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Lemma 7.19. A subset \( A \) of \( S \) is \( \mu \) -measurable precisely when either of the following conditions is satisfied:\n\n(i) \( A \cap U \) is \( \mu \) -measurable for each open set \( U \) of finite measure;\n\n(ii) \( A \cap K \) is \( \mu \) -measurable for each compact set \( K \subseteq S \) . | Naturally, \( A \) ’s \( \mu \) -measurability entails the measurability of \( A \cap U \) and \( A \cap K \) for every open \( U \) and compact \( K \) since all Borel sets are \( \mu \) -measurable.\n\nNow suppose \( A \cap U \) is \( \mu \) -measurable for each open set \( U \) of finite \( \mu \) -measure. Assume \... | Yes |
Lemma 7.21. For any \( f, g,\phi ,\psi \in {\mathcal{K}}^{ + }\left( G\right) \), we have\n\n(i) \( 0 < \left( {f : \phi }\right) < \infty \) ;\n\n(ii) if \( h \in G \), then \( \left( {f : \phi }\right) = \left( {{}_{h}f : \phi }\right) \) ;\n\n(iii) \( \left( {f : \phi }\right) \) is subadditive and positively homoge... | Proof. All of the statements follow easily using the definition of \( \left( {f : \phi }\right) \) . We comment on (v): to show \( \left( {f : \phi }\right) \leq \left( {f : \psi }\right) \left( {\psi : \phi }\right) \), fix \( \epsilon > 0 \), and let \( {c}_{1},\ldots ,{c}_{n} \geq 0 \) and \( {g}_{1},\ldots ,{g}_{n}... | Yes |
Lemma 7.22. For any \( f, g,\phi \in {\mathcal{K}}^{ + }\left( G\right) \)\n\n(i) \( {\mu }_{\phi }\left( f\right) > 0 \) ;\n\n(ii) \( {\mu }_{\phi }\left( {{}_{g}f}\right) = {\mu }_{\phi }\left( f\right) \) for all \( g \in G \) ;\n\n(iii) \( {\mu }_{\phi } \) is subadditive and positively homogeneous; i.e., \( {\mu }... | Proof. We comment on (v): \( \omega, f \) and \( \phi \) are all nonzero, so quantities involved in the inequalities from (v) of Lemma 7.21 are positive. Therefore\n\n\[ \left( {f : \phi }\right) \leq \left( {f : \omega }\right) \left( {\omega : \phi }\right) \]\n\nand\n\n\[ \left( {\omega : \phi }\right) \leq \left( {... | No |
Theorem 7.24 (Bochner and Dieudonné, specially tailored to our purposes). Let \( G \) be a locally compact topological group, let \( K \) be a nonempty compact subset of \( G \), and suppose \( {V}_{1},\ldots ,{V}_{n} \) are open subsets of \( G \) each with compact closure so that \( K \subseteq {V}_{1} \cup \cdots \c... | Proof. In Chapter 3 the \ | No |
Lemma 7.25. Let \( f \in {\mathcal{K}}^{ + }\left( G\right) \), let \( K \) be a compact set in \( G \), and let \( \epsilon > 0 \) be given. Then there exist \( {k}_{1},{k}_{2},\ldots ,{k}_{n} \in K,{h}_{1},\ldots ,{h}_{n} \in {\mathcal{K}}^{ + }\left( G\right) \) so that if \( g \in G, k \in K \), then\n\n\[ \left| {... | Proof. Let \( V \) be a neighborhood of the identity chosen so that if \( {g}_{0} \in {Vg} \) , then\n\n\[ \left| {f\left( g\right) - f\left( {g}_{0}\right) }\right| \leq \epsilon \]\n\nremembering that \( f \in {\mathcal{K}}^{ + }\left( G\right) \) must be (right) uniformly continuous on \( G \) . Now cover \( K \) by... | Yes |
Theorem 7.36. If the unitary representation \( g \rightarrow {U}_{g} \) arises from an elementary continuous positive definite function \( \phi \), then this representation is irreducible. If \( \phi \) is a continuous positive definite function and \( \phi \) gives rise to an irreducible unitary representation, then \... | Briefly if \( P \) is a projection on \( {L}^{2}\left( \phi \right) \) that commutes with each \( {U}_{g}\left( {g \in G}\right) \) (i.e., if \( P \) ’s range is invariant under all the \( {U}_{g} \) ’s), then\n\n\[ \chi \left( g\right) \equiv \left\langle {{U}_{g}P{x}_{0},{x}_{0}}\right\rangle \]\n\nis positive defini... | Yes |
Theorem 7.42 (Key Theorem). Every positive linear functional \( L \) arises from a continuous positive definite function \( \phi \) via the formula\n\n\[ L\left( x\right) = {L}_{\phi }\left( x\right) = \int \phi \left( g\right) x\left( g\right) {d\mu }\left( g\right) . \] | It follows from this that any integrally positive definite function is equal \( \mu \) -almost everywhere to a continuous positive definite function. | No |
Theorem 7.44. The set \( \mathcal{P} \) of continuous positive definite functions \( \phi \) satisfying \( \phi \left( e\right) \leq 1 \) is the smallest weakly closed convex set containing all the normalized elementary continuous positive definite functions and the zero function \( O\left( g\right) \mathrel{\text{:=}}... | Because \( G \) is locally compact, we have a well-tested manner of constructing nontrivial positive definite functions: \( {L}^{2}\left( G\right) \) is a Hilbert space with the usual inner product \( \left( {x, y}\right) \) of \( x, y \in {L}^{2}\left( G\right) \)\n\n\[ \left( {x, y}\right) = \int x\left( g\right) \ov... | Yes |
Theorem 8.1 (R. A. Struble). Let \( G \) be a locally compact group with left Haar measure \( \lambda \) . Let \( \left( {V}_{n}\right) \) be a decreasing sequence of open sets that form a neighborhood basis of the identity \( e \) in \( G \) where \( {\bar{V}}_{n} \) compact for each \( n \) . Then \[ \rho \left( {x, ... | Proof. It’s clear that \( \rho \left( {x, y}\right) \) is well defined and that \( \rho \left( {x, y}\right) = \rho \left( {y, x}\right) \) . Moreover \( \rho \left( {x, y}\right) \geq 0 \) and \( \rho \left( {x, y}\right) < \infty \) regardless of \( x, y \in G \) since each \( {V}_{n} \) is a Borel set with compact c... | Yes |
Lemma 8.4. For any Borel set \( B \subseteq G \) , \[ \lambda \left( B\right) \leq {\nu }^{h}\left( B\right) . \] | Proof. Suppose \( {\chi }_{B} \leq \mathop{\sum }\limits_{j}{c}_{j}{\chi }_{{B}_{j}} \) . Then \( {\chi }_{B} \leq \mathop{\sum }\limits_{j}{c}_{j}{\chi }_{{\bar{B}}_{j}} \) and so \[ \lambda \left( B\right) = {\int }_{B}{d\lambda } \leq \int \sum {c}_{j}{\chi }_{{\bar{B}}_{j}}{d\lambda } \] \[ = \sum {c}_{j}\lambda \l... | Yes |
Lemma 8.6. Let \( E\left( t\right) \) be defined by\n\n\[ E\left( t\right) = \inf \left\{ {\mathop{\sum }\limits_{{j = 1}}^{n}{c}_{j} : n \in \mathbb{N},{\chi }_{A} \leq \mathop{\sum }\limits_{{j = 1}}^{n}{c}_{j}{\chi }_{{B}_{j}},{c}_{j} \geq 0,\operatorname{diam}\left( {B}_{j}\right) \leq t}\right\} .\n\]\n\nIf\n\n\[ ... | Proof. There is a \( c > 0 \) and a sequence \( \left( {t}_{k}\right) ,{t}_{k} > 0 \) with \( {t}_{k} \searrow 0 \) so that\n\n\[ h\left( {t}_{k}\right) E\left( {t}_{k}\right) < c \]\n\nfor all \( k \) . In other words,\n\n\[ E\left( {t}_{k}\right) < \frac{c}{h\left( {t}_{k}\right) } \]\n\nfor all \( k \) . For each \(... | Yes |
The fractional Hausdorff measure is nontrivial; in fact, \[ 0 < {\nu }^{h}\left( A\right) < \infty . \] | Proof. If \( \epsilon > 0 \), then we can choose \( t > 0 \) so that \( t < \min \left\{ {{t}_{0},\epsilon }\right\} \) and \( h \) is continuous at \( t \) . We can do this since \( h \) is monotone and so is continuous at all but countably many points of \( \left( {0,\min \left\{ {{t}_{0},\epsilon }\right\} }\right) ... | Yes |
Lemma 8.10. Let \( G \) be a locally compact, second countable (hence metrizable, separable) group. Then there exists a family \( \left\{ {{U}_{r} : r > 0}\right\} \) such that\n\n(i) for each \( r \), each \( {U}_{r} \) is open and \( {\bar{U}}_{r} \) is compact;\n\n(ii) \( {U}_{r} = {U}_{r}^{-1} \) ;\n\n(iii) \( {U}_... | Proof of Lemma 8.10. Let \( \rho \) be the left invariant metric resulting from Theorem 8.1. We can assume that each of the open balls\n\n\[ {B}_{r} = \{ x \in G : \rho \left( {x, e}\right), r\} \]\n\nhas compact closure for \( 0 < r \leq 2 \) ; after all, there is an \( {r}_{0} \) so that for \( r < {r}_{0},{\bar{B}}_... | Yes |
Theorem 8.11 (Braconnier [12]). If \( G \) is a locally compact topological group and \( G \) admits a bi-invariant metric \( d \) that determines its topology, then \( G \) is unimodular. | Proof. Let \( \lambda \) be left Haar measure on \( G \) and assume that \( G \) is not unimodular. Let \( U \) be an open set containing \( e \) such that \( \lambda \left( U\right) < \infty \) . Let \( B \) be an open ball centered at \( e \) (of radius \( \rho \) ) so that \( \bar{B} \) is compact, and \( \bar{B} \s... | Yes |
The general linear group \( \mathcal{{GL}}\left( {2;\mathbb{R}}\right) \) is unimodular (see §6.3). For each \( m \), let\n\n\[ \left. {{X}_{m} = \left( \begin{array}{ll} \frac{1}{m} & \frac{1}{m} \\ 0 & m \end{array}\right) }\right| \;\text{ and }\left. {{Y}_{m} = \left( \begin{array}{ll} m & \frac{1}{m} \\ 0 & \frac{... | An easy modification of this example shows the same for \( \mathcal{G}\mathcal{L}\left( {n;\mathbb{R}}\right) \), when \( n \geq 2 \) as well. | Yes |
Theorem 8.13. If \( G \) is a locally compact metrizable topological group and \( \rho \) is a left invariant metric that generates \( G \) ’s topology, then \( \left( {G,\rho }\right) \) is a complete metric. | Indeed if \( U \) is an open set with compact closure and if \( e \in U \), then there is an open ball \( B \) centered at \( e \) with \( \bar{B} \) both compact and contained in \( U \) . Suppose \( R \) is the radius of \( B \), and let \( \left( {g}_{n}\right) \) be a \( \rho \) -Cauchy sequence in \( \left( {G,\rh... | Yes |
Theorem 8.15. A compact metrizable topological group \( G \) admits a bi-invariant metric that generates its topology. | Proof. Let \( \rho \) be a left invariant metric on \( G \) that generates \( G \) ’s topology. For \( x, y \in G \) define\n\n\[ d\left( {x, y}\right) = \sup \{ \rho \left( {{xz},{yz}}\right) : z \in G\} . \]\n\nThen \( d \) is finite for all \( x, y \in G \) and is easily seen to be bi-invariant.\n\nSuppose \( \epsil... | Yes |
Theorem 8.16 (V. Klee [67]). Let \( G \) be a topological group with a bi-invariant metric \( \rho \) which generates \( G \) ’s topology. Suppose \( \left( {G,\rho }\right) \) admits a complete metric \( d \) that generates \( G \) ’s topology. Then \( G \) is actually complete under \( \rho \) . | Proof. Let \( \left( {{G}^{ * },{\rho }^{ * }}\right) \) be the completion of \( \left( {G,\rho }\right) \) . Then \( \left( {{G}^{ * },{\rho }^{ * }}\right) \) is a topological group into which \( \left( {G,\rho }\right) \) is naturally isomorphically and isometrically embedded as a dense subgroup. But topological com... | Yes |
Theorem 9.8. Let \( A \subseteq X \) . Then the interior of \( A \) relative to the uniform topology is the set\n\n\[ \{ x \in X : U\left\lbrack x\right\rbrack \subseteq A\text{ for some }U \in \mathcal{U}\} . \] | Proof. If \( O \) is an open set that is contained in \( A \), it’s because for each \( x \in O \) there is a \( U \in \mathcal{U} \) so that \( U\left\lbrack x\right\rbrack \subseteq O \subseteq A \) . It follows that the set\n\n\[ B = \{ x : U\left\lbrack x\right\rbrack \subseteq A\text{ for some }U \in \mathcal{U}\}... | Yes |
Lemma 9.10. If \( V \subseteq X \times X \) is symmetric (i.e., \( V = {V}^{-1} \) ), then for any \( U \subseteq X \times X, \)\n\n\[ V \circ U \circ V = \mathop{\bigcup }\limits_{{\left( {x, y}\right) \in U}}V\left\lbrack x\right\rbrack \times V\left\lbrack y\right\rbrack \] | Proof.\n\n\[ \begin{matrix} V \circ U \circ V & = & \{ \left( {u, v}\right) : \left( {u, x}\right) \in V,\left( {x, y}\right) \in U,\left( {y, v}\right) \in V,\text{for some}\;x, y \in X\} \end{matrix} \]\n\n\[ = \mathop{\bigcup }\limits_{{\left( {x, y}\right) \in U}}\{ \left( {u, v}\right) : \left( {u, x}\right) ,\lef... | Yes |
Theorem 9.11. Let \( A \subseteq X \) . Then the uniform closure of \( A \) is precisely \( \mathop{\bigcap }\limits_{{U \in \mathcal{U}}}U\left\lbrack A\right\rbrack , \) where\n\n\[ U\left\lbrack A\right\rbrack = \{ y \in X : \left( {x, y}\right) \in U\text{ for some }x \in A\} . \] | Proof. Note that \( x \in \overline{A} \) if and only if \( U\left\lbrack x\right\rbrack \cap A \neq \varnothing \) for each \( U \in U \) . But \( U\left\lbrack x\right\rbrack \cap A \neq \varnothing \) if and only if \( x \in {U}^{-1}\left\lbrack A\right\rbrack \) . Since each member of \( \mathcal{U} \) contains a s... | Yes |
Theorem 9.13. Each uniformly continuous function is continuous with respect to the uniform topologies. | Proof. Suppose \( f : \left( {X,\mathcal{U}}\right) \rightarrow \left( {Y,\mathcal{V}}\right) \) is uniformly continuous, and let \( O \) be a neighborhood of \( f\left( x\right) \) . Then there is a \( V \in \mathcal{V} \) so that \( V\left\lbrack {f\left( x\right) }\right\rbrack \subseteq O \) and\n\n\[ \n{f}^{ \left... | Yes |
Theorem 9.15. Let \( \\left( {X,\\mathcal{U}}\\right) \) be a uniform space, and let \( d \) be a pseudo-metric for \( X \) . Then \( d \) is uniformly continuous on \( X \\times X \) relative to the product uniformity if and only if the set\n\n\[ \n\\{ \\left( {x, y}\\right) : d\\left( {x, y}\\right) < r\\} \n\]\n\nis... | Proof. We need to show that \( d \) ’s uniform continuity is equivalent to\n\n\[ \n{V}_{d, r} = \\{ \\left( {x, y}\\right) : d\\left( {x, y}\\right) < r\\} \\in \\mathcal{U} \n\]\n\nfor each \( r > 0 \) .\n\nPreface. Let \( U \\in \\mathcal{U} \) . Then the sets\n\n\[ \n\\{ \\left( {x, y}\\right) ,\\left( {u, v}\\right... | Yes |
Lemma 9.16 (Metrization Lemma). Let \( \left( {U}_{n}\right) \) be a sequence of subsets of \( X \times X \) such that \( {U}_{0} = X \times X \), each \( {U}_{n} \) contains the diagonal \( \Delta \), and \( {U}_{n + 1} \circ {U}_{n + 1} \circ {U}_{n + 1} \subseteq {U}_{n} \) for each \( n \) . Then there is a nonnega... | Proof (Outline). Define \( f : X \times X \rightarrow \mathbb{R} \) by\n\n\[ \begin{array}{r} f\left( {x, y}\right) = \left\{ \begin{array}{ll} {2}^{-n} & \text{precisely when}\;\left( {x, y}\right) \in {U}_{n - 1} \backslash {U}_{n} \\ 0 & \text{when}\;\left( {x, y}\right) \in \mathop{\bigcap }\limits_{n}{U}_{n} \end{... | Yes |
Theorem 9.17. A uniform space is pseudo-metrizable if and only if its uniformity has a countable base. | So suppose \( \left( {X,\mathcal{U}}\right) \) is a uniform space, and let \( \mathcal{P} \) be the family of all pseudo-metrics on \( X \) that are uniformly continuous on \( X \times X \) (with respect to the product uniformity, of course). The uniformity generated by \( \mathcal{P} \) is no bigger than \( \mathcal{U... | Yes |
Corollary 9.21. If \( \left( {X,\mathcal{U}}\right) \) is a compact uniform space and \( f : X \rightarrow \mathbb{R} \) is continuous, then \( f \) is uniformly continuous. | Indeed \( d\left( {x, y}\right) = \left| {f\left( x\right) - f\left( y\right) }\right| \) is a continuous pseudo-metric on \( X \times X \) , and hence uniformly continuous. | Yes |
Theorem 9.24. \( {\mu }^{ * } \) is an outer measure. | Proof. Since \( \varnothing \) is open and \( {\lambda }_{ * }\left( \varnothing \right) = 0,{\mu }^{ * }\left( \varnothing \right) = 0 \) . Clearly \( {\mu }^{ * } \) is monotone nondecreasing.\n\nIf \( \left( {E}_{n}\right) \) is a sequence of subsets of \( S \) with each having \( {\mu }^{ * }\left( {E}_{n}\right) <... | Yes |
Proposition 9.34. Let \( G \) be a group of homeomorphisms from the topological space \( X \) onto itself. Then \( G \) is weakly transitive if and only if for each \( x \in X \), the orbit \[ {Gx} \mathrel{\text{:=}} \{ {gx} : g \in G\} \] of \( x \) is dense in \( X \) . | Proof. Suppose \( G \) is weakly transitive but there is an \( {x}_{0} \in X \) such that \( \overline{G{x}_{0}} \) is not all of \( X \) . Look at the open (nonempty) set \( X \smallsetminus \overline{G{x}_{0}} \) . By weak transitivity there is a \( g \in G \) so that \[ {x}_{0} \in g\left( {X \smallsetminus \overlin... | Yes |
Lemma 9.36. Suppose \( \left( {X,\mathcal{U}}\right) \) is a uniform space and \( G \) is an equicontinuous group of homeomorphisms from \( X \) (in its uniform topology) onto itself. If \( G{x}_{0} \) is dense for some \( {x}_{0} \in X \), then \( G \) is weakly transitive. | Proof. Suppose that \( G \) is not weakly transitive. Then there is an \( x \in X \) and a nonempty open set \( U \subseteq X \) so that \[ x \notin \mathop{\bigcup }\limits_{{g \in G}}{gU} \] Since \( G{x}_{0} \) is dense in \( X \), there is a \( {g}_{1} \in G \) so that \[ G{x}_{0} \cap {g}_{1}U \neq \varnothing \te... | Yes |
Lemma 9.37. Let \( \left( {X,\mathcal{U}}\right) \) be a uniform space, and let \( G \) be a group of homeomorphisms of \( X \) (in its uniform topology) onto itself. Suppose that there is an \( {x}_{0} \in X \) such that \( G \) is equicontinuous at \( {x}_{0} \) and for any open set \( U \) containing \( {x}_{0} \) ,... | Proof. Fix \( U \in \mathcal{U} \) . There is a symmetric \( V \in \mathcal{U} \) such that \[ V \circ V \subseteq U\text{.} \] Since \( G \) is equicontinuous at \( {x}_{0} \) there open set \( O \subseteq X \), containing \( {x}_{0} \) such that \[ {gO} \subseteq V\left\lbrack {g{x}_{0}}\right\rbrack \] for all \( g ... | Yes |
Theorem 9.39. Suppose \( \left( {X,\mathcal{U}}\right) \) is a uniform space whose uniform topology is locally compact and Hausdorff. Let \( G \) be an equicontinuous group of homeomorphisms of \( X \) onto itself. Then there is a nonzero \( G \) -invariant content on \( X \) . | Proof. Let \( {x}_{0} \in X \), and set\n\n\[ \n{X}_{0} = \overline{G{x}_{0}} \n\]\n\nThen \( {X}_{0} \) is a locally compact Hausdorff space in its relative topology which is the uniform topology generated by the relative uniformity \( {\mathcal{U}}_{0} \) on \( {X}_{0} \times {X}_{0} \) inherited from \( \mathcal{U} ... | Yes |
Theorem 9.40 (Steinlage). Let \( G \) be an equicontinuous group of homeomorphisms of the nonempty locally compact space \( X \) onto itself, where \( X \) ’s topology is generated by the uniformity \( \mathcal{U} \). Then the \( G \) -invariant Haar measure on \( X \) is unique if and only if \( G \) is weakly transit... | Proof. If \( G \) is not weakly transitive on \( X \), then by Lemma 9.36 no point of \( X \) has dense orbit. Let \( {x}_{0} \in X \), and set\n\n\[ \n{X}_{0} = \overline{G{x}_{0}}.\n\]\n\nAs in the proof of Theorem \( {9.39},{X}_{0} \) is a locally compact Hausdorff space in its relative topology which is the uniform... | Yes |
Lemma 9.43. Let \( \\left( {X,\\mathcal{U}}\\right) \) be a uniform space, and let \( G \) be an equicontinuous group of homeomorphisms of \( X \) (in its uniform topology) onto itself. For each \( U \\in \\mathcal{U} \) and \( {x}_{0} \\in X \), there is a nonnegative real-valued function \( f \) defined on \( X \) su... | Proof. Fix \( U \\in \\mathcal{U} \), and \( {x}_{0} \\in X \). There is a uniformly continuous pseudometric \( d \) on \( X \) such that for each \( r > 0 \),\n\n\[ \n{U}_{d, r} = \\{ \\left( {x, y}\\right) : d\\left( {x, y}\\right) < r\\} \\in \\mathcal{U},\n\]\n\nand there is an \( {r}_{0} > 0 \) so that \( {U}_{d,{... | Yes |
Lemma 9.44. Suppose \( G \) is a weakly transitive group of homeomorphisms of the locally compact space \( X \) onto itself. Suppose \( f, h \) are test functions defined on \( X \). Suppose that \( h \) is not identically zero. Then there exist \( {g}_{1},\ldots ,{g}_{n} \in G \) and \( c > 0 \) so that\n\n\[ f\left( ... | Proof. Let \( K = \operatorname{supp}f \). There is an \( \eta > 0 \) and a nonempty open set \( O \subseteq X \) so that \( h\left( x\right) > \eta \) for all \( x \in O \). Since \( G \) is weakly transitive,\n\n\[ K \subseteq \mathop{\bigcup }\limits_{{g \in G}}{gO} \]\n\nso there are \( {g}_{1},\ldots ,{g}_{n} \in ... | Yes |
Theorem 10.1 (Oxtoby and Ulam). Let \( G \) be a complete, separable metrizable topological group that is not locally compact, and let \( m \) be any left-invariant Borel measure on \( G \) having at least one nontrivial value. Then every nonempty open set in \( G \) contains uncountably many disjoint congruent compact... | Proof. Suppose \( B \) is a Borel set for which \( 0 < m\left( B\right) < \infty \) . Let \( U \) be a nonempty open subset of \( G \) .\n\nSince \( G \) is separable and \( \{ {xU} : x \in G\} \) is an open cover of \( G \), at least one of the sets \( {xU} \) intersects \( B \) in a set of positive \( m \) -measure, ... | Yes |
Lemma 1.2. If \( {s}_{1},\ldots ,{s}_{n - 1} \) are elements of a group \( G \) satisfying the braid relations, then there is a unique group homomorphism \( f : {B}_{n} \rightarrow G \) such that \( {s}_{i} = f\left( {\sigma }_{i}\right) \) for all \( i = 1,2,\ldots, n - 1 \) . | Proof. Let \( {F}_{n} \) be the free group generated by the set \( \left\{ {{\sigma }_{1},\ldots ,{\sigma }_{n - 1}}\right\} \) . There is a unique group homomorphism \( \bar{f} : {F}_{n} \rightarrow G \) such that \( \bar{f}\left( {\sigma }_{i}\right) = {s}_{i} \) for all \( i = 1,2,\ldots, n - 1 \) . This homomorphis... | Yes |
Lemma 1.3. The group \( {B}_{n} \) with \( n \geq 3 \) is nonabelian. | Proof. The group \( {\mathfrak{S}}_{n} \) with \( n \geq 3 \) is nonabelian because \( {s}_{1}{s}_{2} \neq {s}_{2}{s}_{1} \) . Since the projection \( {B}_{n} \rightarrow {\mathfrak{S}}_{n} \) is surjective, \( {B}_{n} \) is nonabelian for \( n \geq 3 \) . | Yes |
Lemma 1.10. Each \( \beta \in {\mathcal{B}}_{n} \) has a two-sided inverse \( {\beta }^{-1} \) in \( {\mathcal{B}}_{n} \) . | Proof. For \( i = 1,2,\ldots, n - 1 \), we define two elementary braids \( {\sigma }_{i}^{ + } \) and \( {\sigma }_{i}^{ - } \) represented by diagrams with only one crossing shown in Figure 1.9. We claim that the braids \( {\sigma }_{1}^{ + },\ldots ,{\sigma }_{n - 1}^{ + },{\sigma }_{1}^{ - },\ldots ,{\sigma }_{n - 1... | Yes |
Lemma 1.11. The elements \( {\sigma }_{1}^{ + },\ldots ,{\sigma }_{n - 1}^{ + } \in {\mathcal{B}}_{n} \) satisfy the braid relations, that is, \( {\sigma }_{i}^{ + }{\sigma }_{j}^{ + } = {\sigma }_{j}^{ + }{\sigma }_{i}^{ + } \) for all \( i, j = 1,2,\ldots, n - 1 \) with \( \left| {i - j}\right| \geq 2 \), and \( {\si... | Proof. The first relation follows from the fact that its sides are represented by isotopic diagrams. The diagrams representing the sides of the second relation differ by the Reidemeister move \( {\Omega }_{3} \) . | Yes |
Corollary 1.14. The natural inclusion \( \iota : {B}_{n} \rightarrow {B}_{n + 1} \) is injective for all \( n \) . | Proof. In geometric language, \( \iota : {B}_{n} \rightarrow {B}_{n + 1} \) adds to a geometric braid \( b \) on \( n \) strings a vertical string on its right completely unlinked from \( b \) . Denote the resulting braid on \( n + 1 \) strings by \( \iota \left( b\right) \) . If \( {b}_{1},{b}_{2} \) are two geometric... | Yes |
Theorem 1.16. For all \( n \geq 2 \), the group \( {U}_{n} \) is free on the \( n - 1 \) generators \( {\left\{ {A}_{i, n}\right\} }_{i = 1,2,\ldots, n - 1} \) . | A proof of Theorem 1.16 will be given in Section 1.4. | No |
Corollary 1.17. The group \( {P}_{n} \) admits a normal filtration\n\n\[ 1 = {U}_{n}^{\left( 0\right) } \subset {U}_{n}^{\left( 1\right) } \subset \cdots \subset {U}_{n}^{\left( n - 1\right) } = {P}_{n} \]\n\nsuch that \( {U}_{n}^{\left( i\right) }/{U}_{n}^{\left( i - 1\right) } \) is a free group of rank \( n - i \) f... | Proof. Set \( {U}_{n}^{\left( 0\right) } = \{ 1\} \) and for \( i = 1,2,\ldots, n - 1 \) set\n\n\[ {U}_{n}^{\left( i\right) } = \operatorname{Ker}\left( {{f}_{n - i + 1}\cdots {f}_{n - 1}{f}_{n} : {P}_{n} \rightarrow {P}_{n - i}}\right) . \]\n\nThen\n\n\[ {U}_{n}^{\left( i\right) }/{U}_{n}^{\left( i - 1\right) } \cong ... | Yes |
Corollary 1.18. The group \( {P}_{n} \) is torsion free, i.e., it has no nontrivial elements of finite order. | This follows directly from Corollary 1.17, since free groups are torsion free. | Yes |
Corollary 1.19. \( {P}_{n} \) is generated by the \( n\left( {n - 1}\right) /2 \) elements \( {\left\{ {A}_{i, j}\right\} }_{1 \leq i < j \leq n} \) . | This directly follows from formula (1.6) and Theorem 1.16. | No |
Corollary 1.20. We have \( {P}_{n}/\left\lbrack {{P}_{n},{P}_{n}}\right\rbrack \cong {\mathbf{Z}}^{n\left( {n - 1}\right) /2} \) . | Proof. By Corollary 1.19, the abelian group \( {P}_{n}/\left\lbrack {{P}_{n},{P}_{n}}\right\rbrack \) is generated by the elements represented by \( {A}_{i, j} \), where \( 1 \leq i < j \leq n \) . To prove that these elements are linearly independent, it suffices to construct for each pair \( 1 \leq i < j \leq n \) a ... | Yes |
Corollary 1.21. The group \( {B}_{n} \) and all its subgroups are residually finite. | Proof. Recall that a group \( G \) is residually finite if for each \( \beta \in G - \{ 1\} \), there is a homomorphism \( f \) from \( G \) to a finite group such that \( f\left( \beta \right) \neq 1 \) . It is known that free groups are residually finite (see [LS77, Chap. IV, Sect. 4], [MKS66, Sect. 6.5]) and a semid... | Yes |
Corollary 1.22. The group \( {B}_{n} \) and all its finitely generated subgroups are Hopfian. | Proof. A finitely generated residually finite group is Hopfian (see [LS77, Chap. IV, Th. 4.10], [Neu67]). | No |
For \( i = 1,2,\ldots, n \), forgetting the ith string defines a group homomorphism \( {f}_{n}^{i} : {P}_{n} \rightarrow {P}_{n - 1} \). The kernel of \( {f}_{n}^{i} \) is a free group of rank \( n - 1 \) with free generators \( {A}_{1, i},\ldots ,{A}_{i - 1, i},{A}_{i, i + 1},\ldots ,{A}_{i, n} \). | Proof. Set \( {\alpha }_{i, n} = {\sigma }_{n - 1}{\sigma }_{n - 2}\cdots {\sigma }_{i} \) and observe that for any \( \beta \in {P}_{n} \), forgetting the \( n \) th string of \( {\alpha }_{i, n}\beta {\alpha }_{i, n}^{-1} \) yields the braid\n\n\[ {1}_{n - 1}{f}_{n}^{i}\left( \beta \right) {1}_{n - 1} = {f}_{n}^{i}\l... | Yes |
Corollary 1.25. For \( m \neq n \), the groups \( {B}_{m} \) and \( {B}_{n} \) are not isomorphic. | Proof. Theorem 1.24 implies that the image of \( Z\left( {B}_{n}\right) \) in \( {B}_{n}/\left\lbrack {{B}_{n},{B}_{n}}\right\rbrack \cong \mathbf{Z} \) is a subgroup of \( \mathbf{Z} \) of index \( n\left( {n - 1}\right) \) . If \( {B}_{m} \) is isomorphic to \( {B}_{n} \), then we must have \( m\left( {m - 1}\right) ... | Yes |
Lemma 1.27. Let \( M \) be a connected topological manifold of dimension \( \geq 2 \) with \( \partial M = \varnothing \) . For any \( m \geq 0, n > r \geq 1 \), the forgetting map\n\n\[ p : {\mathcal{F}}_{m, n}\left( M\right) \rightarrow {\mathcal{F}}_{m, r}\left( M\right) \]\n\ndefined by \( p\left( {{u}_{1},\ldots ,... | Proof. This lemma is obtained by applying Lemma 1.26 to \( M - {Q}_{m} \) . | No |
Lemma 1.28. For any \( m \geq 0, n \geq 1 \), the manifold \( {\mathcal{F}}_{m, n}\left( {\mathbf{R}}^{2}\right) \) is aspherical. | Proof. Consider the fibration \( {\mathcal{F}}_{m, n}\left( {\mathbf{R}}^{2}\right) \rightarrow {\mathcal{F}}_{m,1}\left( {\mathbf{R}}^{2}\right) = {\mathbf{R}}^{2} - {Q}_{m} \) with fiber \( {\mathcal{F}}_{m + 1, n - 1}\left( {\mathbf{R}}^{2}\right) \) defined above. The homotopy sequence of this fibration gives an ex... | Yes |
For any \( n \geq 1 \), the braid group \( {B}_{n} \) is torsion free. | Proof. Lemma 1.28 with \( m = 0 \) implies that \( {\mathcal{F}}_{n}\left( {\mathbf{R}}^{2}\right) \) is aspherical. Therefore \( {\pi }_{i}\left( {{\mathcal{C}}_{n}\left( {\mathbf{R}}^{2}\right) }\right) = {\pi }_{i}\left( {{\mathcal{F}}_{n}\left( {\mathbf{R}}^{2}\right) }\right) = 0 \) for all \( i \geq 2 \) . The fo... | Yes |
Theorem 1.31. The formula \( {\sigma }_{i} \mapsto {\widetilde{\sigma }}_{i} \) with \( i = 1,2,\ldots, n - 1 \) defines a group isomorphism \( {B}_{n} \rightarrow {\widetilde{B}}_{n} \) . | ## 1.5.2 Proof of Theorem 1.31\n\nThe braid relations for \( {\widetilde{\sigma }}_{1},\ldots ,{\widetilde{\sigma }}_{n - 1} \in {\widetilde{B}}_{n} \) can be verified by a direct computation (they follow also from further arguments in this paragraph). Therefore the formula \( {\sigma }_{i} \mapsto {\widetilde{\sigma }... | Yes |
Theorem 1.33. For any \( n \geq 1 \), the homomorphisms \( \eta \) and \( \rho \) are isomorphisms. The following diagram is commutative: | The commutativity of the diagram (1.14) means that \( \widetilde{\beta } = \rho \left( {\eta \left( \beta \right) }\right) \) for any \( \beta \in {B}_{n} \) . This can be verified at once. Since \( \rho ,\eta \), and \( \beta \mapsto \widetilde{\beta } \) are group homomorphisms, it suffices to verify this equality fo... | No |
Lemma 1.37. For any geometric braid \( b \subset {D}^{ \circ } \times I \) on \( n \) strings, there is a normal isotopy parametrizing \( b \) . | Proof. Consider the evaluation map \( e = {e}_{Q} : \operatorname{Top}\left( D\right) \rightarrow {\mathcal{C}}_{n} = {\mathcal{C}}_{n}\left( {D}^{ \circ }\right) \) sending \( f \in \operatorname{Top}\left( D\right) \) to \( f\left( Q\right) \) . As already observed in Section 1.4.3, the braid \( b \) gives rise to a ... | Yes |
Theorem 1.39. For any geometric braid \( b \) on \( n \) strings, the topological type of the pair \( \left( {{\mathbf{R}}^{2} \times I, b}\right) \) depends only on \( n \) . | Proof. Pick a disk \( D \subset {\mathbf{R}}^{2} \) such that \( b \subset {D}^{ \circ } \times I \) . Then the set \( Q = {Q}_{n} \) defined by (1.16) lies in \( {D}^{ \circ } \) . By Lemma 1.37, there is a normal isotopy \( {\left\{ {f}_{t} : D \rightarrow D\right\} }_{t \in I} \) parametrizing \( b \) . The formula ... | Yes |
Theorem 2.1. For any \( n \geq 1 \) and any \( \beta ,{\beta }^{\prime } \in {B}_{n} \), the closed braids \( \widehat{\beta },{\widehat{\beta }}^{\prime } \) are isotopic in the solid torus if and only if \( \beta \) and \( {\beta }^{\prime } \) are conjugate in \( {B}_{n} \) . | ## 2.2.4 Proof of Theorem 2.1\n\nObserve first that conjugate elements of \( {B}_{n} \) give rise to isotopic closed braids. In other words, \( \widehat{{\alpha \beta }{\alpha }^{-1}} = \widehat{\beta } \) for any \( \alpha ,\beta \in {B}_{n} \) . This is obtained by forming a diagram of \( {\alpha \beta }{\alpha }^{-1... | Yes |
Lemma 2.2. Two closed braid diagrams \( \mathcal{D},{\mathcal{D}}^{\prime } \) in \( {S}^{1} \times I \) represent isotopic closed braids in the solid torus \( {S}^{1} \times I \times I \) if and only if \( \mathcal{D} \) can be transformed into \( {\mathcal{D}}^{\prime } \) by a finite sequence of isotopies (in the cl... | Proof. We need only prove that if \( \mathcal{D},{\mathcal{D}}^{\prime } \) represent isotopic closed braids in the solid torus, then \( \mathcal{D} \) can be transformed into \( {\mathcal{D}}^{\prime } \) by a finite sequence of isotopies and moves \( {\left( {\Omega }_{2}^{\mathrm{{br}}}\right) }^{\pm 1},{\left( {\Om... | Yes |
Theorem 2.3 (J. W. Alexander). Any oriented link in \( {\mathbf{R}}^{3} \) is isotopic to a closed braid. | Proof. By a polygonal link, we shall mean a geometric link in \( {\mathbf{R}}^{3} \) whose components are closed broken lines. By vertices and edges of a polygonal link, we mean the vertices and the edges of its components. It is well known that any geometric link in \( {\mathbf{R}}^{3} \) is isotopic to a polygonal li... | Yes |
Lemma 2.6. An oriented link diagram \( \mathcal{D} \) in \( {\mathbf{R}}^{2} \) with \( h\left( \mathcal{D}\right) = 0 \) is isotopic in the sphere \( {S}^{2} = {\mathbf{R}}^{2} \cup \{ \infty \} \) to a closed braid diagram in \( {\mathbf{R}}^{2} \) . | Proof. Let \( \sum \) and \( {\left\{ {\gamma }_{x}\right\} }_{x} \) be the same objects as in the proof of the previous lemma. Suppose that \( h\left( \mathcal{D}\right) = 0 \) . We must prove that \( \mathcal{D} \) is isotopic in \( {S}^{2} \) to a closed braid diagram in the plane \( {\mathbf{R}}^{2} = {S}^{2} - \{ ... | Yes |
Corollary 2.9. Let \( \mathcal{L} \) be the set of all isotopy classes of nonempty oriented links in \( {\mathbf{R}}^{3} \) . The mapping \( { \coprod }_{n \geq 1}{B}_{n} \rightarrow \mathcal{L} \) assigning to a braid the isotopy class of its closure induces a bijection from the quotient set \( \left( {{ \coprod }_{n ... | Here the surjectivity follows from Alexander's theorem, while the injectiv-ity follows from Markov's theorem. | No |
Lemma 2.12. The move \( {\mathrm{M}}_{3} \) expands as a composition of the moves \( {\mathrm{M}}_{1},{\mathrm{M}}_{2} \) . | Proof. Recall the braid \( {\Delta }_{n} \in {B}_{n} \) defined in Section 1.3.3. By formula (1.8), \[ {\Delta }_{n}{\sigma }_{i}{\Delta }_{n}^{-1} = {\sigma }_{n - i} \in {B}_{n} \] (2.5) for all \( n \geq 1 \) and all \( i = 1,\ldots, n - 1 \) . In particular, \( {\Delta }_{n + 1}{\sigma }_{1}{\Delta }_{n + 1}^{-1} =... | Yes |
Lemma 2.17. Two 0-diagrams in \( {\mathbf{R}}^{2} \) related by a sequence of bendings, tightenings, and isotopies in \( {S}^{2} \) can be related by a sequence of \( \Omega \) -moves. | ## 2.6.4 Proof of Lemma 2.17, part I\n\nWe consider here the simplest case of Lemma 2.17, namely the one in which the sequence relating two 0-diagrams consists solely of isotopies. | No |
Lemma 2.20. For a local maximum \( \mathcal{C}\overset{s}{ \leftarrow }\mathcal{D}\overset{{s}^{\prime }}{ \rightarrow }{\mathcal{C}}^{\prime } \) with \( s \cdot {s}^{\prime } = 0 \), there are sequences of isotopies in \( {S}^{2} \) and bendings \( \mathcal{C} \rightarrow \cdots \rightarrow {\mathcal{C}}_{ * },{\math... | Proof. Let \( c,{c}^{\prime } \) be the reduction arcs of the bendings \( s,{s}^{\prime } \) on \( \mathcal{D} \) . The assumption \( s \cdot {s}^{\prime } = 0 \) implies that \( c \) and \( {c}^{\prime } \) are disjoint. Hence the bendings \( s \) and \( {s}^{\prime } \) are performed in disjoint areas of the plane an... | Yes |
Lemma 2.21. If two braids \( \beta ,{\beta }^{\prime } \) are M-equivalent, then the braids \( \bar{\beta },\overline{{\beta }^{\prime }} \) are M-equivalent. | Proof. We have \( \bar{\beta }{ \sim }_{c}\beta \sim {\beta }^{\prime }{ \sim }_{c}\overline{{\beta }^{\prime }} \) . | No |
Lemma 2.22. Let \( \mu \in {B}_{n + k} \) with \( n \geq 1, k \geq 0 \) . If \( \mu \equiv {1}_{n} \), then \( \bar{\mu }{ \equiv }^{\prime }{1}_{n} \) . | Proof. Pick \( \beta \in {B}_{n + m} \) with \( m \geq 0 \) and set \( \gamma = \left( {{1}_{k} \otimes \beta }\right) \left( {\bar{\mu } \otimes {1}_{m}}\right) \) . We must verify that \( \gamma \sim \beta \) . Obviously, \( \gamma { \sim }_{c}\bar{\gamma } = \left( {\bar{\beta } \otimes {1}_{k}}\right) \left( {{1}_{... | Yes |
Lemma 2.25. Under the assumptions of Lemma 2.24, the M-equivalence class of the braid\n\n\\[ \n\\left( {{1}_{n} \\otimes \\beta }\\right) \\left( {{\\sigma }_{r, n}^{\\varepsilon } \\otimes {1}_{m}}\\right) \\left( {{1}_{r} \\otimes \\gamma }\\right) \\left( {{\\sigma }_{n, r}^{-\\varepsilon } \\otimes {1}_{m}}\\right)... | Proof. This follows from Lemma 2.24 by applying the involution \\( \\mu \\mapsto \\bar{\\mu } \\) and using Lemma 2.21. | Yes |
Theorem 3.1. For any \( n \geq 1 \) and \( A \in {\psi }_{n}\left( {B}_{n}\right) \subset {\mathrm{{GL}}}_{n}\left( \Lambda \right) \), \[ \bar{A}{\Omega }_{n}{A}^{T} = {\Omega }_{n} \] | Proof. If (3.3) holds for a matrix \( A \), then it holds for its inverse: multiplying (3.3) on the left by \( {\bar{A}}^{-1} \) and on the right by \( {\left( {A}^{T}\right) }^{-1} \), we obtain the same formula with \( A \) replaced by \( {A}^{-1} \). If (3.3) holds for two matrices \( {A}_{1},{A}_{2} \), then it hol... | Yes |
Theorem 3.3. \( \operatorname{Ker}{\psi }_{n} \neq \{ 1\} \) for \( n \geq 5 \) . | We point out explicit braids on five and six strings annihilated by the Burau representation. Set\n\n\[ \gamma = {\sigma }_{4}{\sigma }_{3}^{-1}{\sigma }_{2}^{-1}{\sigma }_{1}^{2}{\sigma }_{2}^{-1}{\sigma }_{1}^{-2}{\sigma }_{2}^{-2}{\sigma }_{1}^{-1}{\sigma }_{4}^{-5}{\sigma }_{2}{\sigma }_{3}{\sigma }_{4}^{3}{\sigma ... | Yes |
Lemma 3.4. Let \( \alpha ,\beta \) be transversal spanning arcs on \( \left( {D, Q}\right) \) . If \( \langle \alpha ,\beta \rangle = 0 \) , then \( {\Psi }_{n}\left( {{\tau }_{\alpha }{\tau }_{\beta }}\right) = {\Psi }_{n}\left( {{\tau }_{\beta }{\tau }_{\alpha }}\right) \) . | Proof. To prove the lemma we compute the homological action of the half-twists. As a warmup, we compute the action of \( {\tau }_{\alpha } \) on \( H = {H}_{1}\left( {\sum ;\mathbf{Z}}\right) \) . Consider the loop \( {\alpha }^{\prime } \) on \( D \) drawn in Figure 3.1. This loop has a \ | No |
Lemma 3.5. Let \( c, d \) be simple closed curves on an oriented surface \( \sum \) . The Dehn twists \( {t}_{c},{t}_{d} \) commute if and only if \( c, d \) are isotopic to disjoint simple closed curves. | Proof. If \( c, d \) are disjoint, then they have disjoint cylinder neighborhoods, so that the Dehn twists \( {t}_{c},{t}_{d} \) obviously commute. If \( c, d \) are isotopic to disjoint simple closed curves \( {c}^{\prime },{d}^{\prime } \), then \( {t}_{c} = {t}_{{c}^{\prime }} \) commutes with \( {t}_{d} = {t}_{{d}^... | Yes |
Theorem 3.9. Let \( n \geq 3 \) and \( {V}_{1},{V}_{2},\ldots ,{V}_{n - 1} \) be the \( \left( {n - 1}\right) \times \left( {n - 1}\right) \) matrices over \( \Lambda \) given by\n\n\[ \n{V}_{1} = \left( \begin{matrix} - t & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & {I}_{n - 3} \end{matrix}\right) ,\;{V}_{n - 1} = \left( \begin{ma... | Proof. For \( i = 1,\ldots, n - 1 \), set\n\n\[ \n{V}_{i}^{\prime } = \left( \begin{matrix} {V}_{i} & 0 \\ { * }_{i} & 1 \end{matrix}\right) \n\]\n\nIt suffices to prove that \( {U}_{i}C = C{V}_{i}^{\prime } \) for all \( i \) . Fix \( i \) and observe that for any \( k = 1,\ldots, n \), the \( k \) th column of \( {U}... | Yes |
For \( i = 1,\ldots, n - 1 \), let \( {a}_{i} \) be the ith row of the matrix \( {\psi }_{n}^{\mathrm{r}}\left( \beta \right) - {I}_{n - 1} \). Then\n\n\[ - \left( {1 + t + \cdots + {t}^{n - 1}}\right) { * }_{\beta } = \mathop{\sum }\limits_{{i = 1}}^{{n - 1}}\left( {1 + t + \cdots + {t}^{i}}\right) {a}_{i}. \] | Consider the \( \Lambda \) -module \( {\Lambda }^{n} \) whose elements are identified with rows of length \( n \) over \( \Lambda \). The group \( {\mathrm{{GL}}}_{n}\left( \Lambda \right) \) acts on \( {\Lambda }^{n} \) on the right via the multiplication of rows by matrices. A direct verification shows that the vecto... | Yes |
Lemma 3.14. We have \( \varphi \circ {\widehat{f}}_{\# } = \varphi \) . | Proof. We need to prove that \( w \circ {\widehat{f}}_{\# } = w \) and \( u \circ {\widehat{f}}_{\# } = u \) . The first equality is proven by the same argument as in Section 3.2.2. To prove the second equality, consider the inclusion of configuration spaces \( \mathcal{C} = {\mathcal{C}}_{2}\left( \sum \right) \hookri... | Yes |
Theorem 3.15. The Lawrence-Krammer-Bigelow representation of the braid group \( {B}_{n} \) is faithful for all \( n \geq 1 \) . | This theorem is proven in Sections 3.6 and 3.7. One can give explicit matrices describing the action of the generators \( {\sigma }_{1},\ldots ,{\sigma }_{n - 1} \in {B}_{n} \) on \( \mathcal{H} \) ; see [Kra02], [Big01], [Bud05]. The proof of Theorem 3.15 given below uses neither these matrices nor the isomorphism (3.... | No |
Theorem 3.16. For all \( n \geq 1 \), the braid group \( {B}_{n} \) is linear. | This theorem follows from Theorem 3.15 and the isomorphism (3.16). Indeed, choosing a basis of the \( R \) -module \( \mathcal{H} \), we can identify \( {\operatorname{Aut}}_{R}\left( \mathcal{H}\right) \) with the matrix group \( {\mathrm{{GL}}}_{n\left( {n - 1}\right) /2}\left( R\right) \) . The ring \( R = \mathbf{Z... | Yes |
Lemma 3.17. Let \( L = \mathbf{Z}\left\lbrack {{x}_{1}^{\pm 1},{x}_{2}^{\pm 1}}\right\rbrack \) be the ring of Laurent polynomials in the variables \( {x}_{1},{x}_{2} \) . Let \( C \) be a free L-module of finite rank \( N \geq 1 \) . For an arbitrary \( L \) -submodule \( H \) of \( C \), the group \( {\operatorname{A... | Proof. Let \( Q = \mathbf{Q}\left( {{x}_{1},{x}_{2}}\right) \) be the field of rational functions in the variables \( {x}_{1},{x}_{2} \) with rational coefficients. Clearly, \( Q \) is the field of fractions of \( L \) . Consider the \( Q \) -vector space \( \bar{H} = Q{ \otimes }_{L}H \) . Since \( H \) is a submodule... | Yes |
Lemma 3.18. For any \( {g}_{1},{g}_{2} \in \mathcal{H} \) and \( r \in R \) ,\n\n\[ \n\left\langle {{g}_{2},{g}_{1}}\right\rangle = \overline{\left\langle {g}_{1},{g}_{2}\right\rangle },\;\left\langle {{g}_{1}, r{g}_{2}}\right\rangle = r\left\langle {{g}_{1},{g}_{2}}\right\rangle ,\;\left\langle {r{g}_{1},{g}_{2}}\righ... | Proof. We have\n\n\[ \n\left\langle {{g}_{2},{g}_{1}}\right\rangle = \mathop{\sum }\limits_{{k,\ell \in \mathbf{Z}}}\left( {{q}^{k}{t}^{\ell }{g}_{2} \cdot {g}_{1}}\right) {q}^{k}{t}^{\ell }\n\]\n\n\[ \n= \mathop{\sum }\limits_{{k,\ell \in \mathbf{Z}}}\left( {{g}_{1} \cdot {q}^{k}{t}^{\ell }{g}_{2}}\right) {q}^{k}{t}^{... | Yes |
Lemma 3.19. If there is an isotopy of \( \alpha \) (rel \( \partial \alpha \) ) decreasing the number of points in \( N \cap \alpha \), then the pair \( N,\alpha \) has at least one digon. | Proof. We deduce this lemma from Lemma 3.6 by extending the arcs \( N \) and \( \alpha \) to simple closed curves on a bigger surface. Pick closed disk neighborhoods \( {U}_{1},{U}_{2} \subset D \) of the endpoints of \( \alpha \) such that \( {U}_{1} \cap {U}_{2} = {U}_{i} \cap N = \varnothing \) for \( i = 1,2 \) and... | Yes |
Lemma 3.20. The algebraic intersection \( \langle N,\alpha \rangle \) is invariant under isotopies of \( N \) and \( \alpha \) in \( \sum \) constant on the endpoints. | Proof. It suffices to fix \( N \) and to prove that \( \langle N,\alpha \rangle \) is invariant under isotopies of the spanning arc \( \alpha \) . A generic isotopy of \( \alpha \) in \( \sum \) can be split into a finite sequence of local moves of three types:\n\n(i) an isotopy of \( \alpha \) in \( \sum \) keeping \(... | Yes |
Lemma 3.24. The surface \( \widetilde{F} \) is a closed subset of \( {\widetilde{\mathcal{C}}}^{ \circ } \) . | Proof. Pick an arbitrary point \( a \in {\widetilde{\mathcal{C}}}^{ \circ } - \widetilde{F} \) . Let \( \{ x, y\} \in \mathcal{C} \) be the projection of \( a \) to \( \mathcal{C} \), where \( x, y \) are distinct points of \( \sum \) . The inclusion \( a \in {\widetilde{\mathcal{C}}}^{ \circ } \) implies that \( x, y ... | Yes |
Lemma 3.26. If a self-homeomorphism \( f \) of \( \left( {D, Q}\right) \) represents an element of the kernel \( \operatorname{Ker}\left( {{B}_{n} \rightarrow {\operatorname{Aut}}_{R}\left( \mathcal{H}\right) }\right) \), then \( \langle N, f\left( \alpha \right) \rangle = \langle N,\alpha \rangle \) for any noodle \( ... | Proof. As was already observed above, the homomorphism \( {\widetilde{f}}_{ * } : \mathcal{H} \rightarrow \mathcal{H} \) transforms any \( \alpha \) -class \( v \in \mathcal{H} \) into an \( f\left( \alpha \right) \) -class. Formula (3.31) and the assumption \( {\widetilde{f}}_{ * } = \) id imply that\n\n\[ - {\left( q... | Yes |
Theorem 4.1. For all \( n \geq 1 \), there is a group homomorphism\n\n\[ \n\varphi : {G}_{n} \rightarrow {\mathfrak{S}}_{n}\n\]\n\nsuch that \( \varphi \left( {\dot{s}}_{i}\right) = {s}_{i} \) for all \( i = 1,\ldots, n - 1 \) . The homomorphism \( \varphi \) is an isomorphism. | The definition of \( {G}_{n} \) and relations (4.1) directly imply the existence (and the uniqueness) of \( \varphi \) . The bijectivity of \( \varphi \) will be proved in Section 4.1.2 using Lemmas 4.2 and 4.3 below. | No |
Lemma 4.2. For any \( n \geq 1 \), every element of \( {G}_{n} \) can be written as a word in the letters \( {\dot{s}}_{1},\ldots ,{\dot{s}}_{n - 1} \) with \( {\dot{s}}_{n - 1} \) appearing at most once. | Proof. We proceed by induction on \( n \) . The statement holds for \( n = 1 \) and \( n = 2 \) in view of the computation of \( {G}_{1} \) and \( {G}_{2} \) above. We suppose that the lemma holds for \( n - 1 \geq 2 \) and prove it for \( n \) . Since \( {\dot{s}}_{i}^{2} = 1 \) or, equivalently, \( {\dot{s}}_{i}^{-1}... | Yes |
Lemma 4.3. Any element of \( {G}_{n} \) can be written as a product \( {w}_{1}{w}_{2}\cdots {w}_{n - 1} \) , where \( {w}_{i} \in {\dot{\sum }}_{i} \) for \( i = 1,\ldots, n - 1 \) . | Proof. We prove the lemma by induction on \( n \) . For \( n = 1 \) and \( n = 2 \), the assertion is obvious. We suppose that it holds for \( n - 1 \geq 2 \) and prove it for \( n \) . By Lemma 4.2 it suffices to treat an element \( w \in {G}_{n} \) represented by a word in \( {\dot{s}}_{1},\ldots ,{\dot{s}}_{n - 1} \... | Yes |
For any \( w \in {\mathfrak{S}}_{n} \) and any \( {s}_{i} \in S \) ,\n\n\[ \lambda \left( {{s}_{i}w}\right) = \lambda \left( w\right) \pm 1\;\text{ and }\;\lambda \left( {w{s}_{i}}\right) = \lambda \left( w\right) \pm 1. \] | Proof. By definition of the length, \( \lambda \left( {{s}_{i}w}\right) \leq \lambda \left( w\right) + 1 \) . Replacing in this formula \( w \) by \( {s}_{i}w \), we obtain\n\n\[ \lambda \left( w\right) = \lambda \left( {{s}_{i}^{2}w}\right) \leq \lambda \left( {{s}_{i}w}\right) + 1. \]\n\nTherefore, \( \lambda \left( ... | Yes |
Theorem 4.8. Let \( {s}_{{i}_{1}}\cdots {s}_{{i}_{r}} \) be a reduced expression for \( w \in {\mathfrak{S}}_{n} \), where \( r = \lambda \left( w\right) \) . If \( \lambda \left( {w{s}_{j}}\right) < \lambda \left( w\right) \) for some \( j \in \{ 1,\ldots, n - 1\} \), then there is \( k \in \{ 1,\ldots, r\} \) such th... | Proof. We saw in the proof of Lemma 4.7 (a) that if \( {t}_{1},\ldots ,{t}_{r} \) are the transpositions defined by (4.7), then \( I\left( w\right) = \left\{ {{t}_{1},\ldots ,{t}_{r}}\right\} \) . If \( \lambda \left( {w{s}_{j}}\right) < \lambda \left( w\right) \), then \( {s}_{j} \in I\left( w\right) \) by Lemma 4.7(c... | Yes |
Corollary 4.9. Let \( w \in {\mathfrak{S}}_{n} \) . If \( \lambda \left( {w{s}_{j}}\right) < \lambda \left( w\right) \) for some \( j \in \{ 1,\ldots, n - 1\} \) , then there is a reduced expression for \( w \) ending with \( {s}_{j} \) . If \( \lambda \left( {{s}_{j}w}\right) < \lambda \left( w\right) \) for some \( j... | This is a direct corollary of the previous theorem: if \( \lambda \left( {w{s}_{j}}\right) < \lambda \left( w\right) \), then \( w{s}_{j} = {s}_{{i}_{1}}\cdots \widehat{{s}_{{i}_{k}}}\cdots {s}_{{i}_{r}} \) and \( w = {s}_{{i}_{1}}\cdots \widehat{{s}_{{i}_{k}}}\cdots {s}_{{i}_{r}}{s}_{j} \) is a reduced expression for ... | Yes |
Lemma 4.10. If \( \lambda \left( {{s}_{i}w{s}_{j}}\right) = \lambda \left( w\right) \) and \( \lambda \left( {{s}_{i}w}\right) = \lambda \left( {w{s}_{j}}\right) \) for \( w \in {\mathfrak{S}}_{n} \) and some \( i, j \in \{ 1,\ldots, n - 1\} \), then \( {s}_{i}w = w{s}_{j} \) and \( {s}_{i}w{s}_{j} = w \) . | Proof. (a) Suppose first that \( \lambda \left( {{s}_{i}w}\right) = \lambda \left( {w{s}_{j}}\right) > \lambda \left( {{s}_{i}w{s}_{j}}\right) = \lambda \left( w\right) \) . By Lemma 4.6,\n\n\[ I\left( {{s}_{i}w}\right) = {w}^{-1}I\left( {s}_{i}\right) {w\Delta I}\left( w\right) = \left\{ {{w}^{-1}{s}_{i}w}\right\} {\D... | Yes |
Theorem 4.12. For any monoid \( M \) and any \( {x}_{1},\ldots ,{x}_{n - 1} \in M \) satisfying the relations\n\n\[ \n{x}_{i}{x}_{j} = {x}_{j}{x}_{i}\;\text{ if }\left| {i - j}\right| \geq 2, \]\n\n\[ \n{x}_{i}{x}_{j}{x}_{i} = {x}_{j}{x}_{i}{x}_{j}\;\text{ if }\left| {i - j}\right| = 1, \]\n\nthere is a set-theoretic m... | Proof. Define a monoid homomorphism \( {\rho }^{\prime } : {M}_{n} \rightarrow M \) by\n\n\[ \n{\rho }^{\prime }\left( {{i}_{1},\ldots ,{i}_{k}}\right) = {x}_{{i}_{1}}\cdots {x}_{{i}_{k}} \]\n\nfor all \( \left( {{i}_{1},\ldots ,{i}_{k}}\right) \in {M}_{n} \) . We claim that \( {\rho }^{\prime }\left( S\right) = {\rho ... | Yes |
Lemma 4.13. If \( w \in {\mathfrak{S}}_{n} \) satisfies \( \lambda \left( {w{s}_{i}}\right) < \lambda \left( w\right) \) for all \( i \in \{ 1,\ldots, n - 1\} \) , then \( w = {w}_{0} \) . | Proof. By Lemma 4.7(c), \( {s}_{i} \in I\left( w\right) \) for all \( i \) . Then \( w\left( i\right) > w\left( {i + 1}\right) \) for all \( i \) . The only permutation satisfying these inequalities is \( {w}_{0} \) . | Yes |
Lemma 4.14. For any \( u, v \in {\mathfrak{S}}_{n} \) such that \( {uv} = {w}_{0} \), \[ \lambda \left( u\right) + \lambda \left( v\right) = \lambda \left( {w}_{0}\right) . \] | Proof. The lemma trivially holds for \( u = {w}_{0} \) and \( v = 1 \). We claim that for any \( u \in {\mathfrak{S}}_{n}, u \neq {w}_{0} \), there is a sequence \( {s}_{{i}_{1}},\ldots ,{s}_{{i}_{r}} \) of simple transpositions such that \( u{s}_{{i}_{1}}\cdots {s}_{{i}_{r}} = {w}_{0} \) and \( \lambda \left( {u{s}_{{... | Yes |
Lemma 4.16. (a) For each \( w \in {\mathfrak{S}}_{n} \), there is a unique \( {T}_{w} \in {H}_{n} \) such that \( {T}_{w} = {T}_{{i}_{1}}\cdots {T}_{{i}_{r}} \) whenever \( w = {s}_{{i}_{1}}\cdots {s}_{{i}_{r}} \) is a reduced expression for \( w \) . | Proof. (a) This follows from (4.16), (4.17), and Theorem 4.12. | No |
Theorem 4.17. The R-module \( {H}_{n} \) is free of rank \( n \) ! with basis \( \left\{ {{T}_{w} \mid w \in {\mathfrak{S}}_{n}}\right\} \) . | Proof. Let \( H \) be the \( R \) -submodule of \( {H}_{n} \) spanned by the vectors \( {T}_{w}\left( {w \in {\mathfrak{S}}_{n}}\right) \) . By Lemma 4.16 (b), \( H \) is a left ideal of \( {H}_{n} \) . Since \( 1 = {T}_{1} \in H \), we have \( H = {H}_{n} \) . To prove the theorem, it remains to show that the vectors ... | Yes |
For any reduced expression \( {s}_{{i}_{1}}{s}_{{i}_{2}}\ldots {s}_{{i}_{r}} \) representing \( w \in {\mathfrak{S}}_{n} \) , set \( \mathrm{R} = {\mathrm{R}}_{{i}_{r}}\ldots {\mathrm{R}}_{{i}_{2}}{\mathrm{R}}_{{i}_{1}} \in {\operatorname{End}}_{R}\left( V\right) \) and \( \mathrm{L} = {\mathrm{L}}_{{i}_{1}}{\mathrm{\;... | Proof. The equality \( {e}_{w} = \mathrm{R}\left( {e}_{1}\right) \) is proved by induction on \( r = \lambda \left( w\right) \) . For \( r = 1 \), this equality follows from the definition of \( \mathrm{R} = {\mathrm{R}}_{{i}_{1}} \) . For \( r \geq 2 \), set \( {w}^{\prime } = {s}_{{i}_{1}}{s}_{{i}_{2}}\ldots {s}_{{i}... | Yes |
Lemma 4.20. The endomorphisms \( {\mathrm{L}}_{1},\ldots ,{\mathrm{L}}_{n - 1} \) of the \( R \) -module \( V \) satisfy relations (4.16),(4.17), and (4.18) in which \( {T}_{i} \) is replaced by \( {\mathrm{L}}_{i} \) . | Proof. (a) If \( \lambda \left( {{s}_{i}w}\right) > \lambda \left( w\right) \), then\n\n\[ \n{\mathrm{L}}_{i}^{2}\left( {e}_{w}\right) = {\mathrm{L}}_{i}\left( {e}_{{s}_{i}w}\right) = q{e}_{w} + z{e}_{{s}_{i}w} = z{\mathrm{\;L}}_{i}\left( {e}_{w}\right) + q{e}_{w}.\n\]\n\nIf \( \lambda \left( {{s}_{i}w}\right) < \lambd... | Yes |
Proposition 4.21. The homomorphism \( \iota : {H}_{n} \rightarrow {H}_{n + 1} \) is injective. As a left \( {H}_{n} \) -module, \( {H}_{n + 1} \) is free of rank \( n + 1 \) with basis\n\n\[ \left\{ {1,{T}_{n},{T}_{n}{T}_{n - 1},\ldots ,{T}_{n}{T}_{n - 1}\cdots {T}_{2}{T}_{1}}\right\} . \]\n | Proof. By definition of \( {T}_{w} \), we have \( \iota \left( {T}_{w}\right) = {T}_{w} \) for all \( w \in {\mathfrak{S}}_{n} \), where on the right-hand side \( w \) is considered as an element of \( {\mathfrak{S}}_{n + 1} \) . By Theorem 4.17, \( \iota \) sends a basis of \( {H}_{n} \) to a subset of a basis of \( {... | Yes |
Proposition 4.22. For any \( n \geq 2 \), there is an isomorphism of \( R \) -modules\n\n\[ \varphi : {H}_{n} \oplus \left( {{H}_{n}{ \otimes }_{{H}_{n - 1}}{H}_{n}}\right) \rightarrow {H}_{n + 1} \]\n\ngiven for any \( a \in {H}_{n} \) and any finite family \( {\left\{ {b}_{i},{c}_{i}\right\} }_{i} \subset {H}_{n} \) ... | Proof. Since \( {H}_{n - 1} \) is generated by \( {T}_{1},\ldots ,{T}_{n - 2} \) and \( {T}_{i}{T}_{n} = {T}_{n}{T}_{i} \) for \( i \leq n - 2 \) ,\n\n\[ \varphi \left( {{bh} \otimes c}\right) = {bh}{T}_{n}c = b{T}_{n}{hc} = \varphi \left( {b \otimes {hc}}\right) \]\n\nfor all \( h \in {H}_{n - 1}, b, c \in {H}_{n} \) ... | Yes |
For any oriented link \( L \subset {\mathbf{R}}^{3} \) and any braid \( \beta \in {B}_{n} \) whose closure is isotopic to \( L \), the element\n\n\[ \n{I}_{L}\left( {q, z}\right) = {\tau }_{n}\left( {{\omega }_{n}\left( \beta \right) }\right) \in R \]\n\ndepends only on the isotopy class of \( L \) . For the trivial kn... | The first assertion follows from the theory of Markov functions in Section 2.5.2 and Proposition 4.24. The trivial knot can be realized as the closure of the trivial braid \( 1 \in {B}_{1} = \{ 1\} \) . Therefore,\n\n\[ \n{I}_{O}\left( {q, z}\right) = {\tau }_{1}\left( {{\omega }_{1}\left( 1\right) }\right) = {\tau }_{... | Yes |
Corollary 4.26. There is an isotopy invariant \( L \mapsto {P}_{L}\left( {x, y}\right) \) of oriented links in \( {\mathbf{R}}^{3} \) with values in \( \mathbf{Z}\left\lbrack {x,{x}^{-1}, y,{y}^{-1}}\right\rbrack \) such that its value on the trivial knot \( O \) is 1 and for any Conway triple of oriented links \( \lef... | Proof. Let \( R = \mathbf{Z}\left\lbrack {x,{x}^{-1}, y,{y}^{-1}}\right\rbrack \) be the ring of Laurent polynomials in two variables \( x, y \) with integer coefficients. Set \( {P}_{L}\left( {x, y}\right) = {I}_{L}\left( {q, z}\right) \in R \), where \( {I}_{L}\left( {q, z}\right) \) is the link invariant provided by... | Yes |
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