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Proposition 43.6. For any \( k \neq l \) the topological spaces \( {\mathbb{R}}^{k} \) and \( {\mathbb{R}}^{l} \) are not homeomorphic. | Proof. Clearly we only have to deal with the case \( k, l \geq 1 \) . Let \( k, l \in \mathbb{N} \) and suppose there exists a homeomorphism \( f : {\mathbb{R}}^{k} \rightarrow {\mathbb{R}}^{l} \) . Then we obtain that  the sphere \( {S}^{n - 1} = \partial {\bar{B}}^{n} \) is not a retract of the closed ball \( {\overline{B}}^{n} \) . | Proof. Suppose there exists a retraction \( r : {\bar{B}}^{n} \rightarrow {S}^{n - 1} \) . We denote by \( i : {S}^{n - 1} \rightarrow {\bar{B}}^{n} \) the inclusion map. We consider the following diagram\n\n\n\nTh... | Yes |
Theorem 43.8. (Brouwer \( {}^{689} \) Fixed Point Theorem) Every map \( f : {\bar{B}}^{n} \rightarrow {\bar{B}}^{n} \) admits a fixed point, i.e. for every map \( f : {\bar{B}}^{n} \rightarrow {\bar{B}}^{n} \) there exists a point \( x \in {\bar{B}}^{n} \) such that \( f\left( x\right) = x. | Proof. Suppose there exists a map \( f : {\bar{B}}^{n} \rightarrow {\bar{B}}^{n} \) without a fixed point. Since \( f \) admits no fixed points there exists for each \( x \) precisely one ray which starts at \( f\left( x\right) \) and which goes through \( x \) . We consider the following map which is illustrated in Fi... | Yes |
Theorem 43.9. (Perron-Frobenius Theorem) Let \( A = \left( {a}_{ij}\right) \) be an \( n \times n \) -real matrix. If all entries are positive, then \( A \) has a positive eigenvalue \( \lambda \) with an eigenvector \( v \) such that all entries of \( v \) are non-negative. | Proof. We consider the convex set\n\n\[ T \mathrel{\text{:=}} \left\{ {\left( {{t}_{1},\ldots ,{t}_{n}}\right) \in {\left\lbrack 0,1\right\rbrack }^{n} \mid \mathop{\sum }\limits_{{i = 1}}^{n}{t}_{i} = 1}\right\} .\n\]\n\nThis set is just the standard \( \left( {n - 1}\right) \) -simplex endowed with a different name. ... | Yes |
(1) Let \( 0 \rightarrow {A}_{ * } \rightarrow {B}_{ * } \rightarrow {C}_{ * } \rightarrow 0 \) be a short exact sequence of chain complexes. The connecting homomorphism \( {\partial }_{n} : {\mathrm{H}}_{n}\left( C\right) \rightarrow {\mathrm{H}}_{n - 1}\left( A\right) \) is well-defined and it is indeed a homomorphis... | Proof.\n\nThe proof below is elementary but not very readable. It is much easier to prove the lemma on your own than to try to read the proof.\n\nFor the undeterred reader we now provide the fully gory details.\n\n(1) We start out with the following claim.\n\nClaim. The map\n\n\[{\partial }_{n} : {\mathrm{H}}_{n}\left(... | No |
Lemma 43.14. Let \( X \) be a topological space and let \( n \in {\mathbb{N}}_{0} \) . The following statements hold:\n\n(1) Let \( A \subset X \) . The maps \( {\mathfrak{{sl}}}_{X, A} \) and \( {}_{{\mathrm{K}}_{X}, A} \) are natural in the sense that if we are given a map \( f : \left( {X, A}\right) \rightarrow \lef... | (1) The first statement follows immediately from the definitions.\n\n(2) We will provide the proof of this statement in Exercise 43.13. | No |
Proposition 43.15. Let \( X \) be a topological space and let \( A \subset B \subset X \) be two subsets. We denote by \( i : \left( {B, A}\right) \rightarrow \left( {X, A}\right) \) and by \( p : \left( {X, A}\right) \rightarrow \left( {X, B}\right) \) the obvious maps.\n\n(1) For each \( n \in {\mathbb{N}}_{0} \) the... | Proof. It follows immediately from the definitions and the third isomorphism theorem that\n\n\[ \n0 \rightarrow {\mathrm{C}}_{ * }\left( {B, A}\right) \overset{{i}_{ * }}{ \rightarrow }{\mathrm{C}}_{ * }\left( {X, A}\right) \overset{{p}_{ * }}{ \rightarrow }{\mathrm{C}}_{ * }\left( {X, B}\right) \rightarrow 0 \n\]\n\ni... | No |
Proposition 43.17. Let \( f, g : \left( {X, A}\right) \rightarrow \left( {Y, B}\right) \) be two maps between pairs of topological spaces. If \( f \) and \( g \) are homotopic, then the maps \( {f}_{ * },{g}_{ * } : {\mathrm{C}}_{ * }\left( {X, A}\right) \rightarrow {\mathrm{C}}_{ * }\left( {Y, B}\right) \) are chain h... | Proof. Let \( f, g : \left( {X, A}\right) \rightarrow \left( {Y, B}\right) \) be two homotopic maps between pairs of topological spaces. To prove the proposition we have to look again at the proof of Proposition 42.5, In that proof we explicitly constructed a chain \( {P}_{n} : {\mathrm{C}}_{n}\left( X\right) \rightarr... | Yes |
Corollary 43.18. (1) If \( f : \left( {X, A}\right) \rightarrow \left( {Y, B}\right) \) is a homotopy equivalence of pairs of topological spaces \( {}_{1}^{703} \) then for every \( n \in {\mathbb{N}}_{0} \) the induced map \( {f}_{ * } : {\mathrm{H}}_{n}\left( {X, A}\right) \rightarrow {\mathrm{H}}_{n}\left( {Y, B}\ri... | Proof. Let \( X \) be a topological space. (1) We deduce this statement from Proposition 43.17 the same way we deduced Corollary 42.8 from Proposition 42.5. | No |
Theorem 43.20. (Excision Theorem) Let \( X \) be a topological space.\n\n(1) Let \( K \) be a subset of \( X \) and let \( U \) be a neighborhood of the closure of \( K \) . Then the inclusion \( \left( {U, U \smallsetminus K}\right) \rightarrow \left( {X, X \smallsetminus K}\right) \) induces for each \( n \in {\mathb... | Proof of Theorem 43.20 assuming Theorem 43.19.\n\n(1) We have\n\n\[ \n{\mathrm{H}}_{n}\left( {U, U \smallsetminus K}\right) \underset{ \uparrow }{ = }{\mathrm{H}}_{n}\left( {X \smallsetminus Z, A \smallsetminus Z}\right) \underset{ \uparrow }{\overset{ \cong }{ \rightarrow }}{\mathrm{H}}_{n}\left( {X, A}\right) = {\mat... | Yes |
Lemma 43.21. (*)\n\n(1) Let \( X \) be a topological space and let \( A \subset X \) be a subset. We denote by \( p : X \rightarrow X/A \) the projection. The following diagram commutes.\n\n\n\n(2) We denote by Pai... | Proof \( \left( *\right) \) .\n\n(1) We consider the following diagram\n\n\n\nIt follows immediately from the definitions that all three regions of the diagram commute. This observation gives us the desired stateme... | No |
Proposition 43.22. Let \( X \) be a topological space and let \( A \subset X \) be a subset and let \( n \in {\mathbb{N}}_{0} \) . If \( A \) is good, then the natural homomorphism \( {\mathrm{H}}_{n}\left( {X, A}\right) \overset{\text{ to }}{ \rightarrow }{\widetilde{\mathrm{H}}}_{n}\left( {X/A}\right) \) is an isomor... | Proof of Proposition 4.3.22 assuming the Excision Theorem 4.3.19. Let \( X \) be a topological space, let \( A \subset X \) be a good subset and let \( n \in {\mathbb{N}}_{0} \) .\n\nFirst we need to get the annoying case that \( A = \varnothing \) out of the way. Note that the map \( \left( {X,\varnothing }\right) \ri... | No |
Proposition 43.24. Let \( X \) be a topological space and let \( \mathcal{U} = {\left\{ {U}_{i}\right\} }_{i \in I} \) be a comfortable cover of \( X \) . Then the inclusion map\n\n\[ \n{\mathrm{C}}_{ * }^{\mathcal{U}}\left( X\right) \rightarrow {\mathrm{C}}_{ * }\left( X\right)\n\]\n\ninduces for every \( n \in {\math... | The idea of the proof of Proposition 43.24 is the one we already described in the previous section. We want to replace systematically each singular \( n \) -simplex \( \sigma : {\Delta }^{n} \rightarrow X \) by a singular \( n \) -chain of the form \( \mathop{\sum }\limits_{{j = 1}}^{r}{a}_{j} \cdot {\sigma }_{j} \) su... | No |
Lemma 43.26. The maps \( {u}_{n} : {\mathrm{C}}_{n}\left( X\right) \rightarrow {\mathrm{C}}_{n}\left( X\right) \) are natural \( {}^{712} \) in particular for any map \( f : X \rightarrow Y \) between topological spaces and any \( n \in {\mathbb{N}}_{0} \) we have \[ {f}_{ * } \circ {u}_{n} = {u}_{n} \circ {f}_{ * } : ... | Proof \( \left( *\right) \) . Let \( \sigma : {\Delta }^{n} \rightarrow X \) be a singular \( n \) -simplex. Then \[ \begin{aligned} {f}_{ * }\left( {{u}_{n}\left( \sigma \right) }\right) & = {f}_{ * }\left( {{\sigma }_{ * }\left( {\chi }_{n}\right) }\right) = {\left( f \circ \sigma \right) }_{ * }\left( {\chi }_{n}\ri... | Yes |
Lemma 43.27. Let \( X \) be a topological space. The chain maps \[ {u}_{n} : {\mathrm{C}}_{n}\left( X\right) \rightarrow {\mathrm{C}}_{n}\left( X\right) \] are chain homotopic to the identity. | Proof \( \left( *\right) \) . Showing that the chain maps \( {\left( {u}_{n}\right) }_{n \in \mathbb{N}} \) are chain homotopic to the identity requires us to find maps \( {P}_{n} : {\mathrm{C}}_{n}\left( X\right) \rightarrow {\mathrm{C}}_{n + 1}\left( X\right), n \in {\mathbb{N}}_{0} \) such that for each \( n \in {\m... | Yes |
For each topological space \( X \) and every \( m \in \mathbb{N} \) the maps \( {\left\{ {u}_{n}^{m}\right\} }_{n \geq 0} \) form a chain map \( {\mathrm{C}}_{ * }\left( X\right) \rightarrow {\mathrm{C}}_{ * }\left( X\right) \) that is chain homotopic to the identity. | Recall that in Lemma 43.25 we already saw that the maps \( {\left\{ {u}_{n}\right\} }_{n \geq 0} \) form a chain map \( {\mathrm{C}}_{ * }\left( X\right) \rightarrow {\mathrm{C}}_{ * }\left( X\right) \) . As we remarked on page 1086, the composition of chain maps is again a chain map. Thus for any \( m \in {\mathbb{N}}... | Yes |
Lemma 43.31. Let \( X \) be a topological space and let \( \mathcal{U} = {\left\{ {U}_{i}\right\} }_{i \in I} \) be an open cover of \( X \) . For every singular \( n \) -simplex \( \sigma : {\Delta }^{n} \rightarrow X \) there exists an \( m \) such that \( {u}_{n}^{m}\left( \sigma \right) \) lies in \( {\mathrm{C}}_{... | Proof. Let \( X \) be a topological space let \( \mathcal{U} = {\left\{ {U}_{i}\right\} }_{i \in I} \) be an open cover of \( X \) and let \( \sigma : {\Delta }^{n} \rightarrow X \) be a singular \( n \) -simplex. By the Lebesgue Lemma 2.75 there exists a \( \delta > 0 \) such that for any subset \( A \) of \( {\Delta ... | Yes |
Proposition 44.2. Let \( X \) be an \( n \) -dimensional topological manifold. The following statements hold:\n\n(1) Every point on \( X \) admits either a chart of type (i) or it admits a chart of type (ii). | Proof. Let \( X \) be an \( n \) -dimensional topological manifold.\n\n(1) This statement is precisely the statement of Lemma 44.1 (1). The only reason why we wrote down the statement again is to preserve the analogy with Proposition 6.27. | Yes |
Proposition 44.3. Let \( X \) be an \( m \) -dimensional topological manifold and let \( Y \) be an \( n \) - dimensional submanifold. We assume that \( Y \) is a closed subset of \( X \) . (In most applications \( Y \) will be compact).\n\n(1) \( {\partial }_{0}Y \) and \( {\partial }_{1}Y \) are disjoint and each of ... | Proof. The proposition follows easily from making the obvious modifications to the proofs of Lemma 6.28 and Proposition 6.30. The biggest change is that we need to replace Proposition 6.27 by Proposition 44.2. We leave it to the reader to fill in the details. | No |
Proposition 44.4. Let \( M \) be an \( m \) -dimensional non-empty topological manifold and let \( N \) be an \( n \) -dimensional topological manifold. If \( M \) is homeomorphic to an open subset of \( N \) (e.g. if \( M \) is homeomorphic to \( N \) ), then \( m = n \) . | Proof. By the remark on page 272 it suffices to consider the case that we are given a homeomorphism \( f : M \rightarrow N \) between a non-empty \( m \) -dimensional topological manifold \( M \) and a topological manifold \( N \) . By Proposition 44.2 (5) we know that \( M \smallsetminus \partial M \) is non-empty. In... | Yes |
Theorem 44.5. (Topological Collar Neighborhood Theorem) Given any topological manifold \( M \) there exists a map \( \Phi : \left\lbrack {0,1}\right\rbrack \times \partial M \rightarrow M \) with the following four properties:\n\n(1) The map \( \Phi \) is an embedding.\n\n(2) The image of \( \lbrack 0,1) \times \partia... | Proof of Theorem 44.5 for compact \( M \) . We consider \( M \) with an \ | No |
Proposition 44.6. (*) Let \( M \) be a topological manifold and let \( {f}_{0},{f}_{1} : \left\lbrack {0,1}\right\rbrack \times \partial M \rightarrow M \) be two embeddings that are the identity map on \( \{ 0\} \times \partial M = \partial M \) . If \( \partial M \) is compact, then there exists an isotopy\n\n\[ H : ... | Proof (*). The proof is sketched in [KSi69, Essay I, Theorem A.2]. The proof is also given in Arm70, Theorem 2] 718 | No |
Corollary 44.7. Let \( M \) be an \( n \) -dimensional topological manifold. Let \( \left\lbrack {0,1}\right\rbrack \times \partial M \) be a collar neighborhood.\n\n(1) The subset \( W : = M \smallsetminus \left( {\lbrack 0,1}\right) \times \partial M) \) is an \( n \) -dimensional topological manifold with boundary g... | Proof. The proof of Corollary 44.7 is basically the same as the proof of Corollary 8.14, We just need to replace the Collar Neighborhood Theorem 8.12 by the Topological Collar Neighborhood Theorem 44.5 and we need to replace Proposition 6.27 by Proposition 44.2 . | No |
Proposition 44.8. Let \( M \) be an \( n \) -dimensional topological manifold (we do not assume that \( M \) is connected). Let \( A \) and \( B \) be disjoint unions of boundary components of \( M \) . Furthermore let \( f : A \rightarrow B \) be a homeomorphism. Then the following two statements hold: (1) The topolog... | Proof. For the most part the proof of Proposition 44.8 is almost the same as the proof of Proposition 8.15. For statements (1) to (5) we just need to make the following replacements: (1) We need to replace Proposition 6.27 (3a) by Lemma 6.7, (2) We need to replace the Collar Neighborhood Theorem 8.12 by the Topological... | No |
Theorem 44.10. (Excision Theorem for Topological Manifolds) Let \( X \) be an \( m \) -dimensional topological manifold and let \( A \) be a subset of \( X \) . Furthermore let \( B \) be an m-dimensional submanifold of \( X \) that is a closed subset of \( X \) . We write \( B \mathrel{\text{:=}} B \smallsetminus {\pa... | Proof of Theorem 4.10 Assuming Theorem 4.9. Note that by Proposition 44.3 we know that \( {\partial }_{0}B \) is a union of boundary components of \( B \) and that \( \overset{ \circ }{B} \) is indeed the interior of \( B \) viewed as a subset of \( X \) . In particular we can apply the Topological Collar Neighborhood ... | No |
Lemma 44.11. Let \( M \) be a topological manifold and let \( W \) be the union of some components of \( \partial M \). (1) The natural inclusion \( i : M \rightarrow {\mathrm{D}}_{W}M \) is an embedding. (2) The folding map \( {\mathrm{D}}_{W}M \rightarrow M \) is continuous and it satisfies \( r \circ i = {\mathrm{{i... | Proof. It follows from Lemmas 3.3 and 3.21 (3) that the natural inclusion \( i \) is continuous and that the involution \( \rho \) are continuous. It is clear that \( \rho \circ \rho = \mathrm{{id}} \). Furthermore it follows from Lemmas 3.3 and 3.22 that the folding map \( r \) is continuous. Clearly we have \( r \cir... | Yes |
Lemma 44.12. Let \( M \) be an \( n \) -dimensional topological (smooth) manifold and let \( W \) be the union of some components of \( \partial M \) . The following statements hold:\n\n(1) The double \( {\mathrm{D}}_{W}M \) is also an \( n \) -dimensional topological (smooth) manifold such that \( M \subset {\mathrm{D... | Proof (*). Let \( M \) be an \( n \) -dimensional topological (smooth) manifold. Statements (1),(2) and (4) follow immediately from applying Proposition 8.15 and Proposition 44.8 iteratively to \( \left( {W\times \{ 1\} }\right) \sqcup \left( {W\times \{ 2\} }\right) \subset \left( {M\times \{ 1\} }\right) \sqcup \left... | Yes |
Lemma 44.13. Let \( M \) be a topological manifold and let \( W \) be the union of some components of \( \partial M \). (1) For every \( j \in {\mathbb{N}}_{0} \) the inclusion induced map \( {\mathrm{H}}_{j}\left( M\right) \rightarrow {\mathrm{H}}_{j}\left( {{\mathrm{D}}_{W}M}\right) \) is a monomorphism. (2) If \( \p... | Proof. Let \( i : M \rightarrow {\mathrm{D}}_{W}M \) be the natural inclusion map and let \( r : {\mathrm{D}}_{W}M \rightarrow M \) be the folding map. In Lemma 44.11 we saw that both maps are continuous and that \( r \circ i = {\mathrm{{id}}}_{M} \). The two statements now follow as usual from the functoriality of hom... | Yes |
Proposition 44.14. Let \( M \) be a compact connected \( n \) -dimensional topological manifold, let \( W \) be a component of \( \partial M \) and let \( P \in W \) . If the inclusion induced map \( {\pi }_{1}\left( {W, P}\right) \rightarrow \) \( {\pi }_{1}\left( {M, P}\right) \) is an epimorphism, then the inclusion... | The proof of Proposition 44.14 results on the following group-theoretic lemma, which is precisely the content of Exercise 21.11.\n\nLemma 44.15. Let \( A \) | No |
Lemma 44.15. Let \( A \) and \( G \) be groups and let \( \varphi : A \rightarrow G \) be an epimorphism. Furthermore let \( {G}^{\prime } \) be a copy of \( G \) . We consider the amalgamated product \( G{ * }_{A}{G}^{\prime } \) given by using twice the epimorphism \( \varphi : A \rightarrow G \) and \( \varphi : A \... | Proof of Lemma 4.15. By Proposition 21.21 there exists a unique homomorphism \( \Theta : G{ * }_{A} \) \( {G}^{\prime } \rightarrow {G}^{\prime } \) that makes the following diagram commutes:\n\n\n\nIn particular w... | Yes |
Lemma 44.16. (*) Let \( M \) be a topological manifold and let \( W \) be the union of some components of \( \partial M \). Given \( i \in \{ 1,2\} \), we denote by \( {j}_{i} : M \rightarrow M \times \{ i\} \rightarrow {\mathrm{D}}_{W}M \) the obvious map.\n\n(1) For every \( k \in {\mathbb{N}}_{0} \) the map\n\n\[ \b... | is well-defined. | Yes |
Lemma 45.1. Let \( n \in {\mathbb{N}}_{0} \) . We have\n\n\[ \n{\widetilde{\mathrm{H}}}_{k}\left( {{\bar{B}}^{n},{S}^{n - 1}}\right) \cong \left\{ \begin{array}{ll} \mathbb{Z}, & \text{ if }k = n, \\ 0, & \text{ otherwise. } \end{array}\right.\n\] | Proof. We consider the long exact sequence in reduced homology of the pair \( \left( {{\bar{B}}^{n},{S}^{n - 1}}\right) \) : \n\nThe statement of the lemma is now an immediate consequence of Proposition 43.4. | No |
Lemma 45.2. For any \( n \in {\mathbb{N}}_{0} \) the following two statements hold:\n\n(1) \( {\mathrm{H}}_{n}\left( {{\Delta }^{n},\partial {\Delta }^{n}}\right) \cong \mathbb{Z} \) .\n\n(2) The identity map \( \mathrm{{id}} : {\Delta }^{n} \rightarrow {\Delta }^{n} \) represents a generator of \( {\mathrm{H}}_{n}\lef... | Proof. The first statement is an immediate consequence of Lemmas 45.1 and 41.1, Now we turn to the proof of the second statement. We prove the second statement by induction on \( n \) . It follows easily from Proposition 41.5 that the statement holds for \( n = 0 \) . Now we suppose that the desired statement holds for... | Yes |
Lemma 44.1. Let \( M \) be a \( k \) -dimensional topological manifold. Then for any \( P \in M \) , any \( l \in {\mathbb{N}}_{0} \) and any open neighborhood \( U \) of \( P \) we have\n\n\[ \n{H}_{l}\left( {M, M\smallsetminus \{ P\} }\right) \cong \left\{ \begin{array}{ll} \mathbb{Z}, & \text{ if }P \in M \smallsetm... | Examples. Our goal now is to give explicit generators for the relative homology groups \( {\mathrm{H}}_{k}\left( {M, M\smallsetminus \{ x\} }\right) \cong \mathbb{Z} \) . We first do so for open subsets of \( {\mathbb{R}}^{k} \) and then we \ | No |
Lemma 45.3. Let \( n \geq 1 \) . The above singular \( n \) -chain \( \alpha - \beta \in {\mathrm{C}}_{n}\left( {S}^{n}\right) \) is a cycle and it represents a generator of \( {\widetilde{\mathrm{H}}}_{n}\left( {S}^{n}\right) = {\mathrm{H}}_{n}\left( {S}^{n}\right) \) . | Proof. We start out with the following claim.\n\nClaim. The singular \( n \) -chain \( \alpha - \beta \) is a cycle.\n\nWe denote by \( \rho \) the reflection in the hyperplane defined by \( {x}_{n + 1} = 0 \) . We have\n\n\[ \text{since}\alpha \left( {\partial {\Delta }^{n}}\right) \subset {\mathbb{R}}^{n} \times 0\te... | No |
Lemma 45.4. The singular 1-simplex\n\n\[ \sigma : {\Delta }^{1} \rightarrow {S}^{1} \]\n\n\[ \left( {1 - t, t}\right) \mapsto {e}^{{2\pi }\mathrm{i}t} \]\n\nis a cycle that represents the standard generator \( \left\lbrack {S}^{1}\right\rbrack \) of \( {\mathrm{H}}_{1}\left( {S}^{1}\right) \cong \mathbb{Z} \) . | Proof. We denote by \( \alpha ,\beta : {\Delta }^{1} \rightarrow {S}^{1} \) the singular 1-simplices in the definition of the standard generator of \( {\mathrm{H}}_{1}\left( {S}^{1}\right) \) . We have\n\n\[ \left\lbrack {\alpha - \beta }\right\rbrack = \left\lbrack {\alpha + \bar{\beta }}\right\rbrack = \left\lbrack \... | No |
Proposition 45.5. Let \( n \in \mathbb{N} \) . We denote by \( f : \left( {{\Delta }^{n},\partial {\Delta }^{n}}\right) \rightarrow \left( {{\bar{B}}^{n},{S}^{n - 1}}\right) \) the homeomorphism from Lemma 41.1 and we denote by \( g : {\bar{B}}^{n}/{S}^{n - 1} \rightarrow {S}^{n} \) the homeomorphism from page 182. We ... | Proof of Proposition 45.5 (*).\n\n(1) The triangle commutes by definition of 10, see page 1129. Furthermore it follows from Lemma 43.14 and Corollary 43.16 that the two squares commute.\n\n(3),(4) The calculation of these two signs is delicate and tricky. It becomes somewhat easier after we have developed a few more to... | No |
Lemma 45.6. (*) Let \( n \in \mathbb{N} \) . We consider the following sequence of maps\n\n\[ \n{\mathrm{H}}_{n}\left( {S}^{n}\right) \overset{\text{ a }}{ \rightarrow }{\mathrm{H}}_{n}\left( {{S}^{n},{S}_{ \geq 0}^{n}}\right) \underset{ \uparrow }{\overset{ \cong }{ \leftarrow }}{\mathrm{H}}_{n}\left( {{S}_{ \geq 0}^{... | Proof \( \\left( *\\right) \) . We recall and introduce the following notation:\n\n(1) Let \( f : \\left( {{\\Delta }^{n},\\partial {\\Delta }^{n}}\\right) \\rightarrow \\left( {{\\bar{B}}^{n},{S}^{n - 1}}\\right) \) be the homeomorphism from Lemma 41.1\n\n(2) Let \( {p}_{ + } : {S}_{ \\geq 0}^{n} \\rightarrow {\\bar{B... | Yes |
Proposition 45.7. Let \( X \) be a non-empty topological space and let \( n \in {\mathbb{Z}}^{725} \)\n\n(1) The following maps are isomorphisms:\n\n\[ \n{\widetilde{\mathrm{H}}}_{n}\left( X\right) \underset{ \uparrow }{\overset{\partial }{ \leftarrow }}{\mathrm{H}}_{n + 1}\left( {{C}_{ + }X, X}\right) \underset{ \upar... | Proof (*). Let \( n \in \mathbb{Z} \).\n\n(1) (a) By the same argument as in the proof of Lemma 24.1 we see that both \( {C}_{ - }X \) and \( {C}_{ + }X \) admit a deformation retraction to a point. It follows from Lemma 18.14 that both are contractible, therefore by Lemma 43.1 (7) the reduced homologies of \( {C}_{ - ... | Yes |
Corollary 45.8. (*) Let \( X \) be a topological space that is compact and Hausdorff. Furthermore let \( k \in {\mathbb{N}}_{ \geq - 1} \) . For every \( n \in \mathbb{Z} \) there exists a natural isomorphism\n\n\[ \n\begin{array}{l} {\widetilde{\mathrm{H}}}_{n}\left( X\right) \overset{ \cong }{ \rightarrow }{\widetild... | Proof (*). First note that the case \( X = \varnothing \) follows from the fact that, by definition, \( {S}^{k} * \varnothing = {S}^{k} \), and the calculation of the reduced homology groups of \( {S}^{k} \) that we gave in Proposition 43.4.\n\nNow assume that \( X \neq \varnothing \) . If \( k = - 1 \), then we have, ... | Yes |
Lemma 45.10. For any \( k \in \mathbb{Z} \) the degree of the map\n\n\[ \n{S}^{1} \rightarrow {S}^{1} \]\n\n\[ \nz \mapsto {z}^{k} \]\n\nequals \( k \) . | Proof. We denote by \( \tau : {\Delta }^{1} \rightarrow {S}^{1} \) the singular 1-simplex that is given by \( \tau \left( {1 - t, t}\right) = {e}^{{2\pi }\mathrm{i}t} \) . By Lemma 45.4 we know that \( \left\lbrack \tau \right\rbrack \in {\mathrm{H}}_{1}\left( {S}^{1}\right) \) is a generator.\n\nNow we first consider ... | Yes |
Lemma 45.11. Let \( f, g : {S}^{n} \rightarrow {S}^{n} \) be two maps. Then the following hold:\n\n(1) \( \deg \left( {\operatorname{id}}_{{S}^{n}}\right) = 1 \) . | (1) Since \( {\left( {\operatorname{id}}_{{S}^{n}}\right) }_{ * } = {\operatorname{id}}_{{\mathrm{H}}_{n}\left( {S}^{n}\right) } \) we know that \( \deg \left( {\operatorname{id}}_{{S}^{n}}\right) = 1 \) . | Yes |
Lemma 45.12. Let \( n \in \mathbb{N} \) and let \( f : {S}^{n} \rightarrow {S}^{n} \) be a map. Let \( \sum \left( f\right) : \sum \left( {S}^{n}\right) \rightarrow \sum \left( {S}^{n}\right) \) be the suspension of \( f \) that we defined on page 696. Furthermore let \( \Theta : \sum \left( {S}^{n}\right) \rightarrow ... | Proof. We consider the following diagram \n\nFirst note that it follows from Proposition 45.7 that the square to the left commutes and that the horizontal maps to the left are isomorphisms. It is clear that the squ... | Yes |
Theorem 45.13. Let \( f : {S}^{n} \rightarrow {S}^{n} \) be a map. If \( \deg \left( f\right) \neq {\left( -1\right) }^{n + 1} \), then \( f \) has a fixed point. | Proof. Let \( f : {S}^{n} \rightarrow {S}^{n} \) be a map which admits no fixed point. This means that for all \( x \in {S}^{n} \) we have \( f\left( x\right) \neq x \) . We need to show that \( \deg \left( f\right) = {\left( -1\right) }^{n + 1} \) . We will do this by showing that \( f \) is homotopic to \( - \mathrm{... | Yes |
Theorem 45.14. (Hairy Ball Theorem) Every vector field on an even-dimensional sphere \( {S}^{2k} \) vanishes on at least one point. | Proof. Let \( v \) be a vector field on the sphere \( {S}^{n} \) which is nowhere-vanishing. We have to show that \( n \) is odd. We consider the vector field \( w \) on \( {S}^{n} \) that is given by rescaling \( v \) to length one, i.e. that is given by\n\n\[ {S}^{n} \rightarrow {\mathbb{R}}^{n + 1} \]\n\n\[ x \mapst... | Yes |
Lemma 45.17. Let \( f : U \rightarrow V \) be a diffeomorphism between two open subsets of \( {\mathbb{R}}^{n} \). Let \( P \in U \). We denote by\n\n\[ \lambda : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \]\n\n\[ x \mapsto \left( {\mathrm{D}{f}_{P}}\right) \left( x\right) + f\left( P\right) \]\n\nthe linearization... | Proof. Let \( f : U \rightarrow V \) be a diffeomorphism between two open subsets of \( {\mathbb{R}}^{n} \). To simplify the notation we assume that \( P = 0 \) and that \( f\left( P\right) = 0 \). Furthermore, by restriction to an open ball we might as well assume that \( U \) is convex. We write \( D \mathrel{\text{:... | Yes |
Lemma 45.18. Let \( n \in \mathbb{N} \) . Given \( A \in \mathrm{{GL}}\left( {n,\mathbb{R}}\right) \) we have\n\n\[ \deg \left( {\rho {\left( A\right) }_{ * } : {\mathrm{H}}_{n}\left( {{\mathbb{R}}^{n},{\mathbb{R}}^{n}\smallsetminus \{ 0\} }\right) \rightarrow {\mathrm{H}}_{n}\left( {{\mathbb{R}}^{n},{\mathbb{R}}^{n}\s... | Proof of Lemma 45.18. For simplicity we only prove the first statement of the lemma. Thus let \( A \in \mathrm{{GL}}\left( {n,\mathbb{R}}\right) \) . We start out by picking an orthogonal matrix \( B \in \mathrm{O}\left( n\right) \) such that \( \operatorname{sign}\left( {\det \left( B\right) }\right) = \operatorname{s... | Yes |
Lemma 45.20. The local degree \( \deg \left( {f, x}\right) \) of a map \( f : {S}^{n} \rightarrow {S}^{n} \) at a nice point \( x \in {S}^{n} \) is well-defined. | Proof. Let \( n \geq 1 \), let \( f : {S}^{n} \rightarrow {S}^{n} \) be a map and let \( x \in {S}^{n} \) be a nice point. Let \( U,{U}^{\prime } \) be nice neighborhoods of \( x \) . After replacing \( {U}^{\prime } \) by \( U \cap {U}^{\prime } \) we can without loss of generality assume that \( {U}^{\prime } \subset... | Yes |
Lemma 45.21. Let \( n \geq 1 \), let \( f : {S}^{n} \rightarrow {S}^{n} \) be a map and let \( x \in {S}^{n} \) . If \( f \) is a homeomorphism, then for every nice point \( x \in {S}^{n} \) we have\n\n\[ \deg \left( {f, x}\right) = \deg \left( f\right) \] | Proof. From the hypothesis that \( f \) is a homeomorphism we obtain that \( U = {S}^{n} \) is a nice neighborhood for \( x \) . We take \( V = {S}^{n} \) and we consider the following diagram\n\n\n\nThe statement ... | Yes |
Proposition 45.24. Let \( f \) be a complex polynomial of degree \( \geq 1 \) . The degree of the map\n\n\[ \Theta \left( f\right) : {S}^{2} = \mathbb{C} \cup \{ \infty \} \rightarrow \mathbb{C} \cup \{ \infty \} = {S}^{2} \]\n\n\[ z \mapsto \left\{ \begin{array}{ll} f\left( z\right) , & \text{ if }z \in \mathbb{C}, \\... | Sketch of PROOF. Let \( f \) be a complex polynomial of degree \( n \geq 1 \) . We also consider the polynomial \( g\left( z\right) = {z}^{n} \) . We calculate that\n\nby Lemma 45.11 (3), since by Exercise 18.5 we know that \( \Theta \left( f\right) \) and \( \Theta \left( g\right) \) are homotopic\n\n\[ \deg \left( {\... | No |
Theorem 16.21. (Fundamental Theorem of Algebra) * Every nonconstant polynomial with coefficients in \( \mathbb{C} \) has a zero in \( \mathbb{C} \) . | Proof. Let \( f \) be a nonconstant polynomial with complex coefficients. In other words, \( f \) is a complex polynomial of degree \( n \geq 1 \) . By Proposition 45.24 we know that the degree of the self-map \( \Theta \left( f\right) \) of \( {S}^{2} = \mathbb{C} \cup \{ \infty \} \) equals \( n \geq 1 \) . By Lemma ... | Yes |
Lemma 45.25. (*) Let \( k \in {\mathbb{N}}_{0} \) and let \( f : {\Delta }^{k} \rightarrow {S}^{k} \) be an injective map such that the map \( f \circ \Phi : {\Delta }_{k} \rightarrow {S}^{k} \) is a smooth embedding \( {}^{736} \) Then for every \( Q \in f\left( {\Delta }^{k}\right) \) we have the following equality i... | Proof of Lemma 45.25 (*). First we deal with the special case that \( Q = N \) and that \( f\left( {\Delta }^{k}\right) \subset {S}_{ \geq 0}^{k} \) . We consider the following diagram:\n\n\[ \n{io}{t}_{P} \circ \Psi \left( \begin{matrix} {\mathrm{H}}_{k}\left( {{\mathbb{R}}^{k},{\mathbb{R}}^{k}\smallsetminus \{ 0\} }\... | Yes |
Lemma 46.1. Let\n\n\[ \n0 \rightarrow A\overset{i}{ \rightarrow }B\overset{p}{ \rightarrow }C \rightarrow 0 \]\n\nbe a short exact sequence of abelian groups. If \( C \) is a free abelian group, then the short exact sequence splits. | Proof. Let \( {\left\{ {c}_{i}\right\} }_{i \in I} \) be a basis of the free abelian group \( C \) . For each \( i \in I \) we choose a \( {b}_{i} \in B \) with \( p\left( {b}_{i}\right) = {c}_{i} \) . By Lemma 19.1 there exists a unique homomorphism \( s : C \rightarrow B \) with \( s\left( {c}_{i}\right) = {b}_{i} \)... | Yes |
Lemma 46.2. (Splitting Lemma) Let\n\n\[ \n\{ e\} \rightarrow A\overset{i}{ \rightarrow }B\overset{p}{ \rightarrow }C \rightarrow \{ e\} \n\] \n\nbe a short exact sequence of groups. The following two statements are equivalent:\n\n(1) There exists a homomorphism \( t : B \rightarrow A \) such that \( t \circ i = {\mathr... | Proof.\n\n\( \left( 1\right) \Rightarrow \left( 2\right) \) Let \( t : B \rightarrow A \) be a homomorphism such that \( t \circ i = {\mathrm{{id}}}_{A} \) . We define\n\n\[ \n\Phi : B \rightarrow A \times C \n\] \n\n\[ \nb \mapsto \left( {t\left( b\right), p\left( b\right) }\right) .\n\] \n\nIt follows easily from the... | Yes |
Corollary 46.3. Let\n\n\[ 0 \rightarrow A\overset{i}{ \rightarrow }B\overset{p}{ \rightarrow }C \rightarrow 0 \]\n\nbe a short exact sequence of abelian groups. Then the following hold:\n\n(1) If \( C \) is a free abelian group, then \( B \cong A \oplus C \) .\n\n(2) Suppose \( A \) and \( C \) are free abelian groups.... | Proof. The first statement is an immediate consequence of Lemmas 46.1 and 46.2 The second statement follows easily from Splitting Lemma 46.2 (1) \( \Rightarrow \) (3). \( \overline{\text{We leave the details}} \) to the reader. | No |
Lemma 46.4. Let\n\n\\[ \ldots \rightarrow {A}_{n + 2}\overset{{f}_{n + 2}}{ \rightarrow }{A}_{n + 1}\overset{{f}_{n + 1}}{ \rightarrow }{A}_{n}\overset{{f}_{n}}{ \rightarrow }{A}_{n - 1}\overset{{f}_{n - 1}}{ \rightarrow }{A}_{n - 2} \rightarrow \ldots \\]\n\nbe an exact sequence of abelian groups. Then for any \\( n \... | Proof. The statements follow immediately from the definitions. We leave it to the reader to fill in the details. | No |
Lemma 46.6. Let \( X \) be a topological space and let \( A, B \) be two open subsets of \( X \) with \( X = A \cup B \) . The connecting homomorphism \( {\partial }_{n} : {\mathrm{H}}_{n}\left( X\right) \rightarrow {\mathrm{H}}_{n - 1}\left( {A \cap B}\right) \) equals the map\n\n\[ \n{\mathrm{H}}_{n}\left( X\right) \... | Proof. This equality follows easily from the precise description of the connecting homomorphisms given in the Mayer-Vietoris Theorem 46.5 and in Proposition 43.15. | No |
Theorem 46.7. (Inverted Mayer-Vietoris Theorem) (*) Let \( X \) be a topological space and let \( U \) and \( V \) be open subsets. Then for all \( n \in {\mathbb{N}}_{0} \) there exists a natural homomorphism \( {\partial }_{n} : {\mathrm{H}}_{n}\left( {X, U \cup V}\right) \rightarrow {\mathrm{H}}_{n - 1}\left( {X, U ... | Proof (*). Note that as in the proof of Theorem 46.5 we have a short exact sequence\n\n\[ 0 \rightarrow {\mathrm{C}}_{n}\left( {X, U \cap V}\right) \rightarrow {\mathrm{C}}_{n}\left( {X, U}\right) \oplus {\mathrm{C}}_{n}\left( {X, V}\right) \rightarrow {\mathrm{C}}_{n}\left( X\right) /{\mathrm{C}}_{n}^{\{ U, V\} }\left... | Yes |
Lemma 46.8. Let \( k \in \mathbb{Z} \) . Given any topological space \( X \) we have a natural isomorphism\n\n\[ \n{\sum }_{X} : {\widetilde{\mathrm{H}}}_{k}\left( X\right) \overset{ \cong }{ \rightarrow }{\widetilde{\mathrm{H}}}_{k + 1}\left( {\sum \left( X\right) }\right) \n\] | Proof. Let \( k \in \mathbb{Z} \) .\n\n(1) Let \( X \) be a topological space. We denote by \( p : X \times \left\lbrack {-1,1}\right\rbrack \rightarrow \sum \left( X\right) \) the obvious projection. As on page 1178 we consider the subspaces \( {C}_{ + } \mathrel{\text{:=}} {C}_{ + }X = p\left( {X \times \left( {-1,1}... | Yes |
Theorem 46.10. (Mayer-Vietoris Theorem for Topological Manifolds) Let \( X \) be an \( m \) -dimensional topological manifold and let \( A, B \subset X \) be two \( m \) -dimensional submanifolds such that the following statements hold:\n\n(a) \( X = A \cup B \) ,\n\n(b) \( A \cap B \) is a union of components of \( \p... | Proof (*). Let \( X \) be an \( m \) -dimensional topological manifold and let \( A, B \subset X \) be two \( m \) - dimensional submanifolds such that \( A \) and \( B \) are closed subset of \( M \), such that \( X = A \cup B \) and such that \( A \cap B \) is a union of components of \( \partial A \) and also a unio... | No |
Theorem 46.11. (Mayer-Vietoris Theorem for CW-complexes) Let \( X \) be a CW-complex. If \( A \) and \( B \) are two subcomplexes of \( X \) such that \( X = A \cup B \), then exactly the same conclusion as in the Mayer-Vietoris Theorem 46.10 for Manifolds holds. | Proof (*). Let \( X \) be a CW-complex and let \( A \) and \( B \) be two subcomplexes of \( X \) with \( X = A \cup B \) . By Proposition 36.10 (8) there exist open neighborhoods \( U \) of \( A \) and \( V \) of \( B \) such that the following statements hold:\n\n(1) \( A \) is a deformation retract of \( U \) ,\n\n(... | Yes |
Lemma 46.13. We denote by \( K \) the Klein bottle. Then \[ {\mathrm{H}}_{n}\left( K\right) \cong \left\{ \begin{array}{ll} \mathbb{Z}, & \text{ if }n = 0, \\ \mathbb{Z} \oplus {\mathbb{Z}}_{2}, & \text{ if }n = 1, \\ 0, & \text{ otherwise. } \end{array}\right. \] | Furthermore we can determine explicitly two generators of \( {\mathrm{H}}_{1}\left( K\right) \) . More precisely, we consider Figure 773. There the singular 1-simplex \( d \) generates the \( {\mathbb{Z}}_{2} \) -summand and the singular 1-simplex \( c \) generates a \( \mathbb{Z} \) -summand of \( {\mathrm{H}}_{1}\lef... | No |
Let \( X \) be a topological space and let \( A \subset X \) be a subset. We denote by \( i : A \rightarrow X \) the inclusion map. For any \( n \in {\mathbb{N}}_{0} \) the two maps\n\n\[ \n{\mathrm{H}}_{n}\left( {X, A}\right) \underset{ \uparrow }{ \rightarrow }{\mathrm{H}}_{n}\left( {\operatorname{Cone}\left( {i : A ... | (1) Just for kicks we throw in one more map:\n\n\[ \n{\mathrm{H}}_{k}\left( {X, A}\right) \xrightarrow[]{ \rightarrow }{\mathrm{H}}_{k}\left( {\operatorname{Cone}\left( {i : A \rightarrow X}\right) ,\operatorname{Cone}\left( A\right) }\right) \xleftarrow[\left( \mathrm{c}\right) ]{}{\widetilde{\mathrm{H}}}_{k}\left( {\... | Yes |
Let \( X \) be a topological space and let \( A \subset X \) be a subset. We denote by \( i : A \rightarrow X \) the inclusion map. If \( i \) is a closed cofibration, then for any \( n \in {\mathbb{N}}_{0} \) the natural map to: \( {\mathrm{H}}_{n}\left( {X, A}\right) \rightarrow {\widetilde{\mathrm{H}}}_{n}\left( {X/... | The case that \( A = \varnothing \) is basically trivial and is left to the reader. In the following we assume that \( A \subset X \) is a non-empty subset. Let \( n \in {\mathbb{N}}_{0} \) . We consider the following diagram\n\n Let \( f : A \rightarrow X \) be a map between topological spaces. With these definitions the following sequence is exact\n\n\[ \ldots \rightarrow {\mathrm{H}}_{k}\left( A\right) \overset{{f}_{ * }}{ \rightarrow }{\mathrm{H}}_{k}\left( X\right) \overset{j{\left( f\right) }_{ * }}{ \rightarrow }{\widetilde{\mathrm{H... | (a) We denote by \( j\left( f\right) : X \rightarrow \operatorname{Cone}\left( f\right) = \left( {\operatorname{Cone}\left( A\right) \sqcup X}\right) / \sim \) the obvious inclusion map.\n\n(b) We denote by \( U \) the image of \( A \times \left\lbrack {0,\frac{3}{4}}\right) \) in \( \operatorname{Cone}\left( f\right) ... | Yes |
Lemma 46.19. Let \( X \) be a topological space, let \( n \geq 2 \) and let \( \varphi : {S}^{n - 1} \rightarrow X \) be a map. We denote by \( i : X \rightarrow X{ \cup }_{\varphi }{\bar{B}}^{n} \) the obvious map. (Note that by Lemma 3.43 we know that i is a closed embedding.) There exists a natural exact sequence of... | Proof. On page 991 we made the, almost trivial, observation that there exists a homeomorphism \( \operatorname{Cone}\left( {\varphi : {S}^{n - 1} \rightarrow X}\right) \overset{ \cong }{ \rightarrow }X{ \cup }_{\varphi }{\bar{B}}^{n} \) which is the identity on \( X \) . The conclusion of Lemma 46.19 follows immediatel... | Yes |
Lemma 46.20. (*) Let \( f : A \rightarrow X \) be a map between topological spaces. For every \( k \in {\mathbb{N}}_{0} \) the following diagram commutes:\n\n\[ \n{\widetilde{\mathrm{H}}}_{k}\left( {\operatorname{Cone}\left( {f : A \rightarrow X}\right) }\right) \xrightarrow[]{\;\partial \;}{\widetilde{\mathrm{H}}}_{k ... | Proof (*). The statement follows immediately from the explicit definition of the maps \( \partial \) and \( {\sum }_{A} \) via Mayer-Vietoris sequences, from the explicit definition of the map \( \varphi \), and from the naturality of the connecting homomorphism of the Mayer-Vietoris sequence, see\n\nTheorem 46.5 (2), ... | Yes |
Theorem 46.5 (2), that the isomorphism \( {\sum }_{X} \) is natural. | We leave it to the reader to fill in the details. | No |
Proposition 46.21. Let \( X \) be a topological space and let \( f : X \rightarrow X \) be a map. For every \( n \in {\mathbb{N}}_{0} \) we have a natural short exact sequence \( {}^{761} \n\n\[ \n0 \rightarrow \operatorname{coker}\left( {{\mathrm{H}}_{n}\left( X\right) \xrightarrow[]{{f}_{ * } - \mathrm{{id}}}{\mathrm... | Proof. The proof of the proposition is quite similar to the calculation of the homology groups of the torus \( {}^{762} \) Let \( X \) be a topological space and let \( f : X \rightarrow X \) be a homeomorphism. We compute the homology of the following space that is clearly naturally homeomorphic to the mapping torus: ... | No |
Lemma 46.22. Let \( X \) be a topological space. We denote by \( p : X \times {S}^{1} \rightarrow X \) the natural projection. For every \( n \in {\mathbb{N}}_{0} \) there exists a natural map \( \psi : {\mathrm{H}}_{n}\left( {X \times {S}^{1}}\right) \rightarrow {\mathrm{H}}_{n - 1}\left( X\right) \) such that the map... | Proof. Let \( n \in {\mathbb{N}}_{0} \) . Note that by Lemma 24.21 the map \( \Theta : \operatorname{Tor}\left( {X, f \mathrel{\text{:=}} \mathrm{{id}}}\right) \rightarrow X \times {S}^{1} \) given by \( \left\lbrack \left( {x, t}\right) \right\rbrack \mapsto \left( {x,{e}^{{2\pi }\mathrm{i}t}}\right) \) is a natural h... | Yes |
Lemma 46.23. Let \( X \) be a path-connected topological space and let \( f : X \rightarrow X \) be a map. We denote by \( i : X\xrightarrow[]{x \mapsto \left\lbrack \left( {x,0}\right) \right\rbrack }\operatorname{Tor}\left( {X, f}\right) \) the natural embedding and we denote by \( q : \operatorname{Tor}\left( {X, f}... | Proof. Since \( X \) is path-connected we obtain immediately from the discussion on page 1088 that the map \( {f}_{ * } - \mathrm{{id}} : {\mathrm{H}}_{0}\left( X\right) \rightarrow {\mathrm{H}}_{0}\left( X\right) \) is the zero map. We consider the following maps:\n\n \) be a directed set. Also let \( \\left( {{\\left\\{ {K}_{i}\\right\\} }_{i \\in I},{\\left\\{ {k}_{ij}\\right\\} }_{i \\leq j}}\\right),\\left( {{\\left\\{ {L}_{i}\\right\\} }_{i \\in I},{\\left\\{ {l}_{ij}\\right\\} }_{i \\leq j}}\\right) \) and \( \\left( {{\\lef... | Proof. We write \( f = \\underline{\\lim }{f}_{i} \) and \( g = \\underline{\\lim }{g}_{i} \) . We have to prove that \( g \\circ f = 0 \) and we need to show that \( \\ker \\left( g\\right) \\subset \\operatorname{im}\\left( f\\right) \) . Both statements statements follow our hypothesis together with Observation 47.1... | No |
Corollary 47.3. Let \( \\left( {I, \\leq }\\right) \) be a directed set and let \( \\left( {{\\left\\{ {K}_{i}\\right\\} }_{i \\in I},{\\left\\{ {k}_{ij}\\right\\} }_{i \\leq j}}\\right) \) and \( \\left( {{\\left\\{ {L}_{i}\\right\\} }_{i \\in I},{\\left\\{ {l}_{ij}\\right\\} }_{i \\leq j}}\\right) \) be direct system... | Proof. For each \( i \\in I \) the sequence\n\n\[ \n0 \\rightarrow \\ker \\left( {f}_{i}\\right) \\rightarrow {K}_{i}\\overset{{f}_{i}}{ \\rightarrow }{L}_{i} \\rightarrow \\operatorname{coker}\\left( {f}_{i}\\right) \\rightarrow 0.\n\]\n\nis exact. Evidently these homomorphisms form a homomorphism between the various ... | Yes |
Proposition 47.4. Let \( X \) be a topological space and let \( {X}_{1} \subset {X}_{2} \subset {X}_{3} \subset \ldots \) be a sequence of subsets such that \( X = \mathop{\bigcup }\limits_{{i \in \mathbb{N}}}{X}_{i} \) and such that one the following three conditions holds:\n\n(1) every compact subset of \( X \) is al... | We remark that it follows easily from the Finiteness Theorem 36.14 that hypothesis (3) implies hypothesis (1). Furthermore we point out that it follows from Lemma 25.8 that hypothesis (2) implies hypothesis (1). Thus it suffices to prove Proposition 4.4 under the hypothesis (1). But in that case the proposition is an i... | Yes |
Lemma 47.5. Let \( X \) be a topological space and let \( {X}_{1} \subset {X}_{2} \subset {X}_{3} \subset \ldots \) be a sequence of subsets such that every compact subset of \( X \) is already contained in one of the \( {X}_{i} \). Then the chain complexes \( {\mathrm{C}}_{ * }\left( {X}_{i}\right), i \in \mathbb{N} \... | Proof. Given a topological space \( Y \) and \( n \in {\mathbb{N}}_{0} \) we denote by \( {S}_{n}\left( Y\right) \) the set of singular \( n \) -simplices in \( Y \) . It follows immediately from our hypothesis on the \( {X}_{i} \), the fact that \( {\Delta }^{n} \) is compact and Lemma 2.40 that \n\n\[ \n{S}_{n}\left(... | No |
Lemma 47.6. Let \( \left( {I, \leq }\right) \) be a directed set and furthermore let \( \left( {{\left\{ {\mathrm{C}}_{i}\right\} }_{i \in I},{\left\{ {f}_{ij}\right\} }_{i \leq j}}\right) \) be a direct system of chain complexes. Then for each \( n \in {\mathbb{N}}_{0} \) we obtain an induced direct system \( \left( {... | \[ \mathop{\lim }\limits_{ \rightarrow }{\mathrm{H}}_{n}\left( {\mathrm{C}}_{i}\right) \overset{ \cong }{ \rightarrow }{\mathrm{H}}_{n}\left( {\mathop{\lim }\limits_{ \rightarrow }{\mathrm{C}}_{i}}\right) \] Proof. We have \[ \mathop{\lim }\limits_{ \rightarrow }{\mathrm{H}}_{n}\left( {\mathrm{C}}_{i}\right) = \mathop{... | Yes |
Lemma 47.7. We consider the real line with infinitely many 2-dimensional spheres attached, i.e. we consider\n\n\[ \nX \mathrel{\text{:=}} \left( {\mathbb{R} \cup \left( {{S}^{2} \times \mathbb{Z}}\right) }\right) / \sim \;\text{ where }n \sim \left( {\left( {0,0, - 1}\right), n}\right) \text{ for }n \in \mathbb{Z}. \n\... | In fact we can specify an isomorphism on \( {\mathrm{H}}_{2}\left( X\right) \) . Given \( i \in \mathbb{Z} \) we denote by \( {f}_{i} : {S}^{2} \rightarrow X \) the \ | No |
Let \( A \) and \( B \) be two topological spaces and let \( a \in A \) and \( b \in B \) be good points. We denote by \( A \vee B \) the wedge defined using a and b. Then the natural inclusion maps \( i : A \rightarrow A \vee B \) and \( j : B \rightarrow A \vee B \) that we introduced on page 561 induce for every \( ... | Proof (*). We write \( X = A \vee B \) and \( {x}_{0} = \{ a, b\} \in A \vee B \) . We pick an open neighborhood \( C \) in \( A \) of \( a \) such that \( \{ a\} \) is a deformation retract of \( C \) . Furthermore we pick an open neighborhood \( D \) of \( b \) in \( B \) such that \( b \) is a deformation retract of... | Yes |
Proposition 47.9. Let \( {\left\{ {A}_{k}\right\} }_{k \in K} \) be a family of topological spaces. For each \( k \in K \) suppose that we are given a good point \( {a}_{k} \in {A}_{k}{}^{769} \) We use these points to form the wedge \( \mathop{\bigvee }\limits_{{k \in K}}{A}_{k} \) . Given \( j \in K \) we denote by\n... | FIRST PROOF OF PROPOSITION [47.9] We have the following isomorphisms:\n\n\[ \n{\widetilde{\mathrm{H}}}_{n}\left( {\mathop{\bigcup }\limits_{{k \in K}}{A}_{k}}\right) = {\widetilde{\mathrm{H}}}_{n}\left( {\mathop{\bigcup }\limits_{{k \in K}}{A}_{k}/\mathop{\bigcup }\limits_{{k \in K}}\left\{ {a}_{k}\right\} }\right) \ov... | No |
Second Proof of Proposition 4.9. Let ${\\left\\{ {A}_{k}\\right\\} }_{k \\in K}$ be a family of topological spaces. For each $k \\in K$ suppose that we are given a good point ${a}_{k} \\in {A}_{k}$. We use these points to form the wedge $\\mathop{\\bigvee }\\limits_{{k \\in K}}{A}_{k}$. Recall that the fact that ${a}_{... | We introduce the following notation:\n\n(1) We write $B \\mathrel{\\text{:=}} \\mathop{\\bigvee }\\limits_{{k \\in K}}{A}_{k}$ and we set $U \\mathrel{\\text{:=}} \\mathop{\\bigcup }\\limits_{{k \\in K}}{U}_{k} \\subset B$.\n\n(2) Given a subset $J \\subset K$ we write ${B}_{J} \\mathrel{\\text{:=}} \\mathop{\\bigvee }... | Yes |
Lemma 47.12. Let \( \\left( {X,{x}_{0}}\\right) \) be a pointed topological space, let \( n \\in \\mathbb{N} \), let \( \\omega \\in {\\mathrm{H}}_{n}\\left( {{I}^{n},\\partial {I}^{n}}\\right) \) and let \( f, g : \\left( {{I}^{n},\\partial {I}^{n}}\\right) \\rightarrow \\left( {X,{x}_{0}}\\right) \) be two maps. Then... | Proof \( \\left( *\\right) \) . We write \( J = {I}^{n},{J}_{1} = \\left\\lbrack {0,\\frac{1}{2}}\\right\\rbrack \\times {I}^{n - 1} \) and \( {J}_{2} = \\left\\lbrack {\\frac{1}{2},1}\\right\\rbrack \\times {I}^{n - 1} \) . Furthermore we write \( K = \\partial J \\cup \\left( {\\left\{ \\frac{1}{2}\\right\} \\times {... | Yes |
(1) Let \( f : {S}^{n} \rightarrow {S}_{1}^{n} \vee {S}_{2}^{n} \) be the pinching map that we introduced on page 1060. Let \( i \in \{ 1,2\} \) . If we denote by \( {S}_{1}^{n} \vee {S}_{2}^{n} \rightarrow {S}_{i}^{n} = {S}^{n} \) the obvious projection, then the induced map \( {\left( {p}_{i} \circ f\right) }_{ * } :... | (1) The first statement follows easily from Lemma 47.12. | No |
Lemma 48.2. For all \( n \in {\mathbb{N}}_{0} \) we have \( {d}_{n} \circ {d}_{n + 1} = 0 \) . | Proof. We consider the following commutative diagram of maps\n\n\n\nThe map \( {\partial }_{n} \circ {j}_{n} : {\mathrm{H}}_{n}\left( {X}^{n}\right) \rightarrow {\mathrm{H}}_{n - 1}\left( {X}^{n - 1}\right) \) is t... | Yes |
Proposition 48.5. Let \( X \) be a CW-complex. The following statements hold:\n\n(1) If \( X \) contains precisely \( d \in {\mathbb{N}}_{0} \) cells of dimension \( n \), then \( {\mathrm{H}}_{n}\left( X\right) \) is generated by at most \( d \) elements. In particular, if \( X \) has no cells of dimension \( n \), th... | Proof. Let \( X \) be a CW-complex \( X \) .\n\n(1) If \( X \) contains precisely \( d \in {\mathbb{N}}_{0} \) cells of dimension \( n \), then it follows from Lemma 48.1 that \( {\mathrm{C}}_{n}^{\mathrm{{CW}}}\left( X\right) \cong {\mathbb{Z}}^{d} \) . By Lemma 19.8 any subgroup \( U \) of \( {\mathbb{Z}}^{d} \) is a... | Yes |
Proposition 48.7. Let \( X \) be a CW-complex. We use all the notation that we introduced above. Then for each \( n \) -cell \( \alpha \) we have\n\n\[ \n{d}_{n}\left( \alpha \right) = \mathop{\sum }\limits_{{\beta \in {I}_{n - 1}}}{d}_{\alpha ,\beta } \cdot \beta \n\]\n\nwhere\n\n\[ \n{d}_{\alpha \beta } \mathrel{\tex... | Proof. We write \( I \mathrel{\text{:=}} {I}_{n} \) and \( J \mathrel{\text{:=}} {I}_{n - 1} \) . We consider the following diagram\n\n---\n\n\( {}^{778} \) The strange shape of the diagram is due to space constraints.\n\n---\n\n\n\n we have\n\n\[ \n{\mathrm{H}}_{n}\left( {\sum }_{g}\right) \cong \left\{ {\begin{array}{ll} 0, & \text{ if }n \geq 3, \\ \mathbb{Z}, & \text{ if }n = 2, \\ {\mathbb{Z}}^{2g}, & \text{ if }n = 1, \\ \mathbb{Z}, & \text{ if }n = 0. \end{array}\;\begin{matrix} \text{ F... | Proof. The calculation of the homology groups \( {\sum }_{g, m} \) is a slight generalization of the arguments that we provided above and on page 1270. We leave the details to the reader. | No |
Proposition 48.10. For any \( n \in {\mathbb{N}}_{0} \) we have \( {}^{781} \n\n\[ \n{\mathrm{H}}_{k}\left( {\mathbb{R}{\mathrm{P}}^{n}}\right) \cong \left\{ \begin{array}{ll} \mathbb{Z}, & \text{ if }k = 0, \\ {\mathbb{Z}}_{2}, & \text{ if }k\text{ is odd and }k < n, \\ 0, & \text{ if }k\text{ is even and }0 < k \neq ... | Proof \( \left( *\right) \) . As a reminder we defined\n\n\[ \n{\mathbb{{RP}}}^{n} = {S}^{n}/ \sim \;\text{ where }x \sim - x\text{ for all }x \in {S}^{n}. \n\]\n\nIn Lemma 36.1 we already saw that we can view \( {\mathbb{{RP}}}^{n} \) as a CW-complex with the following two properties:\n\n(1) the CW-structure has exact... | Yes |
Lemma 49.6. Let \( \left( {{\mathrm{C}}_{ * },{c}_{ * }}\right) \) be a free chain complex such that for each \( k \in {\mathbb{N}}_{0} \) the homology group \( {\mathrm{H}}_{k}\left( C\right) \) is finitely generated. Then there exists a chain complex \( \left( {{D}_{ * },{\partial }_{ * }}\right) \) with the followin... | Proof. Let \( k \in {\mathbb{N}}_{0} \) . Recall that we assume that \( {\mathrm{H}}_{k}\left( {C}_{ * }\right) \) is finitely generated. It follows immediately from the classification of finitely generated abelian groups, see Theorem 19.4, that there exists a chain complex \( {D}_{ * }^{k} \) and an isomorphism \( {\v... | No |
Lemma 49.7. Let \( \varphi : {C}_{ * } \rightarrow {D}_{ * } \) be a chain map between two chain complexes \( \left( {{C}_{n},{c}_{n}}\right) \) and \( \left( {{D}_{n},{d}_{n}}\right) \). We denote by \( i : {D}_{n} \rightarrow \mathrm{M}{\left( \varphi \right) }_{n} = {\mathrm{C}}_{n - 1} \oplus {D}_{n} \) the obvious... | Proof. For each \( n \in {\mathbb{N}}_{0} \) we consider the following diagram \n\nWe make the following observations:\n\n(1) the horizontal sequences are exact,\n\n(2) the left-hand square commutes, i.e. the maps ... | Yes |
Lemma 49.8. Let \( \left( {{\mathrm{C}}_{ * },{\partial }_{ * }}\right) \) be a chain complex of free abelian groups such that \( {\mathrm{H}}_{n}\left( {C}_{ * }\right) = 0 \) for all \( n \in {\mathbb{N}}_{0} \) . Then the identity map and the zero map of \( {C}_{ * } \) to itself are chain homotopic. | Proof. For each \( n \in {\mathbb{N}}_{0} \) we write as usual\n\n\[ \n{Z}_{n} = \ker \left( {{\partial }_{n} : {\mathrm{C}}_{n} \rightarrow {\mathrm{C}}_{n - 1}}\right) \;\text{ and }\;{B}_{n} = \operatorname{im}\left( {{\partial }_{n + 1} : {\mathrm{C}}_{n + 1} \rightarrow {\mathrm{C}}_{n}}\right) .\n\]\n\nBy our hyp... | Yes |
Given any CW-complex \( X \) there exists a chain homotopy equivalence \( {\mathrm{C}}_{ * }\left( X\right) \rightarrow {\mathrm{C}}_{ * }^{\mathrm{{CW}}}\left( X\right) \) . | A part of the proof of Corollary 49.9 is pure \ | No |
Corollary 50.2. (Jordan Curve Theorem) For every planar Jordan curve \( C \) the complement \( {\mathbb{R}}^{2} \smallsetminus C \) consists of two path-connected components. | Proof. Let \( h : {S}^{1} \rightarrow {\mathbb{R}}^{2} \) be an injective map. We use the inclusion \( {\mathbb{R}}^{2} \rightarrow {\mathbb{R}}^{2} \cup \{ \infty \} = {S}^{2} \) to view \( {\mathbb{R}}^{2} \) as a subset of \( {S}^{2} \) . Now we see that\n\n\[ \n{\widetilde{\mathrm{H}}}_{0}\left( {{\mathbb{R}}^{2} \... | No |
Theorem 50.4. (Generalized smooth Schönflies Theorem) Let \( h : {S}^{n - 1} \rightarrow {S}^{n} \) be a smooth embedding. If \( n \neq 4 \), then there exists an orientation-preserving diffeomorphism \( H : {S}^{n} \rightarrow {S}^{n} \) with \( {\left. H\right| }_{{S}^{n - 1}} = h \) . | For \( n = 2 \) the statement follows fairly easily from the classification of smooth 2-dimensional smooth manifolds that we gave in the Surface Classification Theorem 23.4, see e.g. Hirs76, p. 207]. For \( n = 3 \) this statement was proved by James Alexander A124. See also [Hat3, Chapter 1] and [Schul14, Theorem 3.2.... | No |
Lemma 50.7. Let \( U \subset {\mathbb{R}}^{n} \) be a subset. If \( U \) is open in \( {\mathbb{R}}^{n} \), then every path-component of \( U \) is also an open subset of \( {\mathbb{R}}^{n} \). | Proof. Let \( C \) be a path-component of the subset subset \( U \subset {\mathbb{R}}^{n} \). Let \( P \in C \). Since \( U \) is open there exists an \( r > 0 \) such that \( {B}_{r}\left( P\right) \subset U \). But all points in \( {B}_{r}\left( P\right) \) are path-equivalent to \( P \), hence \( {B}_{r}\left( P\rig... | Yes |
Corollary 50.8. Let \( M \) be a non-empty closed \( n \) -dimensional topological manifold and let \( N \) be a connected \( n \) -dimensional topological manifold without boundary. Let \( h : M \rightarrow N \) be a map.\n\n(1) If \( h : M \rightarrow N \) is locally injective, then \( h \) is also surjective.\n\n(2)... | Proof. The second statement of the corollary is an immediate consequence of the first statement and Proposition 2.43 (3). So it suffices to prove the first statement.\n\nLet \( h : M \rightarrow N \) be a locally injective map from a non-empty closed \( n \) -dimensional topological manifold \( M \) to a connected \( n... | Yes |
Lemma 51.1. The map\n\n\[ q : \mathrm{{SU}}\left( 2\right) \rightarrow \operatorname{SIsom}\left( {T, g}\right) = \mathrm{{SO}}\left( 3\right) \]\n\n\[ A \mapsto \left( {B \mapsto {AB}{A}^{-1}}\right) \]\n\n is a smooth map between smooth manifolds and it is a well-defined group homomorphism. Furthermore for \( A,{A}^{... | Proof of Lemma \( \left\lbrack {51.1}\right\rbrack \left( *\right) \) . First we have to show that for \( A \in \mathrm{{SU}}\left( 2\right) \) and \( B \in T \) we also have \( {AB}{A}^{-1} \in \overline{T\text{. This is indeed the case since}} \)\n\n\[ \operatorname{tr}\left( {{AB}{A}^{-1}}\right) = \operatorname{tr}... | No |
Lemma 51.3. Let \( q : M \rightarrow N \) be a map between two n-dimensional topological manifolds. Suppose the following conditions are satisfied:\n\n(1) the map \( q \) is surjective,\n\n(2) the map \( q \) is locally injective, i.e. for each \( P \in M \) there exists a neighborhood \( U \) such that \( {\left. q\ri... | Proof (*). Since \( q \) is surjective we only have to show that each \( Q \in N \) admits an open neighborhood that is uniformly covered. So let \( Q \in N \) . We denote its preimages by \( {P}_{1},\ldots ,{P}_{r} \) . Let \( i \in \{ 1,\ldots, m\} \) . It follows easily from (2) and the hypothesis that \( M \) is a ... | Yes |
Corollary 51.4. We have\n\n\[ \n{\pi }_{1}\left( {\mathrm{{SU}}\left( 2\right) }\right) = 0\;\text{ and }\;{\pi }_{1}\left( {\mathrm{{SO}}\left( 3\right) }\right) \cong {\mathbb{Z}}_{2}.\n\]\n\nIn particular \( \mathrm{{SU}}\left( 2\right) \) is the universal covering of \( \mathrm{{SO}}\left( 3\right) . \) | Remark. We can actually give an explicit description of the non-trivial element in the fundamental group \( {\pi }_{1}\left( {\mathrm{{SO}}\left( 3\right) }\right) \cong {\mathbb{Z}}_{2} \) . More precisely, we consider the loop\n\n\[ \n\begin{aligned} \gamma : \left\lbrack {0,1}\right\rbrack & \rightarrow \mathrm{{SO}... | Yes |
Lemma 51.6. The map\n\n\[ \Phi : {S}^{1} \times {S}^{1} \times {S}^{1} \rightarrow \mathrm{{SO}}\left( 3\right) \]\n\n\[ \left( {\alpha ,\beta ,\gamma }\right) \mapsto \overset{{rotation}\;{by}\;\alpha }{{rotation}\;{by}\;\alpha } \circ \overset{{rotation}\;{by}\;\beta }{{round}\;{the}\;y\text{-}{axis}} \circ \n\]\nrot... | Sketch of Proof. Let \( A \in \mathrm{{SO}}\left( 3\right) \) be matrix with columns \( {v}_{1},{v}_{2},{v}_{3} \). After a rotation around the \( z \) -axis we can assume that \( {v}_{1} \) lies in the \( {xz} \) -plane. After a rotation around the \( y \) -axis we can then arrange that \( {v}_{1} \) lies in fact on t... | Yes |
Proposition 51.8. There is no continuous map \( \Psi : \mathrm{{SO}}\left( 3\right) \rightarrow {S}^{1} \times {S}^{1} \times {S}^{1} \) such that \( \Phi \circ \Psi = {\operatorname{id}}_{\mathrm{{SO}}\left( 3\right) } \). | Proof. Let us suppose there exists a continuous map \( \Psi : \mathrm{{SO}}\left( 3\right) \rightarrow {S}^{1} \times {S}^{1} \times {S}^{1} \) such that \( \Phi \circ \Psi = {\operatorname{id}}_{\mathrm{{SO}}\left( 3\right) } \). Then we get the following commutative diagram\n\n![448f61af-e517-4f9c-831f-f6ce5868f6c0_1... | Yes |
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