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Proposition 14.4.2. Let \( H \leq G \) . Then \( {H}^{0}\left( {G,{R}^{H}{\left( G\right) }^{ \frown }}\right) = 0 \) [resp. \( \cong R \) ] iff \( H \) has infinite [resp. finite] index in \( G \) . | From the top line of the above commutative diagram we get a long exact sequence\n\n\[ \cdots \leftarrow {H}^{n}\left( {G, R{G}^{ \frown }}\right) \leftarrow {H}^{n}\left( {G,{R}^{H}{\left( G\right) }^{ \frown }}\right) \leftarrow {H}^{n - 1}\left( {G,{R}^{H}{\left( G\right) }^{e}}\right) \leftarrow {H}^{n - 1}\left( {G... | Yes |
Proposition 14.5.3. \( \widetilde{e}\left( {G, H}\right) \geq e\left( {G, H}\right) \) ; hence, if \( \widetilde{e}\left( {G, H}\right) = 1 \) then we have \( e\left( {G, H}\right) = 1 \) . | Proof. By the (proof of the) Homotopy Lifting Property 2.4.6 the obvious function from filtered ends of \( \left( {G, H}\right) \) to ends of \( \left( {G, H}\right) \) is surjective. For the last part apply 14.5.2. | No |
Proposition 14.5.4. If \( K \leq H \leq G \), where \( \left\lbrack {H : K}\right\rbrack < \infty \), then \( \widetilde{e}\left( {G, H}\right) = \) \( \widetilde{e}\left( {G, K}\right) \) . In particular, if \( H \) is finite, \( \widetilde{e}\left( {G, H}\right) \) is the number of ends of \( G \) . | Proof. In this case the covering projection \( \bar{X}{\left( K\right) }^{1} \rightarrow \bar{X}{\left( H\right) }^{1} \) is proper, so one gets the same filtration of \( {\widetilde{X}}^{1} \) from \( \bar{X}{\left( K\right) }^{1} \) as from \( \bar{X}{\left( H\right) }^{1} \) . | Yes |
For the pair \( \left( {G, H}\right) \) in Example 13.5.13 with \( e\left( {G, H}\right) = n \geq \) 3, we have \( \widetilde{e}\left( {G, H}\right) = \infty \) . | To see this, note that in 13.5.13 we showed that for each \( i \) the inclusion \( {U}_{i} \hookrightarrow \bar{X}\left( H\right) \) induces a monomorphism on fundamental group whose image has infinite index in \( H \) . By 3.4.9, \( \widetilde{e}\left( {G, H}\right) = \infty \) . | No |
Let \( \phi : H \rightarrow H \) be a monomorphism which is not an epimorphism. Let \( G = H{ * }_{\phi } \) be the resulting ascending HNN extension. Then \( G \) has a presentation \( \left\langle {H, t\left| {\;{t}^{-1}{xt\phi }{\left( x\right) }^{-1}}\right. ,\forall x \in H}\right\rangle \) . The standard homomorp... | Inspection of the picture of \( \operatorname{Tel}\left( h\right) \) in Sect. 4.3 shows that there is a basis for the neighborhoods of the end whose path components \( {U}_{i} \) \ | No |
Proposition 14.5.9. If \( N \) is a finitely generated normal subgroup of \( G \) then \( \widetilde{e}\left( {G, N}\right) \) is the number of ends of \( G/N \) . Hence \( \widetilde{e}\left( {G, N}\right) = e\left( {G, N}\right) \) . | Proof. We know that \( \widetilde{e}\left( {G, N}\right) = 0 \) iff \( G/N \) is finite, so we may assume \( N \) has infinite index in \( G \) . Let \( \left( {X, x}\right) \) be a pointed 2-dimensional CW complex whose 1-skeleton is finite and whose fundamental group is (identified with) \( G \) . A finite subcomplex... | No |
Proposition 14.5.11. If \( g \in {\operatorname{Comm}}_{G}\left( H\right) \) then there is a finite set \( F \subset {gH} \) such that \( {gH} \subset {HF} \) . | Proof. The subgroup \( K = {gH}{g}^{-1} \cap H \) has finite index in \( {gH}{g}^{-1} \), so there is a finite set \( {F}_{0} \subset H \) such that \( {gH}{g}^{-1} = {Kg}{F}_{0}{g}^{-1} \subset {Hg}{F}_{0}{g}^{-1} \) . The required \( F \) is \( g{F}_{0} \) . | Yes |
Corollary 14.5.12. If \( L \) is an \( \mathcal{L} \) -bounded subgraph of \( \Gamma \) and \( g \in {\operatorname{Comm}}_{G}\left( H\right) \) , then \( {gL} \) is also \( \mathcal{L} \) -bounded. | Proof. By 14.5.11, \( {gH} \subset {HF} \), so for any \( \bar{g} \in G \) we have \( {gH}\bar{g} \subset {HF}\bar{g} \) . This implies that \( g \) takes the vertices of \( L \) into an \( \mathcal{L} \) -bounded set. It follows easily that \( {gL} \) is \( \mathcal{L} \) -bounded. | Yes |
Proposition 14.5.13. If \( H \) has infinite index in \( {\operatorname{Comm}}_{G}\left( H\right) \), then for any finite set \( F \subset G \) there exists \( g \in {\operatorname{Comm}}_{G}\left( H\right) \) such that \( {gHF} \cap {HF} = \varnothing \) . | Proof. Suppose \( F \) exists such that for every \( g \in {\operatorname{Comm}}_{G}\left( H\right) {gHF} \cap {HF} \neq \) \( \varnothing \) . Then, for each such \( g \), there exists \( f,\bar{f} \in F \) and \( h,\bar{h} \in H \) such that \( g = \bar{h}\left( {\bar{f}{f}^{-1}}\right) {h}^{-1} \) . For pairs \( \le... | Yes |
Theorem 14.5.15. Let \( H \) be a finitely generated subgroup of \( G \) having infinite index in \( {\operatorname{Comm}}_{G}\left( H\right) \) . Then \( \widetilde{e}\left( {G, H}\right) = 2 \) iff there are subgroups \( {G}_{1} \) and \( {H}_{1} \) of finite index in \( G \) and \( H \) respectively such that \( {H}... | For the proof of 14.5.15 we need a variation on 14.5.13: Lemma 14.5.16. For each | No |
Lemma 14.5.16. For each finite set \( F \subset G \) there is a finite set \( {F}_{0} \subset \) \( {\operatorname{Comm}}_{G}\left( H\right) \) such that whenever \( g \in {\operatorname{Comm}}_{G}\left( H\right) - H{F}_{0}H \) then \( {gHF} \cap {HF} = \) 0. | Proof. Suppose \( F \) exists such that for every finite set \( {F}_{0} \subset {\operatorname{Comm}}_{G}\left( H\right) \) there is some \( g \in {\operatorname{Comm}}_{G}\left( H\right) - H{F}_{0}H \) with \( {gHF} \cap {HF} \neq \varnothing \) . Then \( {ghf} = \bar{h}\bar{f} \) for some \( h,\bar{h} \in H \) and \(... | Yes |
Example 14.5.19. Let \( n \geq 3 \) be an integer. In 13.5.13 we saw a pair \( \left( {G, H}\right) \) for which \( e\left( {G, H}\right) = n \) and \( \widetilde{e}\left( {G, H}\right) = \infty \) . Here we discuss a pair \( \left( {G, H}\right) \) such that \( e\left( {G, H}\right) = \widetilde{e}\left( {G, H}\right)... | The proof is sketched as an exercise. | No |
Proposition 15.1.1. The image of the closed embedding\n\n\[ \left| \alpha \right| : \left| {\left( {\operatorname{sd}C\left( e\right) }\right) * b\left( {e}^{\text{dual }}\right) }\right| \rightarrow \left| {\operatorname{sd}X}\right| = X \]\n\nis a neighborhood of every point of \( \overset{ \circ }{e} \) . | Proof. We use the notation and results of Sect. 1.5. Clearly, the image of \( \left| \alpha \right| \) is \( {N}_{\left| \operatorname{sd}X\right| }\left( \widehat{e}\right) \) . Let \( Y = \left| {\operatorname{sd}X}\right| \overset{c}{ \sim }\{ \widehat{e}\} \) . Then \( Y \), being a subcomplex, is closed in \( \lef... | Yes |
Proposition 15.1.2. The underlying space of a CW \( n \) -manifold \( X \) is a topological \( n \) -manifold. Moreover, the boundary of this manifold, \( \partial X \), is the sub-complex consisting of all cells \( e \) of \( X \) for which \( \left| {b\left( {e}^{\text{dual }}\right) }\right| \) is homeomorphic to a ... | Proof. Let \( x \in X \) . By 10.1.25 \( X \) is metrizable. Let \( e \) be the unique cell of \( X \) such that \( x \in e \) . By 15.1.1 and 5.2.7, \( x \) has a neighborhood homeomorphic to \( {\mathbb{R}}^{n} \) or to \( {\mathbb{R}}_{ + }^{n} \), and the latter iff \( \left| {b\left( {e}^{\text{dual }}\right) }\ri... | Yes |
Theorem 15.1.5. Let \( X \) be a CW \( n \) -manifold. There is a regular CW complex structure \( {X}^{ * } \) on \( X \) whose set of \( \left( {n - k}\right) \) -cells is \[ \left\{ {\left| {e}^{\text{dual }}\right| e\text{ is a }k\text{-cell of }X}\right\} \cup \left\{ {{\left| {e}^{\text{dual }}\right| }_{\partial ... | Proof. We show that \( {X}^{ * } \) satisfies (i)-(v) of Proposition 1.2.14. First we note that, by 5.2.7, for every cell \( e \) of \( X \) which is not in \( \partial X \) we have \( {\left| {e}^{\text{dual }}\right| }^{ \circ } \subset \overset{ \circ }{X} \) , and for every cell \( e \) of \( \partial X \) we have ... | Yes |
Proposition 15.1.6. In the \( {CWn} \) -manifold \( X \), let \( {e}_{\beta } \) be a face of \( {e}_{\alpha } \) . Then \( \left| {e}_{\alpha }^{\text{dual }}\right| \) is a face of \( \left| {e}_{\beta }^{\text{dual }}\right| \) . | By 15.1.5, we have two CW complex structures \( X \) and \( {X}^{ * } \) on the same underlying \( n \) -manifold. If these CW complexes \( X \) and \( {X}^{ * } \) are oriented, we get an isomorphism of graded \( R \) -modules \( {\phi }_{k} : {C}_{k}\left( {X;R}\right) \rightarrow {C}_{n - k}\left( {{X}^{ * },\partia... | No |
Proposition 15.1.7. If \( X \) is a path connected \( {CWn} \) -manifold, then we have \( {H}_{n}^{\infty }\left( {X,\partial X;{\mathbb{Z}}_{2}}\right) \cong {\mathbb{Z}}_{2}. \) | Proof. By the last paragraph, \( {H}_{n}^{\infty }\left( {{X}^{ * },\partial {X}^{ * };{\mathbb{Z}}_{2}}\right) \) is isomorphic to \( {H}^{0}\left( {X;{\mathbb{Z}}_{2}}\right) \) . Apply 12.1.2 and 11.1.10. | No |
Corollary 15.1.8. If \( {e}_{\alpha }^{n} \neq {e}_{\beta }^{n} \) are cells of a path connected CW \( n \) -manifold \( X \) , there is a finite sequence \( {e}_{\alpha }^{n} = {e}_{{\alpha }_{0}}^{n},{e}_{{\alpha }_{1}}^{n},\cdots ,{e}_{{\alpha }_{k}}^{n} = {e}_{\beta }^{n} \) of \( n \) -cells of \( X \) such that e... | Proof. Let \( \mathcal{A} \) index the \( n \) -cells of \( X \) . Say \( \alpha ,\beta \in \mathcal{A} \) are \ | No |
Proposition 15.2.1. This orientation on \( \left| {e}_{\alpha }^{\text{dual }}\right| \) is well defined. | Proof. If, instead, we look at \( {d}^{0} \subset {d}^{1} \subset \cdots \subset {d}^{k - 1} \subset {e}^{k} = {e}_{\alpha }^{k} \subset \cdots \subset {e}^{n} = \) \( {e}_{\gamma }^{n} \), and let \( {\tau }^{n} = \left\{ {{\widehat{d}}^{0},\cdots ,{\widehat{d}}^{k - 1},{\widehat{e}}^{k},\cdots ,{\widehat{e}}^{n}}\rig... | Yes |
Proposition 15.2.2. Let \( {e}_{\beta }^{k - 1} \subset {e}_{\alpha }^{k} \) . With these orientations we have \( \left\lbrack {e}_{\alpha }^{k}\right. \) : \( \left. {e}_{\beta }^{k - 1}\right\rbrack = {\left( -1\right) }^{k}\left\lbrack {\left| {e}_{\beta }^{\text{dual }}\right| : \left| {e}_{\alpha }^{\text{dual }}\... | Proof. We saw in 15.1.6 that \( \left| {e}_{\alpha }^{\text{dual }}\right| \) is a face of \( \left| {e}_{\beta }^{\text{dual }}\right| \) . Consider \( {e}^{0} \subset {e}^{1} \subset \) \( \cdots \subset {e}^{k - 1} = {e}_{\beta }^{k - 1} \subset {e}^{k} = {e}_{\alpha }^{k} \subset \cdots \subset {e}^{n} = {e}_{\gamm... | Yes |
Corollary 15.2.9. A formal R-orientation on \( Y \) induces canonical isomorphisms \( {H}_{f}^{n - k}\left( {Y;R}\right) \rightarrow {H}_{k}\left( {Y;R}\right) \) and \( {H}^{n - k}\left( {Y;R}\right) \rightarrow {H}_{k}^{\infty }\left( {Y;R}\right) \) . | Proof. By 13.2.1, the chain complexes \( \left( {R{ \otimes }_{G}{C}_{k}\left( {\widetilde{Y};R}\right) ,\mathrm{{id}} \otimes \partial }\right) \) and \( \left( {{C}_{k}\left( {Y;R}\right) ,\partial }\right) \) are isomorphic; similarly the cochain complexes \( \left( {R{ \otimes }_{G}{C}_{n - k}\left( {\widetilde{Y};... | Yes |
Theorem 15.3.1. The following are equivalent:\n\n(i) \( G \) is an \( n \) -dimensional orientable Poincaré Duality group over \( R \) ;\n\n(ii) (a) \( G \) has type \( {}^{3}{FP} \), and\n\n(b) For each \( k \) and each right \( {RG} \) -module \( M \) there is an isomorphism \( {H}^{n - k}\left( {G, M}\right) \righta... | Proof (of 15.3.1). Let \( 0 \rightarrow {F}_{n}\xrightarrow[]{\partial }\cdots \xrightarrow[]{\partial }{F}_{1}\xrightarrow[]{\epsilon }R \rightarrow 0 \) be a projective \( {RG} \) -resolution of the trivial \( {RG} \) -module \( R \) . Some statements in this proof are more obvious when the modules \( {F}_{k} \) are ... | No |
Proposition 16.1.1. The correspondence \( \tau \mapsto \left( \left\lbrack {\mu }_{i}\right\rbrack \right) \) induces a bijection \( \eta \) : \( \mathcal{{SE}}\left( {Y,\omega }\right) \rightarrow {\mathop{\lim }\limits_{ \leftarrow }}^{1}\left\{ {{\pi }_{1}\left( {Y - {L}_{i},\omega \left( i\right) }\right) }\right\}... | Proof. We indicate the definition of \( {\eta }^{-1} \), leaving it to the reader to check that \( \eta \) and \( {\eta }^{-1} \) are well defined and are mutually inverse. The omitted details are tedious but instructive.\n\nFor each \( i \geq 0 \), let \( {\nu }_{i} \) be an edge loop in \( Y \triangleq {L}_{i} \) bas... | No |
Proposition 16.1.4. The functions \( \eta \) and \( \gamma \) define functions \( a \) and \( b \) in a natural short exact sequence of pointed sets: | \[ {\mathop{\lim }\limits_{ \leftarrow }}^{1}\left\{ {{\pi }_{1}\left( {Y \circeq {L}_{i},\omega \left( i\right) }\right) }\right\} \overset{a}{ \rightarrowtail }{\pi }_{0}^{e}\left( {Y,\omega }\right) \overset{b}{ \twoheadrightarrow }\mathop{\lim }\limits_{ \leftarrow }\left\{ {{\pi }_{0}\left( {Y \circeq {L}_{i},\ome... | No |
Proposition 16.1.5. If \( Y \) has one end, \( \bar{\eta } \) is a bijection. | Proof. The proof is analogous to that of 16.1.1. Alternatively, when \( R = \mathbb{Z} \) or more generally when the homomorphism \( 1 \otimes \cdot \) is surjective, the function \( {h}_{R} \) is surjective by 3.1.19 and 11.3.4, and so the injection \( \bar{\eta } \) is a bijection. | No |
Proposition 16.1.6. \( \bar{\eta } \) is a homomorphism of \( R \) -modules, hence an isomorphism when \( Y \) has one end. | Proof. Let the proper ray \( \tau \) define the same end as \( \omega \) . It is implicit in the proof of 16.1.1 that \( \tau \) is properly homotopic to a proper ray which has the form of an \ | No |
Proposition 16.1.7. \( Y \) is strongly \( R \) -homology connected at the end \( e \) iff \( {\mathop{\lim }\limits_{ \leftarrow }}^{1}\left\{ {{H}_{1}\left( {{Z}_{i};R}\right) }\right\} \) is trivial. | Proof. One defines a bijection \( {\eta }^{\prime } : \mathcal{{SHE}}\left( {Y,\omega ;R}\right) \rightarrow {\underline{\lim }}^{1}\left\{ {{H}_{1}\left( {{Z}_{i};R}\right) }\right\} \) by analogy with the definition of \( \eta \) . The inverse of \( {\eta }^{\prime } \) is defined as in the proof of 16.1.1. The detai... | No |
Proposition 16.1.8. Let \( {H}_{1}\left( {Y;R}\right) \) be finitely generated. The following are equivalent when \( R \) is a PID:\n\n(i) \( Y \) is strongly \( R \) -homology connected at every end;\n\n(ii) \( {H}_{e}^{1}\left( {Y;R}\right) \) is a free \( R \) -module;\n\n(iii) \( \left\{ {{H}_{1}\left( {Y - {L}_{i}... | Proof. The equivalence of (ii),(iii) and (iv) comes \( {}^{2} \) from 12.5.10 and 11.3.2. The equivalence of (i) and (iii) follows from 16.1.22 in the Appendix as explained in 16.1.21. | Yes |
Proposition 16.1.13. Let \( {i}^{\left( n\right) } : {Y}^{n} \hookrightarrow Y \) be the inclusion map. Let \( \omega \) be a cellular base ray (in \( {Y}^{1} \subset Y \) ) which is well parametrized with respect to \( \left\{ {L}_{i}\right\} \) .\n\n(i) \( {i}_{\# }^{\left( 1\right) } : \mathcal{E}\left( {Y}^{1}\righ... | Proof. These follow from the Cellular Approximation Theorems 1.4.3 and 10.1.14. Indeed, (i) is essentially 13.4.1. | No |
Theorem 16.1.22. Let the locally finite pointed tree \( \left( {T, w}\right) \) have no leaves and let \( {\left\{ {M}_{v}\right\} }_{v \in {T}^{0}} \) be a family of \( R \) -modules with given homomorphisms \( {M}_{v} \rightarrow \) \( {M}_{\pi \left( v\right) } \) for all \( v \neq w \) . The total inverse sequence ... | Proof (of 16.1.22). \ | No |
Proposition 16.2.1. The following are equivalent:\n\n(i) \( Y \) is simply connected at \( e \) ;\n\n(ii) \( \left\{ {{\pi }_{1}\left( {Y \triangleq {L}_{i},\omega \left( i\right) }\right) }\right\} \) is pro-trivial;\n\n(iii) \( \lim \left\{ {{\pi }_{1}\left( {Y - {L}_{i},\omega \left( i\right) }\right) }\right\} \) a... | Proof. For (iii) \( \Leftrightarrow \) (ii), use Exercise 10 of Sect. 11.3; (i) \( \Leftrightarrow \) (ii) is clear. | No |
Proposition 16.2.3. Let \( f : Y \rightarrow Z \) be as in 16.1.14, let \( \left\{ {L}_{i}\right\} \) and \( \left\{ {M}_{j}\right\} \) be finite filtrations of \( Y \) and \( Z \) respectively. Then \( \omega \) can be reparametrized by a proper homotopy so that \( {f}_{\# } : \left\{ {{\pi }_{1}\left( {Y - {L}_{i},\o... | Proof. Let \( H : g \circ f \mid \simeq \) inclusion and \( \bar{H} : f \circ g \mid \simeq \) inclusion be proper cellular homotopies. Give \( Y \) the cellular base ray \( \omega \) . By reparametrizing \( \omega \) if necessary we can assume that \( {H}_{t} \circ \omega \) [resp. \( {\bar{H}}_{t} \circ f \circ \omeg... | Yes |
Let \( W = {S}_{1}^{1} \vee {S}_{2}^{1} \), a wedge of two circles with wedge point \( v \) . Let \( f : W \rightarrow W \) be a map taking each circle to itself (hence \( f\left( v\right) = v \) ), agreeing with the map \( {f}_{1,2} \) (of degree 2 - see Sect. 2.4) on \( {S}_{1}^{1} \) and with the identity map on \( ... | Now \( \underline{\lim }{}^{1}\left\{ {{\pi }_{1}\left( {T - c{L}_{i},\omega \left( i\right) }\right) }\right\} \) is non-trivial by 11.4.15 and thus, by 14.1.1, there is a proper ray \( \tau \) in \( T \) which is not properly homotopic to \( \omega \) in \( T \) . Since \( T \) is a retract of \( Y \), such a ray \( ... | No |
Proposition 16.2.6. The function \( b \) is well defined and is an epimorphism of groups. It fits into a natural short exact sequence of groups: | \[ {\mathop{\lim }\limits_{ \leftarrow }}^{1}\left\{ {{\pi }_{2}\left( {Y \triangleq {L}_{i},\omega \left( i\right) }\right) }\right\} \overset{a}{ \mapsto }{\pi }_{1}^{e}\left( {Y,\omega }\right) \overset{b}{ \rightarrow }\mathop{\lim }\limits_{ \leftarrow }\left\{ {{\pi }_{1}\left( {Y \triangleq {L}_{i},\omega \left(... | No |
Proposition 16.2.7. Let \( R \) be a PID and let \( {H}_{1}\left( {Y;R}\right) \) be finitely generated. The following are equivalent:\n\n(i) \( Y \) is 1-acyclic at every end with respect to \( R \) ;\n\n(ii) \( {H}_{e}^{1}\left( {Y;R}\right) = 0 \) ;\n\n(iii) \( \left\{ {{H}_{1}\left( {Y - {L}_{i};R}\right) }\right\}... | Proof. Similar to that of 16.1.8 (including its footnote). Again (i) \( \Rightarrow \) (iii) uses 16.1.22. | No |
Let \( M \) be a compact path connected \( n \) -manifold with path connected (hence non-empty) boundary. Pick a ray \( \omega \) in \( \overset{ \circ }{M} \) approaching \( x \in \) \( \partial M \) . Then the fundamental pro-group of \( \overset{ \circ }{M} \) based at \( \omega \) is stably \( {\pi }_{1}\left( {\pa... | If \( M \) is simply connected and if \( \mathbb{Z} \) acts as covering transformations on \( M \), then 16.3.4 implies that \( {\pi }_{1}\left( {\partial M, x}\right) \) must be a free (and finitely generated) group. | No |
Proposition 16.4.1. For \( n \geq 2 \) every contractible open \( n \) -manifold has one end. | Proof. This follows from Poincaré Duality together with 13.4.11 and the exact sequence in Sec. 12.2. | No |
Proposition 16.4.4. There is a homeomorphism \( f : {\mathbb{R}}^{3} \rightarrow {\mathbb{R}}^{3} \) such that \( f\left( T\right) = L \) and \( f\left( L\right) = T;f \) extends to a homeomorphism of \( {S}^{3} \) . | Proof. Take two circles of string embedded in \( {\mathbb{R}}^{3} \) as illustrated in Fig. 16.5. Stretch and reshape the string \( L \) to occupy the space originally occupied by \( T \) . Then the string \( T \) can easily be moved to occupy the space originally occupied by \( L \) . Since faraway points of \( {\math... | No |
Proposition 16.4.5. Let \( {T}_{1}^{\prime } = \operatorname{cl}\left( {{S}^{3} - {T}_{1}}\right) \) . Then \( {T}_{1}^{\prime } \) is also a solid torus, and \( {T}_{1}^{\prime } \cap {T}_{1} = \partial {T}_{1} \) | Proof. \( {S}^{3} = \partial {B}^{4} \) is homeomorphic to \( \partial \left( {{B}^{2} \times {B}^{2}}\right) = \left( {{B}^{2} \times {S}^{1}}\right) \cup \left( {{S}^{1} \times {B}^{2}}\right) \) , and \( \left( {{B}^{2} \times {S}^{1}}\right) \cap \left( {{S}^{1} \times {B}^{2}}\right) = {S}^{1} \times {S}^{1} \) | No |
Proposition 16.4.6. Let \( {L}_{1,2} = \operatorname{cl}\left( {{T}_{1} - {T}_{2}}\right) \) so that \( \partial {L}_{1,2} = \partial {T}_{1} \cup \partial {T}_{2} \) . There is a homeomorphism of \( {S}^{3} \) mapping \( {L}_{1,2} \) to itself, mapping \( \partial {T}_{1} \) to \( \partial {T}_{2} \), and mapping \( \... | Proof. Interpret Fig. 16.5 as consisting of two solid tori \( T \) and \( L \) . By 16.4.5, there is a homeomorphism \( k : {S}^{3} \rightarrow {S}^{3} \) taking \( T \) to \( {T}_{2} \) and \( L \) to \( \operatorname{cl}\left( {{S}^{3} - {T}_{1}}\right) \) ; in other words, \( k \) maps the copy of \( {S}^{3} \) in F... | Yes |
Proposition 16.4.8. The monomorphisms in 16.4.7 are not epimorphisms. | Proof. \( \operatorname{cl}\left( {{S}^{3} - R}\right) = {L}_{1,2}^{\prime } \cup \operatorname{cl}\left( {{S}^{3} - {T}_{1}}\right) \) and \( {L}_{1,2}^{\prime } \cap \operatorname{cl}\left( {{S}^{3} - {T}_{1}}\right) = \partial {T}_{1} \) . If \( {\pi }_{1}\left( {\partial {T}_{1}, v}\right) \rightarrow {\pi }_{1}\le... | Yes |
Proposition 16.4.9. The inclusion \( {L}_{1} \hookrightarrow {L}_{2} \) is homotopically trivial. | Proof. Fig. 16.5 makes it obvious that the inclusion \( T \hookrightarrow \operatorname{cl}\left( {{S}^{3} - L}\right) \) is ho-motopically trivial. By 16.4.4, it follows that the inclusion \( L \hookrightarrow \operatorname{cl}\left( {{S}^{3} - T}\right) \) is homotopically trivial. So, by means of the homeomorphism \... | Yes |
Corollary 16.4.10. The inclusion \( {L}_{k} \hookrightarrow {L}_{k + 1} \) is homotopically trivial for every \( k \geq 1 \) . | Hence, by 7.1.2, we have | No |
Proposition 16.4.11. \( W \) is contractible. | Choose a base ray \( \omega \) in \( W \) well parametrized with respect to \( \left\{ {L}_{k}\right\} \) . Then for any \( k \) we have \( {\operatorname{cl}}_{W}\left( {W - {L}_{k}}\right) = {L}_{k, k + 1} \cup {L}_{k + 1, k + 2} \cup \cdots \) . We may assume \( \omega \left( i\right) \in \) \( {L}_{i, i + 1} \cap {... | Yes |
Proposition 16.4.12. These inverse sequences are not semistable and are not pro-finitely generated. | Proof. The first statement follows from Exercise 2 and the second from Exercise 3. | No |
Theorem 16.5.1. If \( G \) is semistable at each end then \( {H}^{2}\left( {G,\mathbb{Z}G}\right) \) is a free abelian group. | Proof. By 16.1.8 and 16.1.11, \( {H}_{e}^{1}\left( {\widetilde{X};\mathbb{Z}}\right) \) is free abelian. By 13.2.9 and 13.2.13, \( {H}_{e}^{1}\left( {\widetilde{X};\mathbb{Z}}\right) \cong {H}^{1}\left( {G,\mathbb{Z}{G}^{e}}\right) \cong {H}^{2}\left( {G,\mathbb{Z}G}\right) . | Yes |
Theorem 16.5.2. If \( G \) is simply connected at each end then \( {H}^{2}\left( {G,\mathbb{Z}G}\right) = 0 \) . | Proof. By 16.2.7, \( {H}_{e}^{1}\left( {\widetilde{X};\mathbb{Z}}\right) = 0 \) . But once again \( {H}_{e}^{1}\left( {\widetilde{X};\mathbb{Z}}\right) \cong {H}^{2}\left( {G,\mathbb{Z}G}\right) \) . | Yes |
Theorem 16.5.6. Let the one-ended finitely presented group \( G \) be stable at infinity and assume \( G \) contains an element of infinite order. Then \( G \) is either simply connected at infinity or stably \( \mathbb{Z} \) at infinity. | Proof. By 16.3.4, the stable inverse sequence \( \left\{ {{\pi }_{1}\left( {\widetilde{X} - {K}_{i},\omega \left( i\right) }\right) }\right\} \) must be pro-isomorphic to a finitely generated free group, so \( \left\{ {{H}_{1}\left( {\widetilde{X} - {K}_{i};\mathbb{Z}}\right) }\right\} \) is pro-isomorphic to a finitel... | Yes |
Theorem 16.6.1. The group \( G \) is semistable at infinity; and \( G \) is simply connected at infinity iff \( \left| L\right| \) is simply connected. | Proof. By 16.5.3, these properties are the same for \( G \) and for any torsion free subgroup of finite index; hence they can be checked in \( \left| D\right| \) . The homotopy commutative diagram in the paragraph preceding Theorem 13.9.5 shows that the inclusion map of the \( {\left( n + 1\right) }^{\text{th }} \) nei... | No |
Theorem 16.6.2. \( \left| {D}_{0}\right| \) is a contractible \( {CW4} \) -manifold on which \( G \) acts rigidly with finite cell-stabilizers so that \( G \smallsetminus \left| {D}_{0}\right| \) is finite. | By 9.1.10, \( G \) has a torsion free subgroup \( H \) of finite index. By 16.6.1 and 16.6.2 we have: | No |
Corollary 16.6.3. \( H \smallsetminus \left| {D}_{0}\right| \) is a closed aspherical 4-manifold whose universal cover is not homeomorphic to \( {\mathbb{R}}^{4} \) . | One can build such examples in any dimension \( \geq 4 \) since non-simply connected homology spheres bounding compact contractible \( n \) -manifolds exist for all \( n \geq 4 \) . We call such closed aspherical manifolds Davis manifolds. They first appeared in [46], and until then it was unknown if a contractible ope... | No |
Proposition 16.7.2. The free object in \( \mathcal{D} \) generated by a compact object \( \left( {Z, z}\right) \) in \( \mathcal{C} \) exists. | Proof. First note that the \ | No |
Proposition 16.8.3. The space \( Y \times Z \triangleq {L}_{i} \times {M}_{i} \) is path connected and the embedding \( {f}_{i} \) induces an isomorphism of fundamental groups. | Proof. The paths \( {\sigma }_{ij} \) and \( {\tau }_{ik} \) give path connectedness. The cover of\n\n\[ Y \times Z\overset{c}{ = }{L}_{i} \times {M}_{i} \]\n\nby the sets \( Y \times {V}_{ik} \) and \( {U}_{ij} \times Z \) has the property that no point lies in more than two of those sets. By 6.2.11, the fundamental g... | Yes |
Theorem 16.9.1. Let \( N \rightarrowtail G\overset{\pi }{ \twoheadrightarrow }Q \) be a short exact sequence of infinite finitely generated groups. If \( G \) is finitely presented then \( G \) is semistable at infinity. Hence \( {H}^{2}\left( {G,\mathbb{Z}G}\right) \) is free abelian. | Before proving this we set things up. To begin, we choose a presentation \( \left\langle {{h}_{i},{q}_{j} \mid {r}_{k},{q}_{j}{w}_{ij}{q}_{j}^{-1}{h}_{i}^{-1}}\right\rangle \) of \( G \) where \( i, j \) and \( k \) range over finite sets of indices, and the set of generators \( \left\{ {{h}_{i},{q}_{j}}\right\} \) is ... | No |
Proposition 16.9.2. Given a finite subcomplex \( K \) of \( \widetilde{X} \) there is a finite sub-complex \( M \) such that if the given \( \sigma \) ’s and \( \tau \) ’s in (i)-(iii) all lie in \( \widetilde{X} - M \) , then the proper homotopies constructed in (i)-(iii) all take place in \( \widetilde{X} - K \) . | Proof. Choose \( k \) so that \( {p}_{N}\left( K\right) \subset {L}_{k} \) . Let \( m > 0 \) be such that whenever \( \left( {{\bar{\tau }}_{1},{\bar{\tau }}_{2},\ldots }\right) \) is the image under \( {p}_{N} \) of a selected proper edge ray in \( \widetilde{X} \), then the edges \( {\bar{\tau }}_{m},{\bar{\tau }}_{m... | Yes |
Lemma 16.9.3. Let \( A \) and \( B \) be subcomplexes of a \( {CW} \) complex \( Y \), and let \( s \) be a positive integer. If \( B \subset Y \subseteq {N}^{s}\left( A\right) \), then \( N\left( B\right) \subset Y \subseteq {N}^{s - 1}\left( A\right) \) . Hence \( {N}^{s}\left( B\right) \subset Y\overset{c}{ - }A \),... | Proof. Suppose \( N\left( B\right) \) is not a subset of \( Y - {}^{c}{N}^{s - 1}\left( A\right) \) . Then \( N\left( B\right) \) and \( {N}^{s - 1}\left( A\right) \) share a vertex \( u \) . So there is a cell \( e \) of \( Y \) whose carrier \( C\left( e\right) \) contains both \( u \) and a vertex \( w \) of \( B \)... | Yes |
Let \( {F}_{2} \) be the free group of rank 2 and let \( G = {F}_{2} \times {F}_{2} \). Then \( G \) has one end by 16.8.1, and \( {H}^{2}\left( {G,\mathbb{Z}G}\right) \neq 0 \) by 13.2.9 and 12.6.1; hence \( G \) is not simply connected at infinity. But there is a short exact sequence \( N \rightarrowtail G \twoheadri... | To see this we will now construct \( G \) in another way. Let \( {F}_{2} \) be generated by \( a \) and \( b \), let \( {x}_{i} = {a}^{i}b{a}^{-i} \) and let \( B \vartriangleleft {F}_{2} \) be the subgroup (freely) generated by \( \left\{ {{x}_{i} \mid i \in \mathbb{Z}}\right\} \). The inner automorphism \( \phi : {F}... | No |
Theorem 16.9.5. Let \( H \) be finitely presented, let \( \phi : H \rightarrowtail H \) be a monomorphism, and let \( G = H{ * }_{\phi } \) be the resulting ascending HNN extension. If \( H \) is infinite then \( G \) has one end and is semistable at infinity. If \( H \) has one end then \( G \) is simply connected at ... | Proof. Let \( \left\{ {h}_{i}\right\} \) be a finite set of generators for \( H \) . The group \( G \) has the finite presentation \( \left\langle {H, t \mid {t}^{-1}{h}_{i}{t\phi }{\left( {h}_{i}\right) }^{-1},\forall i}\right\rangle \) (where a finite presentation of \( H \) with generators \( \left\{ {h}_{i}\right\}... | Yes |
Lemma 16.9.6. If every \( {\alpha }_{s} \) is above level \( {m}_{ + } \) then \( \alpha \) is equivalent in \( X - K \) to a trivial edge loop. | Proof. If every \( {\alpha }_{s} \) is an \( H \) -edge then the whole loop lies in a simply connected subcomplex \( g\widetilde{Z} \), and the \( f \) -image of such a subcomplex is an integer \( > {m}_{ + } \) , so \( g\widetilde{Z} \subset \widetilde{X} - K \) . If some \( {\alpha }_{s} \) is a \( t \) -edge, then t... | No |
Proposition 17.1.1. (Proper Whitehead Theorem) Let \( Y \) be finite-dimensional. The following are equivalent:\n\n(i) \( A \) is a proper strong deformation retract of \( Y \) ;\n\n(ii) \( A\overset{i}{ \hookrightarrow }Y \) is a proper homotopy equivalence;\n\n(iii) \( \left( {Y, A}\right) \) is properly \( n \) -con... | Proof. The proof is similar to the proof of the Whitehead Theorem 4.1.4. In an exercise, the reader is asked to consider why one needs \( Y \) to be finite-dimensional. | No |
Proposition 17.1.5. Let \( Y \) be infinite, let \( \omega : \lbrack 0,\infty ) \rightarrow Y \) be a cellular proper ray and an embedding, and let \( n \geq 0 \) . \( Y \) is properly \( n \) -connected iff the inclusion \( {Y}^{n} \hookrightarrow Y \) is properly homotopic to a map into \( \omega \left( {\lbrack 0,\i... | Proof. The case \( n = 0 \) is clear. Let \( Y \) be properly \( n \) -connected. By induction, assume \( {Y}^{n - 1} \hookrightarrow Y \) is properly homotopic to a map into the ray \( \omega \left( {\lbrack 0,\infty }\right) ) \) . Then 17.1.4 implies that \( {Y}^{n} \) has the proper homotopy type of the locally fin... | Yes |
Theorem 17.1.6. (Proper Hurewicz Theorem ) Let \( n \geq 2 \) and let \( Y \) be properly 1-connected. Then \( Y \) is properly \( n \) -connected iff \( Y \) is properly \( n \) - acyclic with respect to \( \mathbb{Z} \) . | Proof. Simply observe that the proof of 4.5.1 gives this; use 17.1.3 and 17.1.5 in place of 7.1.2 and 4.1.8 in that proof. | No |
Hence Thompson’s group \( F \) is \( n \) -connected at infinity for all \( n \). | This follows from 9.3.19,13.10.1,13.3.3 and 16.9.7. | No |
Theorem 17.2.3. Let \( Y \) be a contractible rigid \( G \) -CW complex which is \( {fi} \) - nite mod \( G \), and let the stabilizer of each cell be finite. Then \( G \) is semistable at each end or n-connected at infinity or n-acyclic at infinity iff \( Y \) has the corresponding property. Moreover, for any ring \( ... | Proof. The CW complex \( Y \) is strongly locally finite by 10.1.12. Apply the Borel Construction using a \( K\left( {G,1}\right) \) -complex \( X \) of finite type to get the usual commutative diagram\n\n\n\nHere, t... | Yes |
Theorem 17.2.4. Let \( Y \) be an \( \left( {m - 1}\right) \) -connected \( m \) -dimensional \( G \) -CW complex which is finite \( {modG} \), where the stabilizer of each cell is finite and \( m \geq 2 \) . Then \( G \) satisfies the conclusions of Theorem 17.2.3 for all \( n \leq m - 1 \) . | There is an analog of 7.3.1 in the present context; for a proof see [34]. | No |
Proposition 17.3.1. The map \( \left( {\widetilde{g} \times \mathrm{{id}}}\right) \circ \widetilde{k} \) is a proper homotopy equivalence. | Proof. The maps \( {\widetilde{q}}^{\prime } : {\widetilde{W}}^{\prime } \rightarrow \widetilde{Z} \) and projection: \( \widetilde{Y} \times \widetilde{Z} \rightarrow \widetilde{Z} \) are stacks of CW complexes with every fiber \( \widetilde{Y} \), and the projection map: \( \widetilde{X} \times \widetilde{Z} \rightar... | Yes |
Theorem 17.3.2. Let \( R \) be a PID, and let \( N \) and \( Q \) have type \( F \) . Then \( G \) has type \( F \) and for all \( p \) there are split short exact sequences\n\n\[ 0 \rightarrow {\bigoplus }_{i + j = p}{H}^{i}\left( {N,{RN}}\right) { \otimes }_{R}{H}^{j}\left( {Q,{RQ}}\right) \rightarrow {H}^{p}\left( {... | Applying this with 13.3.3(ii) we get: | No |
Proposition 17.3.4. Under these hypotheses the map \( \left( {\widetilde{g} \times \mathrm{{id}}}\right) \circ \widetilde{k} : {\widetilde{W}}^{\prime } \rightarrow \) \( \widetilde{Y} \times \widetilde{Z} \) is a CW-proper \( n \) -equivalence. | Proof. The proof is similar to that of 17.3.1. That proof depends on 17.1.3, which in turn is merely an observation about the proof of 4.1.7 (in which 17.1.1 and 17.1.2 play the roles of 4.1.4 and 4.1.5). Here we are asserting that a parallel observation, using appropriately modified versions of 17.1.1 and 17.1.2, prov... | No |
Theorem 17.3.5. Let \( R \) be a PID, and let \( N \) and \( Q \) be infinite and have type \( {F}_{n} \) . Then \( G \) has type \( {F}_{n} \) and for all \( p \leq n - 1 \) there are split short exact sequences \[ 0 \rightarrow {\bigoplus }_{i + j = p}{H}^{i}\left( {N,{RN}}\right) { \otimes }_{R}{H}^{j}\left( {Q,{RQ}... | Proof. The CW complexes \( {\widetilde{Y}}^{n} \) and \( {\widetilde{Z}}^{n} \) are locally finite, and \( {\left( {\widetilde{Y}}^{n} \times {\widetilde{Z}}^{n}\right) }^{n} = \) \( {\left( \widetilde{Y} \times \widetilde{Z}\right) }^{n} \) . By 12.6.1, for every \( p \) we have a split short exact sequence \[ 0 \righ... | Yes |
Theorem 17.3.6. Let \( N \rightarrowtail G \twoheadrightarrow Q \) be a short exact sequence of infinite groups of type \( {F}_{n} \), and let \( R \) be a PID. Working with respect to \( R \), let \( N \) be \( s \) -acyclic at infinity and let \( Q \) be \( t \) -acyclic at infinity where \( s \leq n - 1 \) and \( t ... | Proof. By 13.3.2, when \( s \leq n - 1, N \) is \( s \) -acyclic at infinity iff \( {H}^{i}\left( {N,\mathbb{Z}N}\right) = 0 \) for \( i \leq s + 1 \) and \( {H}^{s + 2}\left( {N,\mathbb{Z}N}\right) \) is torsion free; a similar statement holds for \( Q \), replacing \( s \) by \( t \) . It follows from 16.8.5 that \( ... | Yes |
Proposition 17.4.1. If \( \mathcal{L} \mathrel{\text{:=}} \left\{ {L}_{i}\right\} \) is a finite filtration of \( Y \) and if \( \omega \) is well parametrized with respect to \( \mathcal{L} \), then there is a natural short exact sequence\n\n\[ \n{\mathop{\lim }\limits_{ \leftarrow }}^{1}\left\{ {{\pi }_{n + 1}\left( ... | Proof. Similar to the corresponding homological proof indicated in Remark 11.4.9. The details are an exercise. | No |
Proposition 17.4.2. \( \sigma \) is an isomorphism when \( n \geq 1 \) and a bijection when \( n = 0 \) . | Proof (of 17.4.2). There are obvious functions\n\n\[ \n{\pi }_{n}^{e}\left( {Y,\omega }\right) { < }^{\alpha }\;{\pi }_{n}\left( {Y,\mathcal{L},\omega }\right) \overset{\beta }{ \rightarrow }{\pi }_{n}\left( {P{R}_{\mathcal{L}}\left( Y\right) ,\omega }\right) \overset{\gamma }{ \rightarrow }{\pi }_{n}\left( {{PR}\left(... | Yes |
Proposition 17.5.2. If such a \( Y \) admits a \( Z \) -set compactification \( W \), then for any finite filtration \( \mathcal{L} \mathrel{\text{:=}} \left\{ {L}_{i}\right\} \) each \( Y - {L}_{i} \) is finitely dominated and \( \left\{ {Y - {L}_{i}}\right\} \) is equivalent in pro-Homotopy to an inverse sequence of ... | Proof. Write \( C = W - Y \) . The subspace \( {A}_{i} \mathrel{\text{:=}} \left( {Y - {L}_{i}}\right) \cup C \) is a compact neighborhood of \( C \) in \( W \) since \( W - N\left( {L}_{i}\right) \) is compact \( {}^{8} \) (by 1.5.4). Thus \( Y \triangleq {L}_{i} \hookrightarrow {A}_{i} \) is a homotopy equivalence. L... | Yes |
Proposition 17.5.4. \( W \) is a \( Z \) -set compactification of \( {D}_{\infty } \), with \( {C}_{\infty } \) as compactifying space. | Proof. There is an obvious retraction \( {r}_{i} : {D}_{i + 1} \rightarrow {D}_{i} \) under which, for \( i \leq \) \( t \leq i + 1,{\omega }_{t} \mapsto {\omega }_{i} \) . The inverse limit in Spaces of the inverse sequence \( \left\{ {{D}_{0} < \frac{{r}_{0}\;}{}{D}_{1} < \frac{{r}_{1}\;}{}\cdots }\right\} \) is home... | Yes |
Let \( Y \) admit a proper metric \( d \) so that the usual weak topology on \( Y \) is induced by \( d \) . Assume that \( d \) is a \( \operatorname{CAT}\left( 0\right) \) metric or, more generally, a unique-geodesic metric space. Choose a base point \( v \in Y \) and let \( {C}_{\infty } \) be the space of geodesic ... | It follows that each \( {D}_{t} \) is compact and hence that \( {D}_{\infty } \cup {C}_{\infty } \) is compact. The \ | No |
Here we show that every compact metrizable space \( C \) can play the role of \( W - Y \) for a \( Z \) -set compactification of a locally finite CW complex \( Y \) . Let \( C \) be homeomorphic to the inverse limit in Spaces of an inverse sequence \( {W}_{1} \leftarrow {W}_{2} \leftarrow \cdots \) of finite CW complex... | Now \( M\left( {f}_{i}\right) \) is \( {W}_{i}{ \cup }_{{f}_{i}}\left( {{W}_{i + 1} \times I}\right) \) as in Sect. 4.1, so for each \( {w}_{i + 1} \in {W}_{i + 1} \), the path \( t \mapsto \left( {{w}_{i + 1}, t}\right) \) in \( {W}_{i + 1} \times I \) defines a path in \( M\left( {f}_{i}\right) \) from \( {f}_{i}\lef... | No |
To illustrate the differences between \( H,{SS} \) and \( S \) we give two examples of the set of morphisms \( \left( {{W}_{1},{C}_{1}}\right) \rightarrow \left( {{W}_{2},{C}_{2}}\right) \) . The dependence on \( {W}_{1} \) and \( {W}_{2} \) is only a convenience, as we have explained; we are really considering morphis... | The similarity occurs because the solenoid is the inverse limit of fibrations. | No |
Proposition 18.2.2. Let \( Y \) be a path connected rigid \( G \) -graph which is finite mod \( G \) . Assume that the stabilizer of each vertex is finite. Let \( \rho \) denote the CW pseudometric on \( Y \) and let \( d \) be the word metric on \( G \) with respect to some finite set of generators. Then \( \left( {Y,... | Proof. Pick a base vertex \( v \in Y \) and a compact fundamental domain \( C \subset Y \) for the \( G \) -action on \( Y \) so that \( v \in C \) . For the pseudometric \( \rho \), as for metrics, we write \( {B}_{r}\left( y\right) = \left\{ {{y}^{\prime } \in Y \mid \rho \left( {y,{y}^{\prime }}\right) < r}\right\} ... | Yes |
Proposition 18.2.3. Let \( Y \) be a path connected rigid \( G \) -CW complex which is finite mod \( G \) . Let \( \rho \) and \( {\rho }_{1} \) be the \( {CW} \) pseudometrics on \( Y \) and \( {Y}^{1} \) respectively, and let \( i : {Y}^{1} \hookrightarrow Y \) denote the inclusion map. Then \( i \) is a quasi-isomet... | Proof. We first prove that \( \left( {{Y}^{1},{\rho }_{1}}\right) \) and \( \left( {{Y}^{1},\rho \mid }\right) \) are quasi-isometric. There are only finitely many different isomorphism types of carriers \( C\left( e\right) \) of cells \( e \) of \( Y \) . If \( \omega \) is a path in \( Y \) joining two points of \( {... | Yes |
Proposition 18.2.8. If \( Y \) is an \( n \) -dimensional and \( \left( {n - 1}\right) \) -connected rigid \( G \) -CW complex which is finite mod \( G \), then \( {f}_{0} \) extends to a CW-Lipschitz map \( f : {X}^{n} \rightarrow Y \) . The restriction of \( f \) to \( {X}^{n - 1} \) is unique up to \( {CW} \) -Lipsc... | Proof. The map \( {f}_{0} \) is CW-Lipschitz. One extends this map skeleton by skeleton to get the cellular map \( f \) . The connectivity assumptions on \( Y \) make the extensions possible, and the fact that some group \( G \) acts rigidly with finite quotient ensures that the number of cells of \( Y \) involved in e... | No |
Proposition 18.2.10. The map \( f : {X}^{n} \rightarrow Y \) in 18.2.8 is proper. | Proof. \( \left\{ {N\left( v\right) \mid v \in {X}^{0}}\right\} \) is a cover \( {}^{4} \) of \( X \) . Suppose there is a finite subcomplex \( L \) of \( Y \) such that \( {f}^{-1}\left( L\right) \) is not compact. By 11.4.4 and 10.1.12 there is an infinite set \( \left\{ {v}_{i}\right\} \) of vertices of \( X \) such... | Yes |
Theorem 18.2.11. Let \( G \) and \( H \) be finitely generated groups of type \( {F}_{n} \) . Let \( X \) be a \( K\left( {G,1}\right) \) with finite \( n \) -skeleton, and let \( Y \) be a \( K\left( {H,1}\right) \) with finite \( n \) - skeleton. If \( G \) and \( H \) are quasi-isometric then there is a proper \( n ... | Proof. We may assume \( X \) and \( Y \) have one vertex each so we may identify \( {\widetilde{X}}^{0} \) and \( {\widetilde{Y}}^{0} \) with \( G \) and \( H \) . By 18.2.8-18.2.10, there is a proper map \( {\widetilde{X}}^{n} \rightarrow \) \( {\widetilde{Y}}^{n} \) which is a CW-Lipschitz \( n \) -equivalence. By 14... | Yes |
Theorem 18.2.12. If \( G \) and \( H \) are finitely generated quasi-isometric groups and if \( G \) has type \( {F}_{n} \), then \( H \) has type \( {F}_{n} \) . | Proof. Let \( X \) and \( Y \) be \( K\left( {G,1}\right) \) and \( K\left( {H,1}\right) \) complexes respectively, each having one vertex. Since \( G \) and \( H \) are finitely generated they are countable, so, by exercises in Sections 4.5 and 11.4, we may assume \( X \) and \( Y \) are countable\n\n\( {}^{4} \) See ... | No |
Proposition 18.3.2. For any unit vector \( u \) in \( V \) and any vertex \( g \in \Gamma \) there is a ray in \( \Gamma \) starting at \( g \) on which the map \( {h}_{u} \) is strictly increasing. | Proof. The discussion shows that this is true when \( g = 1 \) ; translation by \( g \) preserves this property. | No |
Corollary 18.3.3. The filtered space \( \left( {\Gamma ,{h}^{-1}{\mathcal{F}}_{u}}\right) \) is regular in the sense of Sect. 14.3 and has at least one filtered end. | Proof. The existence of a filtered end follows directly from 18.3.2. Regularity holds because, by 18.3.2, complements have no \( {h}^{-1}{\mathcal{F}}_{u} \) -bounded path components. | Yes |
Proposition 18.3.4. \( \left( {\Gamma ,{h}^{-1}{\mathcal{F}}_{u}}\right) \) is CW compatible. | Proof. This follows from the fact that \( S \) is finite. | No |
Proposition 18.3.5. If \( \left( {\Gamma ,{h}^{-1}{\mathcal{F}}_{u}}\right) \) has one filtered end then for all \( r \in \mathbb{R} \) the subspace \( \Gamma - {h}^{-1}\left( {{H}_{u}\left( r\right) }\right) \) is path connected. | Proof (of 18.3.5). By hypothesis, \( \forall n \in \mathbb{Z}\exists m \geq n \) such that any two points in \( \Gamma - {h}^{-1}\left( {{H}_{u}\left( m\right) }\right) \) can be joined \( {}^{9} \) by a path in \( \Gamma - {h}^{-1}\left( {{H}_{u}\left( n\right) }\right) \) . Let \( x \) and \( y \) be points of \( \Ga... | Yes |
Proposition 18.3.11. \( u \in {\sum }^{1}\left( \rho \right) \) iff there exists \( \lambda \geq 0 \) such that for any \( n \) , any two points in \( \Gamma - {h}^{-1}\left( {{H}_{u}\left( n\right) }\right) \) can be joined by a path in the subspace \( \Gamma - {h}^{-1}\left( {{H}_{u}\left( {n - \lambda }\right) }\rig... | Proof. If \( u \in {\sum }^{1}\left( \rho \right) \) there exists \( \lambda \geq 0 \) such that points in \( \Gamma - {h}^{-1}\left( {{H}_{u}\left( \lambda \right) }\right) \) can be joined by a path in \( \Gamma - {h}^{-1}\left( {{H}_{u}\left( 0\right) }\right) \) . There are elements of \( G \) which translate \( \G... | No |
Theorem 18.3.12. Let \( \rho \left( G\right) \) have discrete orbits in \( V \) . The normal subgroup \( N \mathrel{\text{:=}} \ker \left( \rho \right) \) is finitely generated iff \( {\sum }^{1}\left( \rho \right) = {S}_{\infty }\left( V\right) \) . | Proof. In the commutative diagram\n\n\n\nit is not hard to see that the map \( f \) is proper when the orbits are discrete. Thus the filtration \( {h}^{-1}\mathcal{F} \) of \( \Gamma \) is by sets each of which is co... | No |
Theorem 18.3.13. ( \( {\sum }^{1} \) -Criterion) A unit vector \( u \) in \( V \) lies in \( {\sum }^{1}\left( \rho \right) \) iff there is an equivariant cellular map \( \phi : \Gamma \rightarrow \Gamma \) such that, for all \( x \in \Gamma \) , \( {h}_{u} \circ \phi \left( x\right) - {h}_{u}\left( x\right) > 0. | Proof. We use the notation of Part D. If \( u \in {\sum }^{1}\left( \rho \right) \) choose \( \gamma \in T\left( u\right) \) . Define \( \phi \left( g\right) = {g\gamma } \) for every \( g \in G \) . This is equivariant. Extend \( \phi \) equivariantly to the edges of \( \Gamma \) so that the edge joining 1 to \( {g}_{... | Yes |
Corollary 18.3.16. In the space \( \operatorname{Hom}\left( {G,\operatorname{Transl}\left( V\right) }\right) \) of translation actions of \( G \) on \( V \), with compact-open topology, the condition \( {\sum }^{1}\left( \rho \right) = {S}_{\infty }\left( V\right) \) is an open condition. | Proof. Combine 18.3.1 and 18.3.13. | No |
Theorem 18.3.18. Let \( G \) be finitely presented and let the rank of \( G/{G}^{\prime } \) (as a \( \mathbb{Z} \) -module) be at least 2. If \( G \) has no non-abelian free subgroup, then there is a finitely generated normal subgroup \( L \vartriangleleft G \) with \( G/L \) infinite cyclic. | Proof. Let \( \left( {X, x}\right) \) be the presentation complex for some finite presentation of \( G \) . We may assume that the 1-skeleton of \( \widetilde{X} \) is a Cayley graph with the properties listed in Sect. C.\n\nClaim. \( {S}_{\infty }\left( {V}_{0}\right) = {\sum }^{1}\left( G\right) \cup \left( {-{\sum }... | Yes |
Proposition 18.3.19. If \( G = {G}_{1}{ * }_{{G}_{0}}{G}_{2} \) is a free product with amalgamation, where \( {G}_{0} \) has index \( \geq 2 \) in \( {G}_{1} \) and has index \( \geq 3 \) in \( {G}_{2} \), then \( G \) contains a free non-abelian subgroup. | Proof (Sketch of Proof). The Normal Form Theorem for free products with amalgamation (see \( \left\lbrack {{106}\text{, p. 187}}\right\rbrack \) ) says that a product \( {c}_{1}\ldots {c}_{n} \) is non-trivial in \( G \) if (i) each \( {c}_{i} \) is in \( {G}_{1} \) or \( {G}_{2} \) ,(ii) \( {c}_{i} \) and \( {c}_{i + ... | Yes |
Theorem 1.1 For any graph \( G \) ,\n\n\[ \mathop{\sum }\limits_{{v \in V}}d\left( v\right) = {2m} \] | Proof Consider the incidence matrix \( \mathbf{M} \) of \( G \) . The sum of the entries in the row corresponding to vertex \( v \) is precisely \( d\left( v\right) \) . Therefore \( \mathop{\sum }\limits_{{v \in V}}d\left( v\right) \) is just the sum of all the entries in \( \mathbf{M} \) . But this sum is also \( {2m... | Yes |
Corollary 1.2 In any graph, the number of vertices of odd degree is even. | Proof Consider equation (1.1) modulo 2. We have\n\n\[ d\left( v\right) \equiv \left\{ \begin{array}{ll} 1\left( {\;\operatorname{mod}\;2}\right) & \text{ if }d\left( v\right) \text{ is odd,} \\ 0\left( {\;\operatorname{mod}\;2}\right) & \text{ if }d\left( v\right) \text{ is even. } \end{array}\right. \]\n\nThus, modulo... | Yes |
Proposition 1.3 Let \( G\left\lbrack {X, Y}\right\rbrack \) be a bipartite graph without isolated vertices such that \( d\left( x\right) \geq d\left( y\right) \) for all \( {xy} \in E \), where \( x \in X \) and \( y \in Y \) . Then \( \left| X\right| \leq \left| Y\right| \) , with equality if and only if \( d\left( x\... | Proof The first assertion follows if we can find a matrix with \( \left| X\right| \) rows and \( \left| Y\right| \) columns in which each row sum is one and each column sum is at most one. Such a matrix can be obtained from the bipartite adjacency matrix \( \mathbf{B} \) of \( G\left\lbrack {X, Y}\right\rbrack \) by di... | Yes |
Theorem 2.1 Let \( G \) be a graph in which all vertices have degree at least two. Then \( G \) contains a cycle. | Proof If \( G \) has a loop, it contains a cycle of length one, and if \( G \) has parallel edges, it contains a cycle of length two. So we may assume that \( G \) is simple.\n\nLet \( P \mathrel{\text{:=}} {v}_{0}{v}_{1}\ldots {v}_{k - 1}{v}_{k} \) be a longest path in \( G \) . Because the degree of \( {v}_{k} \) is ... | Yes |
Theorem 2.2 Any simple graph \( G \) with \( \mathop{\sum }\limits_{{v \in V}}\left( \begin{matrix} d\left( v\right) \\ 2 \end{matrix}\right) > \left( \begin{array}{l} n \\ 2 \end{array}\right) \) contains a quadrilateral. | Proof Denote by \( {p}_{2} \) the number of paths of length two in \( G \), and by \( {p}_{2}\left( v\right) \) the number of such paths whose central vertex is \( v \) . Clearly, \( {p}_{2}\left( v\right) = \left( \begin{matrix} d\left( v\right) \\ 2 \end{matrix}\right) \) . As each path of length two has a unique cen... | Yes |
Every tournament has a directed Hamilton path. | Proof Clearly, the trivial tournament (on one vertex) has a directed Hamilton path. Assume that, for some integer \( n \geq 2 \), every tournament on \( n - 1 \) vertices has a directed Hamilton path. Let \( T \) be a tournament on \( n \) vertices and let \( v \in V\left( T\right) \) . The digraph \( {T}^{\prime } \ma... | Yes |
Theorem 2.4 Every loopless graph \( G \) contains a spanning bipartite subgraph \( F \) such that \( {d}_{F}\left( v\right) \geq \frac{1}{2}{d}_{G}\left( v\right) \) for all \( v \in V \) . | Proof Let \( G \) be a loopless graph. Certainly, \( G \) has spanning bipartite subgraphs, one such being the empty spanning subgraph. Let \( F \mathrel{\text{:=}} F\left\lbrack {X, Y}\right\rbrack \) be a spanning bipartite subgraph of \( G \) with the greatest possible number of edges. We claim that \( F \) satisfie... | Yes |
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