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Lemma 62.7. If \( X \) is a finite linear non-empty simplicial complex in \( {\mathbb{R}}^{n} \), then\n\n\[ \mathop{\lim }\limits_{{k \rightarrow \infty }}\max \left\{ {\operatorname{diam}\left( \underset{k}{\underbrace{\operatorname{St}\left( {{\operatorname{sd}}^{k}\left( X\right), v}\right) }}\right) \mid v\text{ i... | Proof (*). Let \( Y \) be a finite linear non-empty \( d \) -dimensional simplicial complex in \( {\mathbb{R}}^{n} \) . We define the mesh of \( Y \) as\n\n\[ \operatorname{mesh}\left( Y\right) \mathrel{\text{:=}} \max \{ \operatorname{diam}\left( s\right) \mid s\text{ is a simplex of }Y\} .\n\nThe lemma follows from L... | No |
Proposition 62.9. If \( X \) is a simplicial complex with countably many simplices, then for every \( n \in \mathbb{N} \) and every \( {x}_{0} \in X \) the group \( {\pi }_{n}\left( {X,{x}_{0}}\right) \) is countable. | Proof. Let \( X \) be a simplicial complex and let \( n \in \mathbb{N} \) . By the discussion on page 1500 we can pick a finite simplicial structure for \( {S}^{n} \) . Let \( * \) be a vertex of \( {S}^{n} \) . By Proposition 40.5 it suffices to show that for a single vertex \( {x}_{0} \) of \( X \) the group \( {\pi ... | Yes |
Lemma 1.3 (4) by choice of \( \varphi \) we have \( f\left( {\operatorname{St}\left( {v}_{i}\right) }\right) \subset \operatorname{St}\left( {\varphi \left( {v}_{i}\right) }\right) \) | \( \Rightarrow \left\{ {\overline{\varphi \left( {v}_{0}\right) },\ldots ,\varphi \left( {v}_{k}\right) }\right\} \) is a simplex in \( T \) . | No |
We claim that \( g \) is the desired map. | First note that it follows immediately from the construction of \( \varphi \) and \( g \) that for any vertex \( v \) of \( \operatorname{sd}\left( X\right) \) we have \( f\left( {\operatorname{St}\left( {X, v}\right) }\right) \subset \operatorname{St}\left( {Y, g\left( v\right) }\right) \) . Thus it remains to show th... | Yes |
Lemma 1.3 (4) by choice of \( \varphi \) we have \( f\left( {\operatorname{St}\left( {v}_{i}\right) }\right) \subset \operatorname{St}\left( {\varphi \left( {v}_{i}\right) }\right) \) | \( \Rightarrow \) there exists a \( t \in T \) which contains \( \varphi \left( s\right) = \left\{ {\varphi \left( {v}_{0}\right) ,\ldots ,\varphi \left( {v}_{k}\right) }\right\} \) and with \( f\left( x\right) \in \Omega \left( {\langle t\rangle }\right) \) | No |
It remains to show that \( g\left( x\right) \in \Omega \left( \left| t\right| \right) \) . | In fact we see that\n\n\[\n\begin{array}{l} x \in \Theta \left( {\langle s\rangle }\right) : \Rightarrow x \in \Theta \left( \left| s\right| \right) : \Rightarrow g\left( x\right) \in \Omega \left( \left| {\varphi \left( s\right) }\right| \right) \Rightarrow g\left( x\right) \in \Omega \left( \left| t\right| \right) . ... | Yes |
Lemma 62.11. Let \( f : K \rightarrow L \) be a simplicial map between two abstract simplicial complexes.\n\n(1) The simplicial mapping cylinder \( \operatorname{Cyl}\left( {f : K \rightarrow L}\right) \) contains \( \operatorname{sd}\left( K\right) \) and \( L \) as subcomplexes. | Sketch of Proof. Let \( f : K \rightarrow L \) be a simplicial map between two abstract simplicial complexes.\n\n(1) By taking \( \tau = \varnothing \) respectively \( \sigma = \varnothing \) one sees immediately that \( \operatorname{sd}\left( K\right) \) and \( L \) are subcomplexes of \( \operatorname{Cyl}\left( {f ... | No |
Lemma 63.1. Let \( n \in \mathbb{N} \) .\n\n(1) The map sign: \( {S}_{n} \rightarrow \{ - 1,1\} \) is a homomorphism.\n\n(2) If \( \sigma \) is a transposition, then \( \operatorname{sign}\left( \sigma \right) = - 1 \) . | Proof. The proof follows immediately from standard properties of the determinant. | No |
Lemma 63.2. Let \( K = \left( {V, S}\right) \) be an ordered abstract simplicial complex. Given any \( n \in {\mathbb{N}}_{0} \) we have \( {\partial }_{n - 1} \circ {\partial }_{n} = 0 \) . In other words, \( \left( {{C}_{ * }^{\operatorname{simp}, \leq }\left( K\right) ,{\partial }_{ * }}\right) \) is a chain complex... | Proof. The proof of the lemma is almost verbatim the same as the proof of Proposition 41.2. We leave it to the reader to make the necessary minute modifications. | No |
Lemma 63.6. Let \( K \) be an abstract simplicial complex.\n\n(1) Given any \( n \in {\mathbb{N}}_{0} \) the group \( {\mathrm{C}}_{n}^{\text{simp }}\left( K\right) \) is a free abelian group where the rank is given by the cardinality of the set \( {S}_{n} \) of n-simplices of \( K \) . In fact, if we choose for each \... | Proof.\n\n(1) It follows from the Well-Ordering Theorem 1.6 that we can equip \( K = \left( {V, S}\right) \) with an order. Thus the desired statement is now an almost immediate consequence of Lemma 63.5 | No |
Lemma 63.8. Let \( K = \left( {V, S}\right) \) be an ordered abstract simplicial complex. The maps \[ {\Omega }_{n} : {\mathrm{C}}_{n}^{\mathrm{{simp}}, \leq }\left( K\right) \rightarrow {\mathrm{C}}_{n}^{\mathrm{{simp}}}\left( K\right) \] \[ \left\{ {{v}_{0} < \cdots < {v}_{n}}\right\} \mapsto \left\lbrack {{v}_{0},\l... | Proof. A very short moment of reflection shows that the given map is a natural chain map. We deduce from Lemma 63.6 (1) that the chain map is in fact an isomorphism of chain complexes. | No |
Lemma 63.9. Let \( K \) be an abstract simplicial complex and let \( m \in {\mathbb{N}}_{0} \cup \{ \infty \} \) . As usual we denote by \( {K}^{m} \) the \( m \) -skeleton of \( K \) . Furthermore let \( n \in {\mathbb{N}}_{0} \) .\n\n(1) If \( K \) has only finitely many n-simplices, then \( {\mathrm{H}}_{n}^{\text{s... | Proof.\n\n(1) This statement follows easily from the definitions together with Lemma 19.8 (2b).\n\n(2) This statement follows from the observation that \( {\mathrm{C}}_{n}^{\operatorname{simp}}\left( {K}^{m}\right) = 0 \) for \( n > m \) .\n\n(3) This statement follows almost immediately from the observation that by de... | Yes |
Lemma 63.10. Let \( K = \left( {V, S}\right) \) be an abstract simplicial complex.\n\n(1) Given any vertex \( v \in V \) the map\n\n\[{\widetilde{\mathrm{H}}}_{0}^{\text{simp }}\left( K\right) \oplus \mathbb{Z} \rightarrow {\mathrm{H}}_{0}^{\text{simp }}\left( K\right)\]\n\n\[ \left\lbrack \sigma \right\rbrack \oplus n... | Proof. The proof is basically identical to the already rather straightforward proof of Lemma 43.1 (4). | No |
Lemma 63.11. Let \( K \) be an abstract simplicial complex. Let \( \operatorname{Cone}\left( K\right) \) be the simplicial cone as defined on page 1506. Given any \( k \in {\mathbb{N}}_{0} \) we have\n\n\[ \n{\widetilde{\mathrm{H}}}_{k}^{\text{simp }}\left( {\operatorname{Cone}\left( K\right) }\right) = 0.\n\] | Proof. Let \( K = \left( {V, S}\right) \) be an abstract simplicial complex. We set \( L \mathrel{\text{:=}} \operatorname{Cone}\left( K\right) \) . Note that it follows from Lemma 42.4 that it remains to prove that the identity map id: \( {\mathrm{C}}_{ * }^{\text{simp }}\left( L\right) \rightarrow {\mathrm{C}}_{ * }^... | No |
Lemma 63.12. Given any \( n \in \mathbb{N} \) and given any \( k \in {\mathbb{N}}_{0} \) we have\n\n\[ \n{\widetilde{\mathrm{H}}}_{k}^{\mathrm{{simp}}}\left( {D}_{n}\right) = 0\;\text{ and }\;{\widetilde{\mathrm{H}}}_{k}^{\mathrm{{simp}}}\left( {S}_{n}\right) \cong \left\{ \begin{array}{ll} \mathbb{Z}, & \text{ if }k =... | (1) Given any \( n \in \mathbb{N} \) and given any \( k \in {\mathbb{N}}_{0} \) we calculate that\n\n\[ \n{\widetilde{\mathrm{H}}}_{k}^{\text{simp }}\left( {D}_{n}\right) \underset{ \uparrow }{ \cong }{\widetilde{\mathrm{H}}}_{k}^{\text{simp }}\left( {\operatorname{Cone}\left( {D}_{n - 1}\right) }\right) \underset{ \up... | No |
Theorem 63.20. There exist two finite linear simplicial complex \( X \) and \( Y \) for which the underlying topological spaces are homeomorphic, but such that there are no subdivisions of \( X \) and \( Y \) which are simplicially isomorphic. | Proof. The theorem is proved in [Miln61, Theorems 1 and 2]. Alternative proofs are given in [Cohe73, p. 84], [Stal65b, p. 252] and [Ran96, Chapter 2]. | No |
Proposition 63.24. Let \( K \) be an abstract simplicial complex. Given an ordered \( n \) -simplex \( s = \left( {{v}_{0},\ldots ,{v}_{n}}\right) \) of \( K \) we denote by\n\n\[{\Phi }_{s}^{ \leq } : {\Delta }^{n} = \left\{ {\left( {{t}_{0},\ldots ,{t}_{n}}\right) \in {\mathbb{R}}_{ \geq 0}^{n + 1}\left| {\;\mathop{\... | Proof of Proposition 63.24. Let \( K = \left( {V, S}\right) \) be an abstract simplicial complex. We write \( X \mathrel{\text{:=}} \left| K\right| \) . Given \( n \in {\mathbb{N}}_{0} \) we denote by \( {S}_{n} \) the set of ordered \( n \) -simplices of \( K \) . Next we consider the following diagram ![448f61af-e517... | Yes |
Theorem 63.25. Let \( K \) be an ordered abstract simplicial complex. The maps\n\n\[ \n{\Theta }_{n} : {\mathrm{C}}_{n}^{\operatorname{simp}, \leq }\left( K\right) \rightarrow {\mathrm{C}}_{n}\left( \left| K\right| \right) \]\n\n\[ s \mapsto \left( {{\Phi }_{s}^{ \leq } : {\Delta }^{n} \rightarrow \left| K\right| }\rig... | Proof of Theorem 63.25. Let \( K \) be an ordered abstract simplicial complex. It follows almost immediately from the definitions that the maps\n\n\[ {\Theta }_{n} : {\mathrm{C}}_{n}^{\operatorname{simp}, \leq }\left( K\right) \rightarrow {\mathrm{C}}_{n}\left( \left| K\right| \right) \]\n\n\[ s \mapsto \left( {{\Phi }... | No |
Lemma 63.29. Let \( K \) be an abstract simplicial complex and let \( L \) be a subcomplex of \( K \). The sequence \[ \ldots \; \rightarrow \;{\mathrm{H}}_{k}^{\mathrm{{simp}}}\left( L\right) \; \rightarrow \;{\mathrm{H}}_{k}^{\mathrm{{simp}}}\left( K\right) \; \rightarrow \;{\mathrm{H}}_{k}^{\mathrm{{simp}}}\left( {K... | \[ \left\lbrack \sigma \right\rbrack \; \mapsto \;\left\lbrack {{\partial }_{k}\left( \sigma \right) }\right\rbrack \] | No |
Lemma 63.31. Let \( K = \left( {V, S}\right) \) be an abstract simplicial complex. The open stars \( \{ \operatorname{St}\left( v\right) {\} }_{v \in V} \) are an open good point-finite cover. | Proof. Let \( K = \left( {V, S}\right) \) be an abstract simplicial complex. Let \( P \in \left| K\right| \) . By Lemma 61.11 (2) we know that there exists a unique \( s \in S \) with \( P \in \langle s\rangle \) . Furthermore, by Lemma 62.5 (2) we know that \( P \in \operatorname{St}\left( v\right) \) if and only if \... | No |
Lemma 62.5 (2), shows that the open stars \( \operatorname{St}\left( v\right) \) form an open point-finite cover of \( \left| K\right| \) . It remains to show that the cover is good. Thus let \( {v}_{0},\ldots ,{v}_{n} \in V \) . We have | \[ \begin{aligned} \operatorname{St}\left( {v}_{0}\right) \cap \cdots \cap \operatorname{St}\left( {v}_{n}\right) & = \overset{\text{contractible by Lemma [62.5] (1f) }}{\overbrace{\operatorname{St}\left( \left\{ {{v}_{0},\ldots ,{v}_{n}}\right\} \right) }} \\ \uparrow & \left\{ \begin{array}{ll} \operatorname{St}\left... | Yes |
Theorem 63.32. (Nerve Theorem) Let \( X \) be a path-connected topological space and let \( \mathcal{U} = {\left\{ {U}_{i}\right\} }_{i \in I} \) be an open good point-finite cover of \( X \) . The following two statements hold:\n\n(1) There exists a weak homotopy equivalence \( \left| {N\left( \mathcal{U}\right) }\rig... | Proof. Statement (2) of the Nerve Theorem is an immediate consequence of Statement (1) together with the Whitehead Theorem 119.9. Thus it remains to prove Statement (1).\n\n--- \n\nIn the discussion below we will only sketch an argument. A full proof of Statement (1) is given in McCor67, Theorem 1].\n\nThus let \( X \)... | No |
Theorem 63.33. (Nerve Theorem) Let \( X \) be a topological space and let \( \mathcal{U} = {\left\{ {U}_{i}\right\} }_{i \in I} \) be an open good cover of \( X \) . If \( X \) is paracompact, then there exists a homotopy equivalence \( \left| {N\left( \mathcal{U}\right) }\right| \rightarrow X \) | Proof. By definition of a paracompact topological space, see page [144, there exists a partition of unity \( {\left\{ {\varphi }_{i}\right\} }_{i \in I} \) subordinate to the covering. We define\n\n\[ G : X \rightarrow \left| {N\left( \mathcal{U}\right) }\right| \]\n\n\[ x \mapsto \left( \begin{aligned} I & \rightarrow... | Yes |
Proposition 63.35. If \( M \) is an open subset of \( {\mathbb{R}}^{n} \), then \( {\mathrm{H}}_{k}\left( M\right) = 0 \) for \( k \geq n \) . | Proof. The proposition follows from a fun argument using the Mayer-Vietoris Theorem [46.5] We invite the reader to fill in the details in Exercise 63.15. | No |
Lemma 63.40. Let \( \left( {K, \leq }\right) \) be an ordered abstract simplicial complex. As before we denote by \( \sigma : \operatorname{sd}\left( K\right) \rightarrow K \) the stretching map introduced in Lemma 63.38,\n\n(1) The induced chain map\n\n\[ \n{\sigma }_{ * } : {\mathrm{C}}_{ * }^{\operatorname{simp}, \l... | Proof (*). Statement (2) of the lemma is an immediate consequence of Statement (1).  and the naturality statements of Lemma 63.38 and Lemma 63.39. Thus it remains to prove Statement (1).\n\nLet \( \left( {K, \leq }... | Yes |
Lemma 63.41. Let \( K \) be an abstract simplicial complex.\n\n(1) For every \( n \in {\mathbb{N}}_{0} \) the map\n\n\[ \n{\mathrm{C}}_{n}^{\text{simp }}\left( K\right) \rightarrow {\mathrm{C}}_{n}^{\text{simp }}\left( {\operatorname{sd}\left( K\right) }\right)\n\]\n\n\[ \n\left\lbrack s\right\rbrack \mapsto \left\lbra... | Proof. The first three statements follow easily from the definitions. The only statement which perhaps needs a little bit of thought is the first statement. We leave it to the reader to fill in the details. Let us turn to the proof of Statement (4). As mentioned before, by the Well-ordering Theorem 1.6 we can pick an o... | No |
Lemma 63.42. Let \( \\left( {K, \\leq }\\right) \) be an abstract simplicial complex. We denote by \( \\sigma : \\operatorname{sd}\\left( K\\right) \\rightarrow \) \( K \) the stretching map from Lemma 63.38. The map \( \\left| \\sigma \\right| \\circ {\\mho }^{-1} : \\left| K\\right| \\rightarrow \\left| K\\right| \) ... | Proof. Since we will not make use of Lemma 63.42 we outsource the proof thereof to Exercise 63.17. | No |
Proposition 63.43. For every ordered abstract simplicial complex \( \left( {K, \leq }\right) \) and every \( n \in \) \( {\mathbb{N}}_{0} \) the following diagram commutes:\n\n\n\nFurthermore all maps form natural ... | The key step in proving Proposition 63.43 is to give an alternative description of the simplicial subdivision map that resembles the definition of the singular subdivision map which we gave on page 1135. We start out with the following definition.\n\nDefinition. Let \( \left( {K, \leq }\right) \) be an ordered abstract... | No |
Lemma 63.44. Let \( \\left( {K, \\leq }\\right) \) be an ordered abstract simplicial complex. Given any \( n \) - simplex \( s \) of \( K \) we have \( {\\widetilde{u}}_{n}\\left( s\\right) = {u}_{n}\\left( s\\right) \) . | Proof of Lemma 6.3.44. Let \( \\left( {K, \\leq }\\right) \) be an ordered abstract simplicial complex. We prove the lemma by induction on \( n \) . For \( n = 0 \) the statement is basically obvious. Now suppose that the statement holds for some \( n \\in {\\mathbb{N}}_{0} \) . Let \( s = \\left\\{ {{v}_{0} < \\cdots ... | Yes |
Proposition 63.45. For every abstract simplicial complex \( K \) and every \( n \in {\mathbb{N}}_{0} \) the following diagram commutes. \n\nFurthermore all maps are natural. | Proof. The proposition follows easily from the fact that every abstract simplicial complex admits an order together with Lemma 63.8 and Proposition 6 Proposition 63.28 | No |
Lemma 64.1. Let \( M \) be a smooth manifold which is equipped with a smooth simplicial structure. If \( M \) is \( n \) -dimensional, then the dimension of each simplex is at most \( n \) . | Proof. Let \( s \) be a \( k \) -simplex of the given smooth simplicial structure. Combining the characteristic map of the simplex with a chart we obtain a map from a non-empty open subset of \( {\mathbb{R}}^{k} \) to \( {\mathbb{R}}^{n} \) . If the simplicial structure is smooth this map is in fact an embedding. It fo... | Yes |
Proposition 64.3. Let \( M \) be a closed smooth submanifold of \( {\mathbb{R}}^{m} \). Given any \( \epsilon > 0 \) there exists a linear simplicial complex \( \left( {L = \left( {W, T}\right) ,\Theta : \left| L\right| \rightarrow X}\right) \) with \( X \subset {\mathbb{R}}^{m} \) and an isotopy \( F : X \times \left\... | Sketch of PROOF. We continue with the notation and results in the proof of Theorem [64.2] The \( \epsilon > 0 \) corresponds to a suitable interpretation of \ | No |
Theorem 64.4. Let \( M \) and \( N \) be smooth manifolds and suppose we are given two smooth simplicial structures \( \left( {K,\Theta : \left| K\right| \rightarrow M}\right) \) and \( \left( {L,\Omega : \left| L\right| \rightarrow N}\right) \) . If \( M \) and \( N \) are diffeomorphic, then there are subdivisions of... | Proof. This statement is shown in [WhdJ40, Theorem 8] and alternatively in [Mun66a], Theorem 10.5 (see also [Mun75, Theorem 10.13]). It is perhaps worth pointing out that we do not need to assume that \( M \) is compact. | No |
Theorem 64.5. Every (compact) n-dimensional smooth manifold \( M \) admits a (finite) CW-structure with the following properties:\n\n(1) every cell has dimension at most \( n \) ,\n\n(2) the boundary \( \partial M \) is a subcomplex.\n\nFurthermore, if \( M = A \cup B \) is a decomposition into two n-dimensional subman... | Proof. The theorem follows immediately from Theorem 64.2 together with Lemma 64.1 and Lemma 61.24. | No |
Proposition 64.6. Let \( M \) be an \( n \) -dimensional 0-connected smooth manifold.\n\n(1) For every \( k > n \) and for every abelian group \( G \) we have \( {\mathrm{H}}_{k}\left( {M;G}\right) = 0 \) .\n\n(2) The higher homotopy groups of \( M \) are countable. | (1) This statement follows from Theorem 64.5 together with Proposition 48.5 (1) and the discussion on page 1402.\n\n(2) This statement follows from Theorem 64.2 together with Proposition 62.9. | Yes |
Lemma 64.7. (*) Let \( X \) be a topological space, let \( A \subset X \) be a subset and let \( k \in {\mathbb{N}}_{0} \) . (1) If the homology groups \( {\mathrm{H}}_{k}\left( X\right) \) and \( {\mathrm{H}}_{k - 1}\left( A\right) \) are finitely generated, then so is the relative homology group \( {\mathrm{H}}_{k}\l... | Proof (*). The first statement follows from the long exact sequence of the pair \( \left( {X, A}\right) \) together with Lemma 19.6. The second statement is proved in a similar fashion. | No |
Proposition 64.8. The composition of two piecewise linear maps between finite simplicial complexes is again piecewise linear. | Proof. The proposition is proved in [Ze63b, Lemma I.2] and [Gla70, Theorem I.6]. | No |
Lemma 64.9. If \( {f}_{1} : {X}_{1} \rightarrow {Y}_{1} \) and \( {f}_{2} : {X}_{2} \rightarrow {Y}_{2} \) are two PL-homeomorphisms between two finite simplicial complexes, then the map\n\n\[ \n{f}_{1} * {f}_{2} : {X}_{1} * {X}_{2} \rightarrow {Y}_{1} * {Y}_{2} \n\]\n\nis also a PL-homeomorphism. | Proof. We leave it to the reader to provide the proof. | No |
Proposition 64.11. Let \( \left( {K = \left( {V, S}\right) ,\Theta : \left| K\right| \rightarrow X}\right) \) be a simplicial complex. The following statements are equivalent:\n\n(1) \( \left( {X,\Theta }\right) \) is an \( n \) -dimensional PL-manifold.\n\n(2) For every vertex \( v \in V \) the link \( \operatorname{L... | Proof.\n\n(1) \( \Rightarrow \) (2) This implication is shown in [Hud69, p. 26] (alternatively see also Ze63b, p. III.2]).\n\n\( \left( 2\right) \Rightarrow \left( 3\right) \) Let \( v \in V \) . First we consider the case that the link \( \operatorname{Lk}\left( v\right) \) is a PL \( \left( {n - 1}\right) \) -sphere.... | Yes |
Proposition 64.12. Let \( \left( {K = \left( {V, S}\right) ,\Theta : \left| K\right| \rightarrow M}\right) \) be a compact PL-manifold and let \( J = \left( {U, R}\right) \) be a subcomplex with \( \Theta \left( \left| J\right| \right) \subset M \smallsetminus \partial M \) .\n\n(1) \( {N}_{K}\left( J\right) \) is a co... | Sketch of Proof. Evidently we can just assume that \( M = \left| K\right| \) and that \( \Theta = \mathrm{{id}} \) . As mentioned above, our definition of the regular neighborhood is an example of a regular neighborhood as defined in [RS72, p. 33]. Thus we can use the results from [RS72].\n\n(1) This statement is expli... | Yes |
Proposition 64.13. Let \( X \) be an \( n \) -dimensional PL-manifold. The following statements hold:\n\n(1) \( X \) is an \( n \) -dimensional topological manifold.\n\n(2) A vertex \( v \) of the PL-structure lies on the boundary of the topological manifold \( X \) if and only if the link \( \operatorname{Lk}\left( v\... | Proof. Let \( \left( {K = \left( {V, S}\right) ,\Theta : \left| K\right| \rightarrow X}\right) \) be an \( n \) -dimensional PL-manifold. Without loss of generality we can assume that \( \Theta = \mathrm{{id}} \), in particular we can assume that \( \left| K\right| = X \) .\n\n(1) In the following we will show that \( ... | Yes |
Theorem 64.15.\n\n(1) For \( n = 1,\ldots ,6 \) the map \( {\mathfrak{V}}_{n} \) is a bijection.\n\n(2) For \( n = 7 \) (and many other dimensions) the map \( {\mathfrak{v}}_{n} \) is not a monomorphism.\n\n(3) For \( n = {10} \) (and many other dimensions) the map \( {\mathbf{v}}_{n} \) is not an epimorphism. In fact ... | Proof. The theorem is a combination of several major results in topology. Later in Section ?? we will discuss this theorem in much greater detail and we will give all the necessary references. | No |
Lemma 64.18. Let \( n \in {\mathbb{N}}_{ \geq 2} \) and let \( M \) be a closed 0-connected \( n \) -dimensional topological manifold. If \( {\pi }_{1}\left( M\right) \) is non-trivial, then the suspension \( \sum \left( M\right) \) is not an \( \left( {n + 1}\right) \) -dimensional topological manifold. | Proof. We outsourced the proof to the very amusing Exercise 45.9 | No |
Lemma 64.19. Let \( K = \left( {V, S}\right) \) be an abstract simplicial complex and let \( s \in S \) . The following statements hold:\n\n(1) \( \operatorname{St}\left( {K, s}\right) = \operatorname{Lk}\left( {K, s}\right) * s \) .\n\n(2) The union of the subcomplexes \( K \smallsetminus \operatorname{St}\left( {K,\w... | Proof. We leave it to the reader to verify that all the statement follow indeed immediately from the definitions. | No |
Lemma 64.20. Let \( K = \left( {V, S}\right) \) be an abstract simplicial complex and let \( s \in S \) . If \( \operatorname{Lk}\left( {K, s}\right) \) is finite, then there exists a natural PL-homeomorphism \( \left| K\right| \rightarrow \left| {{\sigma }_{s}\left( K\right) }\right| \) . | Proof (*). Let \( K = \left( {V, S}\right) \) be a finite abstract simplicial complex and let \( s \in S \) . Recall that on page 1598 we saw that there exists a natural PL-homeomorphism \( \varphi : \left| s\right| \rightarrow \left| {\partial s * \underline{s}}\right| \) which is the identity on the common subset \( ... | Yes |
Theorem 64.21. (Alexander-Newman Theorem) Let \( K \) and \( L \) be two finite abstract simplicial complexes. The following two statements are equivalent:\n\n(1) The abstract simplicial complexes \( K \) and \( L \) are stellar equivalent.\n\n(2) The simplicial complexes \( \left| K\right| \) and \( \left| L\right| \)... | Proof. The \ | No |
Proposition 64.23. Let \( K \) and \( L \) be two finite abstract simplicial complexes. If \( K \) is a (closed) \( n \) -dimensional combinatorial manifold and if \( L \) is combinatorially homeomorphic to \( K \), then \( L \) is a (closed) \( n \) -dimensional combinatorial manifold. | Proof. A short moment's thought shows that this statement does indeed need a proof, it does not follow immediately from the definitions. In fact the proposition is a consequence of Gla70, Lemma on p. 19]. | No |
Lemma 64.24. Let \( K \) be a closed \( n \) -dimensional combinatorial manifold. If \( s \) is a \( k \) - simplex, then the link \( \operatorname{Lk}\left( {K, s}\right) \) is a combinatorial \( \left( {n - k - 1}\right) \) -sphere. | Proof. The proof of this lemma is almost the same as the proof of Proposition 64.11 \( \left( 2\right) \Rightarrow \left( 4\right) \) . The only slight difference is that now we need to use the fact, mentioned above, that every combinatorial \( k \) -sphere is a closed \( k \) -dimensional combinatorial manifold. By de... | No |
Lemma 64.26. Let \( K = \left( {V, S}\right) \) be a closed \( n \) -dimensional combinatorial manifold and let \( s \in S \) be a \( k \) -simplex such that \( \operatorname{Lk}\left( {K, s}\right) \) is a non-trivial \( \left( {n - k - 1}\right) \) -sphere in \( K \) . We denote by \( t \) the dual simplex. The follo... | Proof.\n\n(1) We leave the elementary verification of these statements to the reader. Note that the slightly awkward fact that we need to deal with 0 -simplices separately is due to\n\n---\n\n\( {}^{997} \) Here we view the simplex \( s \) as an abstract simplicial complex in its own right.\n\n\( {}^{998} \) It follows... | No |
Theorem 64.27. (Pachner’s Theorem) Let \( K \) and \( L \) be two closed combinatorial manifolds. The following two statements are equivalent:\n\n(1) The abstract simplicial complexes \( K \) and \( L \) are bistellar equivalent.\n\n(2) The abstract simplicial complexes \( K \) and \( L \) are combinatorially homeomorp... | Proof. The \ | No |
Lemma 65.1. Let \( R \) be a ring and let \( A \) and \( B \) be two \( \left( {n \times n}\right) \) -matrices over \( R \) . If \( R \) is commutative, then we have\n\n\[ \operatorname{tr}\left( {A \cdot B}\right) = \operatorname{tr}\left( {B \cdot A}\right) \] | Proof. We perform the following simple calculation:\n\n\[ \operatorname{tr}\left( {A \cdot B}\right) = \operatorname{tr}\left( {\operatorname{matrix}\text{ with }\left( {i, j}\right) \text{-entry }\mathop{\sum }\limits_{{k = 1}}^{n}{a}_{ik} \cdot {b}_{kj}}\right) = \mathop{\sum }\limits_{{i = 1}}^{n}\mathop{\sum }\limi... | Yes |
Lemma 65.2. Let \( R \) be a commutative domain, let \( V \) be a free \( R \) -module of rank \( n \) and let \( \varphi : V \rightarrow V \) be an endomorphism. We pick a basis for \( V \) . Then \( \operatorname{tr}\left( \varphi \right) \mathrel{\text{:=}} \) trace of the matrix representing \( \varphi \) with the ... | Proof. Let \( R \) be a commutative domain. Let \( \left\{ {{v}_{1},\ldots ,{v}_{m}}\right\} \) and \( \left\{ {{w}_{1},\ldots ,{w}_{n}}\right\} \) be two bases for \( V \) . Since \( R \) is a commutative domain the rank of \( V \) is well-defined, see page 82, Thus we see that \( m = n \) . Now let \( P = \left( {p}_... | No |
Lemma 65.3. Let \( \varphi : V \rightarrow V \) be an endomorphism of a finitely generated free abelian group and let \( R \) be a commutative domain. Let \( \gamma : \mathbb{Z} \rightarrow R \) be the natural ring homomorphism with \( \gamma \left( {1}_{\mathbb{Z}}\right) = {1}_{R} \) from page 490. Then ![448f61af-e5... | Proof. Let \( \left\{ {{v}_{1},\ldots ,{v}_{n}}\right\} \) be a basis for the free abelian group \( V \) and let \( A \) be the \( \left( {n \times n}\right) \) - matrix that represents \( \varphi \) with respect to this basis. It is straightforward to verify that \( {v}_{1} \otimes 1,\ldots ,{v}_{n} \otimes 1 \) is a ... | Yes |
Lemma 65.4. Given a finitely generated abelian group \( H \) the maximal torsion-free quotient \( \mathrm{F}H = H/\operatorname{Tor}\left( H\right) \) of \( H \) is a free abelian group. | Proof. A very elementary exercise shows that \( \mathrm{F}H = H/\operatorname{Tor}\left( H\right) \) is a torsion-free abelian group. Since \( H \) is finitely generated we obtain from Theorem 19.4 that \( \mathrm{F}H \) is in fact a free abelian group. | No |
Lemma 65.5. Let \( H \) be a finitely generated abelian group and let \( \varphi : H \rightarrow H \) be an endomorphism. Then the following equality holds:\n\n\[ \operatorname{tr}\left( {\varphi : H \rightarrow H}\right) = \operatorname{tr}\left( {\varphi \otimes \mathrm{{id}} : H \otimes \mathbb{Q} \rightarrow H \oti... | Proof. By the classification of finitely generated abelian groups, see Theorem 19.4, we can write \( H = F \oplus T \) where \( F \) is a free abelian group and \( T \) is a torsion group. We denote by \( \psi \) the map \( F\xrightarrow[]{f \mapsto f \oplus 0}F \oplus T\xrightarrow[]{\varphi }F \oplus T\xrightarrow[]{... | Yes |
Let \( X \) be a topological space and let \( \varphi : X \rightarrow X \) be a map. (1) Let \( k \in {\mathbb{N}}_{0} \) . We have \( \operatorname{rank}\left( {{\mathrm{H}}_{k}\left( X\right) }\right) = \dim \left( {{\mathrm{H}}_{k}\left( {X;\mathbb{Q}}\right) }\right) \) . Furthermore, if \( {\mathrm{H}}_{k}\left( X... | Proof. Statement (2) is an immediate consequence of Statement (1). Thus let us prove the first statement. Let \( k \in {\mathbb{N}}_{0} \) . The equality \( \operatorname{rank}\left( {{\mathrm{H}}_{k}\left( X\right) }\right) = \dim \left( {{\mathrm{H}}_{k}\left( {X;\mathbb{Q}}\right) }\right) \) was shown in Corollary ... | Yes |
Theorem 45.13. Let \( n \in \mathbb{N} \) and let \( \varphi : {S}^{n} \rightarrow {S}^{n} \) be a map. If \( \deg \left( \varphi \right) \neq {\left( -1\right) }^{n + 1} \), then \( \varphi \) has a fixed point. | Proof. We consider the topological space \( {S}^{n} \) . By the discussion on page 1500 we know that \( {S}^{n} \) admits the structure of a finite simplicial complex. Let \( \varphi : {S}^{n} \rightarrow {S}^{n} \) be a map. We calculate that\n\nby Lemmas 65.8 we have \( \Lambda \left( {\varphi ,\mathbb{Q}}\right) = \... | Yes |
Theorem 65.10. (Lefschetz Fixed Point Theorem II) Let \( M \) be a compact smooth manifold and let \( \varphi : M \rightarrow M \) be a map. If there exists a field \( \mathbb{F} \) such that the \( \mathbb{F} \) -Lefschetz number \( \Lambda \left( {\varphi ,\mathbb{F}}\right) \) is non-zero, then \( \varphi \) has a f... | Proof. Let \( M \) be a compact smooth manifold. By Theorem 64.2 we know that \( M \) admits a finite simplicial structure. Thus we can apply the Lefschetz Fixed Point Theorem 65.9 and we are done. | No |
Corollary 65.11. Let \( M \) be a compact connected smooth manifold such that \( {\mathrm{H}}_{n}\left( M\right) \) is torsion for all \( n \geq 1 \) . Then every map \( \varphi : M \rightarrow M \) has a fixed point. | Proof. Let \( M \) be a compact connected smooth manifold such that \( {\mathrm{H}}_{n}\left( M\right) \) is torsion for all \( n \geq 1 \) . We calculate that\n\n\[ \Lambda \left( {\varphi ,\mathbb{Q}}\right) \overset{\text{ by Lemma }}{ = }\operatorname{tr}\left( \underset{ = 1\text{ by Lemma }\overline{\lbrack {65.6... | Yes |
Lemma 65.13. Let \( \mathbb{F} \) be a field. Suppose we are given a commutative diagram of homomorphisms of finite-dimensional \( \mathbb{F} \) -vector spaces of the following type:\n\n\n\nIf the horizontal sequen... | Proof. Let \( {a}_{1},\ldots ,{a}_{m} \) be a basis for \( A \) and let \( {c}_{1},\ldots ,{c}_{n} \) be a basis for \( C \) . For \( i = 1,\ldots, n \) we pick a \( {\bar{c}}_{i} \in B \) with \( p\left( {\bar{c}}_{i}\right) = {c}_{i} \) . It follows easily from our hypothesis that the horizontal sequences are exact t... | Yes |
Lemma 65.14. Let \( f : M \rightarrow N \) be a map between two smooth manifolds and let \( v \) be a vector field on \( N \) . If \( f \) is a local diffeomorphism, then for each \( x \in M \) we can define \( \left( {{f}^{ * }v}\right) \left( x\right) = {\left( \mathrm{D}{f}_{x}\right) }^{-1}\left( {v\left( {f\left( ... | Proof. The lemma follows easily from the definitions. We do not feel like filling in the details. | No |
Theorem 45.14. (Hairy Ball Theorem) Every vector field on an even-dimensional sphere vanishes on at least one point. | Sketch of PROOF. Let \( v \) be a nowhere vanishing vector field on a sphere \( {S}^{n} \) . By \ | No |
(1) Let \( f : M \rightarrow N \) be a map between smooth manifolds which is a local diffeomorphism.\n\nLet \( v \) be a vector field on \( N \) . The map\n\n\[ P \mapsto {\left( \mathrm{D}{f}_{P}\right) }^{-1}\left( \underset{ \in {\mathrm{T}}_{f\left( P\right) }\left( N\right) }{\underbrace{v\left( {f\left( P\right) ... | (1) The mature reader will have no troubles with providing a proof for this statement. | No |
Proposition 65.18. Let \( M \) be a closed smooth manifold. If there exists a nowhere vanishing vector field, then there exists a map \( g : M \rightarrow M \) that is homotopic to the identity and which has no fixed points. | Sketch of Proof of Proposition 65.18. We sketch two different proofs of the proposition. The first proof is much shorter, but requires more input from analysis and it needs a stronger hypothesis. The second proof is somewhat longer but it is makes fewer demands on analysis.\n\nLet us start with the first proof. To simp... | No |
Lemma 65.6 (1) by the claim and the Lefschetz Fixed Point Theorem II 65.10 | \[ \chi \left( M\right) \overset{ \downarrow }{ = }\Lambda \left( {\mathrm{{id}} : M \rightarrow M}\right) \underset{ \uparrow }{ = }\Lambda \left( {f : M \rightarrow M}\right) \underset{ \uparrow }{ = }\Lambda \left( {f : M \rightarrow M,\mathbb{Q}}\right) \overset{ \downarrow }{ = }0. \] by Lemma 65.7 since \( f \) a... | Yes |
Lemma 65.20. Let \( V \) be a finitely generated free \( \mathbb{Z} \) -module and let \( \rho : V \rightarrow V \) be an isomorphism with the following two properties:\n\n(i) There exists a subset \( T \) of \( V \) such that \( T \cap \rho \left( T\right) = \varnothing \) and such that \( T \cup \rho \left( T\right) ... | Proof. We equip \( V \) with the basis \( T \sqcup \rho \left( T\right) \) . It follows immediately from (ii) that\n\nmatrix representing \( \rho \) with respect to the basis \( T \sqcup \rho \left( T\right) = \frac{T}{\rho \left( T\right) }\left( \begin{matrix} T & \rho \left( T\right) \\ 0 & \text{ id } \\ \text{ id ... | Yes |
Lemma 44.1. Let \( M \) be an \( n \) -dimensional topological manifold and let \( P \in M \) . For every \( j \in {\mathbb{N}}_{0} \) we have\n\n\[ \n{H}_{j}\left( {M, M\smallsetminus \{ P\} }\right) \; \cong \;\left\{ \begin{array}{ll} \mathbb{Z}, & \text{if }P \in M \smallsetminus \partial M\;\text{and}\;j = n, \\ 0... | Proof. Since in a minute we will calculate a different local homology group it is perhaps helpful to remind us of the proof of the lemma. For simplicity we only consider the case that \( P \in M \smallsetminus \partial M \) . In this case we can pick a chart \( \Phi : U \rightarrow {B}^{n} \) for \( P \) with \( \Phi \... | Yes |
Proposition 66.2. Let \( K = \left( {V, S}\right) \) be an abstract simplicial complex and let \( P \in \left| K\right| \) . By Lemma 61.11 there exists a unique simplex \( s \in S \) such that \( P \) is a point in the open simplex \( \langle s\rangle \) . We set \( m \mathrel{\text{:=}} \dim \left( s\right) \) . If \... | Proof. We perform the following calculation:\n\nby the long exact sequence in reduced homology of the pair \( \left( {\operatorname{St}\left( s\right) ,\operatorname{St}\left( s\right) \smallsetminus \{ P\} }\right) \) and the observation, see Lemma 62.5 (1f), that \( \operatorname{St}\left( s\right) \) is contractible... | Yes |
Lemma 66.3. Let \( K = \left( {V, S}\right) \) be an abstract simplicial complex, let \( t \) be a simplex of \( K \) and let \( s \) be a codimension-one face of \( t \) . Any two equivalent orientations of \( t \) induce equivalent orientations of \( s \) . | Proof. This statement follows easily from the properties of the sign of a bijection, see Lemma 63.1. We leave it to the reader to fill in the details. | No |
Lemma 66.5. Let \( A, B, C \) and \( D \) be abelian groups. We denote by \( {p}_{1} : B \oplus C \rightarrow B \) the projection. Now suppose we are given an exact sequence\n\n\[ A\overset{\alpha }{ \rightarrow }B \oplus C\overset{\varphi }{ \rightarrow }D \]\n\nof homomorphisms such that the following conditions are ... | PROOF OF LEMMA 66.5 (*). From our hypotheses (1) and (2) and the exactness of the sequence we easily deduce the following statement:\n\n\( \left( *\right) \) The map \( B\xrightarrow[]{b \mapsto \varphi \left( \left( {b,0}\right) \right) }D \) is the zero map.\n\nNote that \( \left( *\right) \) implies that if \( \varp... | Yes |
Theorem 66.6. Let \( M \) be a connected non-empty \( n \) -dimensional smooth manifold. The following statements hold:\n\n(1) We have an isomorphism as follows:\n\n\[ \n{\mathrm{H}}_{n}\left( {M;\mathbb{Z}}\right) \cong \left\{ \begin{array}{ll} \mathbb{Z}, & \text{ if }M\text{ is closed and orientable,} \\ 0, & \text... | Proof of Theorem [66.6] Let \( M \) be a connected non-empty \( n \) -dimensional smooth manifold. Note that by Theorem 64.2 we know that \( M \) admits a smooth simplicial structure \( \left( {K = \left( {V, S}\right) ,\Theta : \left| K\right| \rightarrow M}\right) \) . By Proposition [66.1] we know that \( K \) is an... | No |
Proposition 66.7. Let \( n \in \mathbb{N} \) and let \( M \) be a connected non-empty \( n \) -dimensional smooth manifold \( {}^{1010} \) Furthermore let \( \left( {K = \left( {V, S}\right) ,\Theta : \left| K\right| \rightarrow M}\right) \) be a smooth simplicial structure. The map ![448f61af-e517-4f9c-831f-f6ce5868f6... | Proof of Proposition 66.7 (*). In the following we will provide a proof of Proposition 66.7. To preserve the author's sanity we will skip a few technical calculations. We leave it to the reader to fill in the details.\n\nThroughout the proof let \( M \) be a non-empty \( n \) -dimensional smooth manifold and let \( \le... | No |
Theorem 66.8. Let \( M \) be a connected non-empty \( n \) -dimensional smooth manifold. The following statements hold:\n\n(1) We have an isomorphism as follows:\n\n\[ \n{\mathrm{H}}_{n}\left( {M,\partial M;\mathbb{Z}}\right) \cong \left\{ \begin{array}{ll} \mathbb{Z}, & \text{ if }M\text{ is orientable and compact,} \... | Proof. Let \( M \) be a connected non-empty \( n \) -dimensional smooth manifold. We dealt with the case \( \partial M = \varnothing \) in Theorem 66.6 so we might as well assume that \( \partial M \neq \varnothing \).\n\nIn the following we first sketch what we think should be the \ | No |
Lemma 66.9. Let \( p \in \mathbb{N} \) and let \( q \in \mathbb{Z} \) be coprime to \( p \) . For the corresponding lens space \( L\left( {p, q}\right) \) we have \[ {\mathrm{H}}_{n}\left( {L\left( {p, q}\right) ;\mathbb{Z}}\right) \cong \left\{ \begin{array}{ll} \mathbb{Z}, & \text{ if }n = 0,3, \\ 0, & \text{ if }n =... | Proof. We start out with the following observations: (1) It follows from Proposition 16.9 and the definition of a lens space that \( L\left( {p, q}\right) \) is finitely covered by \( {S}^{3} \) . It follows from \( \chi \left( {S}^{3}\right) = 0 \) together with Proposition 37.4 that \( \chi \left( {L\left( {p, q}\rig... | Yes |
Lemma 66.10. Let \( M \) be a smooth manifold. If \( M \) is a homology \( n \) -sphere, then \( M \) is connected and orientable. | It follows from the classification of 1-dimensional and 2-dimensional topological manifolds that we had stated in Theorem 7.1 and the Surface Classification Theorem 23.4 and the calculation of homology groups in Proposition 48.9 that for \( n = 1,2 \) every homology \( n \) -sphere is homeomorphic to the standard spher... | No |
Lemma 66.15. Let \( K = \left( {V, S}\right) \) be an n-dimensional abstract simplicial complex that admits a boundary route. There exists a subcomplex \( {K}^{\prime } = \left( {{V}^{\prime },{S}^{\prime }}\right) \) of \( K = \left( {V, S}\right) \) with the following three properties:\n\n(1) \( {K}^{\prime } = \left... | Proof of Lemma 66.15. Let \( K = \left( {V, S}\right) \) be an \( n \) -dimensional abstract simplicial complex that admits a boundary route \( \widetilde{\Gamma } = \left( {W,\widetilde{F},\varphi }\right) \) . Since \( K \) is \( n \) -dimensional we know that it contains at least one \( n \) -simplex and since \( K ... | No |
Proposition 66.16. Let \( M \) be a compact connected \( n \) -dimensional smooth manifold. There exists a CW-structure for \( M \) with a single \( n \) -cell. | In the proof of Proposition 66.16 we will make use of the following lemma.\n\nLemma 66.17. Let \( L \) | No |
Lemma 66.17. Let \( L \) be an \( n \) -dimensional pseudomanifold that contains two subcomplexes \( J \) and \( K \) . If \( K \) consists of a single \( n \) -simplex, if \( J \cap K \) consists of a single \( \left( {n - 1}\right) \) -simplex e and if e has order 1 in \( J \), then there exists a homeomorphism \( \l... | Sketch of A PROOF OF LEMMA 66.17. Let \( w \) be the one \( n \) -simplex of \( J \) that cobounds \( e \) . It is elementary, albeit ever-so-slightly painful, to write down an explicit homeomorphism \( \left| w\right| \cup \left| K\right| \rightarrow \left| w\right| \) that is the identity on \( \left| {\partial w}\ri... | No |
Lemma 67.3. Let \( G \) be an abelian group and let \( K = \left( {V, S}\right) \) be an abstract simplicial complex.\n\n(1) Given any \( g \in G \) the constant cochain \( {\varphi }_{g} : {\mathrm{C}}_{0}^{\operatorname{simp}}\left( K\right) \rightarrow G \) given by \( {\varphi }_{g}\left( v\right) = g \) for every ... | Proof.\n\n(1) Let \( g \in G \) . We need to show that \( {\delta }_{0}\left( {\varphi }_{g}\right) = 0 \in {\mathrm{C}}_{\text{simp }}^{1}\left( K\right) = \operatorname{Hom}\left( {{\mathrm{C}}_{1}^{\text{simp }}\left( K\right), G}\right) \) . In other words, by definition of \( {\mathrm{C}}_{1}^{\text{simp }}\left( ... | Yes |
Lemma 67.4. Let \( K = \left( {V, S}\right) \) be an abstract simplicial complex and let \( \left( {{v}_{0},\ldots ,{v}_{k}}\right) \) be an ordered \( k \) -simplex of \( K \) . We have\n\n\[ \n{\delta }_{k}\left( {\left\lbrack {v}_{0},\ldots ,{v}_{k}\right\rbrack }^{ * }\right) = {\left( -1\right) }^{k + 1} \cdot \ma... | Proof. We need to show that both sides define the same map \( {\mathrm{C}}_{k + 1}^{\text{simp }}\left( K\right) \rightarrow \mathbb{Z} \) . Basically by definition of \( {\mathrm{C}}_{k + 1}^{\text{simp }}\left( K\right) \) this means that we need to show both sides provide the same value for any given \( \left\lbrack... | Yes |
Lemma 67.5. Let \( k \in {\mathbb{N}}_{0} \) and let \( G \) be an abelian group. (1) The maps \[ \left( {K, \leq }\right) \mapsto {\mathrm{H}}_{\operatorname{simp}, \leq }^{k}\left( {K;G}\right) \] \[ \left( {f : \left( {K, \leq }\right) \rightarrow \left( {L, \leq }\right) }\right) \mapsto \left( \begin{matrix} \wide... | Proof. The lemma is an immediate consequence of Lemmas 63.3, 63.7 and 73.8. | Yes |
Lemma 67.10. Let \( R \) be a commutative ring. For every ordered abstract simplicial complex \( \left( {K, \leq }\right) \) the map\n\n\[ \cup : {\mathrm{H}}_{\mathrm{{simp}}, \leq }^{p}\left( {K;R}\right) \times {\mathrm{H}}_{\mathrm{{simp}}, \leq }^{q}\left( {K;R}\right) \rightarrow {\mathrm{H}}_{\mathrm{{simp}}, \l... | Proof. This lemma follows almost immediately from Lemma 67.9. For more details we refer to the proof of Lemma 81.3. In that proof we just need to replace Lemma 81.2 by Lemma 67.9. | No |
Lemma 67.12. Let \( K = \left( {V, S}\right) \) be a closed connected \( n \) -dimensional pseudomanifold that is equipped with an orientation \( {\left\{ \left( {\epsilon }_{w},{Y}_{w}\right) \right\} }_{w \in {S}_{n}} \) in the sense of the definition on page 1652. (1) (a) For any \( s, t \in {S}_{n} \) we have \( \l... | Proof. (1) It follows from Theorem 66.4, and the discussion on page 1659 that the simplicial chain complex of \( K \) is of the form \[ 0 \rightarrow \underset{ = {\mathrm{C}}_{n}^{\text{simp }}\left( K\right) }{\underbrace{\mathbb{Z} \cdot \left\lbrack {\mathop{\sum }\limits_{{w \in {S}_{n}}}{\epsilon }_{w} \cdot {Y}_... | Yes |
Proposition 67.13. Let \( K = \left( {V, S}\right) \) be an abstract simplicial complex and let \( R \) be a commutative ring. The following statements hold:\n\n(1) The cup product on \( {\mathrm{H}}_{\text{simp }}^{ * }\left( {K;R}\right) \) is \( R \) -bilinear and associative.\n\n(2) The abelian group \( {\mathrm{H}... | Proof. We pick an order \ | No |
Let \( f : K \rightarrow L \) be a simplicial map between two abstract simplicial complexes. For any \( \varphi \in {\mathrm{H}}_{\mathrm{{simp}}}^{p}\left( {L;R}\right) \) and \( \psi \in {\mathrm{H}}_{\mathrm{{simp}}}^{q}\left( {L;R}\right) \) we have\n\n\[ \n{f}^{ * }\left( \varphi \right) \cup {f}^{ * }\left( \psi ... | (1) Let \( f : K = \left( {V, S}\right) \rightarrow L = \left( {W, T}\right) \) be a simplicial map between two abstract simplicial complexes. By Exercise 1.2 we can pick total orders \ | No |
Lemma 67.16. Let \( \\left( {K, \\leq }\\right) \) be an ordered abstract simplicial complex. Given any commutative ring \( R \) and given any \( k, n \\in {\\mathbb{N}}_{0} \) the map\n\n\[ \n{\\mathrm{H}}_{\\operatorname{simp}, \\leq }^{k}\\left( {K;R}\\right) \\times {\\mathrm{H}}_{n}^{\\operatorname{simp}, \\leq }\... | Proof. The same argument as in the proof of Lemma 83.1 shows that given an \( n \) -simplex \( s = \\left\\{ {{v}_{0} < \\cdots < {v}_{n}}\\right\\} \), given \( k \\leq n \) and given any simplicial cochain \( \\varphi \\in {\\mathrm{C}}_{\\operatorname{simp}, \\leq }^{k}\\left( {K;R}\\right) \) we have the following ... | Yes |
Lemma 67.18. Let \( K \) be a non-empty connected abstract simplicial complex. Given any \( \varphi \in {\mathrm{H}}_{\text{simp }}^{p}\left( {K;R}\right) \) and \( \sigma \in {\mathrm{H}}_{p}^{\text{simp }}\left( {K;R}\right) \) the following equality holds:\n\n\[ \varphi \cap \sigma = \langle \varphi ,\sigma {\rangle... | Proof. First we pick an order \ | No |
Lemma 67.19. Let \( K \) be an abstract simplicial complex and let \( R \) be a commutative ring. For any \( \varphi \in {\mathrm{H}}_{\text{simp }}^{p}\left( {K;R}\right) \), any \( \psi \in {\mathrm{H}}_{\text{simp }}^{q}\left( {K;R}\right) \) and any \( \sigma \in {\mathrm{H}}_{n}^{\text{simp }}\left( {K;R}\right) \... | Sketch of Proof. We pick an order \ | No |
Lemma 67.20. Let \( R \) be a commutative ring and let \( f : K \rightarrow L \) be a simplicial map between two abstract simplicial complexes. Given any \( p, n \in {\mathbb{N}}_{0} \) the following diagram commutes\n\n\[ \n{\mathrm{H}}_{\mathrm{{simp}}}^{p}\left( {K;R}\right) \times {\mathrm{H}}_{n}^{\mathrm{{simp}}}... | Proof. Let \( f : K = \left( {V, S}\right) \rightarrow L = \left( {W, T}\right) \) be a simplicial map between two abstract simplicial complexes. By Exercise 1.2 we can pick total orders \ | No |
Theorem 68.1. Let \( n \in \mathbb{N} \) and let \( M \) be a compact oriented connected \( n \) -dimensional smooth manifold.\n\n(1) For every \( x \in M \smallsetminus \partial M \) the inclusion induced map \nis... | Proof of Theorem 68.1 (1) for closed manifolds. Let \( n \in \mathbb{N} \) and let \( M \) be a closed oriented connected \( n \) -dimensional smooth manifold. By Theorem 64.2 we can pick a smooth simplicial structure \( \left( {K = \left( {V, S}\right) ,\lambda : \left| K\right| \rightarrow M}\right) \) for the smooth... | Yes |
Lemma 68.3. Let \( M \) be a compact oriented non-empty \( n \) -dimensional smooth manifold and let \( \varphi \in {\mathrm{H}}_{n}\left( {M,\partial M}\right) \) . We denote by \( {N}_{1},\ldots ,{N}_{k} \) the components of \( M \) . The following statements are equivalent:\n\n(1) \( \varphi \) is the fundamental cl... | Proof (*). A short moment of reflection shows that it suffices to prove the lemma in the case that \( M \) is connected. Next we note that the \( \left( 1\right) \Rightarrow \left( 2\right) \) -direction and \( \left( 1\right) \Rightarrow \left( 3\right) \) -direction are of course tautologies. We turn to the proof of ... | Yes |
Proposition 68.4. Let \( M \) be a compact oriented connected \( n \) -dimensional smooth manifold. Let \( {r}_{1} \cdot {\sigma }_{1} + \cdots + {r}_{m} \cdot {\sigma }_{m} \in {\mathrm{C}}_{n}\left( {M,\partial M}\right) \) be a cycle. Suppose there exists a \( j \in \{ 1,\ldots, m\} \) such that the following three ... | Proof of Proposition 68.4 (*). Let \( M \) be a compact oriented connected \( n \) -dimensional smooth manifold, let \( \mathop{\sum }\limits_{{i = 1}}^{m}{r}_{i} \cdot {\sigma }_{i} \in {\mathrm{C}}_{n}\left( {M,\partial M}\right) \) be a cycle and let \( j \in \{ 1,\ldots, m\} \) such that conditions (1)-(3) above ar... | Yes |
Lemma 68.6. Let \( n \in \mathbb{N} \) . The standard generator \( \left\lbrack {S}^{n}\right\rbrack \in {\mathrm{H}}_{n}\left( {S}^{n}\right) \) that we introduced on page 1174 agrees with the fundamental class of the oriented smooth manifold \( {S}^{n} \) . | Proof. It follows from the above discussion, and some elementary arguments that the map \( \alpha : {\Delta }^{n} \rightarrow {S}^{n} \) is an embedding and that it is an orientation-preserving embedding on an open subset of \( {\Delta }^{n} \) . The desired equality now follows almost immediately from Proposition 68.4... | No |
Lemma 68.7. Let \( n \in {\mathbb{N}}_{0} \), let \( M \) and \( N \) be compact oriented connected \( n \) -dimensional smooth manifolds and let \( f : M \rightarrow N \) be a diffeomorphism. We have \( {}^{1034} \n\n\[ \n{f}_{ * }\left( \left\lbrack M\right\rbrack \right) = \left\{ \begin{matrix} \left\lbrack N\right... | Proof (*). If \( n = 0 \), then the statement follows immediately from the definitions. As we will see, for \( n \in \mathbb{N} \) the statement follows basically immediately from the definition property of the fundamental class. Indeed, we set \( \epsilon \mathrel{\text{:=}} + 1 \) if \( f \) is orientation-preserving... | Yes |
Lemma 68.8. For every compact oriented \( n \) -dimensional smooth manifold \( M \) we have\n\n\[ \left\lbrack {-M}\right\rbrack = - \left\lbrack M\right\rbrack \in {\mathrm{H}}_{n}\left( {M,\partial M}\right) . \] | Proof. This statement follows basically immediately from Lemma 68.7 since the identity map \( M \rightarrow - M \) is orientation-reversing. | Yes |
Proposition 68.9. Let \( M \) be a compact oriented \( n \) -dimensional smooth manifold with boundary. The connecting homomorphism\n\n\[ \partial : {\mathrm{H}}_{n}\left( {M,\partial M}\right) \rightarrow {\mathrm{H}}_{n - 1}\left( {\partial M}\right) \]\n\nof the long exact sequence in homology of the pair \( \left( ... | Proof (*). Let \( M \) be a compact oriented \( n \) -dimensional smooth manifold with boundary components \( {N}_{1},\ldots ,{N}_{k} \) . By Theorem 64.2 the smooth manifold \( M \) admits a smooth simplicial structure \( \left( {K = \left( {V, S}\right) ,\lambda : \left| K\right| \rightarrow M}\right) \) . By Proposi... | No |
Let \( M \) be a compact oriented \( n \) -dimensional smooth manifold. Let \( A \) be a union of components of \( \partial M \) . We write \( B \mathrel{\text{:=}} \partial M \smallsetminus A \) . We consider the map \[ {\mathrm{C}}_{n}\left( {M,\partial M}\right) \underset{ = : {\Theta }_{A}}{\underbrace{\overset{{\p... | Proof \( \left( *\right) \) . First note that it is basically clear that it suffices to prove the statement for the case \( A = \partial M \) . Next, pick a singular chain \( \widetilde{\nu } \in {\mathrm{C}}_{n}\left( M\right) \) that represents a fundamental cycle in \( {\mathrm{C}}_{n}\left( {M,\partial M}\right) \)... | Yes |
Corollary 68.11. Let \( M \) be a compact oriented connected \( n \) -dimensional smooth manifold. We denote by \( i : \partial M \rightarrow M \) the inclusion map and we denote by \( {N}_{1},\ldots ,{N}_{k} \) the components of \( \partial M \) .\n\n(1) We have\n\n\[ \ker \left( {{i}_{ * } : \underset{ = \mathbb{Z} \... | Proof. Let \( M \) be a compact oriented \( n \) -dimensional smooth manifold. We denote by \( i : \partial M \rightarrow M \) the inclusion map.\n\n(1) Using Proposition 68.9 and the long exact sequence in homology of the pair \( \left( {M,\partial M}\right) \) we obtain that the following sequence is exact:\n\n\[ \un... | Yes |
Lemma 68.12. Let \( M \) be a compact oriented \( n \) -dimensional smooth manifold. Furthermore let \( W \subset M \) be a compact \( n \) -dimensional submanifold. We write \( \mathring{W} = W \smallsetminus \partial W \) and we equip \( W \) with the orientation given by Lemma 6.46 (5). The following two statements ... | Proof (*).\n\n(1) The first statement is an immediate consequence of the Excision Theorem 44.10.\n\n(2) We denote by \( \Phi : {\mathrm{H}}_{n}\left( {M,\partial M}\right) \rightarrow {\mathrm{H}}_{n}\left( {W,\partial W}\right) \) the map given in statement (2). We have to show that \( \Phi \left( \left\lbrack M\right... | No |
Lemma 68.14. Let \( X \) and \( Y \) be two compact orientable connected \( n \) -dimensional smooth manifolds such that \( \partial X \cong {S}^{n - 1} \) and \( \partial Y \cong {S}^{n - 1} \) . We pick a diffeomorphism \( \varphi : \partial X \rightarrow \partial Y \) . (1) There exists an isomorphism \[ {\mathrm{H}... | Proof of Lemma 68.14. The second statement is an immediate consequence of the first statement and the fact that for every \( k < n \) we have \( {\widetilde{\mathrm{H}}}_{k}\left( {\bar{B}}^{n}\right) = 0 \) . Thus it suffices to prove the first statement. To simplify the notation we write \( {S}^{n - 1} = \partial X =... | Yes |
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