}_{0} \) such that for every \( k \geq {k}_{0} \) every cell of \( {I}_{k}^{n + 1} \) lies in some element of \( \mathcal{U} \) . | Proof. Let \( x \in {U}^{\left( x\right) } \subset {I}^{n + 1} \), where \( {U}^{\left( x\right) } \in \mathcal{U} \) . There are intervals \( {V}_{1}^{\left( x\right) },\ldots ,{V}_{n + 1}^{\left( x\right) } \) which are open sets in \( {I}^{1} \) such that \( x \in {V}_{1}^{\left( x\right) } \times \ldots \times {V}_... | Yes |
Corollary 2.4.13. For each open cover \( \mathcal{U} \) of \( {I}^{n + 1} \) there exists \( {k}_{0} \) such that for every \( k \geq {k}_{0} \) each cell of \( {I}_{k}^{n + 1} \) lies in some element of \( \mathcal{U} \) . | Proof. For each \( U \in \mathcal{U} \) pick \( {U}^{\prime } \) an open subset of \( {I}^{n + 1} \) such that \( {U}^{\prime } \cap {I}^{n + 1} = \) \( U \) . Let \( {\mathcal{U}}^{\prime } \) consist of all the sets \( {U}^{\prime } \) and the set \( {I}^{n + 1} - {I}^{n + 1} \) . Apply 2.4.11 and 2.4.12 to \( {\math... | No |
Proposition 2.4.14. Let \( X \) be an \( n \) -dimensional \( {CW} \) complex, let \( {e}^{n} \) be an \( n \) -cell of \( X \) and let \( U \subset {e}^{n} \) be a neighborhood of \( z \in {e}^{n} \) . There is a homotopy \( H : X \times I \rightarrow X \) such that \( {H}_{0} = \mathrm{{id}},{H}_{t} = \mathrm{{id}} \... | Proof. As in the proof of 1.4.2, pick a characteristic map \( h : \left( {{B}^{n},{S}^{n - 1}}\right) \rightarrow \) \( \left( {{e}^{n},{e}^{n}}\right) \) such that \( h\left( 0\right) = z.{h}^{-1}\left( U\right) \) is a neighborhood of 0 in \( {B}^{n} \), so there is a \( \epsilon > 0 \) such that \( h\left( {B}_{\eps... | Yes |
Proposition 2.4.15. Let \( {x}_{1},\ldots ,{x}_{m},{y}_{1},\ldots ,{y}_{m} \) be distinct points of \( {S}^{n} \) , where \( n \geq 2 \) . There is a homeomorphism \( k : {S}^{n} \rightarrow {S}^{n} \), which is homotopic to \( \mathrm{{id}} \), such that \( k\left( {x}_{i}\right) = {y}_{i} \) for each \( i \) . | Proof. For any \( z \in {S}^{n},{S}^{n} - \{ z\} \) is homeomorphic to \( {\mathbb{R}}^{n} \) . By choosing \( z \) different from each \( {x}_{i} \) and each \( {y}_{i} \), we may work in \( {\mathbb{R}}^{n} \) . Let \( M > 0 \) be such that every \( {x}_{i} \) and every \( {y}_{i} \) lies in \( {\left( -M, M\right) }... | Yes |
Lemma 2.4.17. If \( n \geq 3 \), there exist compact convex pairwise disjoint neighborhoods \( {M}_{i} \) of \( {x}_{i} \) and \( {N}_{i} \) of \( {y}_{i} \), and points \( {x}_{i}^{\prime } \in \operatorname{int}{M}_{i} \) and \( {y}_{i}^{\prime } \in \operatorname{int}{N}_{i} \) such that no four points in \( \left\{... | We now complete the proof of 2.4.15 for \( n \geq 3 \) . By 2.4.16 and 2.4.17, we may assume that the line segments \( {L}_{i} \mathrel{\text{:=}} \left\lbrack {{x}_{i},{y}_{i}}\right\rbrack \) in \( {\mathbb{R}}^{n} \) are pairwise disjoint. Since each \( {L}_{i} \) is compact and convex, \( {C}_{i} \mathrel{\text{:=}... | No |
Corollary 2.5.2. Each cell of a CW complex admits exactly two orientations. If \( n \geq 1 \) and if \( h : \left( {{B}^{n},{S}^{n - 1}}\right) \rightarrow \left( {{e}_{\alpha }^{n},{e}_{\alpha }^{n}}\right) \) determines an orientation of \( {e}_{\alpha }^{n} \) , then hor determines the other orientation, where \( r ... | Proof. First, we observe that \( r \mid : {S}^{n - 1} \rightarrow {S}^{n - 1} \) is not homotopic to id. For \( n = \) 1, this is obvious. For \( n > 1 \), consider the homeomorphism \( {t}_{12} : {S}^{n - 1} \rightarrow {S}^{n - 1} \) \( \left( {{x}_{1},{x}_{2},{x}_{3},\ldots ,{x}_{n}}\right) \mapsto \left( {{x}_{2},{... | Yes |
Lemma 2.5.3. Let \( n \geq 1 \) . Let \( a = \left( {0,\ldots ,0,1}\right) \in {S}^{n} \) and \( b = \left( {0,\ldots ,0, - 1}\right) \in \) \( {S}^{n} \) . Let \( {e}_{ + }^{n} \) and \( {e}_{ - }^{n} \) be the closed upper and lower hemispheres, respectively. Let \( f, g : {S}^{n} \rightarrow {S}^{n} \) be maps such ... | Proof. We consider \( f \) and \( g \) as maps \( \left( {{S}^{n},{S}^{n - 1}}\right) \rightarrow \left( {{S}^{n},{S}^{n}-\{ a, b\} }\right) \) . As in the first part of the proof of 2.4.18, \( f \) and \( g \) are (pairwise) homotopic to maps \( {f}^{\prime } \) and \( {g}^{\prime } \) which preserve hemispheres, henc... | Yes |
Corollary 2.5.4. Let \( f, g : \left( {{B}^{n},{S}^{n - 1}}\right) \rightarrow \left( {{e}_{\alpha }^{n},{\overset{ \bullet }{e}}_{\alpha }^{n}}\right) \) be characteristic maps inducing \( {f}^{\prime },{g}^{\prime } : {B}^{n}/{S}^{n - 1} \rightarrow {e}_{\alpha }^{n}/{e}_{\alpha }^{n} \) . For \( t \in \left( {0,1}\r... | Proof. Since \( {B}^{n}/{S}^{n - 1} \) and \( {e}_{\alpha }^{n}/{\mathbf{e}}_{\alpha }^{n} \) are homeomorphic to \( {S}^{n} \), the result follows from 2.5.3. In fact, \( f\left| { \simeq g}\right| \) as maps from \( {S}_{t} \) to \( {\overset{ \circ }{e}}_{\alpha }^{n} - \{ z\} \) . | Yes |
Lemma 2.5.5. Let \( z \in {\overset{ \circ }{e}}_{\alpha }^{n} \), let \( f \) and \( g \) be two maps \( \left( {{S}^{n - 1} \times I,{S}^{n - 1}\times \{ 1\} }\right) \rightarrow \) \( \left( {{e}_{\alpha }^{n}-\{ z\} ,{\overset{ \bullet }{e}}_{\alpha }^{n}}\right) \), and let \( H : {S}^{n - 1} \times \{ 0\} \times ... | Proof. The map \( \left( {{S}^{n - 1}\times \{ 0\} \times I}\right) \cup \left( {{S}^{n - 1} \times I\times \{ 0,1\} }\right) \rightarrow {e}_{\alpha }^{n} - \{ z\} \), which agrees with \( H \) on \( {S}^{n - 1} \times \{ 0\} \times I \), with \( f \) on \( {S}^{n - 1} \times I \times \{ 0\} \) and with \( g \) on \( ... | Yes |
Proposition 2.5.8. Let \( {e}_{\alpha }^{n} \) and \( {e}_{\beta }^{n - 1} \) be (oriented) cells, \( n \geq 1 \). If \( {e}_{\beta }^{n - 1} \) is not a subset of \( {e}_{\alpha }^{n} \), then \( \left\lbrack {{e}_{\alpha }^{n} : {e}_{\beta }^{n - 1}}\right\rbrack = 0 \). | Proof. When \( n \geq 2 \), the map \( {S}^{n - 1} \rightarrow {S}^{n - 1} \) whose degree defines \( \left\lbrack {{e}_{\alpha }^{n} : {e}_{\beta }^{n - 1}}\right\rbrack \) is not surjective, so, by 1.3.4 and 1.3.7, it is homotopic to a constant map. By 2.4.3 its degree is 0. When \( n = 1 \), the Proposition is obvio... | Yes |
Proposition 2.5.13. Let \( {r}_{i} : {I}^{n} \rightarrow {I}^{n} \) be the homeomorphism \( \left( {{x}_{1},\ldots ,{x}_{n}}\right) \mapsto \) \( \left( {{x}_{1},\ldots ,{x}_{i - 1}, - {x}_{i},{x}_{i + 1},\ldots ,{x}_{n}}\right) \) . Then \( {r}_{i} \) is orientation reversing. | Proof. If \( n = 1 \), this is obvious. Next, let \( n = 2 \) and \( i = 1 \), let \( {M}_{t} \) be the matrix \( \left\lbrack \begin{matrix} \cos \left( {t\pi }\right) & \sin \left( {t\pi }\right) \\ \sin \left( {t\pi }\right) & - \cos \left( {t\pi }\right) \end{matrix}\right\rbrack \), and let \( {a}_{n} : {I}^{n} \r... | No |
Proposition 2.5.14. \( \left\lbrack {{I}^{n} : {F}_{i,1}}\right\rbrack = - \left\lbrack {{I}^{n} : {F}_{i,-1}}\right\rbrack \) . | Proof. This is clear when \( n = 1 \) . Assume \( n \geq 2 \) . Recall our permanent choice of homeomorphism, \( {a}_{n} \), for identifying \( {I}^{n} \) with \( {B}^{n} \) ; see Sect. 1.2. Consider the following commutative diagram:\n\n and \( {c}_{n} \) in the above diagram are homotopy equivalences. Hence the incidence numbers in 2.5.14 are \( \pm 1 \) . | Proof. Let \( {e}^{n}\left( t\right) = \left\{ {x \in {S}^{n} \mid t \leq {x}_{n + 1} \leq 1}\right\} \) . The space \( {e}^{n}\left( 0\right) \) is the upper closed hemisphere, while \( {e}^{n}\left( 1\right) = \{ \) north pole \( \} \) . The reader can easily construct a homotopy \( H : {S}^{n} \times I \rightarrow {... | Yes |
Proposition 2.5.17. \( \left\lbrack {{I}^{n} : {F}_{i,1}}\right\rbrack = {\left( -1\right) }^{i + 1} \) . Hence \( {}^{7}\left\lbrack {{I}^{n} : {F}_{i, - 1}}\right\rbrack = {\left( -1\right) }^{i} \) . | Proof. This is clear when \( n = 1 \) . Assume \( n \geq 2 \) . The orientation on \( {F}_{n,1} \) is given by \( {I}^{n - 1} \rightarrow {F}_{n,1},\left( {{x}_{1},\cdots ,{x}_{n - 1}}\right) \mapsto \left( {{x}_{1},\cdots ,{x}_{n - 1},1}\right) \) ; and the orientation on \( {F}_{n - 1,1} \) is \( {I}^{n - 1} \rightar... | Yes |
Proposition 2.5.18. The coefficient of \( {\bar{h}}_{\beta * }\left( {\lambda }_{n - 1}\right) \) in \( \partial {\bar{h}}_{\alpha * }\left( {\lambda }_{n}\right) \) is the incidence number \( \left\lbrack {{e}_{\alpha }^{n} : {e}_{\beta }^{n - 1}}\right\rbrack \) . | Proof. We leave the case \( n = 1 \) as an exercise. For \( n \geq 2 \) the claim can be read off from the following commutative diagram of singular homology groups ( \( \Delta \) ’s omitted): \n\nThe point here is tha... | No |
Theorem 2.6.1. (Sum Theorem for Degree) Let \( n \geq 1 \) and let\n\n\[ \n{S}^{n}\overset{F}{ \rightarrow }\mathop{\bigvee }\limits_{{\alpha \in \mathcal{A}}}{S}_{\alpha }^{n}\overset{G}{ \rightarrow }{S}^{n} \]\n\nbe maps. The degree of \( G \circ F \) is \( \mathop{\sum }\limits_{{\alpha \in \mathcal{A}}}\deg \left(... | Proof. We use the CW structures with one vertex and one \( n \) -cell for each sphere. We have \( \deg \left( {G \circ {i}_{\alpha }}\right) = \left\lbrack {{S}_{\alpha }^{n} : {S}^{n} : G}\right\rbrack \) and \( \deg \left( {{q}_{\alpha } \circ F}\right) = \left\lbrack {{S}^{n} : {S}_{\alpha }^{n} : F}\right\rbrack \)... | Yes |
Proposition 2.7.7. If \( X \) has finite type and \( R \) is a PID then \( {H}_{n}\left( {X;R}\right) \) is finitely generated for each \( n \) . | Proof. Since \( R \) is a PID the modules of cycles are (free and) finitely generated. | No |
Theorem 2.7.10. (Homotopy Invariance) If \( f, g : X \rightarrow Y \) are homotopic cellular maps, then \( {f}_{ * } = {g}_{ * } : {H}_{n}\left( {X;R}\right) \rightarrow {H}_{n}\left( {Y;R}\right) \) for all \( n \) . | Proof. Let \( z \in {Z}_{n}\left( {X;R}\right) \) represent \( \{ z\} \in {H}_{n}\left( {X;R}\right) \) . Then \( {f}_{ * }\left( {\{ z\} }\right) = \left\{ {{f}_{\# }\left( z\right) }\right\} \) and \( {g}_{ * }\left( {\{ z\} }\right) = \left\{ {{g}_{\# }\left( z\right) }\right\} \) . We must show that \( {f}_{\# }\le... | Yes |
Corollary 2.7.13. If the space \( X \) admits two \( {CW} \) complex structures \( {X}_{1} \) and \( {X}_{2} \), these yield isomorphic homology groups. | Proof. The identity map \( {\operatorname{id}}_{X} \) is a homotopy equivalence \( {X}_{1} \rightarrow {X}_{2} \). | Yes |
Proposition 2.7.14. Let \( F : X \times I \rightarrow Y \) be a cellular homotopy from \( f \) to g. Define \( {D}_{n} : {C}_{n}\left( X\right) \rightarrow {C}_{n + 1}\left( Y\right) \) by \( {D}_{n}\left( {e}_{\alpha }^{n}\right) = {\left( -1\right) }^{n + 1}{F}_{\# }\left( {{e}_{\alpha }^{n} \times I}\right) \) . The... | Proof. We use the proof of 2.7.10 and 2.7.9 to get:\n\n\[ \partial {D}_{n}\left( {e}_{\alpha }^{n}\right) = {\left( -1\right) }^{n + 1}\partial {F}_{\# }\left( {{e}_{\alpha }^{n} \times I}\right) \]\n\n\[ = {\left( -1\right) }^{n + 1}{F}_{\# }\partial \left( {{e}_{\alpha }^{n} \times I}\right) \]\n\n\[ = {F}_{\# }\left... | Yes |
Proposition 2.8.1. The following sequence is exact: | \[ \ldots \rightarrow {H}_{n}\left( {A;R}\right) \overset{{i}_{ * }}{ \rightarrow }{H}_{n}\left( {X;R}\right) \overset{{p}_{ * }}{ \rightarrow }{H}_{n}\left( {X, A;R}\right) \overset{{\partial }_{ * }}{ \rightarrow }{H}_{n - 1}\left( {A;R}\right) \rightarrow \ldots . \] | No |
Proposition 2.8.2. \( {H}_{n}\left( {X, A;R}\right) \cong {Z}_{n}\left( {X, A;R}\right) /{B}_{n}\left( {X, A;R}\right) \) . | Proof. Consider the commutative diagram \n\nHere \( \bar{\partial } \) is the boundary homomorphism. Note that \( {Z}_{n}\left( {X, A;R}\right) = {p}^{-1}\left( {\ker \bar{\partial }}\right) \) , and \( {B}_{n - 1}\lef... | Yes |
Proposition 2.8.3. \( {H}_{n}\left( {X/A;R}\right) \cong \left\{ \begin{array}{ll} {H}_{n}\left( {X, A;R}\right) & \text{ if }n \neq 0 \\ {H}_{0}\left( {X, A;R}\right) \oplus R & \text{ if }n = 0. \end{array}\right. \) | Proof. There is a unique homomorphism \( r \) making the following diagram commute:\n\n\n\nHere, \( q : X \rightarrow X/A \) is the quotient map, and \( X/A \) has the quotient orientation. Since \( p \) and \( {q}_{\#... | Yes |
Proposition 2.8.4. A map \( f : \left( {X, A}\right) \rightarrow \left( {Y, B}\right) \) between \( {CW} \) pairs which is already cellular on the subcomplex \( Z \) of \( X \) is pairwise homotopic, rel \( Z \), to a cellular map. | Proof. The proof of 1.4.3 achieves this. | No |
Proposition 2.8.6. Let \( f : \left( {X, A}\right) \rightarrow \left( {Y, B}\right) \) be a cellular map of oriented CW pairs. Then, for all n, the following diagram commutes:  | Thus, we have a commutative diagram\n\n\n\nwhose horizontal rows are exact. | Yes |
Theorem 2.8.7. Excision maps induce isomorphisms on relative homology groups. | Proof. Let \( i \) be an excision map as above. Then \( Y = X \cup B \) and \( A = X \cap B \) . By elementary algebra\n\n\[ \n{C}_{n}\left( {X;R}\right) /{C}_{n}\left( {A;R}\right) \cong \left\lbrack {{C}_{n}\left( {X;R}\right) + {C}_{n}\left( {B;R}\right) }\right\rbrack /{C}_{n}\left( {B;R}\right) = {C}_{n}\left( {Y;... | Yes |
Proposition 2.9.1. \( {\widetilde{H}}_{n}\left( {\varnothing ;R}\right) = 0 \) if \( n \neq - 1 \), and \( {\widetilde{H}}_{-1}\left( {\varnothing ;R}\right) \cong R \) . When \( X \neq \varnothing ,{\widetilde{H}}_{n}\left( {X;R}\right) \cong {H}_{n}\left( {X;R}\right) \) if \( n \neq 0 \), and there is a short exact ... | \[ 0 \rightarrow {\widetilde{H}}_{0}\left( {X;R}\right) \overset{{q}_{ * }}{ \rightarrow }{H}_{0}\left( {X;R}\right) \rightarrow R \rightarrow 0 \] where \( {q}_{ * } \) is induced by the chain map \( q : {\widetilde{C}}_{ * }\left( {X;R}\right) \rightarrow {C}_{ * }\left( {X;R}\right) \) which is the identity in dimen... | Yes |
Proposition 2.9.4. \( {f}_{\# } : {\widetilde{C}}_{ * }\left( {X;R}\right) \rightarrow {\widetilde{C}}_{ * }\left( {Y;R}\right) \) is a chain map. | Proof. By 2.3.4, it is only necessary to check the commutativity of\n\n\n\nIf \( f\left( {e}_{\alpha }^{0}\right) = {\widetilde{e}}_{\beta }^{0} \), where \( {e}_{\alpha }^{0} \) is a cell of \( X \) oriented by \( {\e... | Yes |
Proposition 3.1.1. \( f \) maps \( {I}_{j} \) onto \( {\overset{ \circ }{e}}_{{\beta }_{j}}^{1} \) for at most finitely many \( j \) . | Proof. Let the 1-cells of \( X \) be \( \left\{ {{e}_{\alpha }^{1} \mid \alpha \in \mathcal{A}}\right\} \) . Pick \( {z}_{\alpha } \in {e}_{\alpha }^{1} \) . There is an open cover of \( {X}^{1} \) consisting of \( {X}^{1} - \left\{ {{z}_{\alpha } \mid \alpha \in \mathcal{A}}\right\} = : U \) and each \( {\overset{ \ci... | Yes |
Proposition 3.1.2. \( \left( {{\tau }_{{\beta }_{1}}^{{i}_{1}},\ldots ,{\tau }_{{\beta }_{k}}^{{i}_{k}}}\right) \) is indeed an edge loop, and is unique up to cyclic permutation. | Proof. Consider \( {}^{1}{I}_{j} \) where \( j > k \) . Since \( f\left( {I}_{j}\right) \) is a proper subset of \( {}_{{\beta }_{j}}^{ \circ }, f\left( {\operatorname{fr}{I}_{j}}\right) \) is a single point of \( {\mathbf{e}}_{{\beta }_{j}}^{1} \) . Thus there is a map \( g : {S}^{1} \rightarrow {X}^{1} \), agreeing w... | No |
Proposition 3.1.6. The definition of equivalence of edge paths is independent of the orientation chosen for \( X \) . In particular, \( {\pi }_{1}\left( {X, v}\right) \) does not depend on the orientation of \( X \) . | Proposition 3.1.6 clarifies the role of the chosen orientation of \( X \) in this section. We must choose an orientation in order to define equivalence of edge loops at \( v \), but the definition of equivalence turns out to be independent of this choice. | No |
Theorem 3.1.8. Let \( X \) be an oriented \( {CW} \) complex having only one vertex, \( v \) . Let \( F \) be the free group generated by the set \( W \) of (oriented) 1-cells of \( X \) , let \( R \) be the set of (oriented) 2-cells of \( X \), for each \( {e}_{\gamma }^{2} \in R \) let \( \tau \mathrel{\text{:=}} \ta... | Proof. Let \( G = \langle W \mid R,\rho \rangle \) . It is enough to define an epimorphism \( \Psi : G \rightarrow \) \( {\pi }_{1}\left( {X, v}\right) \) and a function \( {\Phi }^{\prime } : {\pi }_{1}\left( {X, v}\right) \rightarrow G \) such that \( {\Phi }^{\prime } \circ \Psi = {\operatorname{id}}_{G} \) .\n\nEac... | Yes |
Proposition 3.1.12. Every tree is simply connected. Every simply connected CW complex of dimension \( \leq 1 \) is a tree. Every tree is contractible. | Proof. A tree \( T \) is clearly simply connected, for if \( \left( {{\tau }_{1},\ldots ,{\tau }_{n}}\right) \) is a reduced edge loop in \( T \) at the vertex \( v \) then \( n = 1 \) . Conversely, let \( X \) be simply connected and of dimension \( \leq 1 \) and let \( v \) be a vertex. In the absence of 2 - cells, t... | Yes |
Proposition 3.1.13. Let \( \left( {X, A}\right) \) be a CW pair with \( X \) and \( A \) path connected. Let \( {T}_{A} \) be a maximal tree in \( A \) . There is a maximal tree, \( {T}_{X} \), in \( X \) such that \( {T}_{X} \cap A = {T}_{A} \) | Proof. Let \( \mathcal{T} \) denote the set of trees \( T \) in \( X \) such that \( T \cap A = {T}_{A} \) . This set \( \mathcal{T} \) is non-empty and is partially ordered by inclusion. Let \( \left\{ {T}_{i}\right\} \) be a linearly ordered subset of \( \mathcal{T} \) and let \( \widetilde{T} = \mathop{\bigcup }\lim... | Yes |
Theorem 3.1.15. Tietze transformations of either type applied to a presentation of \( G \) yield another presentation of \( G \) . Conversely, if \( \left\langle {{W}_{1} \mid {R}_{1},{\rho }_{1}}\right\rangle \) and \( \left\langle {{W}_{2} \mid {R}_{2},{\rho }_{2}}\right\rangle \) are presentations of \( G \), then t... | \[ \left\langle {{W}_{1} \mid {R}_{1},{\rho }_{1}}\right\rangle \xrightarrow[]{\text{ Type II }}\left\langle {W \mid {R}^{\prime },{\rho }^{\prime }}\right\rangle \xrightarrow[]{\text{ Type I }}\langle W \mid R,\rho \rangle \xleftarrow[]{\text{ Type I }}\left\langle {W \mid {R}^{\prime \prime },{\rho }^{\prime \prime }... | Yes |
Theorem 3.1.16. Let \( X \) be an oriented path connected \( {CW} \) complex, let \( T \) be a maximal tree in \( X \) and let \( v \) be a vertex of \( X \) . Let \( F \) be the free group generated by the set \( W \) of (oriented) 1-cells of \( X \) . Let \( R \) be the set of (oriented) 2-cells of \( X \) and let \(... | Proof. We claim \( {\pi }_{1}\left( {X, v}\right) \cong {\pi }_{1}\left( {X/T,\bar{v}}\right) \cong \langle W - S \mid R,\bar{\rho }\rangle \cong \langle W \mid R \coprod S,\rho \rangle \) , where \( \bar{v} \) is the only vertex of \( X/T, q : F\left( W\right) \rightarrow F\left( {W - S}\right) \) is the epimorphism o... | Yes |
Theorem 3.1.19. This function \( \widetilde{h} \) induces a homomorphism \( h : {\pi }_{1}\left( {X, v}\right) \rightarrow \) \( {H}_{1}\left( {X;\mathbb{Z}}\right) \) whose kernel is the commutator subgroup of \( {\pi }_{1}\left( {X, v}\right) \) . If \( X \) is path connected, \( h \) is an epimorphism. | Proof. Clearly, \( h \) is a well defined homomorphism. We may assume that \( X \) is path connected: otherwise we could work with the path component containing \( v \) . We first deal with the special case in which \( X \) has only one vertex. Then the result follows from 3.1.8, since the effect of abelianizing the gr... | Yes |
Proposition 3.1.20. With hypotheses as in 3.1.18, \( {j}_{1 * } + {j}_{2 * } : {H}_{1}\left( {{X}_{1};\mathbb{Z}}\right) \oplus \) \( {H}_{1}\left( {{X}_{2};\mathbb{Z}}\right) \rightarrow {H}_{1}\left( {X;\mathbb{Z}}\right) \) is an epimorphism whose kernel is \( \left\{ {{i}_{1 * }\left( z\right) - {i}_{2 * }\left( z\... | Proof (First Proof). Apply 3.1.18 and 3.1.19.\n\nProof (Second Proof). The Mayer-Vietoris sequence (Sect. 2.8) gives an exact sequence\n\n\[ {H}_{1}\left( {{X}_{0};\mathbb{Z}}\right) \xrightarrow[]{\left( {i}_{1 * }, - {i}_{2 * }\right) }{H}_{1}\left( {{X}_{1};\mathbb{Z}}\right) \oplus {H}_{1}\left( {{X}_{2};\mathbb{Z}... | Yes |
Proposition 3.2.1. Let \( Y \) be a free \( G - {CW} \) complex. The quotient map \( q \) : \( Y \rightarrow G \smallsetminus Y = : X \) is a covering projection, and \( X \) admits the structure of a CW complex whose cells are \( \{ q\left( e\right) \mid e \) is a cell of \( Y\} \) . | Proof. Let \( {X}^{n} = \cup \left\{ {q\left( e\right) \mid e}\right. \) is a cell of \( Y \) having dimension \( \left. { \leq n}\right\} \) . \( {X}^{0} \) is discrete. \( X = \bigcup {X}^{n} \) . If \( \mathcal{A} \) indexes the \( n \) -cells of \( Y \), the \( G \) -action on \( Y \) induces a \( G \) -action on \... | No |
Proposition 3.2.2. Let \( Y \) be a rigid \( G - {CW} \) complex and let \( q : Y \rightarrow G \smallsetminus Y \) be the quotient map. Then \( G \smallsetminus Y \) admits a \( {CW} \) complex structure whose cells are \( \{ q\left( e\right) \mid e \) is a cell of \( Y\} \) . | Proof. Similar to the first part of the proof \( {}^{13} \) of 3.2.1. | No |
Proposition 3.2.3. \( \chi \) is well defined and is an isomorphism. | Proof. We first show that \( \chi \) is well defined. Let \( \widetilde{\sigma } \) and \( \widetilde{\tau } \) be edge paths from \( \widetilde{v} \) to \( g.\widetilde{v} \) . Since \( Y \) is simply connected, \( \widetilde{\sigma } \) and \( \widetilde{\tau } \) are equivalent. (To see this [using \( \simeq \) for ... | Yes |
Proposition 3.2.5. Let \( \tau \) be an edge loop at \( v \in X \), and let \( \widetilde{\tau } \) be the lift of \( \tau \) with initial point \( \left( {g, v}\right) \in {\widetilde{X}}^{1} \) . The final point of \( \widetilde{\tau } \) is \( \left( {g\bar{g}, v}\right) \) where \( \bar{g} \) is the element of \( \... | Proof. Let \( \left( {g, u}\right) \in \pi \times {X}^{0} = {\left( {\widetilde{X}}^{1}\right) }^{0} \) . A non-degenerate edge \( {\tau }_{i} \) in \( T \) from \( u \) to \( w \) lifts to an edge in \( {\widetilde{X}}^{1} \) from \( \left( {g, u}\right) \) to \( \left( {g, w}\right) \) . If \( {\tau }_{\beta } \) is ... | Yes |
Proposition 3.2.6. There are maps \( {\widetilde{f}}_{\gamma } : {S}^{1} \rightarrow {\widetilde{X}}^{1} \) such that \( {p}_{1} \circ {\widetilde{f}}_{\gamma } = {f}_{\gamma } \) . If \( {\widetilde{f}}_{\gamma } \) is one such, then the others are \( {d}_{g} \circ {\widetilde{f}}_{\gamma } \) where \( g \in \pi \) . | Proof. Let \( {\mu }_{\gamma } = \left( {{\tau }_{1}^{{i}_{1}},\ldots ,{\tau }_{m}^{{i}_{m}}}\right) \) where \( {\tau }_{j} \) has the chosen orientation, and \( {i}_{j} = \) \( \pm 1 \) . Let \( K \) be the CW complex structure on \( {S}^{1} \) having vertices at the \( {m}^{th} \) roots of unity (compare 1.2.17). By... | Yes |
Proposition 3.2.11. There is a map \( {q}_{H} \) making this diagram commute. Both \( {p}_{H} \) and \( {q}_{H} \) are covering projections. \( \bar{X}\left( H\right) \) admits a \( {CW} \) complex structure whose cells are the \( {p}_{H} \) -images of the cells of \( \widetilde{X} \) . The cells of \( \widetilde{X} \)... | Proof. There is obviously a function \( {q}_{H} \) making the diagram commute; \( {p}_{H} \) is a quotient map by definition, so \( {q}_{H} \) is continuous. It is not hard to show that an open subset of \( X \) evenly covered by \( p \) is evenly covered by \( {q}_{H} \) . By 3.2.1, the CW complex structures on \( \ba... | Yes |
Theorem 3.2.12. Every subgroup of a free group is free. | Proof. Let \( F \) be a free group and let \( H \leq F \) be a subgroup. By 3.1.8, there is a 1-dimensional CW complex \( X \) (having exactly one vertex \( v \) ) such that \( {\pi }_{1}\left( {X, v}\right) \) is isomorphic to \( F \) . Form the covering space \( \bar{X}\left( H\right) \) ; by 3.2.11, it is a CW compl... | Yes |
Proposition 3.3.2. Let \( f : \left( {Y, y}\right) \rightarrow \left( {Z, z}\right) \) be a map such that \( f : Y \rightarrow Z \) is a homotopy equivalence, and let \( z \) be a good base point. Then \( {f}_{\# } : {\pi }_{1}\left( {Y, y}\right) \rightarrow \) \( {\pi }_{1}\left( {Z, z}\right) \) is an isomorphism. | Proof. Let \( g : Z \rightarrow Y \) be a homotopy inverse for \( f \) . Let \( \alpha \) be a path in \( Y \) from \( g\left( z\right) \) to \( y \) . Define \( H : \left( {Z\times \{ 0\} }\right) \cup \left( {\{ z\} \times I}\right) \rightarrow Y \) by \( H\left( {x,0}\right) = g\left( x\right) \) , for \( z \in Z \)... | Yes |
Proposition 3.3.3. \( {\chi }^{\prime } \) is an isomorphism. | Its inverse is explicitly described as follows: let \( \left\lbrack \omega \right\rbrack \in {\pi }_{1}\left( {Y, y}\right) \), let \( \widetilde{z} \in \widetilde{Y} \), and let \( \widetilde{\tau } \) be a path in \( \widetilde{Y} \) from \( \widetilde{y} \) to \( \widetilde{z} \) ; \( {\left( {\chi }^{\prime }\right... | Yes |
Theorem 3.4.1. This association induces an isomorphism \( \alpha : {\pi }_{1}^{\text{edge }}\left( {X, v}\right) \rightarrow \) \( {\pi }_{1}^{\text{top }}\left( {X, v}\right) \) . | Proof. We claim \( {\pi }_{1}^{\text{top }}\left( {\widetilde{X}, v}\right) \) is trivial. Thus the isomorphism \( {\chi }^{\prime } \) of 3.3.3 is defined. By 3.2.3 and 3.2.9, the isomorphism \( \chi \) is well defined. Let \( \alpha = {\chi }^{\prime } \circ {\chi }^{-1} \) . Then \( \alpha \) is indeed induced by th... | No |
Proposition 3.4.2. Let \( {\left\{ \left( {X}_{\alpha },{x}_{\alpha }\right) \right\} }_{\alpha \in \mathcal{A}} \) be a family of pointed spaces and let \( {p}_{\beta } : \mathop{\prod }\limits_{\alpha }{X}_{\alpha } \rightarrow {X}_{\beta } \) be the projection map. Then \( {p}_{\# } : {\pi }_{1}\left( {\mathop{\prod... | Hence, writing \( {T}^{n} \) for the \( n \) -fold product of copies of \( {S}^{1}\left( {T}^{n}\right. \) is the \( n \) -torus), we get \( {\pi }_{1}\left( {{T}^{n}, v}\right) \cong {\mathbb{Z}}^{n} \) . | No |
Proposition 3.4.8. Let \( q : \left( {E, e}\right) \rightarrow \left( {X, v}\right) \) be a covering projection, where \( E \) is path connected and \( X \) is a \( {CW} \) complex. Let \( H = {q}_{\# }\left( {{\pi }_{1}\left( {E, e}\right) }\right) \leq {\pi }_{1}\left( {X, v}\right) \) . Then there is a homeomorphism... | Proof. Apply 3.3.4 to \( q \) and to \( {q}_{H} \) . Uniqueness implies that the resulting lifts are mutually inverse. | No |
Theorem 3.4.10. With notation as in Theorem 3.4.9, let \( \left( {\bar{X}\left( H\right) ,\bar{v}}\right) \) be the pointed covering space corresponding to \( H \) . (i) There is a bijection between the set of path components of \( {q}_{H}^{-1}\left( A\right) \) and the set of double cosets\n\n\[ \left\{ {\text{ H.g. }... | In summary: (i) \( {\pi }_{0}\left( {{q}_{H}^{-1}\left( A\right) }\right) \cong H \smallsetminus {\pi }_{1}\left( {X, v}\right) / \) image \( {i}_{\# } \), and (ii) \( {A}_{\bar{v}} = \) \( \bar{A}\left( {{i}_{\# }^{-1}\left( H\right) }\right) \) . | Yes |
Proposition 4.1.1. Let \( q : Y \coprod X \rightarrow Y{ \cup }_{f}X \) be the quotient map. Then \( Y{ \cup }_{f}X \) admits a CW complex structure whose cells are \( \{ q\left( e\right) \mid e \) is a cell of \( Y \) or a cell of \( X \) which is not in \( A\} \) . | Proof. This is similar to the first part of the proof of 3.2.1. The \( n \) -skeleton is \( {Y}^{n}{ \cup }_{f \mid }{X}^{n} \) | No |
Proposition 4.1.2. The map \( r \) is a homotopy inverse for \( j \), so \( r \) is a homotopy equivalence. Indeed there is a strong deformation retraction \( D \) : \( M\left( f\right) \times I \rightarrow M\left( f\right) \) of \( M\left( f\right) \) onto \( Y \) such that \( {D}_{1} = r \) . | Proof. The required \( D \) is induced by projection: \( Y \times I \rightarrow Y \) and the map \( X \times I \times I \rightarrow X \times I,\left( {x, t, s}\right) \mapsto \left( {x, t\left( {1 - s}\right) }\right) . | Yes |
Theorem 4.1.5. Let \( X = A \cup B \) and \( {X}^{\prime } = {A}^{\prime } \cup {B}^{\prime } \), where \( A \) and \( B \) are subcomplexes of \( X \), while \( {A}^{\prime } \) and \( {B}^{\prime } \) are subcomplexes of \( {X}^{\prime } \). Let \( f : X \rightarrow {X}^{\prime } \) be a cellular map such that \( f\l... | Proof (of 4.1.5). By 4.1.2 \( f \) is a homotopy equivalence iff \( i : X \hookrightarrow M\left( f\right) \) is a homotopy equivalence. Write \( {f}_{1} = f\left| { : A \rightarrow {A}^{\prime },{f}_{2} = f}\right| : B \rightarrow {B}^{\prime } \), \( {f}_{0} = f \mid : A \cap B \rightarrow {A}^{\prime } \cap {B}^{\pr... | Yes |
Theorem 4.1.7. The induced map \( G : Y{ \cup }_{f}X \rightarrow {Y}^{\prime }{ \cup }_{{f}^{\prime }}{X}^{\prime } \) is a homotopy equivalence. | Proof. Consider the commutative diagram\n\n\n\nHere and in what follows, we write \( M\left( f\right) \cup X \) for \( M\left( f\right) { \cup }_{i}X \) where \( i : A \rightarrow \) \( M\left( f\right) \) is the can... | Yes |
Theorem 4.1.8. If the hypotheses of 4.1.7 are weakened from \( k \circ f = {f}^{\prime } \circ g \mid \) to \( k \circ f \simeq {f}^{\prime } \circ g \mid \), it is still the case that \( Y{ \cup }_{f}X \) and \( {Y}^{\prime }{ \cup }_{{f}^{\prime }}{X}^{\prime } \) have the same homotopy type. | Proof. Consider the diagram\n\n\n\nHere \( r \) is the collapse and \( \widetilde{k} = k \circ r \) . The right square homotopy commutes. By 1.3.15, \( \widetilde{k} \) is homotopic to a map \( \bar{k} \) such that \... | Yes |
Corollary 4.1.9. Let \( X \) be a CW complex and \( A \) a contractible subcomplex. The quotient \( q : X \rightarrow X/A \) is a homotopy equivalence. | Proof. Apply 4.1.7 with \( X = {X}^{\prime }, A = {A}^{\prime }, g = \mathrm{{id}}, Y = A,{Y}^{\prime } = \{ q\left( A\right) \} \). | No |
Proposition 4.1.10. Let \( {P}^{\prime } \) be obtained from \( P \) by a Tietze transformation of Type I. Then there is a homotopy equivalence \( h \) making the following diagram commute up to homotopy: | Proof. The attaching maps for the 2-cells of \( {X}_{{P}^{\prime }} \) which are not in \( {X}_{P} \) are homotopic in \( {X}_{P} \) to constant maps. Apply 4.1.8. | No |
Proposition 4.1.11. The map \( {X}_{P} \hookrightarrow {X}_{{P}^{\prime \prime }} \) is a homotopy equivalence. | Proof. \( {\left( {X}_{{P}^{\prime \prime }}\right) }^{1} = {X}_{P}^{1} \vee \left( {\mathop{\bigvee }\limits_{\alpha }{S}_{\alpha }^{1}}\right) \), i.e. the wedge of \( {X}_{P}^{1} \) and a bouquet of circles.\n\nFor each \( \alpha \), the 2-cell \( {e}_{\alpha }^{2} \) of \( {X}_{{P}^{\prime \prime }} \) which is not... | Yes |
Proposition 4.1.12. For \( i = 1 \) and 2, let \( {P}_{i} \mathrel{\text{:=}} \left\langle {{W}_{i} \mid {R}_{i},{\rho }_{i}}\right\rangle \) be presentations of the group \( G \) . There are homotopy equivalent \( {CW} \) complexes \( {Y}_{{P}_{1}} \) and \( {Y}_{{P}_{2}} \) obtained from \( {X}_{{P}_{1}} \) and \( {X... | Proof. If in the proof of 4.1.10 we attach a 3-cell to \( {X}_{P} \vee \left( {\mathop{\bigvee }\limits_{{\alpha \in {R}^{\prime } - R}}{S}_{\alpha }^{2}}\right) \), for each \( \alpha \), by a homeomorphism \( {S}^{2} \rightarrow {S}_{\alpha }^{2} \) we obtain a 3-dimensional complex homotopy equivalent to \( {X}_{P} ... | Yes |
Theorem 4.1.14. Let \( X \) be a path connected \( {CW} \) complex whose fundamental group \( G \) is finitely generated. Then:\n\n(i) \( X \) is homotopy equivalent to a \( {CW} \) complex having finite 1-skeleton.\n\n(ii) If \( G \) is finitely presented, there is a \( {CW} \) complex \( {X}^{\prime } \), obtained fr... | Proof. By 3.1.13, 3.1.12 and 4.1.9, we may assume that \( X \) has only one vertex. Write \( {X}_{{P}_{1}} = {X}^{2} \), and let \( {P}_{2} \) be a presentation of \( G \) which is finite or finitely generated as appropriate. Write \( {P}_{i} = \left\langle {{W}_{i} \mid {R}_{i}}\right\rangle \) . Using 4.1.8 as in the... | No |
Proposition 4.2.1. (Cell Trading Lemma) Let \( \\left( {X, A}\\right) \) be an \( n \) -connected CW pair, where\n\n\[ \nX = A \\cup \\left( {\\mathop{\\bigcup }\\limits_{{\\alpha \\in \\mathcal{A}}}{e}_{\\alpha }^{n}}\\right) \\cup \\left( {\\mathop{\\bigcup }\\limits_{{\\beta \\in \\mathcal{B}}}{e}_{\\beta }^{n + 1}}... | Proof (of 4.2.1). We embed \( X \) as a subcomplex of a CW complex \( {X}^{\\prime } \) ; the required \( {X}^{\\prime \\prime } \) will be a quotient complex of \( {X}^{\\prime } \) by a quotient map which restricts to an embedding of \( A \) in \( {X}^{\\prime \\prime } \) . We will have homotopy equivalences\n\n\[ \... | Yes |
Proposition 4.3.1. \( T\left( h\right) \) is homotopy equivalent to \( T\left( k\right) \) . | Proof. We form an intermediate space \( T\left( {f, g}\right) \), the quotient CW complex of \( M\left( f\right) \coprod M\left( g\right) \) obtained by identifying \( y \in Y \subset M\left( f\right) \) with \( y \in Y \subset M\left( g\right) \) and \( x \in X \subset M\left( f\right) \) with \( x \in X \subset M\lef... | Yes |
Proposition 4.3.3. \( T\left( k\right) \) is homotopy equivalent to \( X \times {S}^{1} \) . | Proof. First, note that \( T\left( {\operatorname{id}}_{X}\right) = X \times {S}^{1} \) and that \( k \simeq {\operatorname{id}}_{X} \) . We saw in Sect. 4.1 that there is a homotopy equivalence \( M\left( k\right) \rightarrow M\left( {\operatorname{id}}_{X}\right) \) which restricts to the identity on the two copies o... | Yes |
Theorem 4.3.5. Let \( X\overset{f}{ \rightarrow }Y\overset{g}{ \rightarrow }X \) be cellular maps between \( {CW} \) complexes, such that \( g \circ f \simeq {\operatorname{id}}_{X} \) . Then \( X \) is homotopy equivalent to \( \operatorname{Tel}\left( {f \circ g}\right) \) , and \( X \times {S}^{1} \) is homotopy equ... | Proof. Combine the commutative diagrams in 4.3.2 and 4.3.4. Pick a base point for \( \operatorname{Tel}\left( {f, g}\right) \), and pick all other base points so that all maps are base point preserving. By 3.3.2, \( {\phi }_{\# },{\psi }_{\# },{\xi }_{\# },{\bar{\phi }}_{\# },{\bar{\psi }}_{\# } \) and \( {\bar{\xi }}_... | No |
Proposition 4.5.5. Let \( n \geq 2 \), let \( \mathcal{A} \) be finite and let the map \( f : {S}^{n} \rightarrow \) \( \bigvee {S}_{\alpha }^{n} \) be concentrated on the archipelago \( \left\{ {{W}_{\gamma } \mid \gamma \in \mathcal{C}}\right\} \) . Then \( f \) is homotopic \( \alpha \in \mathcal{A} \) to a map \( g... | Proof. Pick an archipelago \( \left\{ {{V}_{\alpha } \mid \alpha \in \mathcal{A}}\right\} \) in \( {S}^{n} \) . By a simple application of 2.4.15, there is a homeomorphism \( h : {S}^{n} \rightarrow {S}^{n} \) which is homotopic to the identity map, such that whenever \( f \) is non-constant on the island \( {W}_{\gamm... | Yes |
Consider \( X = {S}^{1} \vee {S}^{2} \) with vertex \( v \) . Its universal cover \( \widetilde{X} \) is a line with a 2-sphere adjoined at each integer point. By 4.5.2 and 4.4.10, | \( {H}_{2}\left( {\widetilde{X};\mathbb{Z}}\right) \cong {\bigoplus }_{n = - \infty }^{\infty }\mathbb{Z} \cong {\pi }_{2}\left( {\widetilde{X},\widetilde{v}}\right) \cong {\pi }_{2}\left( {X, v}\right) \) . Thus \( X \) is an example of a finite CW complex whose second homotopy group is not finitely generated (as an a... | Yes |
Proposition 5.2.2. If \( K \) is a simplicial complex in \( {\mathbb{R}}^{N} \), the weak topology on \( \left| K\right| \) with respect to \( \left\{ {\left| \sigma \right| \mid \sigma \in {S}_{K}}\right\} \) agrees with the topology inherited from \( {\mathbb{R}}^{N} \) . Moreover, \( \left( {\left| K\right| ,\left\{... | Proof. \( {\left| K\right| }_{\text{weak }} \) is a CW complex. Suppose it is not locally finite. Then for some simplex \( \sigma \) of \( K \), there is an infinite collection \( \left\{ {\tau }_{\alpha }\right\} \) of simplexes of \( K \) with \( \left| \sigma \right| \cap \left| {\tau }_{\alpha }\right| \neq \varnot... | No |
Let \( X = \left\{ {\left( {\left( {1 - a}\right), a,0,0}\right) \in {\mathbb{R}}^{4} \mid 0 \leq a \leq 1}\right\} \) and let \( Y = \left\{ {\left( {0,0,\left( {1 - b}\right), b}\right) \in {\mathbb{R}}^{4} \mid 0 \leq b \leq 1}\right\} \). Then \( X * Y \) can be identified with \( \left\{ {\left( {\left( {1 - t}\ri... | Fixing \( a \) and \( b \) we get a line segment joining \( \left( {\left( {1 - a}\right), a,0,0}\right) \in X \) to \( \left( {0,0,\left( {1 - b}\right), b}\right) \in Y \), and the various line segments meet as described above. In this example, \( X * Y \) is the standard 3-simplex in \( {\mathbb{R}}^{4} \). | Yes |
Proposition 5.2.5. If \( K \) or \( L \) is locally finite then \( \phi \) is continuous, and \( \phi \) induces a homeomorphism \( \Phi : \left| K\right| * \left| L\right| \rightarrow \left| {K * L}\right| \) . | Proof. Regard \( I \) as a CW complex in the usual way. By 1.2.19, \( \left| K\right| \times I \times \left| L\right| \) is a CW complex, so in order to show that \( \phi \) is continuous we need only show that for any \( \sigma \in {S}_{K} \) and \( \tau \in {S}_{L},\phi \left| : \right| \sigma \left| {\times I \times... | Yes |
Proposition 5.3.2. Let \( e \) be a cell of the regular \( {CW} \) complex \( X \), and let \( C\left( e\right) \) be the carrier of \( e \) . Then, as spaces, \( C\left( e\right) = e \) . In other words, each cell of \( X \) is a subcomplex of \( X \). | For the proof of 5.3.2 we need two lemmas. | No |
Lemma 5.3.3. There is no embedding of \( {S}^{n} \) in \( {\mathbb{R}}^{n} \) . | Proof. If there were such an embedding \( h : {S}^{n} \rightarrow {\mathbb{R}}^{n} \) then, by 5.1.1 and 1.3.7, \( h \) would map every proper open subset of \( {S}^{n} \) onto an open subset of \( {\mathbb{R}}^{n} \), hence \( h\left( {S}^{n}\right) \) would be open in \( {\mathbb{R}}^{n} \) . But \( h\left( {S}^{n}\r... | Yes |
Lemma 5.3.4. Let \( {e}_{\alpha }^{n - 1} \) and \( {e}_{\beta }^{n} \) be cells of the regular \( {CW} \) complex \( X \) . If \( {\overset{ \circ }{e}}_{\alpha } \cap {\overset{ \bullet }{e}}_{\beta } \neq \varnothing \) then \( {e}_{\alpha } \subset {\overset{ \bullet }{e}}_{\beta } \) . | Proof. We have \( {\overset{ \bullet }{e}}_{\beta }^{n} \subset {X}^{n - 1} \), and \( {\overset{ \circ }{e}}_{\alpha }^{n - 1} \) is open in \( {X}^{n - 1} \), so \( {\overset{ \circ }{e}}_{\alpha }^{n - 1} \cap {\overset{ \bullet }{e}}_{\beta }^{n} \) is open in \( {\overset{ \bullet }{e}}_{\beta }^{n} \) . Now \( {\... | Yes |
Proposition 5.3.5. Let \( X \) be a regular \( {CW} \) complex, and let \( {e}_{\alpha }^{k - 2} \) be a face of \( {e}_{\beta }^{k} \) . Then \( {e}_{\alpha }^{k - 2} \) is a face of exactly two \( \left( {k - 1}\right) \) -dimensional faces of \( {e}_{\beta }^{k} \) . | Proof. Since \( {}^{ \bullet }{}_{\beta }^{k} \) is homeomorphic to \( {S}^{k - 1} \), this follows from the following lemma. | No |
Lemma 5.3.6. Let \( Y \) be a regular CW complex structure on an n-manifold. Every cell of \( Y \) is a face of an \( n \) -cell of \( Y \) . Every \( \left( {n - 1}\right) \) -cell of \( Y \) is a face of at most two \( n \) -cells of \( Y \) . An \( \left( {n - 1}\right) \) -cell, \( e \), of \( Y \) is a face of exa... | Proof. We saw in Sect. 5.1 that every cell of \( Y \) has dimension \( \leq n \) . If some cell were not a face of an \( n \) -cell, there would be \( k < n \) and a \( k \) -cell \( \widetilde{e} \) of \( Y \) which is not a face of any higher-dimensional cell of \( Y \), implying \( \overset{ \circ }{e} \) open in \(... | No |
Proposition 5.3.8. When \( X \) is a regular \( {CW} \) complex, there is a homeomorphism \( h : \left| {\operatorname{sd}X}\right| \rightarrow X \) such that for every simplex \( \left\{ {{e}_{0},\cdots ,{e}_{k}}\right\} \) of \( \operatorname{sd}X \) , \( h\left( {\left| \left\{ {e}_{0},\cdots ,{e}_{k}\right\} \right... | Proof. Observe that for any cell \( e \) of \( X \), sd \( C\left( e\right) \) is the cone \( e * \left\lbrack {\operatorname{sd}C\left( e\right) }\right\rbrack \) where \( C\left( \cdot \right) \) denotes \ | No |
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