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Proposition 5.3.10. Let \( X \) be an oriented regular \( {CW} \) complex. For cells \( {e}_{\beta }^{n - 1} \) and \( {e}_{\alpha }^{n} \) of \( X \), the incidence number \( \left\lbrack {{e}_{\alpha }^{n} : {e}_{\beta }^{n - 1}}\right\rbrack \) is \( \pm 1 \) if \( {e}_{\beta }^{n - 1} \) is a face of \( {e}_{\alpha... | Proof. When \( n = 1 \), this is true by definition; see Sect. 2.5. Let \( n > 1 \) . When \( {e}_{\beta }^{n - 1} \) not a face of \( {e}_{\alpha }^{n} \), the incidence number is 0 by 2.5.8. Let \( Y \) be the subcom-plex \( C\left( {e}_{\alpha }^{n}\right) \) of \( X \), and let \( Z \) be the CW complex structure o... | Yes |
Proposition 5.4.5. Let \( K \) be an ordered abstract simplicial complex and let \( \sigma \) be an \( n \) -simplex of \( K \) . If we write \( \left| \sigma \right| = \left| \left\{ {{v}_{0},\cdots ,{v}_{n}}\right\} \right| \), with vertices listed in order, this orientation convention leads to the following formula ... | \[ \mathop{\sum }\limits_{{i = 0}}^{n}{\left( -1\right) }^{i}\left| \left\{ {{v}_{0},\cdots ,{\widehat{v}}_{i},\cdots ,{v}_{n}}\right\} \right| \] | Yes |
Proposition 6.1.2. For each cell e of \( V,{X}_{e} \) is homeomorphic to \( {G}_{\widetilde{e}} \smallsetminus \widetilde{X} \) , hence \( {\pi }_{1}\left( {{X}_{e},{x}_{e}}\right) \cong {G}_{\widetilde{e}} \) . | Since \( {r}_{e} \) is a quotient map, so is the map \( {r}_{e} \times \) id in the diagram; to see this, apply 1.3.11 \( \left( {n + 1}\right) \) times to conclude that \( {r}_{e} \times \mathrm{{id}} : \widetilde{X} \times \left\{ {\widetilde{u}}_{e}\right\} \times {B}^{n + 1} \rightarrow \) \( {X}_{e} \times {B}^{n ... | No |
Theorem 6.1.3. The map \( {q}^{-1}\left( {V}^{n}\right) \coprod \left( {\mathop{\coprod }\limits_{{e \in {E}_{n + 1}}}{X}_{e} \times {B}^{n + 1}}\right) \rightarrow {q}^{-1}\left( {V}^{n + 1}\right) \) which agrees with inclusion on \( {q}^{-1}\left( {V}^{n}\right) \) and with \( {H}_{e} \) on \( {X}_{e} \times {B}^{n ... | Proof. The function \( {s}_{n + 1} \) is clearly a continuous bijection. Moreover, it maps cells bijectively onto cells. Thus \( {s}_{n + 1}^{-1} \mid \) is continuous on each cell, which, by 1.2.12, is enough to imply continuity of \( {s}_{n + 1}^{-1} \) (exercise). The second part is clear. | No |
Theorem 6.1.5. Let \( Y \) be a simply connected rigid \( G - {CW} \) complex. (i) If \( Y \) has finite 1-skeleton \( {\;\operatorname{mod}\;G} \) and if the stabilizer of each vertex is finitely generated, then \( G \) is finitely generated. (ii) If \( Y \) has finite 2-skeleton mod \( G \) , if the stabilizer of eac... | Proof. By 6.1.1, \( {q}^{-1}\left( {V}^{1}\right) \) is a subcomplex of \( Z \) containing \( {Z}^{1} \) . By hypothesis in (i), for each \( w \in {E}_{0},{\pi }_{1}\left( {{X}_{w},{x}_{w}}\right) \) is finitely generated. By 4.1.14, each \( {X}_{w} \) is homotopy equivalent to a CW complex \( {X}_{w}^{\prime } \) havi... | Yes |
Proposition 6.2.1. (Britton’s Lemma) For each \( {w}_{0} \in {E}_{0} \), the homomorphism \( {\gamma }_{{w}_{0}} : G\left( {w}_{0}\right) \rightarrow {\pi }_{1}\left( {\mathcal{G},\Gamma ;T}\right) \) induced by \( G\left( {w}_{0}\right) \hookrightarrow \left( {\underset{w \in {E}_{0}}{ * }G\left( w\right) }\right) * F... | Remarks on the proof. The version for Cases 1 and 2, above, is found in [106, Chap. IV, Sect. 2]. The general case then follows by the above remarks; a direct proof is found in \( \left\lbrack {{142}\text{, p. 46}}\right\rbrack \) . All these involve reducing the length of a supposedly minimal non-trivial word in the k... | No |
Proposition 6.2.2. Let each \( {p}_{e\# }^{ \pm } \) be a monomorphism. There is an isomorphism \( j \) making the following diagram commute for each vertex \( w \) of \( \Gamma \) : \n\nwhere \( {\beta }_{w} \) is i... | Proof. The tree \( s\left( T\right) \) can be extended, using 3.1.15, to a maximal tree \( {T}^{ + } \) in \( \operatorname{Tot}\left( {\mathcal{X},\Gamma }\right) \) such that, for each vertex \( w,{T}^{ + } \cap X\left( w\right) \) is a maximal tree in \( X\left( w\right) \) (considered as a subcomplex of \( \operato... | No |
Proposition 6.2.4. \( \left( {\overline{\mathcal{G}},\Gamma }\right) \) and \( \left( {\mathcal{G},\Gamma }\right) \) are isomorphic graphs of groups. | Proof. The following diagram commutes, as well as a similar diagram in which \( t\left( e\right) \) replaces \( o\left( e\right), b\left( e\right) \) replaces \( a\left( e\right) \), and + replaces -:\n\n\n\nHere, th... | No |
Proposition 6.2.5. There is a tree, \( \widetilde{T} \), in \( Y \) such that \( p \) maps \( \widetilde{T} \) isomorphically onto \( T \) . | From now on, we choose each \( \widetilde{w} \) to be in \( \widetilde{T} \), and each \( \widetilde{e} \) to be in \( \widetilde{T} \) whenever \( e \) is in \( T \) . The effect is that whenever \( e \) is a cell of \( T, o\left( \widetilde{e}\right) = {\left( o\left( e\right) \right) }^{ \sim } \) and \( t\left( \wi... | Yes |
Proposition 6.2.6. There is an isomorphism \( \psi : {\pi }_{1}\left( {\overline{\mathcal{G}},\Gamma ;T}\right) \rightarrow G \) such that for every vertex \( w \) of \( \Gamma \) the following diagram commutes (where \( {\bar{\gamma }}_{w} \) is analogous to \( {\gamma }_{{w}_{0}} \) in 6.2.1): | Proof. The required \( \psi \) is read off from the following commutative diagram:\n\n\n\nHere, \( {\chi }^{\prime } \) is the isomorphism arising from 6.2.4; \( {\beta }_{w}^{\prime } \) is induced by inclusion. To ... | Yes |
A Baumslag-Solitar group is a group \( {BS}\left( {m, n}\right) \) with presentation \( \left\langle {x, t\left| {\;{t}^{-1}{x}^{m}t{x}^{-n}}\right. }\right\rangle \) where \( m, n \geq 1 \) . | Clearly, \( {BS}\left( {m, n}\right) \) is the HNN extension \( \mathbb{Z}{ * }_{{\phi }_{m, n}} \) where \( {\phi }_{m, n} : m\mathbb{Z} \rightarrow n\mathbb{Z} \) is the isomorphism taking \( m \) to \( n \) . Thus \( {BS}\left( {m, n}\right) \) is the fundamental group of a circle of groups \( \left( {\mathcal{G},\G... | Yes |
Theorem 6.2.11. [Generalized Van Kampen Theorem] Under these hypotheses the fundamental group of \( X \) is isomorphic to \( {\pi }_{1}\left( {\mathcal{G},\Gamma ;T}\right) \) . | Proof. Let \( \left( {\mathcal{X},\Gamma }\right) \) be the generalized graph of path connected CW complexes having \( {X}_{\alpha } \) over \( \alpha ,{Y}_{\alpha \beta \gamma } \) over the edge joining \( \alpha \) to \( \beta \) so indexed, and inclusions as structural maps. Let \( \operatorname{Tot}\left( {\mathcal... | Yes |
Let the \( {CW} \) complex \( X \) be non-empty and let \( n \geq 0 \) . \( X \) is n-connected iff the inclusion \( {X}^{n} \hookrightarrow X \) is homotopic to a constant map. \( X \) is \( n \) -connected for all \( n \) iff \( X \) is contractible. | For \( n = 0 \), the first sentence is clear. Let \( X \) be \( n \) -connected. By induction, \( {X}^{n - 1} \hookrightarrow X \) is homotopically trivial. Hence \( {X}^{n} \hookrightarrow X \) is homotopically trivial iff a certain map \( {X}^{n}/{X}^{n - 1} \rightarrow X \) is homotopically trivial. But \( {X}^{n}/{... | No |
A path connected CW complex \( X \) is n-aspherical iff its universal cover \( \widetilde{X} \) is n-connected. \( X \) is aspherical iff \( \widetilde{X} \) is contractible. | Let \( p : \widetilde{X} \rightarrow X \) be the universal cover, and let \( 2 \leq k \leq n \) . If \( X \) is \( n \) -aspherical and if \( \widetilde{f} : {S}^{k} \rightarrow \widetilde{X} \) is a map then \( p \circ \widetilde{f} \) is homotopically trivial. By 2.4.6, the same is true of \( \widetilde{f} \) . Conve... | Yes |
Proposition 7.1.5. For any group \( G \), there exists a \( K\left( {G,1}\right) \) -complex \( X \) having only one vertex. Moreover, if \( \left( {Y, y}\right) \) is a path connected \( k \) -aspherical pointed \( {CW} \) complex such that \( {\pi }_{1}\left( {Y, y}\right) \) is isomorphic to \( G \), then there exis... | Proof. We will describe the \( K\left( {G,1}\right) \) -complex \( X \) by induction on skeleta. The 2-skeleton \( \left( {{X}^{2}, x}\right) \) is built as in Example 1.2.17, reflecting some chosen presentation of \( G;{X}^{2} \) is 1-aspherical. By induction, assume \( {X}^{n} \) has been constructed and is \( \left(... | Yes |
Theorem 7.1.9. Let \( \left( {\mathcal{X},\Gamma }\right) \) be a graph of pointed \( {CW} \) complexes. Assume each \( X\left( w\right) \) and each \( X\left( e\right) \) is aspherical and that each \( {p}_{e\# }^{ \pm } \) is a monomorphism (on fundamental groups). Then \( \operatorname{Tot}\left( {\mathcal{X},\Gamma... | Proof. We saw in Sect.6.2 (following 6.2.9) that \( U \mathrel{\text{:=}} {\left( \operatorname{Tot}\left( \mathcal{X},\Gamma \right) \right) }^{ \sim } \) is a quotient space obtained by gluing copies of \( \widetilde{X}\left( e\right) \times {B}^{1} \) to copies of \( \widetilde{X}\left( w\right) \) via the pointed l... | Yes |
Theorem 7.1.10. There is a \( K\left( {G,1}\right) \) -complex \( W \) and a stack \( W \rightarrow Z \) all of whose fibers are \( Y \) . | Proof. As in that section, we start with an arbitrary \( K\left( {G,1}\right) \) -complex \( \left( {X, x}\right) \) and we consider the diagonal left action of \( G \) on \( \widetilde{X} \times \widetilde{Z} \) given by \( g\left( {x, z}\right) = \) \( \left( {{gx},\pi \left( g\right) z}\right) \) . The quotient spac... | Yes |
Proposition 7.1.11. Let \( X \) and \( Z \) be homotopy equivalent path connected CW complexes. For each \( k \), let \( X \) and \( Z \) have \( {m}_{k} \) and \( {r}_{k}k \) -cells, respectively. Then \( Z \) is homotopy equivalent to a \( {CW} \) complex \( Y \) such that \( {Y}^{1} = {X}^{1}, Y \) has \( \left( {{r... | Proof. Let \( T \) be a maximal tree in \( X \) . We have \( {\left( X/T\right) }^{2} = {X}_{P} \) where \( P \mathrel{\text{:=}} \) \( \langle W \mid R\rangle \) is a presentation of the fundamental group. By 4.1.16, \( Z \) is homotopy equivalent to a CW complex \( K \) whose 1-skeleton is \( {\left( {X}_{P}\right) }... | Yes |
Proposition 7.1.12. Let \( n \geq 2 \) . Let \( X \) be a \( K\left( {G,1}\right) \) -complex having \( {m}_{k}k \) - cells. Let \( Z \) be a \( K\left( {G,1}\right) \) -complex having \( {r}_{k}k \) -cells, such that \( {Z}^{n - 1} = {X}^{n - 1} \) . Let \( {s}_{2} = {r}_{1} - {r}_{0} + 1 \) and, for \( n \geq 3 \), l... | Proof. We first deal with the case \( n = 2 \) . To begin, assume \( X \) has only one vertex (i.e., \( {m}_{0} = {r}_{0} = 1 \) ). Let \( {P}_{1} \) and \( {P}_{2} \) be presentations of \( G \) such that \( {X}^{2} = {X}_{{P}_{1}} \) and \( {Z}^{2} = {X}_{{P}_{2}} \) . With notation as in the proof of 4.1.12, there a... | No |
Theorem 7.1.13. Let \( n \geq 1 \) . Let \( X \) be a \( K\left( {G,1}\right) \) -complex having \( {m}_{k}k \) -cells. Let \( Z \) be a \( K\left( {G,1}\right) \) -complex having \( {r}_{k}k \) -cells. Then there exists a \( K\left( {G,1}\right) \) - complex \( Y \) with \( {Y}^{n} = {X}^{n} \) such that \( Y \) has \... | Proof. By 7.1.7, all \( K\left( {G,1}\right) \) -complexes are homotopy equivalent. By 7.1.11, there is a \( K\left( {G,1}\right) \) -complex \( {Y}_{1} \) such that \( {Y}_{1}^{1} = {X}^{1} \) and \( {Y}_{1} \) has \( {r}_{k}k \) -cells when \( k \geq 4 \) . Now apply 7.1.12 by induction on \( n \), starting with \( n... | Yes |
Proposition 7.2.1. Every group has type \( {F}_{0};G \) has type \( {F}_{1} \) iff \( G \) is finitely generated; \( G \) has type \( {F}_{2} \) iff \( G \) is finitely presented; for \( n \geq 2 \) , \( G \) has type \( {F}_{n} \) iff there exists a finite pointed \( n \) -dimensional \( \left( {n - 1}\right) \) -asph... | Proof. Every group has type \( {F}_{0} \), by 7.1.5. Let \( G \) have type \( {F}_{n} \) and let \( \left( {Z, z}\right) \) be a \( K\left( {G,1}\right) \) -complex with finite \( n \) -skeleton. For \( n = 1 \) [resp. \( n = 2 \) ], \( G \) is finitely generated [resp. finitely presented], by 3.1.17. For \( n \geq 2,{... | Yes |
Proposition 7.2.3. Let \( H \leq G \) and let \( \left\lbrack {G : H}\right\rbrack < \infty \) . Let \( G \) and \( H \) have type \( {F}_{n - 1} \) . Then \( G \) has type \( {F}_{n} \) iff \( H \) has type \( {F}_{n} \) . | Proof. \ | No |
Corollary 7.2.4. Let \( H \leq G \) and let \( \left\lbrack {G : H}\right\rbrack < \infty \) . For \( 0 \leq n \leq \infty, G \) has type \( {F}_{n} \) iff \( H \) has type \( {F}_{n} \) . | Proof. Apply 7.2.3 inductively. For the case \( n = \infty \) then apply 7.2.2. | No |
Proposition 7.2.6. \( G \) has geometric dimension 0 iff \( G \) is trivial. \( G \) has geometric dimension 1 iff \( G \) is free and non-trivial. If \( G \) has geometric dimension \( d \), every subgroup of \( G \) has geometric dimension \( \leq d \) . | Proof. The dimension 0 statement is clear. By 3.1.16, every 1-dimensional CW complex has free fundamental group, so if \( G \) has geometric dimension 1, \( G \) is free and non-trivial. Conversely, if \( G \) is free and non-trivial, we saw in Example 1.2.17 how to build a 1-dimensional CW complex whose fundamental gr... | Yes |
Theorem 7.2.7. (Serre’s Theorem) Let \( G \) be torsion free, and let \( H \) be a subgroup of finite index having finite geometric dimension. Then \( G \) has finite geometric dimension. | Proof. Let \( Y \) be a finite-dimensional \( K\left( {H,1}\right) \) -complex and let \( H{\bar{g}}_{1},\ldots, H{\bar{g}}_{n} \) be the cosets of \( H \) in \( G \) . Let \( \widetilde{Y} \) be the universal cover of \( Y \) . Let \( \widetilde{X} = \mathop{\prod }\limits_{{i = 1}}^{n}{\widetilde{Y}}_{i} \) where eac... | Yes |
Proposition 7.2.13. \( G \) has type \( {FD} \) iff \( G \) has type \( {F}_{\infty } \) and \( G \) has finite geometric dimension. | Proof. \ | No |
Proposition 7.2.15. If \( G \) has type \( {FD} \), then \( G \times \mathbb{Z} \) has type \( F \) . | Proof. This follows from 4.3.7. In detail, let \( X\overset{f}{ \rightarrow }Y\overset{g}{ \rightarrow }X \) be cellular maps, where \( X \) is a \( K\left( {G,1}\right) \) -complex, \( Y \) is a finite CW complex, and \( g \circ f \simeq {\operatorname{id}}_{X} \) . Then \( X \times {S}^{1} \) is a \( K\left( {G \time... | Yes |
Proposition 7.2.17. If there is a \( K\left( {G,1}\right) \) -complex which is dominated by a \( d \) -dimensional \( {CW} \) complex then \( G \times \mathbb{Z} \) has geometric dimension \( \leq d + 1 \) . | Proof. Let \( Y \) dominate \( X \), where \( X \) is a \( K\left( {G,1}\right) \) -complex and \( Y \) is \( d \) - dimensional. As in the proof of \( {7.2.15}, X \times {S}^{1} \) is homotopy equivalent to a \( \left( {d + 1}\right) \) -dimensional CW complex, which is therefore a \( K\left( {G \times \mathbb{Z},1}\r... | Yes |
Corollary 7.2.18. If there is a \( K\left( {G,1}\right) \) -complex which is dominated by a d-dimensional \( {CW} \) complex then \( G \) has geometric dimension \( \leq d + 1 \) . | Remark 7.2.19. The conclusion of 7.2.18 can be improved to \ | No |
Theorem 7.2.20. Let \( n \geq 1 \), let the group \( G \) have type \( {F}_{n} \), and let \( X \) be a \( K\left( {G,1}\right) \) -complex with finite \( n \) -skeleton. Then \( G \) has type \( {F}_{n + 1} \) iff there is a \( K\left( {G,1}\right) \) -complex \( Y \) with finite \( \left( {n + 1}\right) \) -skeleton ... | Proof. \ | No |
Theorem 7.2.21. Let \( N \rightarrowtail G \twoheadrightarrow Q \) be an exact sequence of groups. If \( G \) has type \( {F}_{n} \) and if \( N \) has type \( {F}_{n - 1} \) then \( Q \) has type \( {F}_{n} \) . | Proof. This is obvious for \( n \leq 2 \) so we assume \( n \geq 3 \) . Let \( Y \) be an \( \left( {n - 2}\right) \) - aspherical finite \( \left( {n - 1}\right) \) -dimensional CW complex whose fundamental group is isomorphic to \( Q \), and let \( X \) be a \( K\left( {G,1}\right) \) -complex. As before, we consider... | Yes |
Theorem 7.3.1. For \( n \geq 1 \), let \( Y \) be an \( \left( {n - 1}\right) \) -connected rigid \( G \) -CW complex having finite \( n \) -skeleton mod \( G \) . If the stabilizer of each \( i \) -cell has type \( {F}_{n - i} \) for all \( i \leq n - 1 \), then \( G \) is of type \( {F}_{n} \) . | Proof. We leave the case \( n = 1 \) as an exercise. Starting with a \( K\left( {G,1}\right) \) - complex \( \left( {X, v}\right) \), we construct a commutative diagram as in Sect. 6.1\n\n\n\nin which \( q : Z \right... | No |
Theorem 7.4.1. (Brown’s Criterion) Let the \( \left( {n - 1}\right) \) -connected free \( G - {CW} \) complex \( Y \) admit a \( G \) -filtration \( \left\{ {K}_{i}\right\} \) where each \( G \smallsetminus {K}_{i} \) has finite \( n \) -skeleton. Then \( G \) has type \( {F}_{n} \) iff \( \left\{ {K}_{i}\right\} \) is... | Proof. \ | No |
Theorem 8.1.1. Let\n\n\[ \cdots \rightarrow {F}_{1} \rightarrow {F}_{0}\overset{\epsilon }{ \rightarrow }R \rightarrow 0 \]\n\nand\n\n\[ \cdots \rightarrow {F}_{1}^{\prime } \rightarrow {F}_{0}^{\prime }\overset{{\epsilon }^{\prime }}{ \rightarrow }R \rightarrow 0 \]\n\nbe free \( {RG} \) -resolutions of \( R \) . For ... | By convention there is a \ | No |
Proposition 8.1.2. The oriented \( n \) -cells \( {\widetilde{e}}_{\alpha }^{n} \) of \( \widetilde{X} \) freely generate \( {C}_{n}\left( {\widetilde{X};R}\right) \) as an \( {RG} \) -module. The boundary \( \partial : {C}_{n}\left( {\widetilde{X};R}\right) \rightarrow {C}_{n - 1}\left( {\widetilde{X};R}\right) \) is ... | \[ \cdots \xrightarrow[]{{\partial }_{3}}{C}_{2}\left( {\widetilde{X};R}\right) \xrightarrow[]{\;{\partial }_{2}}{C}_{1}\left( {\widetilde{X};R}\right) \xrightarrow[]{\;{\partial }_{1}}{C}_{0}\left( {\widetilde{X};R}\right) \xrightarrow[]{\;\epsilon \;}R\xrightarrow[]{\;}0 \] where \( \epsilon \) is defined by \( \epsi... | Yes |
Proposition 8.1.3. Let \( \left( {X, x}\right) \) and \( \left( {Y, y}\right) \) be \( K\left( {G,1}\right) \) -complexes, with pointed universal covers \( \left( {\widetilde{X},\widetilde{x}}\right) \) and \( \left( {\widetilde{Y},\widetilde{y}}\right) \) . Let the groups \( {\pi }_{1}\left( {X, x}\right) \) and \( {\... | Proof. By hypothesis, there is a given isomorphism \( \phi : {\pi }_{1}\left( {X, x}\right) \rightarrow {\pi }_{1}\left( {Y, y}\right) \) inducing id : \( G \rightarrow G \) . By 7.1.7, there is a cellular homotopy equivalence \( f : \left( {X, x}\right) \rightarrow \left( {Y, y}\right) \) inducing \( \phi \), and \( f... | Yes |
Proposition 8.1.4. Let \( \left( {X, v}\right) \) be a \( K\left( {G,1}\right) \) -complex. Then \( {H}_{ * }\left( {G, R}\right) \cong \) \( {H}_{ * }\left( {X;R}\right) \) . | Proof. The chain complex \( \left( {R{ \otimes }_{G}{C}_{ * }\left( {\widetilde{X};R}\right) ,\operatorname{id} \otimes \partial }\right) \) is isomorphic \( {}^{4} \) to the chain complex \( \left( {{C}_{ * }\left( {X;R}\right) ,\partial }\right) \) . | Yes |
Proposition 8.1.5. For \( n \geq 2,{\mathbb{Z}}_{n} \) has infinite geometric dimension. | Indeed, there is a \( K\left( {{\mathbb{Z}}_{n},1}\right) \) -complex \( \left( {X, v}\right) \) such that \( {C}_{ * }\left( {\widetilde{X};R}\right) \overset{\epsilon }{ \rightarrow }R \) is a free \( R{\mathbb{Z}}_{n} \) -resolution of \( R \), but some careful work is needed to describe the attaching maps. The skel... | No |
Theorem 8.2.2. Let \( Z \) be an \( n \) -dimensional free \( G \) -CW complex which is finite mod \( G \), and which is \( \left( {n - 1}\right) \) -acyclic \( {}^{7} \) with respect to \( R \) . Then \( G \) has type \( F{P}_{n} \) ; and \( G \) has type \( F{P}_{n + 1} \) iff \( {H}_{n}\left( {Z;R}\right) \) is a fi... | Proof. There is an obvious exact sequence of \( {RG} \) -modules\n\n\[ 0 \rightarrow {K}_{n} \rightarrow {C}_{n}\left( {Z;R}\right) \rightarrow \cdots \rightarrow {C}_{0}\left( {Z;R}\right) \overset{\epsilon }{ \rightarrow }R \]\n\nand \( {K}_{n} \) is the image of a free module \( {F}_{n + 1} \) . Let \( {K}_{n + 1} =... | Yes |
Proposition 8.3.2. Let \( J \subset {J}^{\prime } \) be closed connected subsets of \( \mathbb{R} \) such that \( {X}_{{J}^{\prime }} - {X}_{J} \) contains no vertices of \( X \) . Then \( {X}_{J} \) is a strong deformation retract of \( {X}_{{J}^{\prime }} \) . | Proof. The general case is easily adapted from the case we consider: \( J = \) \( ( - \infty ,1\rbrack \) and \( {J}^{\prime } = ( - \infty ,2\rbrack \) . If \( e \) is an \( n \) -cell of \( X \) not lying in \( {X}_{J} \) then the set \( e \cap {f}^{-1}\left( \left\lbrack {1,2}\right\rbrack \right) \) inherits a conv... | No |
Proposition 8.3.3. Let \( J \subset {J}^{\prime } \) be closed connected subsets of \( \mathbb{R} \) such that \( \inf J = \inf {J}^{\prime } \) and \( {J}^{\prime } - J \) contains only one point of \( f\left( {X}^{0}\right) \), namely \( t \) . Then \( {X}_{J} \cup \bigcup \left\{ {\text{cone on}{\operatorname{lk}}_{... | Proof. The proof is similar to that of 8.3.2, but the deformations described there are only applied to cells which do not contain a vertex \( v \) such that \( f\left( v\right) = t \) . | No |
Proposition 8.3.5. If each ascending and descending link is \( \left( {n - 1}\right) \) -connected [resp. \( \left( {n - 1}\right) \) -acyclic with respect to \( R \) ], then \( {X}_{\left\lbrack -k, k\right\rbrack } \) is \( \left( {n - 1}\right) \) -connected [resp. \( \left( {n - 1}\right) \) -acyclic with respect t... | Proof. For the \( \left( {n - 1}\right) \) -acyclic case, we apply \( {8.3.4} \) (ii) to the pair of spaces \( \left( {{X}_{( - \infty, t\rbrack },{X}_{( - \infty, k\rbrack }}\right) \) for \( t \geq k \) to deduce that \( {X}_{( - \infty, k\rbrack } \hookrightarrow {X}_{( - \infty, t\rbrack } \) induces isomorphisms o... | Yes |
Theorem 8.3.8. Let \( X \) be a simply connected cubical rigid \( G \) -complex which is finite mod \( G \) . If the simplicial link of every vertex of \( X \) is a flag complex, then \( X \) admits a \( G \) -invariant \( {CAT}\left( 0\right) \) metric which agrees with the (given) Euclidean metric on each cube. | distance between \( a \) and \( b \) is the inf of the lengths of these paths. This is the \( \operatorname{CAT}\left( 0\right) \) metric referred to in 8.3.8. | No |
Proposition 8.3.9. The simplicial link of each vertex of \( X \) is a flag complex. | Proof (of 8.3.9). Let \( v \) be a vertex of \( X \) which lies in the \( \left( {k + 1}\right) \) -dimensional sheet \( \sum \) . Then \( \left| {{\operatorname{simplk}}_{\sum }\left( v\right) }\right| \) is a canonical triangulation of \( {S}^{k} \) . This is the part of \( \left| {{\operatorname{simplk}}_{X}\left( v... | Yes |
Corollary 8.3.10. \( Z \) is a \( K\left( {G,1}\right) \) -complex. | The homomorphism \( \phi : G \rightarrow \mathbb{Z} \) which takes each generator to \( 1 \in \mathbb{Z} \) is induced by a map \( {f}_{0} : Z \rightarrow {S}^{1} \) which takes every 1-cell of \( Z \) homeomorphically to \( {S}^{1} \), and one sees easily that \( {f}_{0} \) can be chosen so that its lift \( f : X \rig... | No |
Proposition 8.3.11. Each ascending and descending link of a vertex of \( X \) is homeomorphic to \( \left| L\right| \) . | Proof. We will show that for every vertex \( v \) of \( X \) both simplk \( {}_{X}^{ \uparrow }v \) and simplk \( {}_{X}^{ \downarrow }v \) are isomorphic to \( L \) as abstract simplicial complexes. Orient the circles \( {S}_{w}^{1} \) of \( Z \) so that \( \phi \) takes the corresponding generator of the fundamental ... | Yes |
Theorem 8.3.12. (Bestvina-Brady Theorem Let \( L \) be a finite non-empty flag complex, let \( G \) be the corresponding right-angled Artin group, let \( \phi : G \rightarrow \) \( \mathbb{Z} \) be the epimorphism taking all generators to \( 1 \in \mathbb{Z} \), and let \( H = \ker \left( \phi \right) \) . Then\n\n(i) ... | For the proof we need:\n\nLemma 8.3.13. Let \( v | No |
Lemma 8.3.13. Let \( v \) be a vertex of \( X \) and let \( U \) be the union of the sheets containing \( v \) .\n\n(a) \( U \) is an open cone; in fact there is a homeomorphism from \( U \) to the space \( {\mathrm{{lk}}}_{X}v \times \lbrack 0,\infty )/{\mathrm{{lk}}}_{X}v \times \{ 0\} \) taking \( v \) to the quotie... | Proof. (a) is clear. For (b) we use 8.3.7. The deformation of all points of \( X \) along geodesics ending at \( v \) gives the required strong deformation retract of \( X - \{ v\} \) onto \( \left| {{\operatorname{simpl}}_{X}v}\right| \) . | No |
Lemma 8.3.14. Let \( K \) be an \( \left( {n - 1}\right) \) -acyclic (with respect to \( R \) ) free \( G \) - \( {CW} \) complex and let \( \left\{ {K}_{i}\right\} \) be a \( G \) -filtration where each \( G \smallsetminus {K}_{i} \) has finite \( n \) -skeleton. If \( G \) has type \( F{P}_{n} \) then \( \left\{ {K}_... | Proof. Let \( \left\{ {{F}_{ * },\partial }\right\} \) be a free \( {RG} \) -resolution of \( R \) which is finitely generated in dimensions \( \leq n \) . By 8.1.1 there are mutually inverse chain homotopy equivalences \( {C}_{ * }\left( {K;R}\right) \underset{g}{\overset{f}{ \leftarrow }}{F}_{ * } \) . Let \( D \) be... | Yes |
Proposition 9.1.1. If \( g = {s}_{1}\cdots {s}_{d} \) with \( d > l\left( g\right) \), there are indices \( i < j \) such that \( g = {s}_{1}\cdots {\widehat{s}}_{i}\cdots {\widehat{s}}_{j}\cdots {s}_{d} \) (i.e., suppress \( {s}_{i} \) and \( {s}_{j} \) ). | Proof. This is a basic algebraic fact about Coxeter systems, in fact a characterizing property. See, for example, p. 53 of [31]. | No |
Corollary 9.1.2. The function \( T \mapsto \langle T\rangle \) from subsets of \( S \) to standard subgroups of \( G \) is a bijection. Moreover, \( {T}_{1} \subset {T}_{2} \) iff \( \left\langle {T}_{1}\right\rangle \subset \left\langle {T}_{2}\right\rangle \) . | Proof. The required inverse is \( H \mapsto H \cap S \) . If \( H \) is a standard subgroup then \( H = \langle H \cap S\rangle \) . If \( T \subset S \), then \( T \subset \langle T\rangle \cap S \) . It remains to show that \( \langle T\rangle \cap S \subset T \) . If \( g \in \langle T\rangle \cap S \) then \( g = {... | Yes |
Theorem 9.1.3. (Davis’ Theorem) The G-CW complex \( \left| D\right| \) is rigid, contractible, and finite mod \( G \) . The stabilizer of each cell is finite. | Proof. Since \( G \) acts by order-preserving simplicial automorphisms, the action on \( \left| D\right| \) is rigid. If \( \left\{ {{g}_{0}\left\langle {T}_{0}\right\rangle ,\cdots ,{g}_{k}\left\langle {T}_{k}\right\rangle }\right\} \) are the vertices, in order, of a simplex of \( D \) then it is easy to see that it ... | Yes |
Lemma 9.1.4. When \( \langle T\rangle \) is non-trivial and finite, \( \left| {F}_{\sigma \left( T\right) }\right| \) is contractible. | Proof. Whenever \( U = \left\{ {{s}_{0},\cdots ,{s}_{r}}\right\} \) is a subset of \( T,\langle U\rangle \) is a finite standard subgroup of \( \langle T\rangle \), so \( {F}_{U} \) is a cone. Thus \( \left\{ {\left| {F}_{\{ s\} }\right| \mid s \in T}\right\} \) is a cover of \( \left| {F}_{\sigma \left( T\right) }\rig... | No |
Proposition 9.1.5. For every \( g \in G \), the standard subgroup \( \langle B\left( g\right) \rangle \) is finite. | Proof. Write \( d = l\left( g\right) \) . It is enough to show that every reduced word in \( \langle B\left( g\right) \rangle \) has length \( \leq d \) . Let \( h = {t}_{1}\cdots {t}_{k} \) be a word of minimal length in the elements of \( B\left( g\right) \) and let \( g = {s}_{1}\cdots {s}_{d} \) be reduced. By 9.1.... | No |
Lemma 9.1.6. \( {A}_{n} \cap {g}_{n + 1}F = {g}_{n + 1}{F}_{\sigma \left( {B\left( {g}_{n + 1}\right) }\right) } \) and therefore \( \left| {A}_{n}\right| \cap \left| {{g}_{n + 1}F}\right| \) is contractible. | Proof. First, \( B\left( {g}_{n + 1}\right) \neq \varnothing \), so the right side is defined. The inclusion \( \supset \) holds because if \( s \in B\left( {g}_{n + 1}\right) \) then \( {g}_{n + 1}s = {g}_{i} \) for some \( i \leq n \), and so any simplex of \( {g}_{n + 1}{F}_{\{ s\} } \) with initial vertex \( {g}_{n... | Yes |
Proposition 9.1.8. Let \( \\left( {G, S}\\right) \) be a Coxeter system and let \( S \) have \( n \) elements. Then \( G \) is isomorphic to a subgroup of \( G{L}_{n}\\left( \\mathbb{R}\\right) \) . | Proof. See [31, Chap. 2, Sect. 5]. In fact there is a \ | No |
Proposition 9.1.9. (Selberg’s Lemma) Every finitely generated subgroup of \( G{L}_{n}\left( \mathbb{C}\right) \) has a torsion free subgroup of finite index. | Proof. See, for example, [132, p. 326]. | No |
Theorem 9.1.11. Let \( \left( {G, S}\right) \) be a Coxeter system and let \( d\left( { \geq 1}\right) \) be the largest number such that there is a d-element subset \( T \) of \( S \) with \( \langle T\rangle \) finite. Then every torsion free subgroup of finite index in \( G \) has geometric dimension \( \leq d \) an... | Proof. The dimension of \( \left| K\right| \) is \( d - 1 \), so the dimension of \( \left| D\right| \) is \( d \) . The torsion free subgroup \( H \) acts freely on \( D \), and \( G \smallsetminus \left| D\right| \) is finite. | No |
Proposition 9.2.2. Every \( x \in F \) can be expressed as a product of the \( {x}_{i} \) ’s and their inverses to satisfy (1) and (2) in exactly one way. | Proof. By 9.2.1 it is enough to show that any \( \bar{x} \in \bar{F} \) has a unique normal form as above. Let \( \bar{x} = {\bar{x}}_{{i}_{1}}\ldots {\bar{x}}_{{i}_{k}}{\bar{x}}_{{j}_{m}}^{-1}\ldots {\bar{x}}_{{j}_{1}}^{-1} \) be the image under \( \rho \) of a normal form in \( F \) (this is what we mean by a normal ... | Yes |
Corollary 9.2.3. The epimorphism \( \rho : F \rightarrow \bar{F} \) is an isomorphism. | Since \( \bar{F} \) is obviously torsion free and \( \bar{\phi } \) is obviously injective (it comes from a conjugation) we have: | No |
Corollary 9.2.4. \( F \) is torsion free and \( \phi \) is injective. | It follows that \( {F}_{1} = \phi \left( F\right) \) is a copy of \( F \) with presentation\n\n\[ \left\langle {{x}_{1},{x}_{2},\ldots \mid {x}_{n}^{{x}_{i}} = {x}_{n + 1}\forall 1 \leq i < n}\right\rangle . \]\n\nHence \( \phi \left( {F}_{1}\right) \leq {F}_{1} \) and we have: | No |
Proposition 9.2.5. \( F \) is the ascending HNN extension of \( {F}_{1} \) by \( \phi \mid {F}_{1} : {F}_{1} \mapsto \) \( {F}_{1} \) with stable letter \( {x}_{0} \) . | Repeating with respect to \( {F}_{2} = \phi \left( {F}_{1}\right) \), etc., we see that \( F \) is an infinitely iterated HNN extension where the intersection of all the base groups \( {F}_{1},{F}_{2},\ldots \) is trivial. | Yes |
Proposition 9.2.6. \( F \) contains a free abelian subgroup of infinite rank. Hence \( F \) has infinite geometric dimension. | Proof. We work in \( \bar{F} \) . The homeomorphisms (of \( \mathbb{R} \) ) \( {\bar{x}}_{2i}{\bar{x}}_{{2i} + 1}^{-1} \) where \( i = \) \( 0,1,2,\ldots \) have disjoint supports and hence generate an abelian subgroup. It is easy to see that they freely generate. The last sentence follows from 7.2.11. | Yes |
Theorem 9.2.7. Every quotient of \( F \) by a non-trivial normal subgroup is abelian, hence is a quotient of \( {\mathbb{Z}}^{2} \) . | Proof. Let \( N \vartriangleleft F \) . Consider a non-trivial element of \( N \) with normal form \( {x}_{{i}_{1}}\ldots {x}_{{i}_{k}}{x}_{{j}_{m}}^{-1}\ldots {x}_{{j}_{1}}^{-1} \) . By conjugating and inverting as needed we may assume either \( {i}_{1} < {j}_{1} \) or \( m = 0 \) (i.e., no \ | No |
Corollary 9.2.8. If \( G \) is a group and \( \rho : F \rightarrow G \) is a homomorphism then either \( \rho \) is a monomorphism or \( \rho \left( {x}_{n}\right) = \rho \left( {x}_{1}\right) \) for all \( n \geq 1 \) . | Proof. The abelianization \( F \rightarrow F/\left\lbrack {F : F}\right\rbrack \) has this property, so the statement follows from 9.2.7. | No |
Theorem 9.2.9. Let \( h : \left( {Y, y}\right) \rightarrow \left( {Y, y}\right) \) be such that \( {}^{4}h \simeq {h}^{2} \) where \( \omega \) is a loop at \( y \) . The following are equivalent (where \( {h}_{\# } : {\pi }_{1}\left( {Y, y}\right) \rightarrow {\pi }_{1}\left( {Y, y}\right) \) is the induced homomorphi... | Proof. (i) \( \Rightarrow \) (ii): Since \( h \) splits there is a homotopy commutative diagram of base point preserving maps\n\n\n\nThus \( k \mathrel{\text{:=}} g \circ f : \left( {X, x}\right) \rightarrow \left( {... | No |
Theorem 9.2.11. (Freyd-Heller Theorem) \( h \) splits iff \( \rho \) is not a monomorphism. In particular, if \( Z \) is a \( K\left( {F,1}\right) \) -complex and \( h : Z \rightarrow Z \) is induced by the shift \( \phi : F \rightarrow F \) then \( h \) is a homotopy idempotent which does not split. | Proof. This follows from 9.2.8 and 9.2.9. | No |
Proposition 9.3.1. If \( h \in P{L}_{2}\left( I\right) \) then \( h \) is defined by some balanced pair of finite binary trees. | Proof. Let \( K \) be a dyadic subdivision of \( I \) such that \( h \) is affine on each interval in \( K \) . Let \( n \) be such that the subdivision \( {K}^{\prime } \) with vertices \( \left\{ {\left. \frac{m}{{2}^{n}}\right| \;0 \leq m \leq {2}^{n}}\right\} \) subdivides \( K \) . Choose \( k \) so that the subdi... | Yes |
Theorem 9.3.2. \( \widetilde{\rho } : F \rightarrow P{L}_{2}\left( I\right) \) is an isomorphism. | Proof. The proof that \( \widetilde{\rho } \) is a monomorphism is similar to the proof of the corresponding statement in 9.2.3. That \( \widetilde{\rho } \) is an epimorphism follows from 9.3.1. | No |
Lemma 9.3.4. Given expansions \( e \) and \( {e}^{\prime } \) there exist expansions \( \bar{e} \) and \( {\bar{e}}^{\prime } \) such that \( \bar{e} \circ e = {\bar{e}}^{\prime } \circ {e}^{\prime } \) . | Proof. When \( e \) and \( {e}^{\prime } \) have length \( \leq 1 \) this follows from 9.3.3. The general case is done by induction on the sum of the lengths of \( e \) and \( {e}^{\prime } \) . | No |
Lemma 9.3.6. The left action of \( F \) on \( F \times \mathcal{T}g\left( {h, T}\right) = \left( {{gh}, T}\right) \) induces a left action of \( F \) on \( B \) . | We write \( \left\lbrack {h, T}\right\rbrack \) for the member of the \( F \) -set \( B \) defined by \( \left( {h, T}\right) \) . The \( F \) - action is \( g\left\lbrack {h, T}\right\rbrack = \left\lbrack {{gh}, T}\right\rbrack \) . That the simple expansion functions \( {e}_{i} : \mathcal{T} \rightarrow \mathcal{T} ... | Yes |
Lemma 9.3.7. If \( \left( {{g}_{1},{S}_{1}}\right) \sim \left( {{g}_{2},{S}_{2}}\right) \) then for all \( i\left( {{g}_{1},{e}_{i}\left( {S}_{1}\right) }\right) \sim \left( {{g}_{2},{e}_{i}\left( {S}_{2}\right) }\right) \) . | Proof. There exist \( e \) and \( T \) such that \( \left( {e\left( {S}_{i}\right), T}\right) \) is a balanced pair representing \( {g}_{i} \) for \( i = 1,2 \) . We write \( \left( {{g}_{1},{S}_{1}}\right) \underset{k}{ \sim }\left( {{g}_{2},{S}_{2}}\right) \) if the length of this \( e \) is \( \leq k \) . The lemma ... | No |
Lemma 9.3.9. Let \( n \geq i \) . The function \( {E}_{i} \) maps \( {f}^{-1}\left( n\right) \) bijectively onto \( {f}^{-1}\left( {n + 1}\right) \) . | Proof. That \( {E}_{i} \) is injective is clear. The proof that \( {E}_{i} \) is surjective should be clear from Example 9.3.10, below. | No |
Proposition 9.3.12. The induced \( F \) -action on \( \left| B\right| \) is a free action. | Proof. The stabilizers of vertices are trivial by 9.3.11. Since the action of \( F \) on \( \left\lbrack {h, S}\right\rbrack \) preserves the number of leaves of \( S \), the stabilizer of each simplex of \( S \) is also trivial. | Yes |
Proposition 9.3.13. The poset \( B \) is a directed set. | Proof. Let \( \left\{ {{b}_{1},\ldots ,{b}_{k}}\right\} \subset B \) . Write \( {b}_{i} = \left\lbrack {{h}_{i},{S}_{i}}\right\rbrack \) . Then \( {S}_{i} \) expands to \( {S}_{i}^{\prime } \) such that \( \left( {{h}_{i},{S}_{i}^{\prime }}\right) \sim \left( {{1}_{F},{T}_{i}^{\prime }}\right) \) for some \( {T}_{i}^{\... | Yes |
Proposition 9.3.14. If a poset \( P \) is a directed set then \( \left| P\right| \) is contractible. | Proof. Whenever \( K \) is a finite subcomplex of \( P \) there exists \( v \in P \) such that the cone \( v * \left| K\right| \) is a subcomplex of \( \left| P\right| \) . Thus the homotopy groups of \( \left| P\right| \) are trivial, so, by the Whitehead Theorem, \( \left| P\right| \) is contractible. | Yes |
Corollary 9.3.15. \( \left| B\right| \) is contractible. | The function \( f : B \rightarrow \mathbb{N} \) extends affinely to a Morse function \( {}^{8} \) (also denoted by) \( f : \left| B\right| \rightarrow \mathbb{R} \) . Then \( \left| {B}_{n}\right| = {f}^{-1}(\left( {-\infty, n\rbrack }\right) \). | No |
Proposition 9.3.16. The \( {CW} \) complex \( F \smallsetminus \left| {B}_{n}\right| \) is finite. | Proof. Let \( \left\lbrack {h, S}\right\rbrack \in {B}_{n} \) . Its \( F \) -orbit contains \( \left\lbrack {1, S}\right\rbrack \) and there are only finitely many finite trees having at most \( n \) leaves. This shows that the 0 -skeleton of \( F \smallsetminus \left| {B}_{n}\right| \) is finite. The rest is clear. | No |
Proposition 9.3.18. For \( n \geq m\left( k\right) ,\left| {B}_{n}\right| \) is \( k \) -connected. Hence \( \left\{ \left| {B}_{n}\right| \right\} \) is essentially \( k \) -connected for all \( k \) . | Proof. By 9.3.17 and 8.3.4 we conclude that \( \left( {\left| {B}_{n}\right| ,\left| {B}_{n - 1}\right| }\right) \) is \( \left( {k + 1}\right) \) -connected if \( n \geq m\left( k\right) \) . When combined with the Whitehead Theorem and 9.3.15 this proves what is claimed. | Yes |
Proposition 9.3.20. If the cover \( \mathcal{U} \) is finite and if \( \mathop{\bigcap }\limits_{{i = 0}}^{k}{X}_{{\alpha }_{i}} \) is contractible whenever it is non-empty, then \( \left| {N\left( \mathcal{U}\right) }\right| \) and \( X \) are homotopy equivalent. | Proof. There is a vertex \( {v}_{{\alpha }_{0},\ldots ,{\alpha }_{k}} \) of the first derived \( {sd}\left| {N\left( \mathcal{U}\right) }\right| \) for each simplex \( \left\{ {{v}_{{\alpha }_{0}},\ldots ,{v}_{{\alpha }_{k}}}\right\} \) of \( N\left( \mathcal{U}\right) \) . Pick a point \( {x}_{{\alpha }_{0},\ldots ,{\... | Yes |
Proposition 9.3.21. For any integer \( k \geq 0 \) there is an integer \( m\left( k\right) \) such that \( \left| {L}_{n}\right| \) is \( k \) -connected when \( n \geq m\left( k\right) \) . | Proof. By induction on \( k \) we prove a sharper statement:\n\nClaim: given \( k \geq 0 \) there are integers \( m\left( k\right) \) and \( q\left( k\right) \) such that when \( n \geq \) \( m\left( k\right) \) the \( k \) -skeleton \( {\left| {L}_{n}\right| }^{k} \) is homotopically trivial by means of a homotopy \( ... | Yes |
Lemma 9.3.23. A set \( \left\{ {{C}_{{i}_{1}}\left( b\right) ,\ldots ,{C}_{{i}_{r}}\left( b\right) }\right\} \) of simple contractions of \( b \), written so that \( {i}_{1} < \ldots < {i}_{r} \), has a lower bound with respect to \( \leq \) iff the pairs \( \left\{ {{i}_{j},{i}_{j + 1}}\right\} \) are pairwise disjoin... | Proof. If there is a lower bound \( \left\lbrack {h, T}\right\rbrack \) then by equivariance we need only consider the case \( h = 1 \) . Then \( b \) is an expansion of \( \left\lbrack {1, T}\right\rbrack \) and so has the form \( \left\lbrack {1, S}\right\rbrack \) . Since \( \left\lbrack {1, T}\right\rbrack \leq {C}... | Yes |
Theorem 9.4.2. The group \( T \) is of type \( {F}_{\infty } \) . | Proof. Since we have a convention \( {\mathbb{R}}^{n} \subseteq {\mathbb{R}}^{n + 1} \), we can write \( {\Delta }^{n} \subseteq {\Delta }^{n + 1} \) and define \( {\Delta }^{\infty } \mathrel{\text{:=}} \mathop{\lim }\limits_{\overrightarrow{n}}{\Delta }^{n} \) to be the \ | No |
Theorem 9.4.3. The group \( T \) is simple. | Proof. Let \( g \neq 1 \in T \) . We show that \( T \) itself is the only normal subgroup of \( T \) containing \( g \) . Since \( g \neq 1 \) there is some \( a \in {S}^{1} \) such that \( g\left( a\right) \neq a \) . Let \( J \) be an open interval in \( {S}^{1} \) containing \( a \) such that \( g\left( J\right) \ca... | Yes |
Proposition 9.5.3. The group \( \operatorname{Out}\left( {F}_{n}\right) \) has a torsion free subgroup of finite index. | Proof. Every homotopy equivalence \( f : \left( {{\Gamma }_{n}, x}\right) \rightarrow \left( {{\Gamma }_{n}, x}\right) \) induces an automorphism of \( {F}_{n} \) on fundamental group and an automorphism of \( {\mathbb{Z}}^{n} \) on first homology (with \( \mathbb{Z} \) -coefficients). This correspondence defines a can... | No |
Proposition 9.5.4. The group \( \operatorname{Out}\left( {F}_{n}\right) \) contains a free abelian subgroup of rank \( {2n} - 3 \) . | Proof. Let \( {y}_{1},\ldots ,{y}_{n} \) denote free generators of \( {F}_{n} \) . The automorphisms in \( \operatorname{Aut}\left( {F}_{n}\right) {y}_{i} \mapsto {y}_{1}{y}_{i} \) and \( {y}_{i} \mapsto {y}_{i}{y}_{1} \) for \( 2 \leq i \leq n \) determine the required free abelian subgroup of \( \operatorname{Out}\le... | No |
Lemma 10.1.1. If \( f : X \rightarrow Y \) is a closed map, if \( A \subset Y \), and if \( U \) is an open subset of \( X \) such that \( {f}^{-1}\left( A\right) \subset U \), then there is an open set \( V \) in \( Y \) such that \( A \subset V \) and \( {f}^{-1}\left( V\right) \subset U \) . | Proof. The required \( V \) is \( Y - f\left( {X - U}\right) \) . | Yes |
Proposition 10.1.2. Let \( f : X \rightarrow Y \) be a perfect surjection, and let the space \( X \) be locally compact. Then \( Y \) is locally compact. | Proof. Let \( y \in Y \) . Since \( X \) is locally compact and \( {f}^{-1}\left( y\right) \) is compact, there is a compact set \( N \subset X \) such that \( {f}^{-1}\left( y\right) \subset \) int \( N \) . By 10.1.1, there is an open neighborhood \( V \) of \( y \) such that \( {f}^{-1}\left( y\right) \subset {f}^{-... | Yes |
Proposition 10.1.3. Let \( f : X \rightarrow Y \) be a proper map where \( X \) and \( Y \) are Hausdorff. If \( Y \) is either first countable or locally compact then \( f \) is perfect. | Proof. We need only prove that \( f \) is closed. First, let \( Y \) be locally compact. Let \( A \) be a closed non-empty subset of \( X \) . Let \( y \in {\operatorname{cl}}_{Y}f\left( A\right) \), and let \( N \) be a compact neighborhood of \( y \) in \( Y \) . Then \( {f}^{-1}\left( N\right) \cap A \) is compact, ... | Yes |
Corollary 10.1.5. Let \( f : X \rightarrow Y \) be a map, where \( X \) and \( Y \) are locally compact Hausdorff. \( f \) is proper iff \( f \) is perfect. | Proof. If \( f \) is proper then, by 10.1.3, \( f \) is perfect . Let \( f \) be perfect, and let \( A \) be a compact subset of \( Y \) . Each \( y \in A \) has a compact neighborhood \( {N}_{y} \) in \( Y \) . Since \( {f}^{-1}\left( y\right) \) is compact, there is a compact subset \( {M}_{y} \) of \( X \) such that... | Yes |
Proposition 10.1.7. This definition is independent of the choice of simultaneous attaching map. When \( A \) is locally compact Hausdorff, \( Y \) is locally compact Hausdorff iff \( f \) is proper. | Proof. For the first part, \( f \) is proper iff each compact subset of \( A \) meets only finitely many \( n \) -cells of \( Y \) . This latter condition is independent of \( f \) .\n\nFor the second part, \( Y \) is Hausdorff, by 1.2.2, and \( {S}^{n - 1}\left( \mathcal{A}\right) \) is clearly Hausdorff. Assume first... | Yes |
Proposition 10.1.8. A CW complex is locally compact iff it is locally finite. | Proof. Let the CW complex \( X \) be locally compact. Suppose \( X \) is not locally finite. Let \( {e}_{\alpha } \) be a cell which meets infinitely many cells. Since \( {e}_{\alpha } \) is compact, there is a compact subset \( N \) of \( X \) such that \( {e}_{\alpha } \subset \) int \( N \) . The weak topology has t... | Yes |
Theorem 10.1.14. [Proper Cellular Approximation Theorem] Let \( f : X \rightarrow Y \) be a proper map between \( {CW} \) complexes, with \( X \) locally finite and \( Y \) strongly locally finite, and let \( A \) be a subcomplex of \( X \) such that \( f \mid A \) is cellular. Then \( f \) is properly homotopic, rel \... | Proof. To see that the proof of 1.4.3 gives this, use 1.4.4 and the following useful criterion 10.1.15. | No |
Proposition 10.1.15. Let \( X \) and \( Y \) be locally compact Hausdorff spaces, let \( F : X \times I \rightarrow Y \) be a homotopy, and let \( {F}_{0} \) be proper. Let \( \mathcal{K} \) be a locally finite cover of \( Y \) by compact sets such that for each \( x \in X \) there exists \( {K}_{x} \in \mathcal{K} \) ... | Proof. Suppose \( {F}^{-1}\left( L\right) \) is not compact, where \( L \) is a compact subset of \( Y \) . Let \( p : X \times I \rightarrow X \) be projection. The closed set \( A \subset X \times I \) is compact iff \( p\left( A\right) \) is compact. Thus \( J \mathrel{\text{:=}} p{F}^{-1}\left( L\right) \) is not c... | Yes |
Lemma 10.1.18. If \( g \) is proper, so is \( \widetilde{g} \) . | Proof. Let \( C \) be a compact subset of \( E \) . Then \( {\widetilde{g}}^{-1}\left( C\right) \) is closed in \( {g}^{ * }E \), and \( {\widetilde{g}}^{-1}\left( C\right) \subset {g}^{-1}\left( {p\left( C\right) }\right) \times C \) . Since \( g \) is proper, \( {g}^{-1}\left( {p\left( C\right) }\right) \times C \) i... | Yes |
Lemma 10.1.19. If \( p \) is a covering projection, so is \( {p}^{\prime } \) . | Proof. Let \( y \in Y \) and let \( U \) be a neighborhood of \( p\left( y\right) \) in \( B \) which is evenly covered by \( p \) . Then \( {p}^{-1}\left( U\right) = \bigcup \left\{ {{U}_{\alpha } \mid \alpha \in \mathcal{A}}\right\} \) where \( \mathcal{A} \) is an indexing set and \( \left\{ {{U}_{\alpha } \mid \alp... | No |
Corollary 10.1.21. With hypotheses as in 10.1.17, let \( F : {X}_{1} \times I \rightarrow {X}_{2} \) be a proper homotopy such that \( {F}_{0} = f \) . Let \( \bar{F} : {\bar{X}}_{1}\left( {H}_{1}\right) \times I \rightarrow {\bar{X}}_{2}\left( {H}_{2}\right) \) be the lift of \( F \) such that \( {\bar{F}}_{0} = \bar{... | Proof. The first part follows from 10.1.17. The second part follows from 2.4.6 | Yes |
Proposition 10.1.22. Let \( f : \left( {{X}_{1},{v}_{1}}\right) \rightarrow \left( {{X}_{2},{v}_{2}}\right) \) be a map of pointed path connected \( {CW} \) complexes, and let \( f : {X}_{1} \rightarrow {X}_{2} \) be a homotopy equivalence [resp. proper homotopy equivalence.] Then \( f : \left( {{X}_{1},{v}_{1}}\right)... | Proof. See the remark following the statement of 4.1.5. The proof in the proper case is similar and is an exercise. | No |
Proposition 10.1.25. Every locally finite CW complex is metrizable. | Proof. It is enough to consider the case where \( X \) is path connected. Then \( X \) is countable by 11.4.3 below. By the Urysohn Metrization Theorem it is enough to show that \( X \) is regular and second countable. Since \( X \) is locally finite each \( {X}^{n} \) is obtained from \( {X}^{n - 1} \) by properly att... | Yes |
Theorem 10.2.3. (CW-proper Cellular Approximation Theorem) Let \( f : X \rightarrow Y \) be a CW-proper map between CW complexes of locally finite type, and let \( A \) be a subcomplex of \( X \) such that \( f \mid A \) is cellular. Then \( f \) is \( {CW} \) - proper homotopic, rel A, to a CW-proper cellular map. | Proof. Similar to the proof of 10.1.14. In other words, the theorem follows from the proof of 1.4.4, using 10.1.15 on each skeleton of \( X \times I \) . | No |
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