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Proposition 11.1.1. The composition\n\n\[ \n{C}_{n}^{\infty }\left( {X;R}\right) \overset{\partial }{ \rightarrow }{C}_{n - 1}^{\infty }\left( {X;R}\right) \overset{\partial }{ \rightarrow }{C}_{n - 2}^{\infty }\left( {X;R}\right) \n\]\n\nis zero, for all \( n \) . | Thus \( \left( {{C}_{ * }^{\infty }\left( {X;R}\right) ,\partial }\right) \) is a chain complex. Its homology modules, denoted \( {}^{2} \) by \( {H}_{ * }^{\infty }\left( {X;R}\right) \), are the cellular homology modules based on infinite chains. | No |
Proposition 11.1.6. Let \( \lbrack 0,\infty ) \) have the usual CW complex structure (vertices \( n,1 \) -cells \( \left\lbrack {n, n + 1}\right\rbrack \) for each \( n \in \mathbb{N} \) ). For any orientation, \( {H}_{n}^{\infty }\left( {\lbrack 0,\infty }\right) ;R) = \) \( 0 \) for all \( n \) . | Proof. For \( n = 0 \) this follows from 11.1.4. For \( n > 1 \) it is trivial. For \( n = 1 \), it is obvious that \( \ker \left( {\partial : {C}_{1}^{\infty }\left( {\lbrack 0,\infty }\right) ;R}\right) \rightarrow {C}_{0}^{\infty }\left( {\lbrack 0,\infty }\right) ;R)) = 0 \) . | Yes |
Theorem 11.1.9. Let \( f, g : X \rightarrow Y \) be CW-proper cellular maps between oriented CW complexes of locally finite type. Assume either (a) \( f \) and \( g \) are \( {CW} \) -proper homotopic or (b) \( f \) and \( g \) are properly homotopic where \( X \) is locally finite and \( Y \) is strongly locally finit... | Proof. Similar to that of 2.7.10: in place of 1.4.3, use 10.2.3 in Case (a) and 10.1.14 in Case (b). | No |
Proposition 11.2.10. Let \( \mathcal{X} \) and \( \mathcal{Y} \) be inverse sequences. A morphism \( f \) : \( \mathcal{X} \rightarrow \mathcal{Y} \) of inv- \( \mathcal{C} \) induces an isomorphism of pro- \( \mathcal{C} \) iff for suitable subsequences as above there are morphisms \( {g}_{{m}_{k}} \) of \( \mathcal{C... |  | No |
If \( \left\{ {G}_{n}\right\} \) is semistable then \( \underline{\lim }{}^{1}\left\{ {G}_{n}\right\} \) is trivial. | Assume \( \left\{ {G}_{n}\right\} \) is semistable and let \( \left( {x}_{n}\right) \in \mathop{\prod }\limits_{n}{G}_{n} \) . With \( \phi \) as in the definition of \ | No |
Proposition 11.3.4. In this situation, there is an exact sequence of pointed sets\n\n\[ \n\{ 1\} \rightarrow \mathop{\lim }\limits_{ \leftarrow }\left\{ {G}_{n}^{\prime }\right\} \overset{\mathop{\lim }\limits_{ \leftarrow }\left\{ {i}_{n}\right\} }{ \rightarrow }\mathop{\lim }\limits_{ \leftarrow }\left\{ {G}_{n}\righ... | Proof. This is a long but straightforward check that \( \delta \) is well defined and kernel \( = \) image in each position. See page 168 of [109] for details. | No |
Proposition 11.3.7. Let \( {C}_{n}^{\prime } = 0 \) for all \( n \) . Then for each \( n \) there is a short exact sequence\n\n\[ 0 \rightarrow {\mathop{\lim }\limits_{ \leftarrow }}_{i}^{1}{H}_{n + 1}\left( {C}^{\left( i\right) }\right) \overset{\bar{a}}{ \rightarrow }{H}_{n}\left( C\right) \overset{\bar{b}}{ \rightar... | Proof. Since \( {C}_{n}^{\prime } = 0 \) for all \( n \) we have a short exact sequence of chain complexes\n\n\[ 0 \rightarrow C \rightarrow \mathop{\prod }\limits_{{i = 1}}^{\infty }{C}^{\left( i\right) }\overset{s}{ \rightarrow }\mathop{\prod }\limits_{{i = 1}}^{\infty }{C}^{\left( i\right) } \rightarrow 0 \]\n\ngivi... | Yes |
Proposition 11.3.8. Let \( {C}_{n} = 0 \) for all \( n \) . Then for each \( n \) there is a short exact sequence\n\n\[ 0 \rightarrow {\mathop{\lim }\limits_{ \leftarrow }}_{i}^{1}{H}_{n}\left( {C}^{\left( i\right) }\right) \overset{a}{ \rightarrow }{H}_{n}\left( {C}^{\prime }\right) \overset{b}{ \rightarrow }\mathop{\... | Remark 11.3.9. Sometimes, one wants explicit descriptions of \( a \) and \( b \) in 11.3.8. The homomorphism \( a \) is induced by inclusion. To describe \( b \), we start with \( \left( {x}_{i}\right) \in \ker {\partial }^{\prime } = {Z}_{n}\left( {C}^{\prime }\right) \) representing an element of \( {H}_{n}\left( {C}... | No |
Proposition 11.3.10. Let \( \left\{ {M}_{n}\right\} \) be an inverse sequence in the category \( R \) - modules, where each \( {M}_{n} \) is finitely generated. If \( R \) is a field, then \( \left\{ {M}_{n}\right\} \) is semistable, hence \( \mathop{\lim }\limits_{ \leftarrow }{}^{1}\left\{ {M}_{n}\right\} = 0 \) . | Proof. For each \( m \), the \( R \) -dimensions of the vector spaces image \( \left( {{M}_{n} \rightarrow {M}_{m}}\right) \) are non-increasing as \( n \) increases. Since \( {M}_{m} \) has finite \( R \) -dimension, this function of \( n \) becomes constant. Hence \( \left\{ {M}_{n}\right\} \) is semistable. Apply 11... | Yes |
Proposition 11.3.11. Let \( R \) be a PID and let \( M \) be a finitely generated \( R \) - module. Then \( M \) is isomorphic to \( F \oplus \operatorname{tor}M \), where \( F \) is the direct sum of \( \rho \) copies of \( R \), tor \( M \) is the direct sum of \( \tau \) cyclic modules \( R/\left( {r}_{i}\right) \),... | Proof. The first part is well known. For the last part, there exists \( r \in R \) which annihilates tor \( M \) . So tor \( M \) is a module over the ring \( R/\left( r\right) \) . This latter ring satisfies the descending chain condition (see page 243 of [156]), hence so does tor \( M \) ; see p. 158 of [156]. | Yes |
Proposition 11.3.12. If \( \left\{ {M}_{n}\right\} \) is an inverse sequence of finitely generated torsion \( R \) -modules, where \( R \) is a PID, then \( \left\{ {M}_{n}\right\} \) is semistable. Hence \( \mathop{\lim }\limits_{n}{}^{1}\left\{ {M}_{n}\right\} = 0. \) | Proof. This is immediate from the last sentence of 11.3.11. | No |
Proposition 11.4.2. Let \( X \) be path connected. Then \( {H}_{n}^{e}\left( {X;R}\right) = 0 \) for all \( n < 0 \) . | Proof. For \( n < - 1 \), this is trivial. For \( n = - 1 \), it follows from 11.1.3 together with the above exact sequence when \( {X}^{1} \) is infinite. When \( {X}^{1} \) is finite then \( {X}^{0} \) is finite, so \( {Z}_{-1}^{e} = {C}_{0}^{\infty } = {C}_{0} = {B}_{-1}^{e} \) . | Yes |
Proposition 11.4.3. A path connected CW complex \( X \) having locally finite type is countable. | Proof. We begin with \( {X}^{1} \) . If \( {X}^{1} \) is infinite then, by 11.1.4, \( {X}^{1} \) is the union of countably many finite subcomplexes. Assume, inductively, that \( {X}^{n} \) is countable. Since \( {X}^{n + 1} \) is locally finite, each cell of \( {X}^{n} \) meets only finitely many \( \left( {n + 1}\righ... | No |
Proposition 11.4.4. If \( X \) is strongly locally finite and \( A \) is finite then the \( {CW} \) neighborhood \( N\left( A\right) \) is finite. | Proof. The collection \( \{ C\left( e\right) \mid e \) is a cell of \( X\} \) is locally finite, and \( A \) is compact. | No |
Proposition 11.4.5. With \( X \) as above, let \( \left\{ {K}_{i}\right\} \) be a finite type filtration of \( X \) . Then, for each \( n,\left\{ {\left( {X - {K}_{i}}\right) \cap {X}^{n}}\right\} \) is a basis for the neighborhoods of the end of \( {X}^{n} \) . | Proof. First, we show that \( {U}_{i} \mathrel{\text{:=}} \left( {X - {K}_{i}}\right) \cap {X}^{n} = {X}^{n} - {K}_{i}^{n} \) is a neighborhood of the end of \( {X}^{n} \) . We have \( {X}^{n} - {U}_{i} \subset {N}_{{X}^{n}}\left( {K}_{i}^{n}\right) \), by 1.5.5. So \( \operatorname{cl}\left( {{X}^{n} - {U}_{i}}\right)... | Yes |
Theorem 11.4.7. Let \( X \) be a countable oriented \( {CW} \) complex having locally finite type. Let \( \left\{ {K}_{i}\right\} \) be a finite type filtration of \( X \) . For each \( n \), there is a natural short exact sequence of \( R \) -modules\n\n\[ 0 \rightarrow {\mathop{\lim }\limits_{ \leftarrow }}_{i}^{1}\l... | Using the companion Theorem 11.3.8, we get a similar short exact sequence for \( {H}_{ * }^{e}\left( {X;R}\right) \) as follows. Consider the short exact sequences\n\n\[ 0 \rightarrow {C}_{n}\left( {X\overset{c}{ \sim }{K}_{i};R}\right) \rightarrow {C}_{n}\left( {X;R}\right) \rightarrow {C}_{n}\left( {X, X\overset{c}{ ... | Yes |
Theorem 11.4.8. With hypotheses as in 11.4.7, we have, for each \( n \), a natural short exact sequence of \( R \) -modules:\n\n\[ 0 \rightarrow {\mathop{\lim }\limits_{ \leftarrow }}_{i}^{1}\left\{ {{H}_{n + 1}\left( {X\overset{c}{ \leftarrow }{K}_{i};R}\right) }\right\} \overset{a}{ \rightarrow }{H}_{n}^{e}\left( {X;... | Remark 11.4.9. Theorems 11.4.7 and 11.4.8 are the fundamental tools for computing \( {H}_{n}^{\infty }\left( {X;R}\right) \) and \( {H}_{n}^{e}\left( {X;R}\right) \) . In Remark 11.3.9, we stated the algebraic meaning of the homomorphisms \( a \) and \( b \) which occur in 11.3.8. These translate into geometric interpr... | No |
Here is an example of a CW complex \( X \) which is not 1- movable at the end (see Fig. 11.3). \( X \) is the graph in \( {\mathbb{R}}^{2} \) having the points \( \{ \left( {i, j}\right) \mid i \in \mathbb{N}, j = 0 \) or 1 \( \} \) as vertices, and having the segments \( \left\lbrack {\left( {i, j}\right) ,\left( {i +... | The reduced Mayer-Vietoris sequence gives an exact sequence\n\n\[ 0 \rightarrow {H}_{1}\left( {X\overset{c}{ \hookrightarrow }{K}_{i + 1};R}\right) \oplus {H}_{1}\left( {{J}_{i};R}\right) \overset{{j}_{ * }}{ \rightarrow }{H}_{1}\left( {X\overset{c}{ \hookrightarrow }{K}_{i};R}\right) \rightarrow 0 \]\n\nThe restrictio... | Yes |
To get an example of the exact sequence of 11.4.8 in which the \( {\underline{\lim }}^{1} \) term does not vanish, apply the above construction with \( {X}_{i} = {S}^{1} \) for all \( i \), and \( {f}_{i} \) the map of degree \( 2,{e}^{2\pi it} \mapsto {e}^{4\pi it} \) . The resulting \( T \) is the dyadic solenoid inv... | \[ 0 \rightarrow {\mathop{\lim }\limits_{ \leftarrow }}^{1}\{ \mathbb{Z}\overset{\times 2}{ \leftarrow }\mathbb{Z}\overset{\times 2}{ \leftarrow }\ldots \} \rightarrow {H}_{0}^{e}\left( {T;\mathbb{Z}}\right) \rightarrow \mathbb{Z} \rightarrow \mathbb{Z} \rightarrow 0. \] Since \( \left\{ {{\mathbb{Z}}^{\times 2}\overse... | Yes |
Proposition 12.1.1. The composition\n\n\[ \n{C}_{n}^{\infty }\left( {X;R}\right) \overset{\delta }{ \rightarrow }{C}_{n + 1}^{\infty }\left( {X;R}\right) \overset{\delta }{ \rightarrow }{C}_{n + 2}^{\infty }\left( {X;R}\right) \n\]\n\nis zero for all \( n \) . | Proof. This follows from 2.3.3 and the corresponding statement for \( {\partial }^{ * } \) . | No |
Theorem 12.1.9. [Homotopy Invariance] If \( f, g : X \rightarrow Y \) are homotopic cellular maps, then \( {f}^{ * } = {g}^{ * } : {H}^{ * }\left( {Y;R}\right) \rightarrow {H}^{ * }\left( {X;R}\right) \) . In particular, cellular cohomology is a topological invariant. | Proof. Although this theorem is analogous to 2.7.10, we prove it using 2.7.14 instead. The reason is explained in Remark 12.1.11 below.\n\nBy 1.4.3 there is a cellular homotopy \( F : X \times I \rightarrow Y \) from \( f \) to \( g \) . Let \( D \) be as in 2.7.14. Consider the diagram:\n\n![3fadc665-adbe-41c9-a331-e3... | Yes |
Proposition 12.3.3. Let \( c \in {Z}_{n}^{\infty }\left( {X,\partial X;R}\right) \), let \( {e}_{\alpha }^{n} \) and \( {e}_{\beta }^{n} \) be \( n \) -cells in the same path component of \( X \), and let \( {r}_{\alpha },{r}_{\beta } \) be the coefficients of \( {e}_{\alpha }^{n} \) and \( {e}_{\beta }^{n} \) in c. Th... | Proof. First assume \( {e}_{\alpha }^{n} \cap {e}_{\beta }^{n} = {e}_{\gamma }^{n - 1} \) . Then \( {e}_{\gamma }^{n - 1} \) is not a face of any other \( n \) -cell. Since \( \partial c = 0 \) we get \( {r}_{\alpha }\left\lbrack {{e}_{\alpha }^{n} : {e}_{\gamma }^{n - 1}}\right\rbrack + {r}_{\beta }\left\lbrack {{e}_{... | Yes |
Proposition 12.3.4. Let \( X \) be path connected. If \( c \) and \( {c}^{\prime } \) are fundamental cycles over \( R \) then, for some unit \( u \in R, c = u{c}^{\prime } \) . | Proof. Let the coefficient of \( {e}_{\alpha }^{n} \) in \( c \) [resp. \( {c}^{\prime } \) ] be the unit \( {u}_{\alpha } \) [resp. \( {u}_{\alpha }^{\prime } \) ]. Then \( c \) and \( {u}_{\alpha }{\left( {u}_{\alpha }^{\prime }\right) }^{-1}{c}^{\prime } \) have the same \( {e}_{\alpha }^{n} \) -coefficient, so the ... | Yes |
Proposition 12.3.7. Let \( X \) be path connected. If \( {H}_{n}^{\infty }\left( {X,\partial X;R}\right) \neq 0 \) then \( X \) has a fundamental cycle over \( R \) ; i.e., \( X \) is \( R \) -orientable. | Proof. Let \( z = \mathop{\sum }\limits_{\alpha }{r}_{\alpha }{e}_{\alpha }^{n} \) be a non-zero relative cycle in \( X \) . Since \( \partial z = 0 \) ,\n\n\( {r}_{\alpha } = \pm {r}_{\beta } \) when \( {e}_{\alpha }^{n} \) and \( {e}_{\beta }^{n} \) share an \( \left( {n - 1}\right) \) -face. Since \( X \) is path co... | Yes |
Proposition 12.3.10. Let \( X \) and \( Y \) be \( {CWn} \) -pseudomanifolds, and let \( f \) : \( \left( {X,\partial X}\right) \rightarrow \left( {Y,\partial Y}\right) \) be a cellular proper homotopy equivalence (of pairs). If \( c \in {Z}_{n}^{\infty }\left( {X,\partial X;R}\right) \) is a fundamental cycle for \( X... | Proof. We may assume that \( X \) is path connected: otherwise work with each path component separately. Clearly, \( {f}_{\# }\left( c\right) \in {Z}_{n}^{\infty }\left( {Y,\partial Y;R}\right) \) . Hence, by 12.3.3, \( {f}_{\# }\left( c\right) = r.d \) where \( r \in R, d \in {C}_{n}^{\infty }\left( {Y;R}\right) \) an... | Yes |
Proposition 12.3.11. If \( {H}_{n - 1}^{\infty }\left( {X,\partial X;{\mathbb{Z}}_{2}}\right) = 0 \) then \( X \) is orientable. | Proof (of 12.3.11). Let \( c = \mathop{\sum }\limits_{\alpha }{e}_{\alpha }^{n} \) where \( \alpha \) ranges over all the \( n \) -cells of\n\n\( X \) . Then \( \partial c = {2d} + e \), where \( d \) has support outside \( \partial X, e \) has support in \( \partial X \), and every coefficient in \( d \) and in \( e \... | Yes |
Proposition 12.3.13. If \( X \) is non-orientable then \( {H}_{n - 1}^{\infty }\left( {X,\partial X;\mathbb{Z}}\right) \) contains an element of order 2. | Proof. We refer to the proof of 12.3.11. We have \( {2d} \in {B}_{n - 1}^{\infty }\left( {X,\partial X;\mathbb{Z}}\right) \) and \( d \in {Z}_{n - 1}^{\infty }\left( {X,\partial X;\mathbb{Z}}\right) \) . If \( d \) were itself a boundary, we would have \( b \) and \( f \) with \( \partial b = d + f \) . So \( \partial ... | No |
Proposition 12.3.15. Let the n-pseudomanifold \( \widetilde{X} \) be orientable. Then \( X \) is orientable iff \( H = G \) . | Proof. Let \( c = \mathop{\sum }\limits_{{g,\alpha }}{r}_{\alpha, g}g{\widetilde{e}}_{\alpha }^{n} \) be a fundamental cycle (over \( \mathbb{Z} \) ) in \( \widetilde{X} \) . Then \( {r}_{\alpha, g} = \) \( \pm 1 \) . First, let \( H = G \) . Then \( {r}_{\alpha, g} = {r}_{\alpha ,1} \) for all \( g \in G \) . Let \( d... | Yes |
Theorem 12.4.2. (Universal Coefficient Theorem in homology) With \( \left( {\left\{ {C}_{n}\right\} ,\partial }\right) \) and \( B \) as above, and each \( {C}_{n} \) free, there is a natural short exact sequence of \( R \) -modules\n\n\[ 0 \rightarrow B{ \otimes }_{R}{H}_{n}\left( C\right) \overset{\beta }{ \rightarro... | This sequence splits, naturally in \( B \) but unnaturally in \( C \) . | No |
Theorem 12.4.5. (Universal Coefficient Theorem in cohomology) With \( \left( {\left\{ {C}_{n}\right\} ,\partial }\right) \) and \( B \) as above, and each \( {C}_{n} \) free, there is a natural short exact sequence of \( R \) -modules\n\n\[ 0 \rightarrow {\operatorname{Ext}}_{R}\left( {{H}_{n - 1}\left( C\right), B}\ri... | Remark 12.4.6. By examining the proofs of 12.4.2 and 12.4.5, one sees that the monomorphism \( \beta \) in 12.4.2 is given by \( \beta \left( {b\otimes \{ z\} }\right) = \{ z \otimes b\} \), where \( \{ \cdot \} \) denotes homology class; and the epimorphism \( \alpha \) in 12.2.5 is given by \( \alpha \left( {\{ f\} }... | No |
Theorem 12.5.1. Let \( X \) be a CW complex, let \( R \) be a PID, and let \( M \) be an \( R \) -module. There is a short exact sequence of \( R \) -modules\n\n\[ 0 \rightarrow {\operatorname{Ext}}_{R}\left( {{H}_{n - 1}\left( {X;R}\right), M}\right) \rightarrow {H}^{n}\left( {X;M}\right) \rightarrow {\operatorname{Ho... | This is natural with respect to homotopy classes of maps. It splits, but unnaturally, with respect to \( M \) . | No |
Proposition 12.5.5. Let \( R \) be a PID. Let \( \left\{ {M}_{n}\right\} \) be a pro-finitely generated inverse sequence of \( R \) -modules. Then \( \left\{ {M}_{n}\right\} \) is semistable iff \( \left\{ {\bar{M}}_{n}\right\} \) is semistable. | Proof. We may assume that each \( {M}_{n} \) is finitely generated. By 11.3.11, \( \left\{ {{M}_{n},{f}_{n}^{m}}\right\} \) can be replaced by \( \left\{ {{\bar{M}}_{n} \oplus \text{tor}{M}_{n},{\bar{f}}_{n}^{m} \oplus \left( {{f}_{n}^{m} \mid \text{tor}{M}_{n}}\right) }\right\} \) . By 11.3.12, \( \left\{ {\text{tor}{... | No |
Proposition 12.5.7. Let \( R \) be a PID. A pro-finitely generated inverse sequence of \( R \) -modules \( \left\{ {M}_{n}\right\} \) is pro-torsion free iff the direct limit of the corresponding direct sequence \( \left\{ {{\operatorname{Ext}}_{R}\left( {{M}_{n}, R}\right) }\right\} \) is trivial. | Proof. This proof uses homological algebra not needed elsewhere in this book. We may assume that each \( {M}_{n} \) is finitely generated. By 11.3.11 and 12.4.3, \( \mathop{\lim }\limits_{ \rightarrow }\left\{ {{\operatorname{Ext}}_{R}\left( {{M}_{n}, R}\right) }\right\} = 0 \) iff \( \mathop{\lim }\limits_{ \rightarro... | Yes |
Proposition 12.5.8. Let \( R \) be a PID. Let the inverse sequence of \( R \) -modules \( \left\{ {M}_{n}\right\} \) be pro-finitely generated.\n\n(a) \( \left\{ {M}_{n}\right\} \) is pro-torsion iff \( \mathop{\lim }\limits_{ \rightarrow }\left\{ {{\operatorname{Hom}}_{R}\left( {{M}_{n}, R}\right) }\right\} = 0 \) .\n... | Proof. The proofs are similar to that of 12.5.6. | No |
Proposition 12.5.9. If \( {H}_{n}\left( {X;R}\right) \) is finitely generated and \( R \) is a PID, then \( {H}_{n}\left( {X - {K}_{i};R}\right) \) is finitely generated for all \( i \) . Hence \( \left\{ {{H}_{n}\left( {X - {K}_{i};R}\right) }\right\} \) is pro-finitely generated. | Proof. Replacing \( X \) by \( {X}^{n + 1} \), we may assume \( X \) is strongly locally finite. Then, by \( {11.4.4}, N\left( {K}_{i}\right) \) is finite. By \( {1.5.5}, X = N\left( {K}_{i}\right) \cup \left( {X - {K}_{i}}\right) \) . And \( N\left( {K}_{i}\right) \cap \left( {X - {K}_{i}}\right) \) is finite. The May... | Yes |
Theorem 12.5.10. Let \( X \) be a countable \( {CW} \) complex having locally finite type, let \( R \) be a PID, let \( {H}_{n}\left( {X;R}\right) \) and \( {H}_{n - 1}\left( {X;R}\right) \) be finitely generated, and let \( \left\{ {K}_{i}\right\} \) be a finite type filtration of \( X \) . (i) \( {H}_{e}^{n}\left( {X... | Proof. By 12.5.9, \( \left\{ {{H}_{k}\left( {X - {K}_{i};R}\right) }\right\} \) is pro-finitely generated for \( k = n - 1 \) and \( n \) . (i) By the previous propositions and Sect. 12.2, \( \left\{ {{H}_{n}\left( {X - {K}_{i};R}\right) }\right\} \) is semistable iff \( \mathop{\lim }\limits_{ \rightarrow }\left\{ {{\... | Yes |
Corollary 12.5.11. Let \( X \) be a path connected \( {CW} \) complex having locally finite type, and let \( R \) be a PID. Then \( {H}_{e}^{0}\left( {X;R}\right) \) is countably generated and free. If \( {H}_{1}\left( {X;R}\right) \) is finitely generated, then \( {H}_{e}^{1}\left( {X;R}\right) \) is torsion free. | Proof. By 11.4.3, \( X \) is countable. Applying 12.5.10 with \( n = 0 \) we find that \( {H}_{e}^{0}\left( {X;R}\right) \) is torsion free. To show that \( {H}_{e}^{0}\left( {X;R}\right) \) is free, we apply 12.5.6 so we must show that \( \left\{ {{H}_{0}\left( {X \circeq {K}_{i};R}\right) }\right\} \) is semistable. ... | Yes |
Lemma 13.2.2. Let \( {e}_{\alpha }^{k} \) and \( {e}_{\beta }^{k - 1} \) be cells of \( X \), and let \( g,\bar{g} \in G \) . Then \( \left\lbrack {{Hg}{\widetilde{e}}_{\alpha }^{k}}\right. \) : \( \left. {H\bar{g}{\widetilde{e}}_{\beta }^{k - 1}}\right\rbrack = \mathop{\sum }\limits_{{h \in H}}\left\lbrack {g{\widetil... | Proof.\n\n\[ \partial \left( {g{\widetilde{e}}_{\alpha }^{k}}\right) = \mathop{\sum }\limits_{{h \in H}}\left\lbrack {g{\widetilde{e}}_{\alpha }^{k} : h\bar{g}{\widetilde{e}}_{\beta }^{k - 1}}\right\rbrack h\bar{g}{\widetilde{e}}_{\beta }^{k - 1} + \text{ other terms. } \]\n\n\[ {\left( {p}_{H}\right) }_{\# }\partial \... | Yes |
Corollary 13.2.3. Let \( X \) be a \( K\left( {G,1}\right) \) -complex. Then, for all \( k \) ,\n\n\[ \n{H}_{k}\left( {\bar{X}\left( H\right) ;R}\right) \cong {H}_{k}\left( {G, R\left( {H \smallsetminus G}\right) }\right) .\n\]\n\nIf \( X \) has finite type then\n\n\[ \n{H}_{k}^{\infty }\left( {\bar{X}\left( H\right) ;... | Proof. \( {C}_{ * }\left( {\widetilde{X};R}\right) \) gives a free \( {RG} \) -resolution of \( R \) . Apply 13.2.1. | No |
Proposition 13.2.4. Let \( X \) be \( \left( {n - 1}\right) \) -aspherical. Then for all \( k \leq n - 1 \) , \( {H}_{k}\left( {\bar{X}\left( H\right) ;R}\right) \cong {H}_{k}\left( {G, R\left( {H \smallsetminus G}\right) }\right) \) . If \( {X}^{n} \) is finite, the other conclusions of 13.2.3 hold for \( k \leq n - 1... | Proof. By 7.1.5 there is a \( K\left( {G,1}\right) \) -complex \( Y \) with \( {Y}^{n} = {X}^{n} \) . The first claim follows from 13.2.1. If \( {X}^{n} \) is finite, \( \partial : {C}_{k}^{\infty }\left( {\bar{Y}\left( H\right) ;R}\right) \rightarrow {C}_{k - 1}^{\infty }\left( {\bar{Y}\left( H\right) ;R}\right) \) is... | Yes |
Proposition 13.2.5. Let \( H \leq G \) . Then, for all \( k \) ,\n\n\[ \n{H}_{k}\left( {G, R\left( {H \smallsetminus G}\right) }\right) \cong {H}_{k}\left( {H, R}\right) \n\] | Proof. Let \( X \) be a \( K\left( {G,1}\right) \) -complex. By 13.2.1, we have \( {H}_{k}\left( {\bar{X}\left( H\right) ;R}\right) \cong \) \( {H}_{k}\left( {G, R\left( {H \smallsetminus G}\right) }\right) \) and (applying 13.2.1 with \( H \) replacing \( G \) ) \( {H}_{k}\left( {\bar{X}\left( H\right) ;R}\right) \con... | Yes |
Corollary 13.2.7. For \( k > 1,{H}_{k}\left( {G, R{G}^{ * }}\right) \cong {H}_{k}\left( {G, R{G}^{e}}\right) \) . If \( G \) is infinite and finitely generated then \( {H}_{1}\left( {G, R{G}^{e}}\right) \cong {H}_{1}\left( {G, R{G}^{ \frown }}\right) \oplus R \) . | Proof. For \( k > 1 \), the claim follows from 13.2.6 and the preceding remarks. The claim for \( k = 1 \) also follows from 13.2.6, because, by 13.2.3 (with \( n = 0 \) ) and 11.1.3, \( {H}_{0}\left( {G, R{G}^{ \sim }}\right) = 0 \) when \( G \) is infinite and finitely generated. | No |
Proposition 13.2.11. Let \( H \leq G.{H}^{0}\left( {G, R\left( {H \smallsetminus G}\right) }\right) = 0 \) [resp. \( \cong R \) ] iff \( H \) has infinite [resp. finite] index in \( G \) . In particular, \( {H}^{0}\left( {G,{RG}}\right) = 0 \) [resp. \( \cong R \) ] iff \( G \) is infinite [resp. finite]. | Proof. Let \( \left( {X, v}\right) \) be a \( K\left( {G,1}\right) \) -complex having one vertex. By 8.1.2 we see that \( {H}^{0}\left( {G, R\left( {H \smallsetminus G}\right) }\right) \) is the module of cocycles in \( {\operatorname{Hom}}_{G}\left( {{C}_{0}\left( {\widetilde{X};R}\right), R\left( {H \smallsetminus G}... | Yes |
Corollary 13.2.13. For \( k > 1,{H}^{k}\left( {G,{RG}}\right) \cong {H}^{k - 1}\left( {G, R{G}^{e}}\right) \) . If \( G \) is infinite, \( {H}^{0}\left( {G, R{G}^{e}}\right) \cong {H}^{1}\left( {G,{RG}}\right) \oplus R \) . | Proof. The first part is clear. For the last part, the short exact sequence splits. | No |
Proposition 13.2.14. Let \( H \leq G \) where \( H \) has type \( {F}_{n} \) . If \( H \) has finite index in \( G \) then for \( k \leq n,{H}^{k}\left( {G,{RG}}\right) \cong {H}^{k}\left( {H,{RH}}\right) \) . | Proof. By 7.2.4, \( G \) has type \( {F}_{n} \) iff \( H \) has type \( {F}_{n} \) . Let \( X \) be as in 13.2.9 with \( {X}^{n} \) finite and \( {\pi }_{1}\left( {X, v}\right) \cong G \) . By 13.2.9, \( {H}^{k}\left( {G,{\left( RG\right) }^{e}}\right) \cong {H}_{e}^{k}\left( {\widetilde{X};R}\right) \cong \) \( {H}^{k... | Yes |
Proposition 13.2.17. If \( R \) is a PID there is a free \( R \) -module \( F \) such that \( {H}_{f}^{n}\left( {{\widetilde{X}}^{n};R}\right) \cong {H}^{n}\left( {G,{RG}}\right) \oplus F \) . Thus \( {H}^{n}\left( {G,{RG}}\right) \) is free over \( R \) iff \( {H}_{f}^{n}\left( {{\widetilde{X}}^{n};R}\right) \) is fre... | Proof. We will find \( F \) such that \( {H}^{n - 1}\left( {G, R{G}^{e}}\right) \oplus F \cong {H}_{f}^{n}\left( {{\widetilde{X}}^{n};R}\right) \) ; by 13.2.13 this is enough. For any PID \( R,{H}^{n - 1}\left( {G, R{G}^{e}}\right) \cong {H}_{e}^{n - 1}\left( {{\widetilde{X}}^{n};R}\right) \) by 13.2.9. From Sect. 12.2... | No |
Proposition 13.2.19. Let \( X \) be a \( K\left( {G,1}\right) \) -complex with finite \( n \) -skeleton. As \( R \) -modules, \( {H}_{e}^{n}\left( {{\widetilde{X}}^{n};R}\right) \) is torsion free iff \( {H}^{n + 1}\left( {G,{RG}}\right) \) is torsion free. | Proof. By 13.2.13, \( {H}^{n + 1}\left( {G,{RG}}\right) \) is torsion free iff \( {H}^{n}\left( {G, R{G}^{e}}\right) \) is torsion free. By 13.2.8 and 13.2.9, there is a short exact sequence of cochain complexes\n\n\[ 0 \rightarrow A \rightarrow {C}_{n}^{\infty }\left( {\widetilde{X};R}\right) /{C}_{n}\left( {\widetild... | Yes |
Proposition 13.3.1. If \( G \) is finite, \( {H}^{n}\left( {G,{RG}}\right) = 0 \) for all \( n > 0 \) . | Proof. By 7.2.5, there is a \( K\left( {G,1}\right) \) -complex \( X \) of finite type. Since \( G \) is finite, 3.2.13 implies that \( \widetilde{X} \) also has finite type, so \( {H}_{f}^{n}\left( {\widetilde{X};R}\right) = {H}^{n}\left( {\widetilde{X};R}\right) = 0 \) for all \( n > 0 \) . Apply 13.2.9. | No |
Theorem 13.3.2. Let \( n \geq 0 \) and let \( G \) be a group of type \( {F}_{n} \) ; when \( n = 0 \) assume \( G \) is countable. Let \( X \) be an \( \left( {n - 1}\right) \) -aspherical \( {CW} \) complex with finite \( n \) -skeleton whose fundamental group is isomorphic to \( G \) . Let \( \left\{ {K}_{i}\right\}... | Proof. Combine 12.5.11 (applied to the appropriate skeleton), 13.2.9 and 13.2.13. For the case \( k = n + 1 \) in (ii), use 13.2.19. There are a few details to be worked out in (iii) and (iv) when \( k = 1 \) (details concerning the behavior of \( {\widetilde{H}}_{0} \) ); we leave these as an exercise.\n\nThe hypothes... | No |
Proposition 13.4.7. This correspondence induces a bijection \( \mathcal{E}\left( Y\right) \rightarrow \mathop{\lim }\limits_{ \leftarrow }\left\{ {{\pi }_{0}\left( {Y - {L}_{i}}\right) }\right\} \) | By 13.4.4, \( \left\{ {{\pi }_{0}\left( {Y - {L}_{i}}\right) }\right\} \) is an inverse sequence of finite sets. If each is regarded as a discrete space and the inverse limit is interpreted in the category Spaces then, via the bijection of 13.4.7, \( \mathcal{E}\left( Y\right) \) becomes a compact totally disconnected ... | No |
Proposition 13.4.8. Let \( Y \) have \( m \) ends where \( 0 \leq m \leq \infty \) . For all \( i, Y - {L}_{i} \) has finitely many path components. When \( m < \infty \), there exists \( {i}_{0} \) such that whenever a finite subcomplex \( K \) contains \( {L}_{{i}_{0}}, Y\overset{c}{ \sim }K \) has exactly \( m \) un... | Proof. The first sentence follows from 13.4.4 (or from the discussion preceding 11.1.4). The rest follows from 13.4.7; to see this, observe that a compact path component of \( Y \triangleq {L}_{i} \) lies in \( {L}_{j} \) for some \( j \) . | No |
Proposition 13.4.10. Let \( Y \) have \( m \) ends, where \( m < \infty \) . Let \( K \) and \( L \) be finite subcomplexes of \( Y \) such that each of \( Y\overset{c}{ \backsim }K \) and \( Y\overset{c}{ \backsim }L \) has \( m \) unbounded path components \( {C}_{1},\cdots ,{C}_{m} \) and \( {D}_{1},\cdots ,{D}_{m} ... | Proof. By 13.4.9, we may assume \( Y - K \) and \( Y - L \) have no bounded path components. If \( m = 1, Y - \left( {{C}_{1} \cap {D}_{1}}\right) = K \cup L \) which is compact. The subcomplex \( {C}_{1} - c\left( {{C}_{1} \cap {D}_{1}}\right) \) is therefore a closed subset of a compact set. Now let \( m \geq 2 \) . ... | Yes |
Proposition 13.4.11. Let \( R \) be a PID. \( Y \) has \( m \) ends iff \( {H}_{e}^{0}\left( {Y;R}\right) \) is a free \( R \) - module of rank \( m \) . \( Y \) has infinitely many ends iff \( {H}_{e}^{0}\left( {Y;R}\right) \) is free of countably infinite rank. | Proof. We saw in 12.5.11 that \( {H}_{e}^{0}\left( {Y;R}\right) \) is free. First, let \( m \) be finite. By 12.5.10 (iv), \( {H}_{e}^{0}\left( {Y;R}\right) \) has rank \( m \) iff \( \left\{ {{H}_{0}\left( {Y - {L}_{i};R}\right) }\right\} \) is stable with free inverse limit of rank \( m \), iff \( \left\{ {{\pi }_{0}... | Yes |
Proposition 13.4.13. If \( Z \) is a compact totally disconnected metrizable space, there is an inverse sequence \( \left\{ {Z}_{n}\right\} \) of finite discrete spaces whose inverse limit is homeomorphic to \( Z \) . | Proof. Pick a metric for \( Z \) . By 13.4.12, for each \( n \geq 1 \) there is a finite cover \( {\mathcal{Z}}_{n} \) of \( Z \) by pairwise disjoint closed-and-open sets of diameter \( \leq \frac{1}{n} \), and these \( {\mathcal{Z}}_{n} \) ’s can be chosen inductively so that each member of \( {\mathcal{Z}}_{n + 1} \... | Yes |
Proposition 13.5.2. If \( H \) has finite index in \( G \), then \( H \) and \( G \) have the same number of ends. | Proof. \( X \) and its finite covering space \( \bar{X}\left( H\right) \) have the same universal cover. | No |
Proposition 13.5.3. The R-module \( {H}^{1}\left( {G,{RG}}\right) \) is countably generated and free. | Proof. Apply the first two parts of 13.3.2 and the following lemma. | No |
Lemma 13.5.4. Let \( Y \) be a path connected countable \( {CW} \) complex of locally finite type, and let \( \left\{ {L}_{i}\right\} \) be a finite type filtration of \( Y \) . Then \( \left\{ {{H}_{0}\left( {Y - {L}_{i};R}\right) }\right\} \) is semistable. | Proof. By 12.5.9, each \( {H}_{0}\left( {Y \triangleq {L}_{i};R}\right) \) is finitely generated and free. The image of each bond \( {H}_{0}\left( {Y \circeq {L}_{j};R}\right) \rightarrow {H}_{0}\left( {Y \circeq {L}_{i};R}\right) \) is a direct summand generated by the finite number of path components of \( Y - {L}_{i... | Yes |
The number of ends of a finite group is 0 . The number of ends of an infinite finitely generated group \( G \) is well defined and equals \( 1 + {\operatorname{rank}}_{R}\left( {{H}^{1}\left( {G,{RG}}\right) }\right) \) . | For a finite group, the result follows from 3.2.13 and 13.4.2. Assume \( G \) is infinite. Let \( \rho \) be a non-negative integer. We use both the result and the notation of 13.3.2: \( {H}^{1}\left( {G,{RG}}\right) \) has rank \( \rho \) iff \( \left\{ {{\widetilde{H}}_{0}\left( {{\widetilde{X}}^{1} - {K}_{i};R}\righ... | Yes |
The groups \( {\mathbb{Z}}^{n}\left( {n > 1}\right) \) have one end. The group \( \mathbb{Z} \) has two ends. A free group of rank \( n > 1 \) has infinitely many ends. | To see this, consider the universal covers of \( {T}^{n},{S}^{1} \) and the \( n \) -fold wedge of circles; the last is an infinite tree every vertex of which has valence \( {2n} \) . | No |
Theorem 13.5.7. The number of ends of a finitely generated group is 0,1,2 or \( \infty \) . Hence for \( G \) infinite and finitely generated, the \( R \) -module \( {H}^{1}\left( {G,{RG}}\right) \) is isomorphic to 0 or \( R \) or \( {\bigoplus }_{1}^{\infty }R \) . | Proof. Suppose \( G \) has \( m \) ends where \( 3 \leq m < \infty \) . Let \( \Gamma \) be the Cayley graph of \( G \) with respect to a finite set of generators. By 13.5.5, \( G \) is infinite, hence \( \Gamma \) is infinite. By 13.4.9, there is a finite path connected subgraph \( K \) of \( \Gamma \) such that \( {}... | Yes |
Proposition 13.5.11. If \( N \) is a normal subgroup of \( G \) then the number of ends of the pair \( \left( {G, N}\right) \) is equal to the number of ends of the group \( G/N \) . | Proof. Let \( f : \left( {X, x}\right) \rightarrow \left( {Y, y}\right) \) be a cellular map between 2-dimensional complexes having the following properties: each of \( X \) and \( Y \) has just one vertex, the base point; \( f \mid : {X}^{1} \rightarrow {Y}^{1} \) is an isomorphism of finite graphs (i.e., a homeomorph... | Yes |
Corollary 13.5.12. The number of ends of a finitely generated group \( G \) is the number of ends of any path connected free \( {}^{20} \) G-CW complex whose quotient by the action of \( G \) is a finite \( {CW} \) complex. | Proof. Let \( \bar{Z} \) be the free \( G \) -CW complex and \( Z \) the finite quotient. Write \( H = {\pi }_{1}\left( {Z, z}\right) \) . By covering space theory (see Sect. 3.4), there is a short exact sequence of groups \( N \rightarrowtail H \twoheadrightarrow G \) such that \( \bar{Z} = \bar{Z}\left( N\right) \) .... | Yes |
For each \( n \geq 3 \) we describe a pair of groups \( \left( {G, H}\right) \) having \( n \) ends. Let \( X \) be a closed path connected orientable surface of genus \( g \) and let \( Y \) be a compact subsurface (i.e., \( {\operatorname{cl}}_{X}\left( {X - Y}\right) \) is also a surface) whose boundary consists of ... | Consider the covering projection \( {q}_{H} : \bar{X}\left( H\right) \rightarrow X \), with base point \( \bar{x} \) over \( x \) . By 3.4.10 the path component \( {Y}_{\bar{x}} \) of \( {q}_{H}^{-1}\left( Y\right) \) is a copy of \( Y \) which \ | No |
Proposition 13.5.15. If \( N \) is a finite normal subgroup of \( G \), then the number of ends of \( \left( {G, N}\right) \) is equal to the number of ends of \( G \) . Hence \( G \) and \( G/N \) have the same number of ends. | Proof. If \( G \) is finite, both numbers are 0 . Assume \( G \) is infinite. Then \( {p}_{N} \) : \( {\widetilde{X}}^{1} \rightarrow \bar{X}{\left( N\right) }^{1} \) is a finite-to-one covering projection, hence a proper map. The induced function \( {}^{21}{p}_{N\# } : \mathcal{E}\left( {\widetilde{X}}^{1}\right) \rig... | Yes |
Proposition 13.5.17. The R-module \( {H}^{1}\left( {G, R\left( {H \smallsetminus G}\right) }\right) \) is free. | Proof. By 13.2.12 we have an exact sequence\n\n\[ \n{H}^{1}(G, R({H\smallsetminus G}{)}^{\hat{} })\overset{i}{\leftarrow }{H}^{1}(G, R({H\smallsetminus G}))\overset{j}{\leftarrow }{H}^{0}(G, R({H\smallsetminus G}{)}^{e})\overset{k}{\leftarrow }{H}^{0}(G, R({H\smallsetminus G}{)}^{\hat{} })\overset{r}{\leftarrow }{H}^{0... | Yes |
Proposition 13.5.19. Let \( H \), generated by \( h \), be an infinite cyclic subgroup of the one-ended finitely generated group \( G \) . Then the number of ends of \( \left( {G, H}\right) \) equals \( 1 + {\operatorname{rank}}_{\mathbb{Z}}\left( {{H}^{1}\left( {G,\mathbb{Z}\left( {H \smallsetminus G}\right) }\right) ... | Proof. First, assume \( G \) is finitely presented. Then \( X \) can be taken to have finite 2-skeleton and, by 13.2.9 and 13.5.18, we are to show that\n\n\[ i : {H}_{f}^{1}\left( {\bar{X}\left( H\right) ;\mathbb{Z}}\right) \rightarrow {H}^{1}\left( {\bar{X}\left( H\right) ;\mathbb{Z}}\right) \]\n\nis zero. By 13.2.10 ... | No |
Proposition 13.5.20. Let \( H \) be a subgroup of infinite index in the finitely generated group \( G \) . Assume that whenever there is a short exact sequence \( N \rightarrowtail H \rightarrow \mathbb{Z} \) then the pair \( \left( {G, N}\right) \) has one end. Then the number of ends of \( \left( {G, H}\right) \) equ... | Proof. The proof is the same as that of 13.5.19. In that proof, we used only the facts that \( H \) had infinite index in \( G \) and that the covering space \( \bar{X} \) (ker \( \phi \) ) had one end. The details are an exercise. | No |
Theorem 13.6.2. The graph \( T \) is a tree. The function \( \left( {E, \leq }\right) \mapsto T \) from Tree Posets to Trees is inverse to \( \Phi \) . | Proof. For distinct edges \( \sigma \) and \( \tau \) of \( T \) we have \( \sigma \vdash \tau \) iff there is a two-edge edge path \( \left( {\sigma ,\tau }\right) \) . By 13.6.1 it follows that, for general \( \sigma \) and \( \tau ,\sigma \leq \tau \) iff there is a reduced edge path in \( T \) starting with \( \sig... | No |
Corollary 13.6.4. There are almost invariant sets \( U \) such that both \( U \) and \( {U}^{c} \) are infinite. | Proof. Since \( \Gamma \) has more than one end, we can pick a finite path connected subgraph \( \Delta \) of \( \Gamma \) and let \( {Z}_{1},\ldots ,{Z}_{m} \) be the unbounded path components of \( \Gamma \overset{c}{ \sim }\Delta \) with \( m \geq 2 \) . Let \( U = {Z}_{1}^{0} \), the set of vertices of \( {Z}_{1} \... | Yes |
Proposition 13.6.5. Let \( {U}_{0} \) and \( {U}_{1} \) be almost invariant. Then for almost all \( g \in G \) at least one of the following is true: \( g{U}_{1} \subseteq {U}_{0}, g{U}_{1}^{c} \subseteq {U}_{0}, g{U}_{1} \subseteq {U}_{0}^{c} \) , \( g{U}_{1}^{c} \subseteq {U}_{0}^{c} \) | Proof. If \( {U}_{0} \) or \( {U}_{0}^{c} \) is empty the Proposition is trivial, so we assume they are non-empty. For \( i = 0 \) or 1 let \( {\Delta }_{i} \) be a finite path connected full subgraph of \( \Gamma \) containing the support of \( \delta {U}_{i} \) . Then for almost all \( g \in {U}_{0} \) we have (i) \(... | Yes |
Proposition 13.6.6. Let \( H = \{ g \in G \mid {gU} \triangleq U\} \) . If \( G \) has infinitely many ends then \( H \) is finite. | Proof. Recall that both \( U \) and \( {U}^{c} \) are infinite. We will show that if \( H \) is infinite then \( G \) has an infinite cyclic subgroup of finite index, implying \( G \) has two ends by 13.5.9.\n\nWe may assume that \( H \cap U \) is infinite and that \( 1 \in U \) . By (the proof of) 13.6.5 we have \( {g... | Yes |
Proposition 13.6.7. Let \( G \) have infinitely many ends. If \( V, W \) and \( X \) are infinite almost invariant sets with infinite complements, then \( \{ g \in G \mid V \subseteq \) \( {gW}\overset{a}{ \subseteq }X\} \) is finite. | Proof. We may assume \( V \subseteq X \) and, enlarging \( V \) if necessary, that \( V ⊄ X \) . Let \( V \subseteq {gW} \subseteq X \) . Then either \( {gW} ⊄ X \) or \( V ⊄ {gW} \) . It is enough to show that \( \{ g \in G \mid {gW}\overset{a}{ \subseteq }X \) and \( {gW} ⊄ X\} \) and \( \{ g \in G \mid V\overset{a}{... | Yes |
Proposition 13.6.8. If \( {U}_{1} \supseteq {U}_{2} \supseteq \cdots \) are narrow sets, and if \( V \mathrel{\text{:=}} \bigcap {U}_{n} \) is non-empty, then the sequence stabilizes, i.e., for some \( N,{U}_{n} = {U}_{N} \) for all \( n \geq N \) . | Proof. If an edge \( e \) lies in the support of \( {\delta V} \) then one vertex of \( e \) lies in every \( {U}_{n} \), while there is an integer \( N \) such that the other vertex of \( e \) does not lie in \( {U}_{n} \) when \( n \geq N \) . Thus \( e \) is in the support of \( \delta {U}_{n} \) when \( n \geq N \)... | Yes |
Proposition 13.6.9. Fix \( {g}_{0} \in G \) . Let \( U \) be a minimal narrow set containing \( {g}_{0} \) . For any narrow set \( V \) at least one of the following holds: \( U \subseteq V, U \subseteq {V}^{c} \) , \( {U}^{c} \subseteq V,{U}^{c} \subseteq {V}^{c} \) | Proof. Write \( {W}_{1} = U \cap V,{W}_{2} = U \cap {V}^{c},{W}_{3} = {U}^{c} \cap V \), and \( {W}_{4} = {U}^{c} \cap {V}^{c} \) . We are to prove that one of the \( {W}_{i} \) is finite. The support of \( \delta {W}_{i} \) lies in the union of the supports of \( {\delta U} \) and \( {\delta V} \) . Since the sets \( ... | Yes |
Theorem 13.6.10. Let \( G \) be a finitely generated group with infinitely many ends. There is an almost invariant set \( U \subseteq G \) with respect to which the \( G \) -poset \( \left( {E, \leq }\right) \) is a \( G \) -(tree poset). | Let \( T \) be the \( G \) -tree determined by \( \left( {E, \leq }\right) \) in 13.6.10 via the function \( \Phi \) . The stabilizer of \( \left\lbrack U\right\rbrack \in E \) is \( \left\{ {g \in G \mid {gU} \triangleq U}\right\} \) ; this is finite by 13.6.6. So edge stabilizers in \( T \) are finite.\n\nFrom the de... | Yes |
Proposition 13.7.2. \( Z \) is the mapping torus of \( j : Y \rightarrow Y \) . | Proof. Apply 6.1.3. The space \( {q}_{1}^{-1}\left( v\right) \) is a copy of \( Y \), and \( Z \) is obtained by attaching \( Y \times {B}^{1} \) via the attaching maps id on \( Y \times \{ - 1\} \) and \( j \) on \( Y \times \{ 1\} \) . | Yes |
Proposition 13.7.4. There is a homeomorphism \( h : \mathbb{R} \times \left( {J \smallsetminus Y}\right) \rightarrow Z \) which is fiber preserving; i.e., \( {q}_{2} \circ h \) is projection on the \( J \smallsetminus Y \) factor. | Proof (of 13.7.4). There is certainly a fiber preserving homeomorphism \( {h}_{0} \) : \( \mathbb{R} \times {\left( J \smallsetminus Y\right) }^{0} \rightarrow {q}_{2}^{-1}\left( {\left( J \smallsetminus Y\right) }^{0}\right) \) . Moreover, we can choose \( {h}_{0} \) to be order preserving on each fiber \( \left( {\ma... | Yes |
Lemma 13.7.5. Let \( H : {S}^{n} \times \mathbb{R} \rightarrow {S}^{n} \times \mathbb{R} \) be a homeomorphism of the form \( H\left( {x, t}\right) = \left( {x,{H}_{x}\left( t\right) }\right) \) where each homeomorphism \( {H}_{x} : \mathbb{R} \rightarrow \mathbb{R} \) is order preserving. Then \( H \) extends to a hom... | Proof. Note that when \( n > 0 \) the hypothesis on each \( {H}_{x} \) holds iff it holds on one \( {H}_{x} \) ; but we also need this lemma for \( n = 0 \) . The required \( \widehat{H} \) is given by the formula \( \widehat{H}\left( {x, t}\right) = \left( {x,\left( {1 - \left| x\right| }\right) t + \left| x\right| {H... | Yes |
Proposition 13.7.6. If \( W \) is a countable \( {CW} \) complex of locally finite type, then for all \( n,{H}_{f}^{n}\left( {W;R}\right) \cong {H}_{f}^{n + 1}\left( {W \times \mathbb{R};R}\right) \) . | Proof. This follows from 12.6.1. Here is a short direct proof. Let \( \left\{ {K}_{i}\right\} \) be a finite type filtration of \( W \) . Then \( \left\{ {{K}_{i} \times \left\lbrack {-i, i}\right\rbrack }\right\} \) is a finite type filtration of \( W \times \mathbb{R} \) . Fix \( i \), and let \( {X}_{i} = \left( {W ... | Yes |
Proposition 13.7.8. Assume \( {Y}^{2} \) does not have two ends. If \( {j}^{ * } : {H}_{e}^{1}\left( {Y;R}\right) \rightarrow \) \( {H}_{e}^{1}\left( {Y;R}\right) \) agrees with the identity on a non-trivial \( R \) -submodule, \( A \), then either \( A \cong R \) or \( J \smallsetminus Y \) has more than one end. | Proof. Recall that \( Y \) is assumed to be simply connected. The space \( J \smallsetminus Y \) is not compact (i.e., does not have 0 ends) because \( Y \) does not have two ends; see 13.5.9. Assuming \( J \smallsetminus Y \) has one end, we must show \( A \cong R \) . We have an exact sequence\n\n\[ \n{H}^{1}\left( {... | Yes |
Proposition 13.7.9. Assume \( {Y}^{2} \) has one end. If \( {j}^{ * } : {H}_{e}^{1}\left( {Y;R}\right) \rightarrow {H}_{e}^{1}\left( {Y;R}\right) \) agrees with the identity on a non-trivial \( R \) -submodule \( A \), then \( A \) is a free \( R \) -module. | Proof. Consider the first exact sequence given in the proof of 13.7.8. The monomorphism \( \alpha \) splits by 12.2.3, and we have seen that \( {H}^{1}\left( {J \smallsetminus Y;R}\right) \) is isomorphic to \( R \) . Moreover, \( {H}_{e}^{0}\left( {J \smallsetminus Y;R}\right) \) is free, by 12.5.11. Thus \( {H}_{f}^{... | Yes |
Theorem 13.7.12. (Farrell’s Theorem) Let the finitely presented group \( G \) act freely on \( \widetilde{X} \) with compact quotient. Assume \( G \) has an element of infinite order. \( {}^{26} \) Then the abelian group \( {H}_{e}^{1}\left( {\widetilde{X};\mathbb{Z}}\right) \) is trivial or is isomorphic to \( \mathbb... | Proof. By 12.4.8 and 12.2.2, there is a monomorphism of \( {\mathbb{Z}}_{2} \) -modules \( \beta \) : \( {\mathbb{Z}}_{2}{ \otimes }_{\mathbb{Z}}{H}_{e}^{1}\left( {\widetilde{X};\mathbb{Z}}\right) \rightarrow {H}_{e}^{1}\left( {\widetilde{X};{\mathbb{Z}}_{2}}\right) \) . In fact, by naturality of the universal coeffici... | Yes |
Proposition 13.8.3. Let \( f : X \rightarrow {\Delta }^{n} \) be an aspherical model over \( {\Delta }^{n} \) and let \( \left( {K,\pi }\right) \) be a simplicial complex over \( \mathbf{n} \) . Then \( X\bigtriangleup \left| K\right| \) is non-empty and each of its path components is aspherical. Moreover, for any non-... | Proof (of 13.8.3). First we handle the special case in which \( K \) is a simplex, i.e., \( K = \bar{\sigma } \) for some \( \sigma \in K \) . Then for any non-empty subcomplex \( L \) of \( \bar{\sigma } \) , \( X\bigtriangleup \left| L\right| = {f}^{-1}\left( {\left| \pi \right| \left( \left| L\right| \right) }\right... | Yes |
Lemma 13.9.2. \( {F}_{\sigma \left( T\right) } = {N}_{K}\left( {K\left( T\right) }\right) \) . | Proof. Let \( \tau \) be a simplex of \( {F}_{\sigma \left( T\right) } \) . Then \( \tau \) is a simplex of \( {F}_{\{ s\} } \) for some \( s \in T \) , so \( \tau \) is a face of a simplex \( \mu \) whose initial vertex is \( \langle \{ s\} \rangle \) . Thus \( \tau \) is a face of a simplex which shares a vertex with... | Yes |
Proposition 13.9.3. When \( \langle T\rangle \) is a non-trivial finite standard subgroup of \( G,\left| {F}_{\sigma \left( T\right) }\right| \) is a \( {PL}\left( {d - 1}\right) \) -ball. | Proof (Sketch). This requires knowledge of piecewise linear topology, in particular, of regular neighborhoods in PL manifolds; we have set things up so that references are easily given. By 13.9.2, \( {F}_{\sigma \left( T\right) } \) is the simplicial neighborhood of \( K\left( T\right) \) in the closed combinatorial \(... | Yes |
Proposition 13.9.4. Each \( {A}_{n} \) is a finite connected closed combinatorial manifold of dimension \( d - 1 \) . \( {A}_{n} \) is orientable iff \( \left| L\right| \) is orientable. | Proof. The proof is by induction on \( n \), starting with \( {A}_{0} = K = \operatorname{sd}L \) . Assume the Proposition for \( {A}_{n} \) . Let \( {B}_{n + 1} = {g}_{n + 1}{F}_{\sigma \left( {B\left( {g}_{n + 1}\right) }\right) } \) . The full subcomplex of \( D \) generated by \( {A}_{n} \) and \( {g}_{n + 1}F \) c... | Yes |
Corollary 13.9.6. The group \( G \) has one end, \( {H}^{2}\left( {G,\mathbb{Z}G}\right) \) is free abelian, and every torsion free subgroup of finite index in \( G \) has geometric dimension \( d \) . | Proof. By 13.5.2 and 13.5.5, \( G \) has one end. We have \( {H}^{2}\left( {G,\mathbb{Z}G}\right) \) free abelian because \( {H}^{1}\left( {\left| L\right| ;\mathbb{Z}}\right) \) is (finitely generated and) free abelian. Let \( H \) be a torsion free subgroup of finite index (see 9.1.10). By 9.1.11, \( H \) has geometr... | Yes |
Proposition 13.9.7. (a) If \( {M}_{1} \) and \( {M}_{2} \) are orientable then \( {M}_{1}\# {M}_{2} \) is orientable; \( {f}^{ * } : {H}^{n}\left( {M}_{1}\right) \rightarrow {H}^{n}\left( {{M}_{1}\# {M}_{2}}\right) \) is an isomorphism, and for \( i \leq n - 1 \) the following diagram commutes | Proof. We make some comments, leaving the rest to the reader.\n\n(i) \( {H}^{n}\left( {M}_{2}\right) \) is infinite cyclic if \( {M}_{2} \) is orientable, and has order 2 if \( {M}_{2} \) is nonorientable. Indeed, the map \( {f}^{ * } : {H}^{n}\left( {{B}_{2}^{n}, S}\right) \rightarrow {H}^{n}\left( {M}_{2}\right) \) c... | No |
Theorem 13.11.1. \( {H}^{ * }\left( {F,{RF}}\right) = 0 \) . | Proof. The proof requires a return to the notation of Sect. 9.3. We sketch it, leaving the reader to fill in the details. \( {}^{31} \)\n\nFor \( b \in B \) let \( \lambda \left( b\right) \) be the smallest integer such that an expansion of \( b \) of length \( \lambda \left( b\right) \) has the form \( \left\lbrack {{... | No |
Proposition 14.1.2. Let \( \left( {X,\mathcal{K}}\right) \) and \( \left( {Y,\mathcal{L}}\right) \) be topologically well filtered Hausdorff spaces where each \( {K}_{i} \) and each \( {L}_{j} \) is compact. A map \( f : X \rightarrow Y \) is filtered iff it is proper. | Proof. Use Lemma 14.1.1 and the fact that compact subsets of Hausdorff spaces are closed. Note that two of the four conditions in the definition of filtered map hold trivially. | No |
Proposition 14.1.4. If \( X \) is a strongly locally finite \( {CW} \) complex and \( \left( {X,\mathcal{K}}\right) \) is topologically well filtered then \( E\left( \mathcal{K}\right) \) and \( S\left( \mathcal{K}\right) \) are filtrations of \( X \) . | Proof. The only non-trivial part in the envelope case involves showing that \( \mathop{\bigcap }\limits_{i}E\left( {K}_{i}\right) = \varnothing \) . Suppose otherwise. Since this intersection is a subcomplex of \( X \) it contains a vertex \( v \) . It is not hard to see that \( E\left( {K}_{i}\right) = \bigcup \left\{... | Yes |
Let \( X \) be a rigid \( G \) -CW complex such that \( G \smallsetminus X \) is finite, let \( M = \mathbb{R} \) where a left action of \( G \) on \( \mathbb{R} \) by translations is given, let \( \mathcal{L} = \left\{ {L}_{i}\right\} \) where \( i \in \mathbb{Z} \) and \( {L}_{i} \mathrel{\text{:=}} \left( {-\infty, ... | A particular case is this. Assume \( G \) has type \( {F}_{n}, X \) is the \( n \) -skeleton of the universal cover of a \( K\left( {G,1}\right) \) -complex which has finite \( n \) -skeleton, \( \chi : G \rightarrow \mathbb{R} \) is a character (i.e., a homomorphism into the additive group of real numbers), and the \(... | No |
Lemma 14.1.9. Let \( \\left( {X,\\mathcal{K}}\\right) \) be a well filtered \( {CW} \) complex and let \( F : X \\times I \\rightarrow \) \( X \) be a (cellular) CW-Lipschitz homotopy with \( {F}_{0} = {\\mathrm{{id}}}_{X} \). Then \( F \) is CW-filtered. | Proof. We need only check that \( F \\mid {X}^{n} \\times I \) is filtered for given \( n \). Let \( m \) be such that for every cell \( e \) of \( {X}^{n} \) the carrier \( C\\left( {F\\left( {e \\times I}\\right) }\\right) \) contains at most \( m \) cells. Since \( F\\left( {e \\times I}\\right) \) is path connected... | Yes |
Lemma 14.1.10. Let \( f : \left( {X,\mathcal{K}}\right) \rightarrow \left( {Y,\mathcal{L}}\right) \) be a (cellular) CW-filtered map and let \( g : Y \rightarrow X \) be a cellular map such that \( g \circ f \) and \( f \circ g \) are \( {CW} \) -filtered maps which are CW-filtered homotopic to the appropriate identity... | Proof. Given \( {L}_{i} \) we seek \( {K}_{j} \) such that \( g\left( {L}_{i}\right) \subset {K}_{j} \) . There exists \( k \) such that \( {fg}\left( {L}_{i}\right) \subset {L}_{k} \) and there exists \( j \) such that \( f\left( {X - {K}_{j}}\right) \subset Y - {L}_{k} \) . So \( g\left( {L}_{i}\right) \subset {K}_{j... | Yes |
Proposition 14.1.12. If \( {h}_{1} : {X}_{1} \rightarrow \mathbb{R} \) and \( {h}_{2} : {X}_{2} \rightarrow \mathbb{R} \) both satisfy these conditions (for \( X \) and \( h \) above), there is a cellular map \( f : {X}_{1} \rightarrow {X}_{2} \) which is a CW-filtered \( \left( {n - 1}\right) \) -equivalence \( \left(... | Proof. By 7.1.8 and the \( n \) -equivalence version of Example 14.1.7, there is a CW-Lipschitz (cellular) \( \left( {n - 1}\right) \) -equivalence \( f : {X}_{2} \rightarrow {X}_{1} \), and \( f \) is CW-filtered as a map \( \left( {{X}_{2},{f}^{-1}{h}_{1}^{-1}\mathcal{L}}\right) \rightarrow \left( {{X}_{1},{h}_{1}^{-... | Yes |
Proposition 14.2.3. The homology and cohomology theories \( {H}_{ * }^{\mathcal{K}}\left( {X;R}\right) \), \( {H}_{ * }^{\mathcal{K}, e}\left( {X;R}\right) ,{H}_{\mathcal{K}}^{ * }\left( {X;R}\right) \) and \( {H}_{\mathcal{K}, e}^{ * }\left( {X;R}\right) \) are filtered homotopy invariants. | The more complete version of 14.2.3 in which the categories and functors are described explicitly is left to the reader. | No |
Example 14.2.5. We continue the discussion of Example 14.1.5. In Sect. 13.2 we met the right \( {RG} \) -module \( {\left( RG\right) }^{ \sim } \) . Elements of \( {\left( RG\right) }^{ \sim } \) are written as \( \mathop{\sum }\limits_{{g \in G}}{r}_{g}g \) with each \( {r}_{g} \in R \) . Those where all but finitely ... | One checks that the multiplication in \( {RG} \) does extend to \( {\left( RG\right) }_{\chi } \), making the latter a ring called the Novikov ring defined by \( \chi \) . Regarding this ring as a right \( G \) -module (via right multiplication by elements of \( G \) ) and using the notation of 14.1.5 , it is clear tha... | Yes |
Proposition 14.3.2. \( \\left( {Y,\\mathcal{L}}\\right) \) is regular iff the inverse sequence of sets \( \\left\\{ {{\\pi }_{0}\\left( {Y \\triangleq {L}_{i}}\\right) }\\right\\} \) is semistable. | Proof. For each \( i \), we write \( {\\pi }_{0}\\left( {Y - {L}_{i}}\\right) = {B}_{i} \\cup {U}_{i} \) where \( {B}_{i} \) is the set of \( \\mathcal{L} \) - bounded path components and \( {U}_{i} \) is the set of \( \\mathcal{L} \) -unbounded path components. If \( \\left( {Y,\\mathcal{L}}\\right) \) is regular then... | Yes |
Proposition 14.3.7. Let \( R \) be a PID and let \( \mathcal{L} \) be regular. For \( m < \infty ,\left( {Y,\mathcal{L}}\right) \) has \( m \) filtered ends iff \( {H}_{\mathcal{L}, e}^{0}\left( {Y;R}\right) \) is a free \( R \) -module of rank \( m;\left( {Y,\mathcal{L}}\right) \) has \( \infty \) filtered ends iff \(... | Proof. First, assume \( m < \infty \) . By 14.3.4, the following are equivalent:\n\n(i). \( \left( {Y,\mathcal{L}}\right) \) has \( m \) filtered ends;\n\n(ii). \( \left\{ {{\pi }_{0}\left( {Y - {L}_{i}}\right) }\right\} \) is stable with inverse limit having \( m \) elements;\n\n(iii). \( \left\{ {{H}_{0}\left( {Y - {... | Yes |
Proposition 14.4.1. \( {\psi }_{ * }^{H} \) and \( {\psi }_{ * }^{H, e} \) induce isomorphisms\n\n\[ \n{H}_{\mathcal{L}}^{ * }\left( {\widetilde{X};R}\right) \rightarrow {H}^{ * }\left( {G,{R}^{H}{\left( G\right) }^{ \frown }}\right) \]\n\nand\n\n\[ \n{H}_{\mathcal{L}, e}^{ * }\left( {\widetilde{X};R}\right) \rightarro... | By a proof similar to that of 13.2.11 one shows: | No |
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