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Theorem 9. In a triangulated 3-manifold \( K \), let \( S \) be a polyhedral 2-sphere, lying in a set \( \left| {\mathrm{{St}}v}\right| \) . Then \( S \) is the boundary of a combinatorial 3-cell. | Proof. Let \( h : \left| {\mathrm{{St}}v}\right| \rightarrow {\mathbf{R}}^{3} \) be a PLH. By the PL Schönflies theorem (Theorem 17.12), \( h\left( S\right) \) bounds a combinatorial 3-cell \( {D}^{3} \) in \( {\mathbf{R}}^{3} \), and it is easy to check that \( {D}^{3} \subset h\left( \left| {\mathrm{{St}}v}\right| \r... | Yes |
Theorem 10. In a triangulated 3-manifold \( K \), let \( {C}^{3} \) be a polyhedral 3-cell, lying in a set \( \operatorname{Int}\left| {\mathrm{{St}}v}\right| \) . Let \( {D}_{1} \) and \( {D}_{2} \) be polyhedral 2-cells such that \( {D}_{1} \cup {D}_{2} = \operatorname{Bd}{C}^{3} \) and \( {D}_{1} \cap {D}_{2} = J = ... | Proof. We may assume that \( N \subset \) Int St \( v \) . Let \( g : \left| {\operatorname{St}v}\right| \rightarrow {\mathbf{R}}^{3} \) be a PLH. By Theorem 17.8 there is a PLH \( f : {\mathbf{R}}^{3} \leftrightarrow {\mathbf{R}}^{3}, g\left( {D}_{1}\right) \leftrightarrow g\left( {D}_{2}\right) \), such that \( f\par... | Yes |
Theorem 11. In a triangulated 3-manifold \( K \), let \( {C}_{1}^{3} \) and \( {C}_{2}^{3} \) be combinatorial 3-cells such that (1) each \( {C}_{i}^{3} \) lies in a set Int \( \left| {\mathrm{{St}}{v}_{i}}\right| \) and (2) \( {C}_{1}^{3} \cap {C}_{2}^{3} \) is a polyhedral 2-cell \( D = \operatorname{Bd}{C}_{1}^{3} \... | Proof. By the preceding theorem, there is a PLH \( h : {C}_{1}^{3} \cup {C}_{2}^{3} \leftrightarrow {C}_{1}^{3} \) . | No |
Theorem 12. Let \( K \) be a triangulated 3-manifold, and let \( L \) be a finite subcomplex of \( K \), such that \( L \) is a triangulated 3-manifold with boundary. Then \( \left| L\right| \) and \( N\left( L\right) \) are combinatorially equivalent. | Proof. For each \( {\sigma }^{2} \in \partial L \), the set \( {C}^{3} = \mathrm{{Cl}}\left\lbrack {{N}^{\prime }\left( {\sigma }^{2}\right) - L}\right\rbrack \) is a combinatorial 3-cell, intersecting \( \left| L\right| \cup N\left( {L}^{1}\right) \) in a 2-cell. By repeated applications of Theorem 10, there is a PLH ... | Yes |
Theorem 13. Let \( K \) be a triangulated 3-manifold with boundary. Then \( K \) is combinatorially equivalent to a set \( N\left( L\right) \), where \( N\left( L\right) \) is the regular neighborhood of a subcomplex \( L \) of a triangulated 3-manifold \( {K}^{\prime } \) . | Proof. Let \( M = \left| K\right| \) . As in the proof of Theorem 6, the theorem reduces to the case in which every simplex of \( K \) intersects Bd \( M \) in a simplex. Let \( {M}^{\prime } \) be a \ | No |
Theorem 16. Let \( K \) be an orientable triangulated 3-manifold with boundary. Then \( K \) is combinatorially equivalent to the regular neighborhood \( N\left( L\right) \) of a subcomplex \( L \) of an orientable triangulated 3-manifold \( {K}^{\prime } \) . | Proof. By Theorem 15, the \( {K}^{\prime } \) used in the proof of Theorem 13 is orientable. | No |
Theorem 18. In an orientable triangulated 3-manifold \( K \), let \( L \) be a subcom-plex, of dimension \( \leq 1 \), let \( N = N\left( L\right) \), and let \( B = \mathrm{{Bd}}N \). Then\n\n\[ h\left( B\right) = {p}^{1}\left( N\right) . \] | Proof. (1) \( h\left( B\right) \) and \( {p}^{1}\left( N\right) \) are additive over the components of \( N \). Therefore we may suppose, with no loss of generality, that \( L, N \), and \( B \) are connected.\n\n(2) Suppose that \( L \) is acyclic. By induction, based on Theorem 11, it follows that \( N \) is a 3-cell... | Yes |
Theorem 19. Let \( K \) be an orientable triangulated 3-manifold with boundary. Then\n\n\[ \n{p}^{1}\left( K\right) \geq h\left( {\operatorname{Bd}\left| K\right| }\right) \n\] | Proof. (1) By Theorem 16, the theorem reduces to the case in which \( \left| K\right| = N\left( L\right) \), where \( L \) is a finite subcomplex of a triangulated 3-manifold \( {K}^{\prime } \) .\n\n(2) We shall show that the theorem reduces to the case in which \( L \) is at most 2-dimensional. Suppose that we have g... | Yes |
Theorem 1. Let \( g : \widetilde{M} \rightarrow M \) be a covering, and let \( f : \Delta \rightarrow M \) be a mapping of a 2-cell into \( M \) . Let \( {Q}_{0} \in \Delta \), let \( {P}_{0} = f\left( {Q}_{0}\right) \), and let \( {\widetilde{P}}_{0} \in {g}^{-1}\left( {P}_{0}\right) \) . Then there is one and only on... | Proof. (1) Let \( U \) and \( \left\{ {\widetilde{U}}_{i}\right\} \) be as in the definition of a covering, and let \( {\widehat{U}}_{k} \) be the set \( {\widetilde{U}}_{i} \) that contains \( {\widetilde{P}}_{0} \) . Suppose that \( f\left( \Delta \right) \subset U \) . Then the desired \( \widetilde{f} \) exists: we... | Yes |
Theorem 2. Let \( g : \widetilde{M} \rightarrow M \) be a covering, let \( {P}_{0} \in M \), and let \( {\widetilde{P}}_{0} \in {g}^{-1}\left( {P}_{0}\right) \) . Then the induced homomorphism\n\n\[ \n{g}_{0}^{ * } : \pi \left( {\widetilde{M},{\widetilde{P}}_{0}}\right) \rightarrow \pi \left( {M,{P}_{0}}\right)\n\]\n\n... | Proof. This follows from Theorem 1. | No |
Theorem 4. If \( g : \widetilde{M} \rightarrow M \) is a \( k \) -fold covering, then \( k \) is the index of \( {\pi }_{0} \) in \( \pi \left( {M,{P}_{0}}\right) \), for every choice of \( {\widetilde{P}}_{0} \) in \( {g}^{-1}\left( {P}_{0}\right) \) . | Proof. Let \( {g}^{-1}\left( {P}_{0}\right) = \left\{ {{\widetilde{P}}_{0},{\widetilde{P}}_{1},\ldots }\right\} \), with \( {\widetilde{P}}_{i} \neq {\widetilde{P}}_{j} \) for \( i \neq j \) . For each \( i \), let \( {\widetilde{p}}_{i} \) be a path in \( \widetilde{M} \), from \( {\widetilde{P}}_{0} \) to \( {\wideti... | Yes |
Theorem 5. Let \( M \) be a connected polyhedron. If \( \pi \left( M\right) \) has a subgroup \( \pi \), of index \( k \), then there is a \( k \) -fold covering of \( M \) . | Proof. Use the standard construction, using \( {\pi }_{g} = \pi \) . | No |
Theorem 6. Let \( g : \widetilde{M} \rightarrow M \) be a covering. Then \( \widetilde{M} \) is a polyhedron. In fact, every triangulation \( K \) of \( M \) can be lifted so as to give a triangulation \( \widetilde{K} \) of \( \widetilde{M} \) . That is, there is a triangulation \( \widetilde{K} \) of \( \widetilde{M}... | Proof. This is obvious in the case in which \( K \) is a sufficiently fine triangulation so that each \( \sigma \in K \) lies in a set \( U \) of the sort described in the definition of a covering. The general case is reducible to this case; the point is that every \( n \) -simplex is expressible as the union of a sequ... | No |
Theorem 7. Let \( M \) be a connected polyhedral 3-manifold with boundary, and suppose that \( M \) is not orientable. Then \( M \) has a 2-fold covering. | Proof. Let \( K \) be a triangulation of \( M \), and let \( {P}_{0} \) be a vertex of \( K \) . Take a fixed orientation \( {C}^{3} = \sum {\alpha }_{i}{\sigma }_{i}^{3} \) of the complex St \( {P}_{0} \), as in the discussion just before Theorem 23.14. If \( {P}_{0}v \) is an edge of \( K \), then \( {C}^{3} \) gives... | Yes |
Theorem 8. Let \( M \) be a compact, connected, orientable polyhedral 3-manifold with boundary, and suppose that some component of \( \mathrm{Bd}M \) is not a 2-sphere. Then \( M \) has a 2-fold covering. | Proof. By Theorem 23.19, we have \( {p}^{1}\left( M\right) > 0 \) . Therefore \( {H}_{1}\left( M\right) \approx \mathbf{Z} + G \) , where the structure of \( G \) does not concern us. Evidently there is a surjective homomorphism \( \mathbf{Z} + G \rightarrow {\mathbf{Z}}_{2} \), where \( {\mathbf{Z}}_{2} \) is the addi... | Yes |
Theorem 9. Let \( K \) be a triangulated 3-manifold, let \( M = \left| K\right| \), and let \( S \) be a polyhedral solid torus in \( M \) . Then \( S \) is a CST if and only if there is a PL mapping \( \phi : {\sigma }^{2} \times \left\lbrack {0,1}\right\rbrack \rightarrow S \), such that \( {\sigma }^{2} \times \{ 0\... | Proof. (1) Given that \( S = \mathop{\bigcup }\limits_{{i = 1}}^{n}{C}_{i}^{3} \), as in the definition of a CST, decompose \( \left\lbrack {0,1}\right\rbrack \) into \( n \) linear intervals \( {I}_{i} \), end to end, with end-points \( {x}_{i} = i/n\left( {0 \leq i \leq n}\right) \) . Then define \( \phi \mid {\sigma... | No |
Theorem 10. Every two combinatorial solid tori are combinatorially equivalent. | Proof. Use the apparatus of the preceding proof. | No |
Theorem 11. Let \( K \) be an orientable triangulated 3-manifold, let \( M = \left| K\right| \), let \( J \) be a polygon in \( M \), and let \( S \) be the regular neighborhood of \( J \) in a subdivision \( {K}^{\prime } \) of \( K \) in which \( J \) forms a subcomplex. Then \( S \) is a CST. | Proof. There is no loss of generality in supposing that \( {K}^{\prime } = K \) . Let the vertices and edges of \( J \) be \( {v}_{1},{e}_{1},{v}_{2},\ldots ,{v}_{n},{e}_{n} \), in the cyclic order of their appearance on \( J \) . As in the discussion just after Theorem 23.11, let \( N\left( {v}_{i}\right) \) be the re... | Yes |
Theorem 12. Let \( M = \\left| K\\right| \) be a triangulated 3-manifold, let \( J \) be a polygon in \( M \), and suppose that \( J \) is contractible in \( M \) . Let \( N \) be a regular neighborhood of \( J \), in a subdivision of \( K \) in which \( J \) forms a subcomplex. Then \( N \) is \( a \) CST. | Proof. We may suppose, with no loss of generality, that \( J \) forms a subcomplex of \( K \), and that \( N \) is the union of all simplexes of \( {b}^{2}K \) that intersect \( J \) . Thus \( N = \\mathop{\\bigcup }\\limits_{i}\\left| {\\text{St}{v}_{i}}\\right| \), where the points \( {v}_{i} \) are the vertices of \... | No |
Theorem 2 (Loop theorem, first form; C. Papakyriakopoulos). Let \( K \) be an orientable triangulated 3-manifold with boundary, and let \( M = \left| K\right| \) . Let \( B \) be a component of \( \mathrm{{Bd}}M \), and suppose that there is a loop \( L \) in \( B \) such that \( L \) is contractible in \( M \) but not... | This is the classical Loop theorem. It was, of course, proved first. Stallings’s proof of Theorem 1, given in \( \left\lbrack {\mathrm{S}}_{2}\right\rbrack \), was the final stage in a long development, to which many authors contributed in various ways. For a general account of the history, see the end of this section.... | Yes |
Lemma 1. Suppose that \( B \) is a 2-sphere, let \( {B}^{\prime } \) be a regular neighborhood of \( \left| L\right| \) in \( B \), suppose that the base point \( {P}_{0} \) lies in Int \( {B}^{\prime } \), and let \( {N}^{\prime } \) be a normal subgroup of \( \pi \left( {{B}^{\prime },{P}_{0}}\right) \), such that \(... | Proof. Since \( B \) is a 2-sphere and \( \left| L\right| \) is connected, \( {B}^{\prime } \) must be a \( k \) -annulus for some \( k \) . Therefore \( \pi \left( {{B}^{\prime },{P}_{0}}\right) \) is freely generated by a finite set \( \left\{ {{\bar{p}}_{1},{\bar{p}}_{2},\ldots ,{\bar{p}}_{k}}\right\} \), where each... | Yes |
Lemma 2. Suppose that \( D \) is PL, is locally a homeomorphism, and is at most two-to-one at each point of \( \Delta \) . Let \( {B}^{\prime } \) be a connected 2-manifold with boundary which forms a neighborhood of \( \left| L\right| \) in \( B \), suppose that \( {P}_{0} \in \) Int \( {B}^{\prime } \), and let \( {N... | Proof. Here the special hypothesis for \( D \) means that (1) each point of \( \Delta \) has a neighborhood on which \( D \) is a homeomorphism and (2) each point \( {Q}^{\prime } \) of \( \left| D\right| \) is \( = D\left( Q\right) \) for at most two points \( Q \) of \( \Delta \) . If \( {D}^{-1}\left( Q\right) \) co... | No |
Lemma 3. For each normal system there is a nonsingular PL 2-cell \( {D}^{\prime } : \Delta \rightarrow \) \( {M}_{1} \) such that for \( {L}^{\prime } = \mathrm{{Bd}}{D}^{\prime } \) we have \( \left| {L}^{\prime }\right| \subset {B}_{1} \) and \( {L}^{\prime }\left( {B}_{1}\right) \cap {N}_{1} = \varnothing \) . | Proof of LEMMA. Suppose that the lemma is false. Let \[ \left\lbrack {{M}_{i},{K}_{1}, D, K\left( \Delta \right) ,{B}_{1},{N}_{1}}\right\rbrack \] be a normal system for which the lemma fails. Let \( k \) be the complexity of this system. We may suppose, as an induction hypothesis, that the lemma holds for every normal... | Yes |
Theorem 1. Let \( {M}^{3} = \left| K\right| \) be a triangulated 3-manifold with boundary, and let \( {M}^{2} \) be a polyhedral 2-manifold lying in Int \( {M}^{3} \) . Suppose that \( {M}^{2} \) is the union of a collection of components of the boundary \( \mathrm{{Bd}}N \) of a 3-manifold \( N \) with boundary, lying... | Proof. We need to show that every component \( B \) of \( {M}^{2} \) is two sided. Let \( W \) be any connected neighborhood of \( B \) which intersects no other component of Bd \( N \) . Then\n\n\[ W - B = W - \operatorname{Bd}N = \left( {W \cap \operatorname{Int}N}\right) \cup \left( {W - N}\right) ,\]\n\nwhere the t... | Yes |
Theorem 2. Let \( {M}^{3} = \left| K\right| \) be a triangulated 3-manifold with boundary, let \( B = \operatorname{Bd}{M}^{3} \), and suppose that \( B \) is compact. Then there is a PLH\n\n\[ \rho : B \times \left\lbrack {0,1}\right\rbrack \leftrightarrow W \subset {M}^{3} \]\n\nsuch that (1) \( W \) is a neighborhoo... | Proof. Let \( {d}_{1},{d}_{2},\ldots ,{d}_{n} \) be a sequence of polyhedral 2-cells in \( B \), with disjoint interiors, such that \( B = \mathop{\bigcup }\limits_{i}{d}_{i} \) and such that the union of any subcollection of these 2-cells is a 2-manifold with boundary. Then there is a polyhedral 3-cell \( {C}_{1} \) i... | Yes |
Theorem 3. Let \( {M}^{3} = \left| K\right| \) be a triangulated 3-manifold with boundary, and let \( {M}^{2} \) be a compact polyhedral 2-manifold in Int \( {M}^{3} \), such that \( {M}^{2} \) is two sided in Int \( {M}^{3} \) . Then there is a PLH \[ \rho : {M}^{2} \times \left\lbrack {-1,1}\right\rbrack \leftrightar... | Proof of Theorem 3. Suppose (without loss of generality) that \( {M}^{2} \) is connected. Let \( N \) be a regular neighborhood of \( {M}^{2} \) in Int \( M \) . Since \( {M}^{2} \) is two sided, it follows that \( N - {M}^{2} \) is not connected; and it is easy to check geometrically that \( N - {M}^{2} \) has only tw... | Yes |
Theorem 5. Suppose that in Theorem \( 4,{M}^{2} \) is a closed set in \( {M}^{3} \), but is not necessarily compact. Then the conclusion of Theorem 4 still holds. | Proof. We observe the following.\n\n(1) In the proof of Theorem 2, if \( {M}^{2} \) is not known to be compact, then the construction of \( \rho \) may require an infinite process. But this process is always locally finite, in the sense that it terminates on every finite polyhedron in \( {M}^{2} \) . Thus the more gene... | Yes |
Theorem 6. Let \( {M}^{2} \) be a compact connected polyhedral 2-manifold in \( {\mathbf{R}}^{3} \) . Then \( {M}^{2} \) is 2-sided in \( {\mathbf{R}}^{3} \) . In fact, \( {\mathbf{R}}^{3} - {M}^{2} \) is the union of two connected sets \( I \) and \( E \), with \( {M}^{2} \) as their common frontier. | Proof. Let \( P \) be a point in the unbounded component \( E \) of \( {\mathbf{R}}^{3} - {M}^{2} \), and let \( B \) be a broken line \( {PQ} \) such that \( B \cap {M}^{2} \) is a point which lies in the interior of an edge of \( B \) and in the interior of a 2-simplex of \( {M}^{2} \) . We shall show that \( Q \) li... | Yes |
In \( {\mathbf{R}}^{3} \), let \( {M}_{1}^{2},{M}_{2}^{2} \), and \( {M}_{3}^{2} \) be connected polyhedral 2-manifolds with boundary, such that the sets \( \operatorname{Bd}{M}_{i}^{2} \) are all the same, and the sets Int \( {M}_{i}^{2} \) are disjoint. Let \( E \) be the unbounded component of \( {\mathbf{R}}^{3} - ... | Let \( e \) be an edge of Bd \( {M}_{i}^{2} \), and let \( {C}^{2} \) be a small circular region which is orthogonal to \( e \) at an interior point of \( e \) . Now proceed as in the proof of Theorem 2.7. | No |
Theorem 8. Let \( K \) be a triangulation of \( {\mathbf{R}}^{3} \), and let \( {M}^{2} \) be a compact connected 2-manifold which forms a polyhedron in \( \left| K\right| \) . Then \( {M}^{2} \) is orientable. | Proof. We know by Theorem 6 that \( {\mathbf{R}}^{3} - {M}^{2} \) has exactly two components. Let \( {C}_{1}^{3} \) be a combinatorial 3-cell in \( {\mathbf{R}}^{3} \), containing \( {M}^{2} \) in its interior, and let \( {C}_{2}^{3} \) be another combinatorial 3-cell, such that \( {C}_{1}^{3} \cap {C}_{2}^{3} = \opera... | Yes |
Theorem 1. Let \( {M}^{3} \) be a PL 3-manifold, let \( N \) be a polyhedral 3-manifold with boundary, lying in \( {M}^{3} \), and let \( {M}^{2} \) be a polyhedral 2-manifold (not necessarily compact) lying in \( \mathrm{{Bd}}N \) . Then every cellular PLH \( h : {M}^{2} \leftrightarrow {M}^{2} \) has a PLH extension ... | Proof. Let \( d \subset {M}^{2} \) be as in the definition of a cellular PLH. Let \( {C}_{1} \) and \( {C}_{2} \) be as in Problem 26.3, with \( {C}_{1},{C}_{2} \subset W \) . Define \( {h}^{\prime } \) as the identity on CL \( \left\lbrack {{M}^{3} - \left( {{C}_{1} \cup {C}_{2}}\right) }\right\rbrack \), and define \... | Yes |
Theorem 2. Let \( A \) be a PL annulus, and let \( J \) and \( {J}^{\prime } \) be polygons in \( \operatorname{Int}A \) , neither of which bounds a 2-cell in \( A \) . Then there is PLH \( h : A \leftrightarrow A \) such that (1) \( h\left( J\right) = {J}^{\prime },\left( 2\right) h \mid \mathrm{{Bd}}A \) is the ident... | Proof. There is no loss of generality in supposing that \( {J}^{\prime } \) forms the mid-line in a rectangular diagram of \( A \), as in Figure 27.1. We then proceed as follows.\n\n\n\nFigure 27.1\n\n(1) By a finite... | Yes |
Theorem 4. Let \( N \) be a polyhedral 3-manifold with boundary, in a PL 3-manifold \( {M}^{3} \), let \( A \) be a polyhedral annulus in \( \mathrm{{Bd}}N \), and let \( J \) and \( {J}^{\prime } \) be polygons in Int \( A \), neither of which bounds a 2-cell in Int \( A \) . Let \( W \) be a neighborhood of \( A \) (... | Proof. Let \( h \mid A \) be the \( h \) given by Theorem 2. Then extend \( h \) to all of \( {M}^{3} \) by repeated applications of Theorem 1. | No |
Theorem 5 (The Dehn lemma, C. Papakyriakopoulos). Let \( {M}^{3} \) be a PL 3-manifold, and let \( D : \Delta \rightarrow {M}^{3} \) be a PL singular 2-cell with no singularities on its boundary. Then the polygon \( D\left( {\mathrm{{Bd}}\Delta }\right) \) is the boundary of a polyhedral 2-cell \( {\Delta }_{1} \) in \... | Proof. Let \( {M}^{3} = \left| K\right| \) . We may suppose that \( \left| D\right| = D\left( \Delta \right) \) is a subcomplex of \( K \) . Let \( L = \mathrm{{Bd}}D \), so that \( \left| L\right| = D\left( {\mathrm{{Bd}}\Delta }\right) \) ; let \( N\left( \left| D\right| \right) \) be the regular neighborhood of \( \... | Yes |
Theorem 2. Let \( {J}_{x} \) be latitudinal in \( S \), and let \( J \) be a polygon in \( T \) . Then there is a PLH \( h : {M}^{3} \leftrightarrow {M}^{3}, S \leftrightarrow S \), such that \( h\left( J\right) \) is in standard position relative to \( {J}_{x} \) . And given any neighborhood \( W \) of \( T, h \) can ... | Proof. Evidently \( J \) can be moved into general position by a finite sequence of cellular PL homeomorphisms. By Theorem 27.1, \( J \) can be moved into general position by a PLH \( {h}_{1} : {M}^{3} \leftrightarrow {M}^{3}, S \leftrightarrow S \), such that \( {h}_{1} \mid \left( {{M}^{3} - W}\right) \) is the ident... | Yes |
Theorem 3. Let \( {J}_{x} \) be latitudinal on \( S \), and let \( {J}_{1},{J}_{2},\ldots ,{J}_{n} \) be disjoint polygons in \( T \) . Then there is a PLH \( h : {M}^{3} \leftrightarrow {M}^{3}, S \leftrightarrow S \), such that each set \( h\left( {J}_{i}\right) \) is in standard position relative to \( {J}_{x} \) . ... | Proof. First move \( {J}_{1} \) into standard position, as in the preceding proof, by a PLH \( {h}_{1} \) . Then move \( {h}_{1}\left( {J}_{2}\right) \) into standard position, by a PLH \( {h}_{2} \), chosen so that \( {h}_{2}{h}_{1}\left( {J}_{1}\right) \) is in standard position. (Note that \( {h}_{1}\left( {J}_{1}\r... | Yes |
Theorem 4. Let \( J \) and \( {J}_{x} \) be polygons in \( T \), such that \( {J}_{x} \) is latitudinal and \( J \) is in standard position relative to \( {J}_{x} \) . Let \( n \) be the number of points in \( J \cap {J}_{x} \) . Then \( {Z}^{1}\left( J\right) \sim n{Y}^{1} \) on \( S \), where \( {Y}^{1} \) is a gener... | Proof. Let \( {J}_{v} \) be a polygon in \( T \) which appears in the cylindrical diagram as a broken line with its end-points in the two bases. Let \( {Y}^{1} \) be a 1-cycle defined by an orientation of \( {J}_{v} \) . Either orientation makes \( {Y}^{1} \) a generator of \( {H}_{1}\left( S\right) \), and one of them... | No |
Theorem 5. Let \( J \) be a polygon in \( T \) . If \( J \sim 0 \) on \( S \) but not on \( T \), then \( J \) is latitudinal in \( S \) . | Proof. Let \( {J}_{x} \) be a latitudinal polygon in \( S \) . Let \( h \) be as in Theorem 2, so that \( h\left( J\right) \) is in standard position relative to \( {J}_{x} \) . Let \( n \) be the number of points in \( h\left( J\right) \cap {J}_{x} \) . Since \( J \sim 0 \) on \( S \), it follows that \( n = 0 \) . Le... | Yes |
Theorem 6. Let \( {J}_{1},{J}_{2},\ldots ,{J}_{n}\left( {n > 1}\right) \) be disjoint polygons in \( T \), such that \( {J}_{i} \nsim 0 \) on \( T \) for each \( i \) . Let \( U \) be a component of \( T - \cup {J}_{i} \), and let \( A = \bar{U} \) . Then \( A \) is an annulus, and \( \operatorname{Bd}A = {J}_{i} \cup ... | Proof. By Theorem 3, we may suppose that all the polygons \( {J}_{i} \) are in standard position relative to a latitudinal polygon \( {J}_{x} \) . There are now two cases.\n\nCASE 1. \( A \) appears in the boundary of the cylindrical diagram as a finite union of disjoint 2-cells, each of which intersects each of the ba... | Yes |
Theorem 8. Let \( {J}_{1},{J}_{2},\ldots ,{J}_{n} \) be disjoint polygons on \( T \), such that \( {J}_{i} \nsim 0 \) on \( S \) for each \( i \), and such that \( \cup {J}_{i} \) carries a generator of \( {H}_{1}\left( S\right) \) . Then each \( {J}_{i} \) carries a generator of \( {H}_{1}\left( S\right) \) . | Proof. As in the proof of Theorem 6, we may suppose that all the polygons \( {J}_{i} \) are in standard position relative to a latitudinal polygon \( {J}_{x} \) . And obviously we may assume that \( n > 1 \) . Now each component \( A \) of \( T - \bigcup {J}_{i} \) is as in Case 1 in the proof of Theorem 6. It follows ... | Yes |
Theorem 9. Let \( J \) be a polygon in \( T \). If \( J \sim 0 \) on \( T \), then \( J \) bounds a 2-cell in \( T \). | Proof. By a PLH as in Theorem \( 2, J \) can be moved onto a polygon \( {J}^{\prime } \) which is in standard position relative to a latitudinal polygon \( {J}_{x} \). Since \( J \sim 0 \) on \( T \), we have \( {J}^{\prime } \sim 0 \) on \( T \). By Theorem \( 4,{J}^{\prime } \) intersects neither of the bases of the ... | Yes |
Theorem 10. Let \( K \) be a polyhedron in \( T \), such that \( K \) carries a generator of \( {H}_{1}\left( S\right) \). Let \( J \) be a polygon in \( T - K \), such that \( J \) does not bound a 2-cell in T. Then \( J \) carries a generator of \( {H}_{1}\left( S\right) \). | Proof. Let \( B \) be a regular neighborhood of \( J \), sufficiently small so that \( B \cap K = \varnothing \), and let \( A = \mathrm{{Cl}}\left( {T - B}\right) \). By Theorems 7 and \( 9, A \) is an annulus; and \( K \subset A \). Let \( \operatorname{Bd}A = {J}_{1} \cup {J}_{2} = \operatorname{Bd}B \). Since \( {J... | Yes |
Theorem 11. Let \( {K}_{1} \) and \( {K}_{2} \) be complexes whose union is a complex \( K \) . Let \( {Z}^{n} \) be a cycle on \( {K}_{1} \), such that \( {Z}^{n} \sim 0 \) on \( K \) . Then there is a cycle \( {Y}^{n} \), on \( {K}_{1} \cap {K}_{2} \), such that (1) \( {Y}^{n} \sim {Z}^{n} \) on \( {K}_{1} \) and (2)... | Proof. Let \( {C}^{n + 1} \) be an \( \left( {n + 1}\right) \) -chain on \( K \), such that \( \partial {C}^{n + 1} = {Z}^{n} \) . Let \( {C}^{n + 1} \land {K}_{1} \) be the sum of all terms \( {\alpha }_{i}{\sigma }_{i}^{n + 1} \) of \( {C}^{n + 1} \) such that \( {\sigma }_{i}^{n + 1} \in {K}_{1} \) . Let\n\n\[ \n{Y}... | Yes |
Theorem 12. Let \( S \) be a CST in \( {\mathbf{R}}^{3} \) (or \( {\mathbf{S}}^{3} \) ), and let \( T = \mathrm{{Bd}}S \) . Let \( \Delta \) be a polyhedral 2-cell in \( \mathrm{{Cl}}\left( {{\mathbf{R}}^{3} - S}\right) \), such that \( J = \mathrm{{Bd}}\Delta = \Delta \cap T \), and such that \( J \sim 0 \) on \( T \)... | Proof. Let \( {Z}^{1} \) be a generator of \( {H}_{1}\left( S\right) \) ; let \( N \) be a regular neighborhood of \( \Delta \), and triangulate \( {\mathbf{R}}^{3} \) in such a way that \( S,\Delta, N \), and \( J \) form subcomplexes of the triangulation. Let\n\n\[ D = S \cup N = S \cup \mathrm{{Cl}}\left( {N - S}\ri... | Yes |
Theorem 13. Let \( S \) be a regular neighborhood of a polygon \( {J}_{0} \) in \( {\mathbf{R}}^{3} \) (or \( {\mathbf{S}}^{3} \) ). Let \( T = \mathrm{{Bd}}S \), let \( J \) be a polygon in \( T \), such that \( J \mathrel{\text{\sim \not{} }} 0 \) on \( T \), and let \( \Delta \) be a polyhedral 2-cell such that \( \... | Proof. Let \( {Z}^{1} = {Z}^{1}\left( J\right) \) be an orientation of \( J \) . By the preceding theorem, \( {Z}^{1} \) generates \( {H}_{1}\left( S\right) \) . Let \( {J}_{x} \) be a latitudinal polygon in \( T \) . By Theorem 2, there is a PLH \( {\mathbf{R}}^{3} \leftrightarrow {\mathbf{R}}^{3}, S \leftrightarrow S... | Yes |
Theorem 14. Let \( J \) be a polygon in \( {\mathbf{R}}^{3} \) (or \( {\mathbf{S}}^{3} \) ). Then \( {H}_{1}\left( {{\mathbf{R}}^{3} - J}\right) \approx \mathbf{Z} \) . And if \( S \) is a regular neighborhood of \( J \), with boundary \( T \), and \( {J}_{x} \) is latitudinal on \( T \), then \( {Z}^{1}\left( {J}_{x}\... | Proof. In the fundamental group of \( {\mathbf{R}}^{3} - J \), we have generators \( {g}_{1},{g}_{2},\ldots ,{g}_{n} \), and relations of the form \( {g}_{i}{g}_{k}^{-1}{g}_{j}^{-1}{g}_{k} \cong e \) . When we make the group commutative, passing from \( \pi \left( {{\mathbf{R}}^{3} - J}\right) \) to \( {H}_{1}\left( {{... | Yes |
Theorem 15. Let \( J \) be a polygon in \( {\mathbf{R}}^{3} \) (or \( {\mathbf{S}}^{3} \) ). Let \( S \) be a regular neighborhood of \( J \) . Then there is a polygon \( {J}_{v} \) in \( T = \mathrm{{Bd}}S \) such that \( {Z}^{1}\left( {J}_{y}\right) \) generates \( {H}_{1}\left( S\right) \) and \( {Z}^{1}\left( {J}_{... | Proof. Let \( {J}_{x} \) be a latitudinal polygon in \( T \) . Now \( {Z}^{1}\left( {J}_{x}\right) \) generates \( {H}_{1}\left( {{\mathbf{R}}^{3} - J}\right) \), and \( \mathrm{{Cl}}\left( {{\mathbf{R}}^{3} - S}\right) \) is a deformation retract of \( {\mathbf{R}}^{3} - J \) . Since the inclusion \( \mathrm{{Cl}}\lef... | Yes |
Theorem 16. Let \( J \) be a polygon in \( {\mathbf{R}}^{3} \) (or \( {\mathbf{S}}^{3} \) ), and suppose that \( \pi \left( {{\mathbf{R}}^{3} - J}\right) \) is commutative. Then \( J \) is unknotted. | Proof. If \( \pi \left( {{\mathbf{R}}^{3} - J}\right) \) is commutative, then \( \pi \left( {{\mathbf{R}}^{3} - J}\right) \approx {H}_{1}\left( {{\mathbf{R}}^{3} - J}\right) \approx \mathbf{Z} \) . Let \( S \) be a regular neighborhood of \( J \), let \( {J}_{x} \) be latitudinal in \( T = \mathrm{{Bd}}S \) , and take ... | Yes |
Theorem 17 (Henri Poincaré). There is a compact connected triangulated 3-manifold which has the homology groups of a 3-sphere, but is not simply connected (and hence is not a 3-sphere). | Proof. Let \( J \) be a knotted polygon in \( {\mathbf{S}}^{3} \), let \( S \) be a regular neighborhood of \( J \), with Bd \( S = T \), let \( {J}_{x} \) be latitudinal in \( T \), and let \( {J}_{v} \) be as in Theorem 15, so that \( {Z}^{1}\left( {J}_{y}\right) \) generates \( {H}_{1}\left( S\right) \) and \( {Z}^{... | Yes |
Lemma 1. \( K \) is orientable. | Proof. Assign to \( {K}_{1} \) and \( {K}_{2} \) orientations which induce opposite orientations of \( {K}_{1} \cap {K}_{2} \) . This gives an orientation of \( K \) . | Yes |
Lemma 2. \( {H}_{0}\left( K\right) \approx {H}_{0}\left( {\mathbf{S}}^{3}\right) \) . | Proof. Because both \( K \) and \( {\mathbf{S}}^{3} \) are connected. | No |
Lemma 3. \( {H}_{1}\left( K\right) = 0\left( { = {H}_{1}\left( {\mathrm{\;S}}^{3}\right) }\right) \) . | Proof. Let \( {Z}^{1} \) be a 1-cycle on \( K \), and let \( {Y}^{0} = \partial \left( {{Z}^{1} \land {K}_{1}}\right) \) . Since \( {K}_{1} \cap {K}_{2} \) is connected, \( {Y}^{0} \sim 0 \) on \( {K}_{1} \cap {K}_{2} \), and \( {Y}^{0} = \partial {C}^{1} \), where \( {C}^{1} \) is a 1-chain on\n\n\( {K}_{1} \cap {K}_{... | Yes |
Lemma 4. \( {H}_{2}\left( K\right) = 0\left( { = {H}_{2}\left( {\mathrm{\;S}}^{3}\right) }\right) \) . | Proof. By Lemmas 1 and 3, together with the Poincaré duality theorem. [ST], p. 245. | No |
Lemma 5. \( {H}_{3}\left( K\right) \approx \mathbf{Z} \approx {H}_{3}\left( {\mathbf{S}}^{3}\right) \) . | Proof. Because both \( K \) and \( {\mathbf{S}}^{3} \) are orientable.\n\nThus \( K \) has the same homology groups as \( {\mathbf{S}}^{3} \) . Therefore Lemma 6 will complete the proof of Theorem 17. | No |
Lemma 6. \( \left| K\right| \) is not simply connected. | Proof. If \( \left| K\right| \) is simply connected, then the homomorphism\n\n\[ \n{i}^{ * } : \pi \left( T\right) \rightarrow \pi \left( \left| K\right| \right)\n\]\n\ninduced by the inclusion \( T \rightarrow \left| K\right| \) has a nontrivial kernel. By Theorem 26.4 it follows that there is a 2-cell \( \Delta \) in... | No |
Theorem 18. Let \( {M}^{2} \) be a polyhedral projective plane, in an orientable PL 3-manifold \( M \), and let \( N \) be a regular neighborhood of \( {M}^{2} \) . Then \( \mathrm{{Bd}}N \) is a 2-sphere. | Proof. Let \( K \) be a triangulation of \( {M}^{3} \) in which \( {M}^{2} \) forms a subcomplex. Let \( \psi \) be the usual PL identification mapping \( {\left\lbrack 0,1\right\rbrack }^{2} \rightarrow {M}^{2} \), and let \( J = \) \( \psi \left( {\mathrm{{Bd}}{\left\lbrack 0,1\right\rbrack }^{2}}\right) \), so that ... | Yes |
Theorem 19. Let \( {M}^{2} \) be a polyhedral 2-manifold, in a PL 3-manifold \( {M}^{3} = \left| K\right| \) . Let \( \Delta \) be a polyhedral 2-cell such that \( \mathrm{{Bd}}\Delta = J = \Delta \cap {J}^{2} \) . Then \( J \) has an annular neighborhood in \( {M}^{2} \) . | Proof. We may suppose that \( {M}^{2} \) and \( \Delta \) form subcomplexes of \( K \) . Let \( N \) be the regular neighborhood of \( J \) relative to \( K \) . By a very special case of Theorem 24.12, \( N \) is a CST. Now \( \Delta \cap \mathrm{{Bd}}N \) is a polygon, and \( \Delta \cap N \) is an annulus, so that \... | Yes |
Theorem 20. Let \( {M}^{2} \) be a polyhedral 2-manifold, in a PL 3-manifold \( {M}^{3} \) , and let \( \Delta \) be as in Theorem 19. Suppose that \( {M}^{2} \rightarrow {M}_{1}^{2} \), under an operation which splits \( {M}^{2} \cup \Delta \) apart at \( \Delta \) . Then \( \chi \left( {M}_{1}^{2}\right) = \chi \left... | Proof. By Theorem 21.10, when \( {M}^{2} \) is split apart at \( J = \mathrm{{Bd}}\Delta \), the Euler characteristic is unchanged. When we add two new 2-cells, we have \( \chi \left( {M}^{2}\right) \rightarrow \chi \left( {M}^{2}\right) + 2 \) | Yes |
Theorem 1. Let \( X \) be a simply connected and locally connected topological space in which every connected open set is pathwise connected. Let \( H, K \) , \( C \), and \( D \) be disjoint closed sets, and suppose that both \( H \) and \( K \) are connected. If \( C \cup D \) separates \( H \) from \( K \) (in \( X ... | Proof. Suppose not, and let \( U \) be the component of \( X - C \) that contains \( H \) . Since \( X - C \) is locally connected, all components of \( X - C \) are open. Let \( V \) be the union of all components of \( X - C \) other than \( U \) . If \( K \subset V \) , then \( C \) separates \( H \) from \( K \), c... | Yes |
Theorem 2. Let \( X, H \), and \( K \) be as in Theorem 1. Let \( C \) be a closed set which separates \( H \) from \( K \) (in \( X \) ), and suppose that \( C \) has only a finite number of components. Then some component of \( C \) separates \( H \) from \( K \) . | Proof. This follows from Theorem 1, by induction. | No |
Theorem 3. Let \( M \) be a connected PL 3-manifold, let \( H \) and \( K \) be disjoint closed sets in \( M \), and let \( C \) be a closed set which separates \( H \) from \( K \) . Let \( \Delta \) be a polyhedral 2-cell in \( C \), let \( J = \mathrm{{Bd}}\Delta \), and suppose that \( \Delta \) has a neighborhood ... | Proof. Here the splitting operation, for \( C \) at \( \Delta \), leaving \( {D}_{2} \) fixed, is defined in exactly the same way as for \( {M}^{2} \cup \Delta \), in the discussion just before Theorem 28.20. We choose a regular neighborhood \( N\left( \Delta \right) \) such that \( N\left( \Delta \right) \cap \) \( \l... | Yes |
Theorem 4. Let \( X \) be a spherical shell in \( {\mathbf{R}}^{3} \) (or \( {\mathbf{S}}^{3} \) ). Then there is a polyhedral 2-sphere \( B \) in \( \operatorname{Int}X \) such that \( B \) separates \( {B}_{0} \) from \( {B}_{1} \) in \( {\mathbf{R}}^{3} \) (and hence in \( X \) ). | Proof. Let \( N \) be a finite polyhedral closed neighborhood of \( {B}_{0} \) (in \( {\mathbf{R}}^{3} \) ) such that \( N \) is a 3-manifold with boundary and \( N \cap {B}_{1} = \varnothing \) . By Theorem 23.8, Bd \( N \) is the frontier of \( N \) in \( {\mathbf{R}}^{3} \) ; and since Bd \( N \) separates \( {B}_{0... | Yes |
Theorem 5. Let \( {C}_{1} \) and \( {C}_{2} \) be topological 3-cells in \( {\mathbf{R}}^{3} \) (or \( {\mathbf{S}}^{3} \) ) such that \( {C}_{1} \subset \) Int \( {C}_{2} \) and such that \( \mathrm{{Cl}}\left( {{C}_{2} - {C}_{1}}\right) \) is a spherical shell. Then there is a polyhedral 3-cell \( C \) such that\n\n\... | Proof. Let \( B \) be a polyhedral 2-sphere in Int \( \mathrm{{Cl}}\left( {{C}_{2} - {C}_{1}}\right) \), such that \( B \) separates Bd \( {C}_{1} \) from Bd \( {C}_{2} \) (in \( {\mathbf{R}}^{3} \) or \( {\mathbf{S}}^{3} \) ). Let \( C \) be the closure of the component of \( {\mathbf{R}}^{3} - B \) (or \( {\mathbf{S}... | Yes |
Theorem 7. Let \( {S}_{1} \) and \( {S}_{2} \) be (topological) solid tori in \( {\mathbf{R}}^{3} \) (or \( {\mathbf{S}}^{3} \) ) such that \( {S}_{1} \subset \operatorname{Int}{S}_{2} \) and \( \mathrm{{Cl}}\left( {{S}_{2} - {S}_{1}}\right) \) is a toroidal shell. Then there is a combinatorial solid torus (CST) \( S \... | Proof. By Theorem 6 there is a polyhedral torus \( T \) in Int \( \mathrm{{Cl}}\left( {{S}_{2} - {S}_{1}}\right) \) , separating Bd \( {S}_{1} \) from Bd \( {S}_{2} \) . Let \( S \) be the closure of the component of \( {\mathbf{R}}^{3} - T \) that contains \( {S}_{1} \) . (This is of course the bounded component.) We ... | Yes |
Theorem 8. Let \( {S}_{1}, S \), and \( {S}_{2} \) be as in Theorem 7, let \( J \) be a spine of \( {S}_{1} \), and let \( {P}_{0} \in J \) . Then \( {p}_{J} \) generates \( \pi \left( S\right) = \pi \left( {S,{P}_{0}}\right) \) . | Proof. Let \( q \) be a closed path which generates \( \pi \left( S\right) = \pi \left( {S,{P}_{0}}\right) \) . Then \( {p}_{J} \cong {q}^{n} \) in \( \pi \left( S\right) \) for some \( n \), and so \( {p}_{J} \cong {q}^{n} \) in \( \pi \left( {S}_{2}\right) \) . Now \( {p}_{J} \) generates \( \pi \left( {S}_{2}\right)... | Yes |
Theorem 2. In a canonical configuration, each of the sets \( {J}_{j}^{\prime } \) and \( {J}_{j + 1}^{\prime } \) carries a generator of \( \pi \left( {S}_{j}^{\prime \prime }\right) \) . | Proof. This follows by repeated applications of Theorem 30.8. | No |
Theorem 3. In a canonical configuration, \( {S}_{i}^{\prime \prime } \cap {S}_{i + 2}^{\prime \prime } = \varnothing \) . | Proof. By hypothesis, \( {D}_{i} \cap {D}_{i + 2} = \varnothing \) . Therefore \( {S}_{i} \cap {S}_{i + 2} = \varnothing = {S}_{i}^{\prime } \cap \) \( {S}_{i + 2}^{\prime } \) . Since \( {S}_{j}^{\prime \prime } \subset {S}_{j}^{\prime } \), the theorem follows. | Yes |
Theorem 1. Let \( N, K \), and \( h : N \leftrightarrow {N}^{\prime } \subset {\mathbf{R}}^{3} \) be as in the definition of a tube. Let \( D \) be a splitting disk of \( N \), and let \( \{ P\} = D \cap \left| K\right| \) . Let \( {C}_{1} \) and \( {C}_{2} \) be the dual cells of \( N \) such that \( {C}_{1} \cap {C}_... | Proof. In the proof, we shall be concerned not with all of \( N \), but merely with \( {C}_{1} \cup {C}_{2} \) . Therefore, without loss of generality, we may regard \( D \) as a closed circular region, with center at the origin \( P \), in the \( {xz} \) -plane in \( {\mathbf{R}}^{3} \) . We decompose Int \( D - \{ P\... | Yes |
Lemma 1. Let \( X \) be a subset of \( \mathop{\bigcup }\limits_{i}{S}_{i}^{\prime } \cup \left\{ {P}^{\prime }\right\} \), containing \( {P}^{\prime } \), such that each set \( X \cap {S}_{i}^{\prime } \) is closed. Then (1) C1 \( X \subset X \cup \) Bd \( {D}^{\prime } \), so that (2) \( X \) is closed in Int \( \lef... | Proof. Let \( Q \in \mathrm{{Cl}}X - X \) . Then \( Q \) is not a limit point of any one set \( X \cap {S}_{i}^{\prime } \) . Therefore \( Q \notin \mathop{\bigcup }\limits_{i}{S}_{i}^{\prime } \cup \left\{ {P}^{\prime }\right\} \) . It follows that \( Q \in \mathrm{{Bd}}{D}^{\prime } \), so that (1) holds. Since Bd \(... | Yes |
Lemma 3. \( {M}_{1} \) is closed in \( \operatorname{Int}\left( {{C}_{1}^{\prime } \cup {C}_{2}^{\prime }}\right) \), and separates \( {v}_{1}^{\prime } \) from \( {v}_{2}^{\prime } \) in Int \( \left( {{C}_{1}^{\prime } \cup {C}_{2}^{\prime }}\right) \) . | Proof. The first of these conclusions follows from Lemma 1. To prove the second, we note that Int \( D \) separates \( {v}_{1} \) from \( {v}_{2} \) in Int \( \left( {{C}_{1} \cup {C}_{2}}\right) \) ; and this property is preserved by \( h \) . Therefore Int \( {D}^{\prime } \) separates \( {v}_{1}^{\prime } \) from \(... | Yes |
Under the conditions of Theorem 1, E can be chosen so that \( \left( {{C}_{1}^{\prime } \cup {C}_{2}^{\prime }}\right) - E \) has exactly two components \( {U}_{1},{U}_{2} \), and such that for \( i = 1,2 \) we have (5) \( E \subset \operatorname{Fr}{U}_{i} \) and (6) Bd \( {C}_{i}^{\prime } \cap \) Bd \( {N}^{\prime }... | Proof. Evidently \( {v}_{i}^{\prime } \) can be joined to a point of Bd \( {C}_{i}^{\prime } \cap \) Bd \( {N}^{\prime } \) by an arc \( {B}_{i} \) in \( {C}_{i}^{\prime } - {D}^{\prime } \) . For appropriate choice of \( W \), we have \( {B}_{i} \cap E = \varnothing \) . Now the sets Int (Bd \( {C}_{i}^{\prime } \cap ... | Yes |
Theorem 3. Under the conditions just stated, the sets \( {W}_{e} \) can be chosen so that (7) each set \( {C}_{i}^{\prime \prime } \) contains only one vertex of \( {K}^{\prime },\left( 8\right) \) the sets \( {C}_{i}^{\prime \prime } \) lie in arbitrarily small neighborhoods of the sets \( {C}_{i}^{\prime },\left( 9\r... | Proof. For each \( e \), let \( {C}_{e,1} \) and \( {C}_{e,2} \) be two 3-cells intersecting in \( {D}_{e} \), such that \( {C}_{e,1} \cup {C}_{e,2} \) is a neighborhood of \( {D}_{e} \) in \( N \) . We choose these so that they lie in small neighborhoods of the sets \( {D}_{e} \), and so that the closure of \( N - \bi... | No |
Theorem 4. Let \( E \) be a pseudo-cell with center \( {P}^{\prime } \), in \( {\mathbf{R}}^{3} \), and let \( \delta \) be a positive number. Then there is a polyhedral 2-cell \( {\Delta }_{1} \), lying in \( N\left( {{P}^{\prime },\delta }\right) \), such that (1) \( J = \mathrm{{Bd}}{\Delta }_{1} = {\Delta }_{1} \ca... | Proof. Let \( {C}^{3} \) be a polyhedral 3-cell in \( N\left( {{P}^{\prime },\delta }\right) \), containing \( {P}^{\prime } \) in its interior, such that \( {C}^{3} \cap E \) lies in a 2-cell in \( E \cap N\left( {{P}^{\prime },\delta }\right) \), and such that Bd \( {C}^{3} \) is in general position relative to \( E ... | No |
Theorem 1. Let \( K \) be a finite connected 1-dimensional polyhedron in \( {\mathbf{R}}^{3} \), such that \( K \) has no end-points. Let \( U \) be an open set containing \( K \), and let \( h \) be a homeomorphism \( U \rightarrow {\mathbf{R}}^{3} \). Let \( \varepsilon \) be a positive number. Then there is a regula... | Proof. We shall regard \( K \) as a finite complex. For each vertex \( v \) of \( K \), let St \( v \) be the star of \( v \) in \( K \). Evidently, by a suitable subdivision, we can make the diameters \( \delta \left| {\mathrm{{St}}v}\right| \) as small as we please. Since \( K \) is compact, \( h \mid K \) is uniform... | Yes |
Lemma 3. \( X \) can be chosen so that if \( J \) is a component of a set \( E \cap \operatorname{Bd}X \) , then \( J \) does not bound a disk in Int \( E - \left\{ {P}^{\prime }\right\} \) . | Proof. Given \( X \) as in Lemma 2, suppose that some such \( J \) bounds a disk \( {D}_{J} \) in \( E - \left\{ {P}^{\prime }\right\} \) . Then some such \( J \) is inmost in \( E \), in the sense that \[ \text{Bd}X \cap \operatorname{Int}{D}_{J} = \varnothing \text{.} \] We split \( {D}_{J} \cup \operatorname{Bd}X \)... | Yes |
Lemma 4. We may also assume that each set \( E \cap \operatorname{Bd}X \) is a single polygon. | Proof. The components of \( E \cap \operatorname{Bd}X \) appear in \( E \) in topologically the same way as concentric circles in a circular disk. Therefore, if some set \( E \cap \mathrm{{Bd}}X \) contains more than one polygon, it must contain two polygons \( {J}_{1} \) and \( {J}_{2} \) such that \( {J}_{1} \cup {J}... | Yes |
Lemma 5. \( X \) can also be chosen so that both \( X \) and \( \operatorname{Bd}X \) are connected. | Proof. The first part is trivial: delete from \( X \) all components of \( X \) except the one which contains \( {K}^{\prime } \) . (We recall that \( K \) is connected.) To get the second part, add to \( X \) all components of \( {\mathbf{R}}^{3} - X \) except the unbounded component (which contains Bd \( {N}^{\prime ... | Yes |
Lemma 6. Under the conditions of the preceding lemmas, each set \[ {C}_{v}^{\prime \prime } \cap \operatorname{Bd}X \] is a connected polyhedral 2-manifold \( {A}_{v}^{\prime } \) with boundary, and \( \operatorname{Bd}{A}_{v}^{\prime } \) lies in the union of the pseudo-cells \( E \) that lie in \( {C}_{v}^{\prime \pr... | Proof. All this is trivial, except perhaps for connectivity. Let \( {X}_{v} \) be the component of \( X \cap {C}_{v}^{\prime \prime } \) that contains \( {v}^{\prime } \). Then \( {v}^{\prime } \) lies in \( {X}_{v} - \operatorname{Fr}{X}_{v} \), and so Fr \( {X}_{v} \) separates \( {v}^{\prime } \) from the \ | No |
Lemma 7. Under the conditions of the preceding lemmas, we may also assume that no set \( {C}_{v}^{\prime \prime } \) contains an LTD. | Proof. Suppose that \( {C}_{v}^{\prime \prime } \) contains an LTD \( \Delta \) . If \( \Delta \cap E \neq \varnothing \) for some \( E \) , then \( \Delta \) can be moved slightly off of \( E \) into \( {C}_{v}^{\prime \prime } - E \), preserving the LTD property. We may therefore assume that \( \Delta \cap E = \varno... | Yes |
Lemma 8. Given a set \( {C}_{{v}_{1}}^{\prime \prime } \), and a pseudo-cell \( {E}_{1} \) lying in \( {C}_{{v}_{1}}^{\prime \prime } \). Let \( \Delta \) be a polyhedral disk lying in \( {C}_{{v}_{1}}^{\prime \prime } \cap \operatorname{Int}{N}^{\prime } \), such that (1) Bd \( \Delta = \Delta \cap {E}_{1} \), (2) Bd ... | Proof. Suppose that \( \Delta \cap {K}^{\prime } = \varnothing \). Let \( {D}_{1} \) be the disk in \( {E}_{1} \), bounded by Bd \( \Delta \), so that \( {P}_{1}^{\prime } \in \operatorname{Int}{D}_{1} \). Let \( {P}_{2}^{\prime } \) be the center of some pseudo-cell \( {E}_{2} \neq {E}_{1} \) that lies in \( {C}_{{v}_... | Yes |
Lemma 9. Under the conditions of Lemmas 1-7, Int \( {N}^{\prime } \) contains no LTD. | Proof. Suppose that \( \Delta \) is an LTD lying in Int \( {N}^{\prime } \) . We may suppose that \( \Delta \) is in general position relative to the pseudo-cells \( E \), in the sense that each component of each set \( E \cap \Delta \) is either a polygon \( J \) lying in Int \( E \cap \) Int \( \Delta \) or a broken ... | Yes |
Lemma 11. Bd \( X \) and Bd \( N \) are homeomorphic. | Proof. By Lemma 10, \( \pi \left( {\operatorname{Bd}X}\right) \approx \pi \left( {\operatorname{Bd}N}\right) \) . Therefore \( {H}_{1}\left( {\operatorname{Bd}X}\right) \approx \) \( {H}_{1}\left( {\mathrm{{Bd}}N}\right) \), so that \( {p}^{1}\left( {\mathrm{{Bd}}X}\right) = {p}^{1}\left( {\mathrm{{Bd}}N}\right) \) . B... | No |
Lemma 12. Every set \( {A}_{v}^{\prime } \) is either a disk or a disk with holes. | Proof. Evidently each \( {A}_{v}^{\prime } \) is a sphere with holes and possible handles. If no set \( {A}_{v}^{\prime } \) has any handles, then a direct computation gives \( {p}^{1}\left( {\operatorname{Bd}X}\right) = \) \( {p}^{1}\left( {\mathrm{{Bd}}N}\right) \), which we know to be correct. If some \( {A}_{v}^{\p... | No |
Theorem 1. Let \( K \) be a polyhedral 3-cell in \( {\mathbf{R}}^{3} \), let \( h \) be a homeomorphism \( K \rightarrow {\mathbf{R}}^{3} \), and let \( \varepsilon \) be a positive number. Then there is a PLH \( f : K \rightarrow {\mathbf{R}}^{3} \) such that \( f \) is an \( \varepsilon \) -approximation of \( h \) . | Proof. Given such a \( K \), there is a PLH \( \phi : K \rightarrow \) Int \( K \) ; and \( \phi \) can be chosen so as to be arbitrarily close to the identity. Now the homeomorphism \( h \mid \phi \left( K\right) \) can be extended to a neighborhood of the polyhedral 3-cell \( \phi \left( K\right) \) . If \( {f}^{\pri... | No |
Lemma 1. There is a regular neighborhood \( N \) of \( {K}^{1} \), and a PLH\n\n\[ \n{f}_{1} : N \leftrightarrow {N}^{\prime \prime } \subset {\mathbf{R}}^{3} \]\n\n such that\n\n(1) \( {N}^{\prime \prime } \) is a neighborhood of \( h\left( {K}^{1}\right) \) .\n\n(2) \( {D}_{e}^{\prime \prime } \cap {\sigma }^{\prime ... | The proof is by Theorem 33.1. Conditions (2)-(5) hold whenever \( {f}_{1} \) is a sufficiently close approximation of \( h \mid N \) . Note that in Lemma \( 1, N \) can be chosen so as to lie in any given neighborhood of \( {K}^{1} \), and so \( N \) and \( {f}_{1} \) can be chosen so that \( {N}^{\prime \prime } \) li... | Yes |
Lemma 3. Each set \( {\sigma }^{\prime } \) has arbitrarily small polyhedral 3-cell neighborhoods \( {C}_{\sigma } \) such that \( \mathrm{{Bd}}{C}_{\sigma } \) is in general position relative to \( \mathrm{{Bd}}{N}^{\prime \prime } \) and such that Bd \( {C}_{\sigma } \cap \) Bd \( {N}^{\prime \prime } \) is in genera... | Proof. Each \( \sigma \) has arbitrarily small 3-cell neighborhoods \( {C}_{1} \) and \( {C}_{2} \) such that \( \mathrm{{Cl}}\left( {{C}_{2} - {C}_{1}}\right) \) is a spherical shell. Therefore \( {\sigma }^{\prime } \) has the same property. By Theorem 30.5, \( {\sigma }^{\prime } \) has arbitrarily small polyhedral ... | No |
Lemma 4. Let \( C \) be a polyhedral 3-cell neighborhood of a set \( {\sigma }^{\prime } \) . If \( C \) lies in a sufficiently small neighborhood of \( {\sigma }^{\prime } \), then the set\n\n\[ \n\text{Bd}C \cap \text{Bd}{N}_{\sigma }^{\prime \prime } \cap \text{Bd}{N}^{\prime \prime }\n\]\n\ncarries a generator of \... | Proof. Let \( \tau \) be a polyhedral disk in \( K \), such that \( \tau \cap {K}^{2} = \operatorname{Bd}\sigma = \operatorname{Bd}\tau \) . Then\n\n\[ \n{\sigma }^{\prime } \cap {\tau }^{\prime } = \operatorname{Bd}{\sigma }^{\prime } = \operatorname{Bd}{\tau }^{\prime }.\n\]\n\nLet \( C \) and \( {C}^{\prime } \) be ... | Yes |
Lemma 5. Under the conditions of Lemmas 1 and 2, there is a collection \( \left\{ {C}_{\sigma }\right\} \) of polyhedral 3-cells (one for each \( \sigma \) ) such that (1) \( {C}_{\sigma } \) is a neighborhood of \( \mathrm{{Bd}}{\sigma }^{\prime },\left( 2\right) {C}_{\sigma } \cap {C}_{v}^{\prime \prime } \neq \varno... | The proof is by Lemmas 3 and 4. Note that by these lemmas we easily get the following stronger conditions as well: \( \left( {1}^{\prime }\right) {C}_{\sigma } \) is a neighborhood of \( {\sigma }^{\prime } \), and \( \left( {8}^{\prime }\right) \) the sets \( {C}_{\sigma } \) can be chosen so as to lie in arbitrarily ... | Yes |
Lemma 6. Operation 1 preserves the conditions of Lemmas 1, 2, and 5. | Proof. The less trivial verifications are as follows. (2) of Lemma 5 holds because Bd \( {C}_{\sigma }^{\prime } \cap {C}_{v}^{\prime \prime } \neq \varnothing \) only if \( v \) is a vertex of \( \sigma \) . (5) of Lemma 5 holds because different sets Bd \( {C}_{\sigma }^{\prime },{C}_{\tau }^{\prime } \) intersect on... | No |
Lemma 7. Operation 2 preserves the conditions of Lemmas 1, 2, and 5. | Proof. The verifications are all trivial. | No |
Lemma 9. No set \( \mathrm{{Bd}}{C}_{\sigma } \cap \mathrm{{Bd}}{C}_{v}^{\prime \prime } \cap \mathrm{{Bd}}{N}^{\prime \prime } \) contains a polygon. | Proof. Suppose that such a set contains a polygon \( J \) . By general position, \( J \) lies in the interior of the \( k \) -annulus\n\n\[ \n{A}_{v}^{\prime \prime } = \operatorname{Bd}{C}_{v}^{\prime \prime } \cap \operatorname{Bd}{N}^{\prime \prime }.\n\]\n\n\( J \) bounds a disk \( {D}_{J} \) in Bd \( {C}_{v}^{\pri... | Yes |
Lemma 10. No set \( \operatorname{Bd}{C}_{a} \cap {A}_{v}^{\prime \prime } \) contains a broken line \( B \) both of whose end-points \( x \) and \( y \) lie in the same set \( {D}_{e}^{\prime \prime } \) . | Proof. Suppose that there is such a \( B \) . By general position, \( B \) intersects no other splitting disk. Now Bd \( {D}_{e}^{\prime \prime } \) is the union of two broken lines \( {B}_{1} \) and \( {B}_{2} \), with end-points \( x \) and \( y \) . One of the sets \( B \cup {B}_{1}, B \cup {B}_{2} \) bounds a disk ... | Yes |
Lemma 11. For each \( \sigma \), each component \( J \) of \( \mathrm{{Bd}}{C}_{\sigma } \cap \mathrm{{Bd}}{N}_{\sigma }^{\prime \prime } \) crosses each set Bd \( {D}_{e}^{\prime \prime } \) (where \( e \) is an edge of \( \sigma \) ) exactly once. | Proof. By (6) of Lemma 5, the union of the polygons \( J \) carries a generator of \( {H}_{1}\left( {N}_{\sigma }^{\prime \prime }\right) \) . By Theorem 28.8 it follows that each such \( J \) either carries a generator of \( {H}_{1}\left( {N}_{\sigma }^{\prime \prime }\right) \) or bounds a disk in Bd \( {N}_{\sigma }... | Yes |
Lemma 1. Every polygon \( J \) in \( {A}_{e}^{\prime } \cap {B}_{e}^{\prime } \) either bounds a disk in \( {A}_{e}^{\prime } \) and a disk in \( {B}_{e}^{\prime } \) or carries a generator of \( {H}_{1}\left( {A}_{e}^{\prime }\right) \) and a generator of \( {H}_{1}\left( {B}_{e}^{\prime }\right) \) . | Proof. Obviously each component of Bd \( {A}_{e}^{\prime } \) carries a generator of \( {H}_{1}\left( {T}_{e}^{\prime }\right) \) . By (4), Bd \( {f}_{w}\left( {C}_{w}^{\prime }\right) \) separates these components from one another in \( {M}_{2} \), and hence in \( {A}_{e}^{\prime } \) . By (3) it follows that Int \( {... | Yes |
Lemma 2. Let \( J \) be a component of a set \( {A}_{e}^{\prime } \cap {B}_{e}^{\prime } \) . Then \( J \) carries a generator of \( {H}_{1}\left( {A}_{e}^{\prime }\right) \) and a generator of \( {H}_{1}\left( {B}_{e}^{\prime }\right) \) . | Proof. Suppose not. Then \( J \) bounds a disk \( {D}_{J} \) in \( {A}_{e}^{\prime } \) and a disk \( {E}_{J} \) in \( {B}_{e}^{\prime } \) . We may suppose that \( {D}_{J} \) is inmost in \( {A}_{e}^{\prime } \), in the sense that Int \( {D}_{J} \cap {B}_{e}^{\prime } = \) (c). We replace \( {E}_{J} \) by \( {D}_{J} \... | Yes |
Lemma 3. Every set \( {A}_{e}^{\prime } \cap {B}_{e}^{\prime } \) is connected. | Proof. Suppose not. Then there is an annulus \( {B}^{\prime \prime } \subset \) Int \( {B}_{e}^{\prime } \), such that Bd \( {B}^{\prime \prime } \subset {A}_{e}^{\prime } \) and Int \( {B}^{\prime \prime } \cap {A}_{e}^{\prime } = \varnothing \) . Now Bd \( {B}^{\prime \prime } \) is the boundary of an annulus \( {A}^... | Yes |
Theorem 2. Let \( {M}_{1} \) and \( {M}_{2} \) be PL 3-manifolds, let \( K \) be a polyhedral 3-manifold with boundary in \( {M}_{1} \), let \( h \) be a homeomorphism \( K \rightarrow {M}_{2} \), and let \( \phi \) be a strongly positive function on \( K \) . Then there is a PLH \( f : K \rightarrow {M}_{2} \) , such ... | Proof. The proof is virtually a repetition of that of Theorem 34.1. As before, \( K \) can be moved into Int \( K \) by a PLH which is as close to the identity as we please. Thus the theorem reduces to the case in which \( h \) and \( \phi \) are defined on an open set \( U \) containing \( K \) . Just as in the proof ... | Yes |
Theorem 3 (The triangulation theorem for 3-manifolds). Let \( M \) be a 3- manifold. Then there is a complex \( K \) such that \( M \) and \( \left| K\right| \) are homeomorphic. | The proof is by a straightforward analogy with the proof that was used in the 2-dimensional case, in Section 8. For an \ | No |
Theorem 3. In \( {\mathbf{R}}^{3} \), or in a PL 3-manifold \( M \), every semi-locally tame set is tame. In fact, if \( L \) is semi-locally tame, then for every open set \( V \) containing \( L \), and for every \( \psi \gg 0 \) on \( V \), there is a homeomorphism \( {g}^{\prime } : M \leftrightarrow M \) such that ... | Proof. We have given \( V \) and \( \psi \) . Let \( U \) and \( g : U \leftrightarrow {U}^{\prime } \subset M \) be as in the definition of semi-local tameness, so that \( U \) is open, \( L \subset U \), and \( g\left( L\right) \) is a polyhedron. Evidently we may choose \( U \) so that \( U \subset V \) . For each p... | Yes |
Theorem 4. In \( {\mathbf{R}}^{3} \), or in a PL 3-manifold, every locally tame set is tame. | This was proved first by Bing \( \left\lbrack {\mathrm{B}}_{1}\right\rbrack \) and independently by the author \( \left\lbrack {\mathrm{M}}_{7}\right\rbrack ,\left\lbrack {\mathrm{M}}_{8}\right\rbrack \) . For the proofs, we refer the reader to the original sources; the old proofs have not been simplified, as far as we... | No |
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