Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
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Theorem 3. Given \( g \sim h, S \subset \left| g\right| = \left| h\right| \) . If \( S \) forms a face of \( g \), then \( S \) forms a face of \( h \) . | Proof. Given \( S = g\left( \tau \right) \left( {\tau \text{a face of}{\sigma }_{g}}\right) \), let \( \rho = {h}^{-1}\left( {g\left( \tau \right) }\right) = {h}^{-1}\left( S\right) \) . Then \( \rho \) is a face of \( {\sigma }_{h} \) . | No |
Theorem 4. Equivalent coordinate mappings induce the same barycentric coordinate systems in their common image. | Proof. Given \( g \sim h,{\sigma }_{g} = {v}_{0}{v}_{1}\ldots {v}_{k} \), let\n\n\[ \n{w}_{i} = g\left( {v}_{i}\right) ,\;{x}_{i} = {h}^{-1}\left( {w}_{i}\right) .\n\]\n\nSince \( {h}^{-1}\left( g\right) \) is simplicial, we have \( {\sigma }_{h} = {x}_{0}{x}_{1}\ldots {x}_{k} \) . Thus if \( w = g\left( {\sum {\alpha ... | Yes |
Theorem 5. Let \( \mathcal{K} \) be a finite-dimensional PL complex. Then there is a Euclidean complex \( K \) such that there is a simplicial homeomorphism \[ f : \left| K\right| \leftrightarrow \left| \mathcal{K}\right| \] And for each such \( K \) and \( f \) we have \[ \mathcal{K} = \{ \left\lbrack {f \mid \sigma }... | Proof. For each \( \left\lbrack h\right\rbrack \in \mathcal{K} \), let \( {\phi }_{h} \) be the set of all vertices of \( \left\lbrack h\right\rbrack \), and let \( \Phi \left( \mathcal{K}\right) = \left\{ {\phi }_{h}\right\} \) . Then \( \Phi \left( \mathcal{K}\right) \) is a finite-dimensional abstract complex. By Th... | Yes |
Theorem 6. Let \( {\mathcal{K}}_{1} \) and \( {\mathcal{K}}_{2} \) be PL complexes, in the same space \( \left\lbrack {X,\mathcal{O}}\right\rbrack \) . Suppose that if \( \left\lbrack g\right\rbrack \in {\mathcal{K}}_{1},\left\lbrack h\right\rbrack \in {\mathcal{K}}_{2} \), and \( S = \left| \left\lbrack g\right\rbrack... | Proof. Conditions (K.1) and (K.2) hold by hypothesis, and the verification of Condition (K.3) is trivial. | No |
Theorem 1. Let \( M \) be an \( n \) -manifold. Then there is a sequence\n\n\[ \left( {{N}_{1},{N}_{1}^{\prime }}\right) ,\left( {{N}_{2},{N}_{2}^{\prime }}\right) ,\ldots \]\n\nof ordered pairs of open sets in \( M \), such that (1) for each \( i \) there is a homeomorphism\n\n\[ {h}_{i} : {\bar{N}}_{i} \leftrightarro... | Proof. Since \( M \) is locally Euclidean, for each point \( P \) there are open sets \( {N}_{P},{N}_{P}^{\prime } \), with \( P \in {N}_{P}^{\prime } \), and a homeomorphism \( {h}_{P} : {\bar{N}}_{P} \leftrightarrow \bar{D},{\bar{N}}_{P}^{\prime } \leftrightarrow {\bar{D}}^{\prime } \) . Thus the collection \( \left\... | Yes |
Theorem 4. Let \( {K}_{1} \) and \( {K}_{2} \) be triangulated 2-manifolds, let \( U \) be an open set in \( \left| {K}_{1}\right| \), let \( h \) be a homeomorphism \( U \rightarrow \left| {K}_{2}\right| \), and let \( \phi \) be a strongly positive function on \( U \) . Then there is a PLH \( f : U \rightarrow \left|... | Proof. By Theorem \( 2, U \) is a polyhedron, \( = \left| {K}_{U}\right| \), where every \( \sigma \in {K}_{U} \) is a rectilinear subsimplex of a simplex of \( {K}_{1} \) . Thus, in Theorem 6.4, we may take \( {K}_{1} = {K}_{U} \) . By Theorem 6.4 it follows that there is a PLH \( f : U \rightarrow \left| {K}_{2}\righ... | Yes |
Theorem 1. Let \( J \) be a 1-sphere in \( {\mathbf{R}}^{2} \) which is the union of an arc \( A \) and a broken line \( B \), intersecting in their end-points \( P \) and \( Q \) . Let \( I \) be the interior of \( J \) . Let \( R \) and \( S \) be points of Int \( A \) and Int \( B \) respectively. Let \( M \) be the... | Proof. Since \( I \) is connected, there is a broken line \( S{R}^{\prime } \), lying in \( I \) except for its end-points \( S \in \operatorname{Int}B \) and \( {R}^{\prime } \in \operatorname{Int}A \) . (See Figure 9.1.) We take \( S{R}^{\prime } \) in general position relative to \( M \), in the sense that no vertex... | No |
Theorem 2. Let \( J \) be a 1-sphere in \( {\mathbf{R}}^{2} \), with interior \( I \), and let \( A \) be an arc in \( J \) . Then there is a linear interval \( v{v}^{\prime } \), with \( v \in \operatorname{Int}A \) and \( v{v}^{\prime } - v \subset I \) . | Proof. Take \( w \in \operatorname{Int}A \), and take \( \varepsilon > 0 \) such that \( J \cap N\left( {w,\varepsilon }\right) \subset \operatorname{Int}A \) . Since \( w \in \operatorname{Fr}I, N\left( {w,\varepsilon }\right) \) contains a point \( {v}^{\prime } \) of \( I \), and \( {v}^{\prime }w \cap J \subset \op... | Yes |
Theorem 3. Let \( J \) be a 1-sphere in \( {\mathbf{R}}^{2} \), with interior \( I \) . Then there is a sequence \( {G}_{1},{G}_{2},\ldots \) such that (1) for each \( i,{G}_{i} \) is a finite decomposition of \( J \) into arcs intersecting only in their end-points,(2) for each \( i,{G}_{i + 1} \leq \) \( {G}_{i} \) ,(... | Proof. By definition of a 1-sphere, \( J \) is the image of a circle \( C \) under a homeomorphism \( f \) . Let \( E \) be the set of all points of \( J \) that are linearly accessible from \( I \) . By Theorem \( 2, E \) is dense in \( J \) . Therefore \( {f}^{-1}\left( E\right) \) is dense in \( C \) . For every \( ... | Yes |
Theorem 4. Let \( J, I \), and \( {G}_{1},{G}_{2},\ldots \) be as in Theorem 3. Then there is a sequence \( {H}_{1},{H}_{2},\ldots \) of collections of linear intervals \( v{v}^{\prime }\left( {v \in J}\right) \), such that (1) if \( v{v}^{\prime } \in {H}_{i} \), then \( v{v}^{\prime } - v \subset I \), and \( v \) is... | Proof. Every \( {G}_{i} \) is a finite collection. Under the conditions for \( {G}_{1} \), there is an \( {H}_{1} \) satisfying (1) and (2). If the elements of \( {H}_{1} \) are sufficiently short, then \( {H}_{1} \) will also satisfy (3). Now proceed recursively to define \( {H}_{2},{H}_{3},\ldots \) . (At each stage,... | Yes |
Theorem 5. Let \( J \) be a 1-sphere in \( {\mathbf{R}}^{2} \), let \( I \) be its interior, and let \( A \) be an arc in \( J \), with end-points \( {v}_{0} \) and \( {v}_{1} \) . Let \( {v}_{0}{v}_{0}^{\prime } \) and \( {v}_{1}{v}_{1}^{\prime } \) be linear intervals such that \( {v}_{i}{v}_{i}^{\prime } - {v}_{i} \... | Proof. Let \( b \) be a broken line from \( {v}_{0}^{\prime } \) to \( {v}_{1}^{\prime } \), lying in \( I \) . We may assume that \( b \cap {v}_{i}{v}_{i}^{\prime } = {v}_{i}^{\prime } \), since if this condition does not hold, we can replace \( b \) , \( {v}_{0}{v}_{0}^{\prime } \), and \( {v}_{1}{v}_{1}^{\prime } \)... | Yes |
Theorem 1. Let \( J \) be a 1-sphere in a 2-sphere \( {S}^{2} \) . Then \( {S}^{2} - J \) is the union of two disjoint connected open sets \( U \) and \( V \), such that \( J = \operatorname{Fr}U = \operatorname{Fr}V \) . | Proof. Delete any point of \( {S}^{2} - J \), apply the Jordan curve theorem, and then reinstate the deleted point. | No |
Theorem 3. Let \( J \) be a 1-sphere in a 2-sphere \( {S}^{2} \), and let \( h \) be a homeomorphism \( J \leftrightarrow {J}^{\prime } \subset {S}^{2} \). Then \( h \) can be extended to give a homeomorphism \( {S}^{2} \leftrightarrow {S}^{2} \) | Proof. Let \( {C}_{1} \) and \( {C}_{2} \) be the 2-cells whose common boundary is \( J \) ; and similarly \( {C}_{1}^{\prime },{C}_{2}^{\prime } \) for \( {J}^{\prime } \). By Theorem 5.6, \( h \) can be extended to give \( {C}_{i} \leftrightarrow {C}_{i}^{\prime } \). | No |
Theorem 4 (The Schönflies theorem, second form). Let \( J \) be a 1-sphere in \( {\mathbf{R}}^{2} \). Then every homeomorphism of \( J \) into \( {\mathbf{R}}^{2} \) can be extended to give a homeomorphism of \( {\mathbf{R}}^{2} \) onto \( {\mathbf{R}}^{2} \). | Proof. This is deducible from the preceding theorem. | No |
Theorem 6 (The frame theorem). Let \( M \) be a compact set in \( {\mathbf{R}}^{2} \), and let \( U \) be an open set containing \( M \) . Then there is a compact polyhedral 2-manifold \( N \) with boundary such that (1) \( N \) is a neighborhood of \( M,\left( 2\right) N \subset U,\left( 3\right) \) every component of... | Proof. Evidently we can get an \( N \) satisfying (1) and (2) by using a sufficiently fine brick-decomposition of \( {\mathbf{R}}^{2} \), as in Section 4. To get (3) is trivial: we delete useless components.\n\nTo ensure that (4) holds, it is sufficient to choose \( N \) in such a way as to minimize the number of compo... | Yes |
Lemma 1. \( K \) has a subdivision \( {K}_{1} \) such that the sets \( h\left( {\tau }^{1}\right) \left( {{\tau }^{1} \in {K}_{1}}\right) \) form a contracting collection. | Proof. We observed, at the beginning of Section 7, that for each complex \( K,{K}^{0} \) is countable. Therefore so also is \( {K}^{1} \) . Let \( {\sigma }_{1}^{1},{\sigma }_{2}^{1},\ldots \) be the edges of \( K \), with \( {\sigma }_{i}^{1} \neq {\sigma }_{j}^{1} \) for \( i \neq j \) . Since each mapping \( \phi \m... | Yes |
Lemma 2. \( K \) has a subdivision \( {K}_{2} \) such that if \( {\tau }^{1} \in {K}_{2} \), then the set \( {e}^{\prime } = h\left( {\tau }^{1}\right) \) has a neighborhood \( {N}_{{e}^{\prime }} \subset U \) such that\n\n\[ \delta {N}_{{e}^{\prime }} < \operatorname{Inf}\left( {\phi \mid {N}_{{e}^{\prime }}}\right) \... | Proof. Hereafter, the images of the edges and vertices of \( K \) will be called edges and vertices of \( M \) . Obviously every edge \( e = h\left( {\sigma }^{1}\right) \) of \( M \) has a\n\ncompact neighborhood \( {N}_{e} \subset U \) ; and by definition of a strongly positive function we have\n\n\[ \inf \left( {\ph... | Yes |
Lemma 3. The sets \( {I}_{i} \) are disjoint. | Proof. If \( {I}_{i} \) intersects \( {I}_{j}\left( {i \neq j}\right) \), then there is a broken line from an interior point of \( {a}_{i} \) to an interior point of \( {a}_{j} \), lying in \( {N}_{v} \), and not intersecting the arcs \( {e}_{k}^{\prime } \) which lie in Fr \( {I}_{i} \) . This contradicts Theorem 4.2. | Yes |
Lemma 4. \( {N}_{v} = \cup {\bar{I}}_{i} = \cup {J}_{i} \cup \cup {I}_{i} \) . | Proof. If this is false, then \( {N}_{v} - \cup {e}_{i}^{\prime } \) has a component \( V \), different from each of the sets \( {I}_{i} \) . Therefore \( \operatorname{Fr}V \subset \cup {e}_{i}^{\prime } \) . But \( \operatorname{Fr}V \) cannot lie in any one set \( {e}_{j}^{\prime } \), because no arc separates \( {\... | Yes |
Theorem 9. Let \( {C}^{2} \) be a 2-cell, and let \( P, Q, R, S \) be points of Bd \( {C}^{2} \), such that \( \{ P, R\} \) separates \( Q \) from \( S \) in \( \mathrm{{Bd}}{C}^{2} \) . Let \( {M}_{1} \) and \( {M}_{2} \) be disjoint closed sets in \( {C}^{2} \), such that \( {M}_{1} \cap \mathrm{{Bd}}{C}^{2} = \{ P\}... | Proof. Evidently there is a homeomorphism \( h \), of Bd \( {C}^{2} \) onto the boundary \( J \) of a rectangular region \( \bar{I} \) in \( {\mathbf{R}}^{2} \), and this can be chosen so that the images of \( P \) and \( R \) (and of \( Q \) and \( S \) ) are the mid-points of the vertical (and the horizontal) sides o... | Yes |
Theorem 10. Let \( {C}^{2} \) be a 2-cell, and let \( J = \operatorname{Bd}{C}^{2} \) . Then \( J \) is not a retract of \( {C}^{2} \) . | Proof. As in the proof of Theorem 9, we may suppose that \( {C}^{2} \) is a rectangular region in \( {\mathbf{R}}^{2} \) . Let \( P, Q, R, S \) be as in Theorem 9, and suppose that there is a retraction \( r : {C}^{2} \rightarrow J \) . Let\n\n\[ \n{M}_{1} = {r}^{-1}\left( P\right) ,\;{M}_{2} = {r}^{-1}\left( R\right) ... | Yes |
Theorem 11. Let \( J \) be the unit circle \( {\mathbf{S}}^{1} \) in \( {\mathbf{R}}^{2} \), and let \( {C}^{2} \) be a 2-cell in \( {\mathbf{R}}^{2} \) such that \( \mathrm{{Bd}}{C}^{2} = J \) . Then \( {C}^{2} \) is the unit disk \( {\mathbf{B}}^{2} \) . | Proof. Let \( I \) and \( E \) be the interior and exterior of \( J \) in \( {\mathbf{R}}^{2} \) . Since Int \( {C}^{2} \) is connected, Int \( {C}^{2} \) lies either in \( E \) or in \( I \) . If Int \( {C}^{2} \subset E \), then an obvious construction shows that Bd \( {C}^{2} \) is a retract of \( {C}^{2} \), which ... | Yes |
Theorem 12. Let \( {C}^{2} \) be a 2-cell in \( {\mathbf{R}}^{2} \) . Then Int \( {C}^{2} \) is the interior \( I \) of Bd \( {C}^{2} \) in \( {\mathbf{R}}^{2} \) . | Proof. By Theorem 4 there is a homeomorphism \( h : {\mathbf{R}}^{2} \leftrightarrow {\mathbf{R}}^{2} \), mapping Bd \( {C}^{2} \) onto the unit circle \( {\mathbf{S}}^{1} \) . Now \( h\left( {\operatorname{Int}{C}^{2}}\right) = \operatorname{Int}h\left( {C}^{2}\right) \), and the interior of \( h\left( {\mathrm{{Bd}}{... | Yes |
Theorem 13. Let \( M \) be a triangulable set in \( {\mathbf{R}}^{2} \). Then \( M \) is tame. In fact, for each open set \( U \) containing \( M \), and every strongly positive function \( \phi : U \rightarrow \mathbf{R} \), there is a homeomorphism \( h : {\mathbf{R}}^{2} \leftrightarrow {\mathbf{R}}^{2} \) such that... | Proof. By hypothesis for \( M \), we have a complex \( K \) and a homeomorphism \( f : \left| K\right| \leftrightarrow M \). By Theorem 8 there is an \( h \) such that \( \left( {1}^{\prime }\right) h\left( {f\left( \left| {K}^{1}\right| \right) }\right) \) is a polyhedron and such that (2) and (3) hold. Consider a \( ... | Yes |
In \( {\mathbf{R}}^{n} \), let \( {\mathbf{B}}^{n} = \left\{ {P \mid \parallel P\parallel \leq 1}\right\} ,{\mathbf{S}}^{n - 1} = \operatorname{Fr}{\mathbf{B}}^{n} \) \( = \left\{ {P \mid \parallel P\parallel = 1}\right\} \) . Let \( {f}_{1} \) be a homeomorphism \( {\mathbf{B}}^{n} \leftrightarrow {\mathbf{B}}^{n} \),... | Proof. Define \( \phi : {\mathbf{B}}^{n} \times \left\lbrack {0,1}\right\rbrack \rightarrow {\mathbf{B}}^{n} \) as follows:\n\n\[ \phi \left( {P,0}\right) = P\text{ for every }P \]\n\n\[ \phi \left( {P, t}\right) = P\;\text{ for }\parallel P\parallel \geq t > 0 \]\n\n\[ \phi \left( {P, t}\right) = t{f}_{1}\left( {\frac... | Yes |
Theorem 2. Let \( {f}_{1} \) be a stable homeomorphism \( {\mathbf{R}}^{n} \leftrightarrow {\mathbf{R}}^{n} \) . Then \( {f}_{1} \) is isotopic to the identity. | Proof. We may suppose that \( {f}_{1} \) is the identity on \( {\mathbf{B}}^{n} \), since \( {f}_{1} \) is isotopic to a homeomorphism which has this property. Let inv be the inversion\n\n\[ \n{\mathbf{R}}^{n} - \{ 0\} \leftrightarrow {\mathbf{R}}^{n} - \{ 0\} ,\;P \mapsto P/\parallel P{\parallel }^{2}, \n\] \n\nwhere ... | Yes |
Theorem 1. Let \( {M}_{1},{M}_{2},\ldots \) be a descending sequence of compact sets, in a metrizable space \( \left\lbrack {X,\mathcal{O}}\right\rbrack \), and let \( A \) and \( B \) be disjoint closed sets in \( X \). If \( A \) and \( B \) are inseparable in each set \( {M}_{i} \), then \( A \) and \( B \) are inse... | Proof. Suppose not. Then \( {M}_{\infty } = {M}_{A} \cup {M}_{B} \), where \( {M}_{A} \) and \( {M}_{B} \) are disjoint closed sets containing \( {M}_{\infty } \cap A \) and \( {M}_{\infty } \cap B \) respectively. The distance between \( {M}_{A} \cup A \) and \( {M}_{B} \) is positive; that is,\n\n\[ \inf \left\{ {d\l... | Yes |
Theorem 2. Let \( M \) be a compact set, in a metrizable space \( \left\lbrack {X,\mathcal{O}}\right\rbrack \), and let \( A \) and \( B \) be disjoint closed sets in \( X \), such that \( A \) and \( B \) are inseparable in \( M \). Then there is an \( {M}^{\prime } \subset M \) such that (1) \( {M}^{\prime } \) is cl... | Proof. We shall regard \( M \) as a space. Since \( M \) is compact and metrizable, it follows that \( M \) has a countable basis; that is, there is a countable neighborhood system \( \mathcal{N} = \left\{ {{U}_{1},{U}_{2},\ldots }\right\} \) for \( M \) such that \( \mathcal{O}\left( \mathcal{N}\right) \) is the given... | Yes |
Theorem 3. Let \( M \) be a compact set, in a metrizable space \( \left\lbrack {X,\mathcal{E}}\right\rbrack \), and let \( A \) and \( B \) be disjoint closed sets in \( X \) . Then either (1) \( M \) contains a connected set which intersects both \( A \) and \( B \) or (2) \( A \) and \( B \) are separable in \( M \) ... | Proof. Suppose that (2) is false. We shall show that (1) is true. Let \( {M}^{\prime } \) be as in the preceding theorem. Suppose that \( {M}^{\prime } \) is not connected. Then \( {M}^{\prime } \) is the union of two disjoint nonempty sets \( H \) and \( K \) . Since \( {M}^{\prime } \) is irreducible, \( A \) and \( ... | Yes |
Theorem 4. Let \( C \) be a totally disconnected compact set, in a metric space, and let \( \varepsilon \) be a positive number. Then \( C \) is the union of a finite collection \( {G}_{\varepsilon } = \left\{ {{g}_{1},{g}_{2},\ldots ,{g}_{n}}\right\} \) of disjoint nonempty closed sets, with \( \delta {g}_{i} < \varep... | Proof. \( C \) is covered by the set of all neighborhoods of the form\n\n\[ N\left( {P,\frac{\varepsilon }{4}}\right) \;\left( {P \in C}\right) . \]\n\n(Hereafter, we regard \( C \) as a space.) Therefore \( C \) is covered by a finite collection\n\n\[ \left\{ {{N}_{1},{N}_{2},\ldots ,{N}_{n}}\right\} \]\n\nof such nei... | Yes |
Theorem 6. Let \( \left\lbrack {X,\mathcal{O}}\right\rbrack \) and \( \left\lbrack {Y,{\mathcal{O}}^{\prime }}\right\rbrack \) be metrizable spaces. If \( X \) is compact, and \( f \) is a bijective mapping \( X \leftrightarrow Y \), then \( f \) is a homeomorphism. | Proof. We need to show that \( {f}^{-1} \) is a mapping. Given \( {Q}_{1},{Q}_{2},\ldots \) in \( Y \) , with \( \lim {Q}_{i} = Q \), let \( {P}_{i} = {f}^{-1}\left( {Q}_{i}\right) \) and \( P = {f}^{-1}\left( Q\right) \) . We need to know that \( \lim {P}_{i} = P \) . If not, we have \( \lim {P}_{{n}_{i}} = {P}^{\prim... | No |
Theorem 7. Let \( C \) be a Cantor set, and let \( {C}^{\prime } \) be a compact metrizable space. Let \( {G}_{1},{G}_{2},\ldots \) be a sequence of finite coverings of \( C \) by disjoint nonempty open (and therefore closed) sets, such that (1) \( {G}_{i + 1} \leq {G}_{i} \) for each \( i \) and (2) \( \begin{Vmatrix}... | Proof. For each \( i \), and each \( P \in C \), let \( {g}_{i, P} \) be the element of \( {G}_{i} \) that contains \( P \) . Then\n\n\[ \n\{ P\} = \mathop{\bigcap }\limits_{{i = 1}}^{\infty }{g}_{i, P} \n\]\n\nEvidently the sequence \( {g}_{1, P},{g}_{2, P},\ldots \) is descending. Therefore so also is the sequence \(... | Yes |
Theorem 8. Every two Cantor sets are homeomorphic. | Proof. Let the sets be \( C \) and \( {C}^{\prime } \) . By Theorem \( 4, C \) is the union of a collection \( {G}_{1} = \left\{ {{g}_{11},{g}_{12},\ldots ,{g}_{1,{n}_{1}}}\right\} \) of disjoint closed sets of diameter \( < \) 1. Now apply Theorem 4 to \( {C}^{\prime } \), choosing \( \varepsilon \) sufficiently small... | Yes |
Theorem 2. Let \( A \) be a \( k \) -annulus in \( {\mathbf{R}}^{2} \), and let \( B \) be the union of some or all of the boundary components \( {J}_{1},{J}_{2},\ldots ,{J}_{k} \) . Then there is a 2-cell \( C \) such that (1) Bd \( C \subset \operatorname{Int}A \) ,(2) \( B \subset \operatorname{Int}C \), and (3) \( ... | Proof. By Theorem 1, Theorem 2 reduces to the case in which all the sets \( {J}_{i} \) are circles, and in which \( {J}_{i} \) is a very small circle for every \( i > 0 \) . In the latter case, the construction of \( C \) is trivial. | No |
Theorem 3. Let \( {C}^{2} \) be a 2-cell, with \( \operatorname{Bd}{C}^{2} = J = {B}_{1} \cup {B}_{2} \), where \( {B}_{1} \) and \( {B}_{2} \) are arcs with common end-points \( Q \), S. Let \( {M}_{1} \) and \( {M}_{2} \) be disjoint closed sets in \( {C}^{2} \), such that \( {M}_{i} \cap J \subset \operatorname{Int}... | Proof. The proof is the same as that of Theorem 10.9. | No |
Theorem 4. Let \( M \) be a totally disconnected compact set in \( {\mathbf{R}}^{2} \), and let \( U \) be a connected open set containing \( M \) . Then \( U - M \) is connected. | Proof. Let \( Q \) and \( S \) be points of \( U - M \) . Then \( Q \) and \( S \) can be joined by a broken line in \( U \) . It follows, by an easy construction, that there is a (polyhedral) 2-cell \( {C}^{2} \), with Bd \( {C}^{2} = {B}_{1} \cup {B}_{2} \) and \( {B}_{1} \cap {B}_{2} = \{ Q, S\} \) . For \( i = 1,2 ... | Yes |
Theorem 5. Let \( M \) be a totally disconnected compact set in \( {\mathbf{R}}^{2} \), and let \( N \) be a frame \( {}^{1} \) of \( M \) . Then every component of \( N \) is a 2-cell. | Proof. We know that different components of \( {\mathbf{R}}^{2} - N \) lie in different components of \( {\mathbf{R}}^{2} - M \) . Since \( {\mathbf{R}}^{2} - M \) is connected, so also in \( {\mathbf{R}}^{2} - N \) . Therefore each component \( C \) of \( N \) has a connected boundary. Therefore Bd \( C \) is a 1-sphe... | Yes |
Theorem 6. Let \( M \) and \( N \) be as in Theorem 5, and let \( \varepsilon \) be a positive number.\n\nIf \( N \) lies in a sufficiently small neighborhood of \( M \), then every component of \( N \) has diameter less than \( \varepsilon \) . | Proof. Let \( M = \mathop{\bigcup }\limits_{{i = 1}}^{n}{g}_{i} \), as in Theorem 12.4, with \( \delta {g}_{i} < \varepsilon /3 \) . If \( \alpha > 0 \), and \( \alpha \) is sufficiently small, then \( N\left( {{g}_{i},\alpha }\right) \cap N\left( {{g}_{j},\alpha }\right) = \varnothing \) for \( i \neq j \) . Take such... | Yes |
Theorem 7. Let \( M \) and \( {M}^{\prime } \) be totally disconnected compact sets in \( {\mathbf{R}}^{2} \), and let \( f \) be a homeomorphism \( M \leftrightarrow {M}^{\prime } \). Then \( f \) has an extension \( F : {\mathbf{R}}^{2} \leftrightarrow {\mathbf{R}}^{2} \). | Proof. (1) Let \( A \) and \( {A}^{\prime } \) be 2-cells containing \( M \) and \( {M}^{\prime } \) respectively in their interiors. Let \( {N}_{1} \) be a frame of \( M \), lying in Int \( A \), and lying in a sufficiently small neighborhood of \( M \) so that every component of \( {N}_{1} \) has diameter \( < 1 \). ... | Yes |
Theorem 3. \( \left\lbrack {\pi \left( {X,{P}_{0}}\right) , \cdot }\right\rbrack \) is a group. | The identity in \( \pi \left( {X,{P}_{0}}\right) \) is \( \bar{e} \), where \( e \) is the constant path \( \left\lbrack {0,1}\right\rbrack \rightarrow \left\{ {P}_{0}\right\} \) . If \( \pi \left( {X,{P}_{0}}\right) = \{ \bar{e}\} \), then \( X \) is simply connected. | No |
Theorem 8. For every complex \( K \), the canonical homomorphism\n\n\[ h : \pi \left( {\left| K\right| ,{P}_{0}}\right) \rightarrow {H}_{1}\left( K\right) = {H}_{1}\left( {K,\mathbf{Z}}\right) \]\n\n is surjective. Its kernel \( \ker h \) is the commutator subgroup of \( \pi \left( {\left| K\right| ,{P}_{0}}\right) \) ... | See Seifert and Threlfall [ST], pp. 171-174. For an outline of the proof, see Problems 14.4-14.13 below. | No |
Theorem 2. Let\n\n\\[ \np = \\mathop{\\prod }\\limits_{{i = 1}}^{m}{g}_{{j}_{i}}^{{\\alpha }_{i}}\\;\\left( {{\\alpha }_{i} = \\pm 1}\\right) .\n\\]\n\nIf \\( p \\cong e \\), then the generator word on the right can be reduced to e by a finite sequence of operations, each of which inserts or deletes an expression of on... | Proof. Let\n\\[ \nf : {\\left\\lbrack 0,1\\right\\rbrack }^{2} \\rightarrow {\\mathbf{R}}^{3} - L\n\\]\n\nbe a PL mapping under which \\( p \\cong e \\) . We choose \\( f \\) so that \\( f \\) is linear on every simplex of a triangulation \\( K \\) of \\( {\\left\\lbrack 0,1\\right\\rbrack }^{2} \\), as in Figure 15.9.... | Yes |
Theorem 3. \( \ker {\phi }^{ * } = N\left( \left\lbrack R\right\rbrack \right) \). | Proof. (1) It is easy to see that \( N\left( \left\lbrack R\right\rbrack \right) \) is the set of all elements of \( F\left( G\right) \) that are obtainable from elements of \( \left\lbrack R\right\rbrack \) in a finite number of steps by multiplication, inversion, and conjugation. Since \( {\phi }^{ * }\left( \left\lb... | Yes |
Theorem 4. Let \( L \) be a link in \( {\mathbf{R}}^{3} \), in general position relative to the axes. Let \( G = \left\{ {{\bar{g}}_{1},{\bar{g}}_{2},\ldots ,{\bar{g}}_{n}}\right\} \) and \( R = \left\{ {{r}_{1},{r}_{2},\ldots ,{r}_{n}}\right\} \) be the generating set and the set of crossing words derived from the dia... | \[ \phi : W\left( G\right) \rightarrow \pi \left( {{\mathbf{R}}^{3} - L,{P}_{0}}\right) ,\] \[ \phi : {\bar{g}}_{{j}_{1}}^{{\alpha }_{1}},{\bar{g}}_{{j}_{2}}^{{\alpha }_{2}},\ldots ,{\bar{g}}_{{j}_{m}}^{{\alpha }_{m}} \mapsto \mathop{\prod }\limits_{{i = 1}}^{m}{\bar{g}}_{{j}_{i}}^{{\alpha }_{i}} \] induces a homomorph... | Yes |
Theorem 1. Let \( A \) be an annulus. Then \( \pi \left( A\right) \approx \mathbf{Z} \), where \( \mathbf{Z} \) is the additive group of integers. | Proof. A brute-force computation of \( \pi \left( A\right) \) is easy. We may assume, as in Figure 16.1, that \( A \) is a polyhedron in \( {\mathbf{R}}^{2} \) . Here \( {g}_{1} \) generates \( \pi \left( A\right) \) : given a (PL) closed path in \( \mathrm{{CP}}\left( {A,{P}_{0}}\right) \), we can reduce it to a produ... | Yes |
Theorem 3. Let \( A \) be a \( k \) -annulus. Then \( \pi \left( A\right) \) is a free group on \( k \) generators. | Proof. We recall, from the beginning of Section 13, that a \( k \) -annulus is a compact connected 2-manifold \( A \) with boundary, imbeddable in \( {\mathbf{R}}^{2} \), such that \( \operatorname{Bd}A \) has \( k + 1 \) components. Since \( \pi \left( A\right) \) is a topological invariant of \( A \), we may assume t... | Yes |
Theorem 4. Let \( L \) be a link in \( {\mathbf{R}}^{3} \), with \( k \) components, and suppose that the components of \( L \) are polygons which form the boundaries of disjoint polyhedral 2-cells. Then the group of \( L \) is a free group on \( k \) generators. | Proof. We arrange the diagram so as to get \( k \) generators and no relations. | No |
Theorem 5. Let \( {J}_{1},{J}_{2},{J}_{3} \) be plane polygons, simply linked in series, as in Figure 16.3, let \( D \) be the plane 2-cell bounded by \( {J}_{2} \), and suppose that \( D \) is simply punctured by \( {J}_{1} \) and \( {J}_{3} \). Let \( p \) be a closed path in \[ U = D - \left( {{J}_{1} \cup {J}_{2} \... | Proof. In \( U \), take generators \( {g}_{1},{g}_{2} \) of \( \pi \left( {U,{P}_{0}}\right) \), as in the proof of Theorem 3. Then \( \left\{ {{g}_{1},{g}_{2}}\right\} \) freely generates \( \pi \left( {{\mathbf{R}}^{3} - \left( {{J}_{1} \cup {J}_{3}}\right) }\right) \), as in the proof of Theorem 4; and Theorem 5 fol... | No |
Theorem 6. The group of the trefoil knot is not commutative. | Proof. The trefoil is the knot defined by Figure 16.4. From the diagram\n\n\n\nFigure 16.4\n\nwe read off the relation\n\n\[ \n{r}_{1} = {g}_{1}{g}_{3}^{-1}{g}_{2}^{-1}{g}_{3} \cong e.\n\]\n\nThe figure is invariant ... | Yes |
Theorem 7. \( \pi \left( V\right) \) is isomorphic to the group of the trefoil knot. | Note that Theorem 6 furnishes us with a proof of the \ | No |
Theorem 1. Let \( M \) be a 3-manifold with boundary, lying in \( {\mathbf{R}}^{3} \). If \( M \) is closed, then\n\n\[ \text{Bd}M = \operatorname{Fr}M\text{.} \] | Proof. Let \( U = M - \mathrm{{Fr}}M \) . Thus \( U \) is the topological interior of \( M \) in \( {\mathbf{R}}^{3} \) , that is, the union of all open sets in \( {\mathbf{R}}^{3} \) that lie in \( M \) . Obviously \( M \) is locally Euclidean at every point of \( U \) . Therefore we have\n\n\[ U \subset \operatorname... | Yes |
Theorem 2. Let \( K \) be a cell-decomposition of a 2-cell, and suppose that \( K \) has more than one 2-cell. Then at least two of the 2-cells of \( K \) are free in \( K \) . | Proof. Here \( \left| K\right| \) is being regarded as a space. We may assume, however, that \( \left| K\right| \subset {\mathbf{R}}^{2} \) ; and by the Tame imbedding theorem for linear graphs (Theorem 10.8) we may suppose that all edges of \( K \) are polyhedra. It follows that all 2-cells of \( K \) are polyhedra.\n... | Yes |
Theorem 3. Let \( K \) be as in Theorem 1. Let \( D \) be a 2-cell which forms a proper subcomplex of \( K \). Then there is a 2-cell which is free in \( K \) and does not lie in \( D \). | Proof. Let \( C \) be any 2-cell of \( K \) which does not lie in \( D \), and suppose that \( C \) is not free. Let \( {D}_{1},{D}_{2},{K}_{1} \), and \( {K}_{2} \) be as in the preceding proof. Then \( D \) lies in one of the sets \( {D}_{i} \), say, \( {D}_{2} \). Let \( {C}^{\prime } \) be a 2-cell, other than \( C... | Yes |
Theorem 4. Let \( {\sigma }^{3} \) be a 3-simplex in \( {\mathbf{R}}^{3} \), and let \( {\sigma }^{2} \) be a face of \( {\sigma }^{3} \) . Then \( {\sigma }^{3} \) has the push property at \( {\sigma }^{2} \) . | The proof is by a direct geometric construction. See the proof of Theorem 3.4, where a PLH of an analogous sort, in \( {\mathbf{R}}^{2} \), is described explicitly. | No |
Theorem 5. Given \( {\sigma }^{3} \subset {\mathbf{R}}^{3} \). Let \( D \) be a polyhedral 2-cell in \( \mathrm{{Bd}}{\sigma }^{3} \), and let \( W \) be an open set containing \( {\sigma }^{3} \). Then there is a PLH \[ f : {\mathbf{R}}^{3} \leftrightarrow {\mathbf{R}}^{3},\;{\sigma }^{3} \leftrightarrow {\sigma }^{3}... | Proof. First we take a (rectilinear) triangulation \( K \) of Bd \( {\sigma }^{3} \), such that \( D \) forms a subcomplex of \( K \), and sufficiently fine so that for each \( {\tau }^{2} \in K \) , \( \left| {\text{St}{\tau }^{2}}\right| \) avoids a 2-face of \( {\sigma }^{3} \). (Here St \( {\tau }^{2} \) is the set... | No |
Theorem 6. Every 3-simplex in \( {\mathbf{R}}^{3} \) has the push property. | Proof. Let \( {D}_{1} \) be a polyhedral 2-cell in Bd \( {\sigma }^{3} \), let\n\n\[ \n{D}_{2} = \mathrm{{Cl}}\left( {\mathrm{{Bd}}{\sigma }^{3} - {D}_{1}}\right) \n\] \n\nlet \( J = \mathrm{{Bd}}{D}_{1} = \mathrm{{Bd}}{D}_{2} \), and let \( N \) be a closed polyhedral neighborhood of \( {\sigma }^{3} - J \) . By the p... | Yes |
Theorem 7. The push property, for polyhedral 3-cells in \( {\mathbf{R}}^{3} \), is preserved by every PLH. | Proof. Use transforms. | No |
Theorem 9. Let \( {C}^{3} \) be a convex polyhedral 3-cell in \( {\mathbf{R}}^{3} \). Then \( \operatorname{Bd}{C}^{3} \) is simply imbedded. | Proof. We have given a triangulation of Bd \( {C}^{3} \). We form the join of this complex with any point \( v \) of Int \( {C}^{3} \), getting a triangulation \( K \) of \( {C}^{3} \). If \( {\sigma }^{3} \in K \), then \( {\sigma }^{3} \) is free in \( K \), in the sense that \( \operatorname{Bd}{\sigma }^{3} \cap \o... | Yes |
Theorem 11. Let \( {S}_{1} \) and \( {S}_{2} \) be polyhedral 2-spheres in \( {\mathbf{R}}^{3} \), such that \( {S}_{1} \cap {S}_{2} \) is a plane 2-cell D. Let\n\n\[ S = \left( {{S}_{1} \cup {S}_{2}}\right) - \text{Int}D\text{.} \]\n\nIf \( {S}_{1} \) and \( {S}_{2} \) are simply imbedded, then so also is \( S \) . | Proof. Let \( W \) be a convex open set containing \( S \) . For \( i = 1,2 \), let\n\n\[ {D}_{i} = \mathrm{{Cl}}\left( {{S}_{i} - D}\right) \]\n\nLet\n\n\[ J = \operatorname{Bd}{D}_{1} = \operatorname{Bd}{D}_{2}. \]\n\nSince \( W \) is convex, \( J \subset S \subset W \), and \( D \) is a plane 2-cell, it follows that... | Yes |
Theorem 12 (The PL Schönflies theorem). Let \( S \) be a polyhedral 2-sphere in \( {\mathbf{R}}^{3} \), and let \( W \) be a convex open set containing \( S \) . Then there is a PLH\n\n\[ f : {\mathbf{R}}^{3} \leftrightarrow {\mathbf{R}}^{3},\;S \leftrightarrow \operatorname{Bd}{\sigma }^{3}, \]\n\nwhere \( {\sigma }^{... | Proof. Given such an \( S \) and \( W \), we choose the axes in general position, in the sense that no horizontal plane \( E \) contains more than one vertex of \( S \) . Thus there are three possibilities for \( E \cap S \) . (1) \( E \cap S \) may be a single point, or the union of a singleton and a finite union of d... | No |
Lemma 1. \( n = 0 \) . | Proof. Suppose that \( n > 0 \) . Let \( E \) be a horizontal plane containing a singular point \( P \), let \( J \) be a polygon in \( E \cap S \), and suppose that \( J \) is inmost in \( E \cap S \), in the sense that \( J \) is the boundary of a 2-cell \( {D}_{J} \) in \( E \), such that Int \( {D}_{J} \cap S = \va... | Yes |
Lemma 2. \( {E}_{1} \cap S \) and \( {E}_{m} \cap S \) are singletons. | Proof. \( {E}_{1} \cap S \) contains at most one vertex of \( S \), and no plane below \( {E}_{1} \) intersects \( S \) . Therefore \( {E}_{1} \cap S \) contains no polygon, so that each point of \( {E}_{1} \cap S \) is isolated. Therefore \( {E}_{1} \cap S \) is a singleton. Similarly, so is \( {E}_{m} \cap S \) . | Yes |
Lemma 3. For \( 1 < i < m,{E}_{i} \cap S \) is a polygon. | Proof. Since no finite set separates \( S,{E}_{i} \cap S \) contains a polygon \( J \) . We shall show that \( {E}_{i} \cap S \) is connected. It will follow that \( {E}_{i} \cap S = J \) .\n\nFor each \( k,{k}^{\prime } \), with \( k < {k}^{\prime } \), let\n\n\[ N\left( {k,{k}^{\prime }}\right) = \left\{ {\left( {x, ... | Yes |
Lemma 4. Bd \( {M}_{1} \) is simply imbedded. | Proof. The theorem follows from the fact that \( {M}_{1} \) can be triangulated so as to form the join of the point \( {E}_{1} \cap S \) and the polyhedral 2-cell \( {E}_{2} \cap \left| K\right| \) (Theorem 10). | No |
Lemma 6. For \( 1 < i < m - 1 \), Bd \( {M}_{i} \) is simply imbedded. | Proof. We know that one of the planes \( {E}_{i},{E}_{i + 1} \) contains no vertex of \( K \) . Suppose that \( {E}_{i} \) contains a vertex \( v \) of \( K \) . Let \( \mathrm{{St}}v \) be the closed star of \( v \) in \( K \) . Then for each \( {\tau }^{3} \in K \) with \( v \) as a vertex, \( {E}_{i} \cap {\tau }^{3... | Yes |
Theorem 1. Let the components \( {C}_{i} \) and the spanning 2-cells \( {D}_{i}\left( {i \leq k}\right) \) be as in the definition of \( {T}_{2} \) . Then \( \mathrm{{Bd}}{T}_{1} \) is a retract of the set \[ {T}_{1} - \left\lbrack \begin{matrix} k & k \\ \mathop{\bigcup }\limits_{{i = 1}}{C}_{i} \cup & \mathop{\bigcup... | The proof is by straightforward carpentry. | No |
Theorem 3. Let \( p \) be a closed path in \( {\mathbf{R}}^{3} - {T}_{1} \), and suppose that \( p \cong e \) in \( {\mathbf{R}}^{3} - \mathcal{Q} \) . Then \( p \cong e \) in \( {\mathbf{R}}^{3} - {T}_{1} \) . | Proof. We may suppose that \( p \) is PL, and that there is a PL contraction\n\n\[ \phi : {\left\lbrack 0,1\right\rbrack }^{2} \rightarrow {\mathbf{R}}^{3} - \mathcal{Q} \]\n\nof \( p \) . If it were true that \( \left| \phi \right| \cap {T}_{n} \neq \varnothing \) for each \( n \), then it would follow that \( \left| ... | Yes |
Theorem 4. \( {\mathbf{R}}^{3} - \mathcal{Q} \) is not simply connected. | Proof. Since \( \pi \left( {{\mathbf{R}}^{3} - {T}_{1}}\right) \) is nontrivial, this follows from Theorem 3. | No |
Theorem 5. There are Cantor sets \( {C}_{1} \) and \( {C}_{2} \) in \( {\mathbf{R}}^{3} \) such that no homeomorphism \( {C}_{1} \leftrightarrow {C}_{2} \) has a homeomorphic extension \( {\mathbf{R}}^{3} \leftrightarrow {\mathbf{R}}^{3} \) . | Proof. Let \( {C}_{1} = \mathcal{Q} \), and let \( {C}_{2} \) be the standard \ | No |
Theorem 7 (Louis Antoine). \( {\mathbf{R}}^{3} \) contains a wild 2-sphere. | Proof. By Theorem 6 there is a 2-sphere \( S \) such that \( \mathcal{Q} \subset S \subset \) Int \( {T}_{1} \) . Let \( U \) be the unbounded component of \( {\mathbf{R}}^{3} - S \) . Then there is a closed path \( p \) , in \( {\mathbf{R}}^{3} - {T}_{1} \), such that \( p \) is not contractible in \( {\mathbf{R}}^{3}... | Yes |
Theorem 8 (Louis Antoine). \( {\mathbf{R}}^{3} \) contains a wild arc. | Proof. Let \( S \) be as in the proof of the preceding theorem. By the result of Problem 13.1, \( \mathcal{Q} \) lies in an arc \( A \) in \( S \) . Let \( V = {\mathbf{R}}^{3} - A \) . As in the preceding proof, we show that \( V \) is not simply connected. But the complement of a broken line in \( {\mathbf{R}}^{3} \)... | No |
Theorem 1. Let \( B \) be a broken line in \( {\mathbf{R}}^{3} \), with end-points \( P \) and \( Q \) . Then there is a polyhedral 3-cell \( C \) such that (1) Int \( B \subset \operatorname{Int}C \) ,(2) \( P, Q \in \) Bd \( C \), and (3) there is a PLH \( \phi : C \leftrightarrow {\sigma }^{2} \times \left\lbrack {0... | This can be proved by induction on the number of edges of \( B \) . Since \( \phi \) can then be extended to the rest of \( {\mathbf{R}}^{3} \), it follows that \( B \) is imbedded in \( {\mathbf{R}}^{3} \) in the same way as a linear interval. | No |
Theorem 2. If \( {B}_{i} \) is unknotted in \( {C}_{i}\left( {i = 1,2}\right) \), then every PLH\n\n\[ h : \mathrm{{Bd}}{C}_{1} \leftrightarrow \mathrm{{Bd}}{C}_{2},\;\mathrm{{Bd}}{B}_{1} \leftrightarrow \mathrm{{Bd}}{B}_{2} \]\n\ncan be extended to give a PLH\n\n\[ {h}^{\prime } : {C}_{1} \leftrightarrow {C}_{2},\;{B}... | Proof. For \( i = 1,2 \) we have a PLH\n\n\[ {f}_{i} : {C}_{i} \leftrightarrow {\sigma }^{2} \times \left\lbrack {0,1}\right\rbrack ,\;{B}_{i} \leftrightarrow {R}_{i} \times \left\lbrack {0,1}\right\rbrack . \]\n\nAnd \( {f}_{2} \) can be chosen so that \( {R}_{2} = {R}_{1} \) . Thus the theorem reduces to the case in ... | Yes |
Theorem 3. \( {A}_{1} \) is tame. | Proof. In each spherical shell \( \mathrm{{Cl}}\left( {{C}_{i} - {C}_{i + 1}}\right) \), take a polyhedral 3-cell \( {D}_{i} \) , such that \( {B}_{i} \) is unknotted in \( {D}_{i} \), and \( {D}_{i} \) intersects Bd \( {C}_{i} \) and Bd \( {C}_{i + 1} \) in 2-cells \( {d}_{i} \) and \( {d}_{i + 1} \) . (The notation c... | Yes |
Theorem 4. \( A \) is wild. | Proof. If \( A \) is tame, then there is a homeomorphism \( \phi : {\mathbf{R}}^{3} \leftrightarrow {\mathbf{R}}^{3} \), such that \( \phi \left( A\right) \) is a linear interval \( I \) . We shall show that this is impossible. Let \( {P}^{\prime } = \phi \left( P\right) \), and let \( U = {\mathbf{R}}^{3} - I \) . (Se... | No |
Theorem 5. \( {\mathbf{R}}^{3} - A \) is homeomorphic to the complement of a point. | Proof. First show that \( {\mathbf{R}}^{3} - A \) is the intersection of a sequence \( {C}_{1}^{3},{C}_{2}^{3},\ldots \) of polyhedral 3-cells, with \( {C}_{i + 1}^{3} \subset \) Int \( {C}_{i}^{3} \) for each \( i \) . Then use the result of Problem 17.6. | No |
In \( {\mathbf{R}}^{3} \), let \( {P}_{0} = \left( {0,0,0}\right) ,{P}_{1} = \left( {0,0,\frac{3}{2}}\right) ,{P}_{2} = \left( {0,0,2}\right) \), and \( {P}_{3} = \left( {0,1,2}\right) \) ; let \( T \) be the 2 -simplex \( {P}_{0}{P}_{2}{P}_{3} \), and let \( {D}^{3} \) be the solid of revolution of \( T \) about the \... | Proof. The construction of such an \( f \) is straightforward. See Figure 20.1. | No |
Theorem 2. Let \( A \) be an arc in \( {\mathbf{R}}^{3} \), with \( \operatorname{Bd}A = \{ P, Q\} \), such that \( A - \{ Q\} \) is an (infinite) polyhedron. Then \( {\mathbf{R}}^{3} - A \) is homeomorphic to \( {\mathbf{R}}^{3} - \{ Q\} \) . | INDICATION OF PROOF. In the proof, the question whether \( A \) is tame does not arise. Let \( {C}^{3} \) be a 3-cell neighborhood of \( A - \{ Q\} \), of the sort suggested by Figure 20.2. Thus \[ {C}^{3} = \mathop{\bigcup }\limits_{{i = 1}}^{\infty }{C}_{i}^{3} \cup \{ Q\} \] where for each \( i,{C}_{i}^{3} \) is a 3... | Yes |
Theorem 4. \( {A}_{2} \) is wild. | Proof. We shall show that \( {\mathbf{R}}^{3} - {A}_{2} \) is not locally simply connected at \( Q \) . Theorem 4 will then follow from Theorem 3.\n\nLet \( V \) be the interior of a cubical neighborhood of \( {A}_{2} - S \), as indicated schematically by the dotted square in Figure 20.4, so that \( \operatorname{Fr}V ... | Yes |
Theorem 3. For open cell-complexes, the Euler characteristic is preserved by Operations \( \alpha ,\beta ,\gamma \), and \( \delta \) . | For example, under Operation \( \alpha, V \mapsto V + 1 \), and \( E \mapsto E + 1 \) ; and \( V \) and \( E \) appear with opposite signs in the formula for \( \chi \left( \mathcal{C}\right) \) . Preservation by Operations \( \beta ,\gamma \), and \( \delta \) is verified similarly. | No |
Theorem 4. For open cell-complexes, the Euler characteristic is preserved under subdivision, and hence is a combinatorial invariant. | Proof. Suppose that \( {\mathcal{C}}_{2} < {\mathcal{C}}_{1} \) . Let \( {C}^{2} \in {\mathcal{C}}_{1} \), and let \( M \) be the union of all edges of \( {\mathcal{C}}_{2} \) that lie in \( {\bar{C}}^{2} \) . We assert that \( M \) is connected. Suppose not.\n\nThen \( M = H \cup K \), where \( H \) is the component o... | No |
Theorem 5. All triangulations of the same compact 2-manifold have the same Euler characteristic. | Proof. Let \( {K}_{1} \) and \( {K}_{2} \) be triangulations of the compact 2-manifold \( M \) . By the Hauptvermutung (Theorem 8.5), \( {K}_{1} \) and \( {K}_{2} \) are combinatorially equivalent. Now use Theorem 4. | Yes |
Theorem 6. If \( J \) is a polygon, then \( \chi \left( J\right) = 0 \) . | Proof. Let \( P \in J \), and let \( \mathcal{C} = \{ \{ P\}, J - \{ P\} \} \) . Then \( \chi \left( J\right) = \chi \left( \mathcal{C}\right) = 1 - 1 \) \( + 0 = 0 \) . | Yes |
Theorem 7. Let \( K \) be any triangulation of a 2-cell. Then \( \chi \left( K\right) = 1 \) . | Proof. Let \( v \) be a vertex of \( K \), lying in Bd \( \left| K\right| \), and let\n\n\[ \n{\mathcal{C}}_{1} = \{ \{ v\} ,\text{ Bd }K - \{ v\} ,\text{ Int }\left| K\right| \} .\n\]\n\nLet \( {\mathcal{C}}_{2} \) be the set whose elements are the vertices of \( K \) and the interiors of the edges and 2 -faces of \( ... | Yes |
Theorem 9. Let \( M \) be a compact 2-manifold with boundary. Then all triangulations \( K \) of \( M \) have the same Euler characteristic. | Proof. Let the components of Bd \( M \) be \( {J}_{1},{J}_{2},\ldots ,{J}_{n} \) ; and for each \( i \), let \( {D}_{i} \) be a 2-cell, with \( \operatorname{Bd}{D}_{i} = {J}_{i} \), such that \( {M}^{\prime } = M \cup \cup {}_{\mathrm{i}}{D}_{i} \) is a compact 2-manifold. Let \( r = \chi \left( {M}^{\prime }\right) \... | Yes |
Theorem 10. When a 2-manifold with boundary is split apart at a 1-sphere lying in its interior, and separating a connected neighborhood of itself, the Euler characteristic is unchanged. | The theorem is a consequence of Theorem 6. | No |
Theorem 11. If a 2-manifold \( M \) with boundary is split apart as in Theorem 10, and the new boundary components are spanned by 2-cells, then the Euler characteristic is increased by 2: | This is obvious. Of course, the 2-cells are supposed to be disjoint, and intersect \( M \) only in their boundaries. | No |
Theorem 1. There is an open cell-decomposition \( \mathcal{C} \) of \( M \) such that (1) \( \mathcal{C} \) has exactly one face \( {C}^{2} \) and (2) every edge of \( \mathcal{C} \) is an edge of \( K \) . | Proof. Evidently there are finite sequences \( {\sigma }_{1},{\sigma }_{2},\ldots ,{\sigma }_{n} \) of 2 -faces of \( K \) such that (a) the 2 -faces \( {\sigma }_{i} \) are all different and (b) \( {\sigma }_{i + 1} \) has an edge \( {e}_{i} \) in common with \( \mathop{\bigcup }\limits_{{j \leq i}}{\sigma }_{j} \) . ... | Yes |
Theorem 2. Let \( L \) be a connected acyclic linear graph which is a subcomplex of \( K \) . Then \( N\left( L\right) \) is a 2-cell. | Proof. If \( L \) consists of a single edge of \( K \), this is clear. The proof proceeds by induction on the number of edges of \( L \) . Let \( e \) be any edge of \( L \) . Then the complex \( L - e \) is \( = {L}_{1} \cup {L}_{2} \), where \( {L}_{1} \) and \( {L}_{2} \) are disjoint, connected, and acyclic, and ha... | Yes |
Theorem 3. \( M \) can be expressed as a union\n\n\[ M = D \cup {D}^{\prime } \cup \mathop{\bigcup }\limits_{{i = 1}}^{n}{S}_{i} \]\n\nof polyhedral 2-cells with disjoint interiors, such that (1) for each \( i \), each of the sets \( {S}_{i} \cap D \) and \( {S}_{i} \cap {D}^{\prime } \) is the union of two disjoint ar... | The sets \( {S}_{i} \) will be called strips, and \( {M}^{\prime } \) will be called a 2-cell with strips. Evidently such an \( {M}^{\prime } \) can always be imbedded in \( {\mathbf{R}}^{3} \), and thus can be described by a figure such as Figure 22.1. Under the conditions of\n\n![0d8d6f7c-790d-4773-9cb3-7748c6b57409_... | Yes |
Theorem 4. Let \( M \) be a compact connected 2-manifold. Then \( M \) is a 2-sphere with \( h \) handles and \( m \) cross-caps \( \left( {h \geq 0,0 \leq m \leq 2}\right) \) . | We now define a new open cell-decomposition of \( M \), as follows. As indicated in Figure 22.10, we choose a point \( v \) of Int \( D \), and we define a collection \( \left\{ {{J}_{i},{J}_{j}^{\prime }}\right\} \) of polyhedral 1-spheres (one \( {J}_{i} \) for each annular strip, and one \( {J}_{j}^{\prime } \) for ... | No |
Theorem 5. Let \( M \) be a 2-sphere with \( h \) handles and \( m \) cross-caps \( (h \geq 0 \) , \( 0 \leq m \leq 2) \) . Then\n\n\[ \chi \left( M\right) = 2 - \left( {{2h} + m}\right) \] | Proof. \( V - E + F = 1 - \left( {{2h} + m}\right) + 1 \) . | No |
Theorem 7. If \( M \) is orientable, then\n\n\[ \chi \left( M\right) = 2 - {p}^{1}\left( M\right) \]\n\nIf \( M \) is not orientable, then\n\n\[ \chi \left( M\right) = 1 - {p}^{1}\left( M\right) \] | Proof. Let \( h \) be the number of handles in \( M \), and let \( m \) be the number of cross-caps, with \( 0 \leq m \leq 2 \) . For \( m = 0 \), we have \( \chi \left( M\right) = 2 - {2h} = 2 - \) \( {p}^{1}\left( M\right) \) . For \( m = 1,\chi \left( M\right) = 2 - \left( {{2h} + 1}\right) ,{p}^{1}\left( M\right) =... | Yes |
Theorem 8. For \( i = 1,2 \), let \( {M}_{i} \) be a 2-sphere with \( {h}_{i} \) handles and \( {m}_{i} \) cross-caps, with \( 0 \leq {m}_{i} \leq 2 \) . Then (1) \( {M}_{1} \) and \( {M}_{2} \) are homeomorphic if and only if \( \left( 2\right) {h}_{1} = {h}_{2} \) and \( {m}_{1} = {m}_{2} \) . | The proof that \( \left( 2\right) \Rightarrow \left( 1\right) \) requires the construction of a homeomorphism. For suggestions, see the problems below. To show that \( \left( 1\right) \Rightarrow \left( 2\right) \), we observe that since \( \chi \) is a topological invariant, we have\n\n\[ \chi \left( {M}_{1}\right) = ... | No |
Theorem 11. Let \( M \) be a compact connected 2-manifold. If \( M \) is simply connected, then \( M \) is a 2-sphere. | Proof. If \( \pi \left( M\right) = 0 \), then \( {H}_{1}\left( M\right) = 0 \) . (Theorem 14.8.) By Theorem 6 it follows that \( M \) is a 2-sphere with 0 handles and 0 cross-caps. | No |
Theorem 12. Let \( M \) be a compact connected 2-manifold. If \( \chi \left( M\right) = 1 \), then \( M \) is a projective plane. | Proof. \( M \) is a 2-sphere with \( h \) handles and \( m \) cross-caps, with \( 0 \leq m \leq 2 \) . By Theorem 5 we have \( 1 = 2 - \left( {{2h} + m}\right) \), so that \( {2h} + m = 1 \) . Therefore \( m = 1 \) and \( h = 0 \) . Therefore \( M \) is a projective plane (Problem 21.1). | Yes |
Theorem 2. Let \( M \) be a 3-manifold with boundary, and let \( P \in \mathrm{{Bd}}M \) . Then there is a homeomorphism \( f : {\sigma }^{3} \leftrightarrow {C}^{3} \subset M \) such that (1) \( {C}^{3} \) is a neighborhood of \( P \) in \( M,\left( 2\right) {C}^{3} \cap \operatorname{Bd}M = f\left( {\sigma }^{2}\righ... | Proof. Let \( {D}^{3} \) be a 3-cell neighborhood of \( P \) in \( M \) . Then \( {D}^{3} = g\left( {\tau }^{3}\right) \) , where \( g \) is a homeomorphism. Let \( \operatorname{Fr}{D}^{3} \) be the frontier of \( {D}^{3} \) in the space \( M \), and let \( U = {D}^{3} - \operatorname{Fr}{D}^{3} \) . Then \( U \) is o... | Yes |
Theorem 5. Let \( M \) be a 3-manifold with boundary, and for \( i = 1,2 \) let \( {h}_{i} : M \leftrightarrow {M}_{i} \) be a homeomorphism, such that \( {M}_{1} \cap {M}_{2} = \varnothing \) . Let \( {M}^{\prime } \) be the space obtained by identifying every pair \( {h}_{1}\left( P\right) ,{h}_{2}\left( P\right) \) ... | Proof. To get a Cartesian neighborhood \( U \) of a point \( {f}_{1}\left( P\right) = {f}_{2}\left( P\right) \) in \( {M}^{\prime } \) , we take \( {g}_{i} : {\sigma }_{i}^{3} \leftrightarrow {C}_{i}^{3} \subset {f}_{i}\left( M\right) ,{\sigma }_{i}^{2} \leftrightarrow d\left( {i = 1,2}\right) \), as in Theorems 2 and ... | Yes |
Theorem 6. Every triangulated 3-manifold \( K \) with boundary is a combinatorial 3-manifold with boundary. | Proof. Let \( M = \left| K\right| \), let \( {bK} \) be the first barycentric subdivision of \( K \), and for each vertex \( v \) of \( K \) let \( {\mathrm{{St}}}_{b}v \) be the star of \( v \) in \( {bK} \) . It is then easy to verify that \( {\mathrm{{St}}}_{b}v \) is combinatorially equivalent to \( \mathrm{{St}}v ... | No |
Theorem 8. Let \( {M}^{\prime } \) be a 3-manifold with boundary, lying in a 3-manifold. If \( {M}^{\prime } \) is closed, then \( \mathrm{{Bd}}{M}^{\prime } = \operatorname{Fr}{M}^{\prime } \) . | Proof. (1) Evidently \( {M}^{\prime } - \operatorname{Fr}{M}^{\prime } \subset \operatorname{Int}{M}^{\prime } = {M}^{\prime } - \operatorname{Bd}{M}^{\prime } \) . Therefore \( \mathrm{{Bd}}{M}^{\prime } \subset \operatorname{Fr}{M}^{\prime }. \n\n(2) Since \( {M}^{\prime } \) is closed, \( \operatorname{Fr}{M}^{\prim... | Yes |
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