Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
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Lemma 68.16. For any knot \( K \subset {S}^{3} \) the following statements hold:\n\n(1) The exterior \( {X}_{K} \) of the knot \( K \) is a compact orientable connected 3-dimensional smooth manifold and the boundary \( \partial {X}_{K} \) is diffeomorphic to the torus.\n\n(2) The diffeomorphism type of \( {X}_{K} \) as... | Proof. Let \( K \subset {S}^{3} \) be an oriented knot and let \( P \in K \) . We pick a tubular neighborhood \( {\bar{B}}^{2} \times K \) for \( K \) and we pick \( Q \in {S}^{1} \) . We write \( {\mu }_{K} = {S}^{1} \times \{ P\} \) and \( \gamma = \{ Q\} \times K \) .\n\n(1) This statement is basically a consequence... | Yes |
Proposition 68.18. Let \( M \) be a compact connected \( n \) -dimensional smooth manifold. Let \( \left( {{\sigma }_{1} + \cdots + {\sigma }_{m}}\right) \otimes 1 \in {\mathrm{C}}_{n}\left( {M,\partial M;{\mathbb{F}}_{2}}\right) = {\mathrm{C}}_{n}\left( {M,\partial M}\right) \otimes {\mathbb{F}}_{2} \) be a cycle. Sup... | Proof. The proof of this proposition is almost the same as the proof of Proposition 68.4, We leave it to the reader to make the necessary modifications. | No |
Proposition 68.19. Let \( M \) be a compact \( n \) -dimensional smooth manifold. The connecting homomorphism\n\n\[ \partial : {\mathrm{H}}_{n}\left( {M,\partial M;{\mathbb{F}}_{2}}\right) \rightarrow {\mathrm{H}}_{n - 1}\left( {\partial M;{\mathbb{F}}_{2}}\right) \]\n\nof the long exact sequence in homology with \( {\... | Proof. The proof is the obvious modification of the previous proof. | No |
Corollary 68.20. Let \( M \) be a compact \( n \) -dimensional smooth manifold. We denote by\n\n\( i : \partial M \rightarrow M \) the inclusion map.\n\n(1) We have\n\n\[ \ker \left( {{i}_{ * } : {\mathrm{H}}_{n - 1}\left( {\partial M}\right) \rightarrow {\mathrm{H}}_{n - 1}\left( M\right) }\right) = \mathbb{Z} \cdot {... | Proof. Once again the proof is basically the same as the proof of the analogous statement for ordinary homology groups. | No |
Lemma 69.1. Let \( n \in \mathbb{N} \) and let \( L, M \) and \( N \) be compact oriented connected non-empty \( n \) -dimensional smooth manifolds.\n\n(1) \( \deg \left( {\operatorname{id}}_{M}\right) = 1 \) . | (1) This statement is trivial. | Yes |
Proposition 69.3. Let \( n \in \mathbb{N} \) and let \( M \) be a closed oriented connected non-empty \( n \) -dimensional smooth manifold. For every \( k \in \mathbb{Z} \) there exists a map \( f : M \rightarrow {S}^{n} \) of degree \( k \) . | Proof. Let \( M \) be a closed oriented connected non-empty \( n \) -dimensional smooth manifold and let \( k \in \mathbb{Z} \) . As we mentioned on page 1184, by Lemmas 45.10 and 45.12 there exists a map \( g : {S}^{n} \rightarrow {S}^{n} \) of degree \( k \) . It follows from this fact and Lemma 69.1 (4) that it suff... | Yes |
Lemma 69.5. Let \( k \in {\mathbb{N}}_{0} \) . For every \( g \geq k \) there exists a degree-one map from \( {\sum }_{g} \) to \( {\sum }_{k} \) . | Proof. We prove the lemma for \( g = 2 \) and \( k = 1 \) . It should be clear how to modify the discussion for other values of \( g \geq k \) . Let us consider the map from the surface \( {\sum }_{2} \) of genus two to the torus \( {\sum }_{1} \) that is shown in Figure 1072. In fact in Figure 1072 we show the map twi... | No |
Proposition 69.7. Let \( f : M \rightarrow N \) be a map between two compact oriented connected non-empty \( n \) -dimensional smooth manifolds such that \( {f}^{-1}\left( {\partial N}\right) = \partial N \) .\n\n(1) Let \( y \in N \smallsetminus \partial N \) such that the following condition is satisfied:\n\n\( \left... | Proof of Proposition 69.7 (1) (*). Let \( f : M \rightarrow N \) be a map between two compact oriented connected non-empty \( n \) -dimensional smooth manifolds such that \( {f}^{-1}\left( {\partial N}\right) = \partial M \) . Let \( y \in N \smallsetminus \partial N \) such that \( {f}^{-1}\left( {\{ y\} }\right) \) c... | Yes |
Proposition 69.8. Let \( p : \widetilde{M} \rightarrow M \) be a finite covering of compact oriented connected non-empty smooth manifolds. If \( p \) is orientation-preserving, then\n\n\[ \deg \left( p\right) = \left\lbrack {\widetilde{M} : M}\right\rbrack \] | Proof. It turns out that there are two quick proofs for the proposition. In fact the desired statement follows immediately from Proposition 69.7. Alternatively, if we denote\n\nby \( {p}^{ * } : {\mathrm{H}}_{n}\left( {M,\partial M}\right) \rightarrow {\mathrm{H}}_{n}\left( {\widetilde{M},\partial \widetilde{M}}\right)... | No |
Corollary 69.10. Let \( g, k \in {\mathbb{N}}_{0} \) . If there exists a degree-one map from \( {\sum }_{g} \) to \( {\sum }_{k} \), then \( g \geq k \) . | Proof. Suppose that there exists a degree-one map from \( {\sum }_{g} \) to \( {\sum }_{k} \) . By Proposition 69.9 we obtain an epimorphism \( {\pi }_{1}\left( {\sum }_{g}\right) \rightarrow {\pi }_{1}\left( {\sum }_{k}\right) \) . In particular we obtain an epimorphism from the abelianization \( {\pi }_{1}\left( {\su... | Yes |
Proposition 70.2. Let \( X \) be a topological space. Every class in \( {\mathrm{H}}_{1}\left( X\right) \) is realized by a smooth manifold. | Proof. The proposition is basically an immediate consequence of Proposition 52.3. Nonetheless we prefer to do prove the proposition here \ | Yes |
Proposition 70.3. Let \( X \) be a topological space. Every class in \( {\mathrm{H}}_{2}\left( X\right) \) is realized by a smooth manifold. | Proof. The basic idea of the proof of Proposition 70.3 is similar to the proof of Proposition 70.2. Nonetheless, ironing out the details is much more messy, hence we postpone the proof of the case \( n = 2 \) to Section 70.2, Alternatively see also [Cal09, p. 3] for a quick sketch of the proof. | No |
Proposition 70.4. Let \( k \in \mathbb{N} \) . (1) Let \( X \) be a topological space and let \( A \subset X \) be a subset. We suppose that \( X \) is path-connected. If \( A = \varnothing \) or if \( k \geq 2 \), then every homology class in \( {\mathrm{H}}_{k}\left( {X, A}\right) \) that can be realized by a smooth ... | Proof. In order not to disrupt the flow of the conversation too much we postpone the somewhat technical proof of Proposition 70.4 to Section 70.3. | No |
Theorem 70.5. (Thom's Theorem I)\n\n(1) If \( X \) is a topological space, then every homology class of dimension \( \leq 6 \) can be realized by a smooth manifold.\n\n(2) Given any \( n \geq 7 \) there exists a topological space \( X \) and a homology class in \( {\mathrm{H}}_{n}\left( X\right) \) that is not realized... | Proof. As mentioned before, the theorem is proved in Tho54b, Theorem III.3, III.4 and III.3.9]. A more modern account of the proof is given in Theorems IV.7.35, IV.7.36, IV.7.37 and IV.7.38 together with Remark IV.7.40 of [Rudy98]. In fact the topological space \( X \) in (2) is given by the \( \left( {n - 7}\right) \)... | Yes |
Theorem 70.6. (Thom’s Theorem II) Let \( X \) be a path-connected topological space and let \( A \subset X \) be a subset. Given any \( \varphi \in {\mathrm{H}}_{n}\left( {X, A;{\mathbb{F}}_{2}}\right) \) there exists a compact connected \( n \) -dimensional smooth manifold \( N \) and a map \( f : N \rightarrow X \) w... | Proof. First let us consider the case \( A = \varnothing \) and for the time being let us allow possibly disconnected smooth manifolds. In this setting, for \( n = 1 \) and \( n = 2 \) the statement follows from straightforward modifications of the proofs of Propositions 7.0.2 and 7.0.3. In fact, much more interestingl... | No |
Proposition 70.3. Let \( X \) be a topological space. Every class in \( {\mathrm{H}}_{2}\left( X\right) \) is realized by a smooth manifold. | The proof of Proposition 70.3 is modelled on the proof of Proposition 70.2. But, as we will see, the proof is technically more difficult.\n\nWe start out our proof with the more general setting of some \( n \) -dimensional class. Later on we will specialize to the case that \( n = 2 \) . It is even worthwhile keeping t... | No |
(1) \( \mu \in {\mathrm{C}}_{n}\left( {Y\left( \Theta \right) }\right) \) is a cycle. | Proof. First we show that \( \mu \) is a cycle. This follows from the following straightforward calculation:\n\n\[ \partial \mu = \partial \left( {\mathop{\sum }\limits_{{j = 1}}^{k}{a}_{j} \cdot p \circ {\Psi }_{j}}\right) = \mathop{\sum }\limits_{{j = 1}}^{k}\mathop{\sum }\limits_{{r = 0}}^{n}{\left( -1\right) }^{r} ... | Yes |
Lemma 70.8. We continue with the above notation. If \( n = 1 \) or if \( n = 2 \), then the topological space \( Y\left( \Theta \right) \) admits the structure of a closed oriented 2-dimensional smooth manifold such that \( \left\lbrack \mu \right\rbrack \in {\mathrm{H}}_{2}\left( {Y\left( \Theta \right) }\right) \) is... | Sketch of A proof of Lemma 70.8. The proof of Lemma 70.8 for the case \( n = 1 \) is very elementary and thus left to the reader.\n\nIn the following we sketch a proof of Lemma 70.8 for the case \( n = 2 \) . As we will see, the proof is quite similar to the proof, given in Propositions 6.8 and 6.21, that the surface \... | No |
Lemma 70.9. Given any \( n \in {\mathbb{N}}_{ \geq 3} \) there exists a topological space \( X \) together with a cycle \( c = {a}_{1} \cdot {\sigma }_{1} + \cdots + {a}_{k} \cdot {\sigma }_{k} \in {\mathrm{C}}_{n}\left( X\right) \) such that for any choice of \( \Theta \) the topological space \( Y\left( \Theta \right... | Sketch of PROOF. Let \( n \in {\mathbb{N}}_{ \geq 3} \) and let \( M \) be the \( \left( {n - 1}\right) \) -dimensional torus. Furthermore let \( \left( {K,\Xi : \left| K\right| \rightarrow M}\right) \) be a smooth simplicial structure for \( M \), which we know exists by Theorem 64.2, or by some more down-to-earth arg... | No |
Lemma 70.10. Let \( M \) be a smooth manifold, let \( A \) be a union of components of \( \partial M \) and let \( \varphi \in {\mathrm{H}}_{k}\left( {M, A}\right) \) . If \( N \) is a proper submanifold that represents \( \varphi \), then \( \partial N \) represents the class \( {\partial }_{k}\left( \varphi \right) \... | Proof (*). We consider the following diagram:\n\n\n\nwhere the horizontal maps are given by the connecting homomorphisms and where the vertical maps are induced by the inclusion maps. By Proposition 43.15 we know t... | Yes |
Proposition 70.12. Let \( k \in \mathbb{N} \) . Suppose that for every (orientable) \( n \) -dimensional smooth manifold every homology class in \( {\mathrm{H}}_{k}\left( M\right) \) can be represented by a submanifold. Then also for every (orientable) \( n \) -dimensional smooth manifold \( M \) and every union \( A \... | Proof (*). Let \( A \) be the union of some components of \( \partial M \) and let \( \sigma \in {\mathrm{H}}_{k}\left( {M, A}\right) \) . Let \( {M}^{\prime } \) be a second copy of \( M \) . As on page 1163 we consider the double of \( M \) along \( A \), i.e. we consider \( {\mathrm{D}}_{A}M = M{ \cup }_{A = {A}^{\p... | Yes |
Lemma 70.15. Let \( T = {S}^{1} \times {S}^{1} \) be the torus. We denote by \( x = \left\lbrack {{S}^{1}\times \{ 1\} }\right\rbrack \) and \( y = \left\lbrack {\{ 1\} \times {S}^{1}}\right\rbrack \) the obvious basis for \( {\mathrm{H}}_{1}\left( T\right) \) . A non-zero class \( r \cdot x + s \cdot y \) can be repre... | Proof. We, i.e. you, will provide the proof in Exercise 70.4. For fun we will deal with the analogous problem for surfaces of higher genus in Exercise 70.5, | No |
Proposition 98.4. Let \( M \) be a compact orientable \( n \) -dimensional smooth manifold and let \( A \subset \partial M \) be a union of boundary components. Every class in \( {\mathrm{H}}_{n - 1}\left( {M, A}\right) \) can be represented by a submanifold. | SKETCH OF PROOF. As the number already suggests, we will not prove this statement in this chapter. In the following we will just provide the flavor of the argument which is meant to serve as an amuse-bouche. For simplicity we assume that \( M \) is closed. Now suppose we are given a homology class \( \sigma \in {\mathr... | No |
Theorem 70.20. (Thom's Theorem IV) Let \( M \) be a compact \( n \) -dimensional smooth manifold and let \( A \) be a union of boundary components.\n\n(1) If \( n \leq 5 \), then any class in \( {\mathrm{H}}_{n - 2}\left( {X, A;{\mathbb{F}}_{2}}\right) \) can be represented by a submanifold.\n\n(2) If \( n \leq 7 \), t... | Proof. In the case \( A = \varnothing \) the three statements are precisely the content of Tho07, Théorème II.26]. The relative case follows from the \( {\mathbb{F}}_{2} \) -analogue of Proposition 70.12 . Finally note that Statement (3) also follows from Theorem 70.5 together with the \( {\mathbb{F}}_{2} \) - analogue... | No |
Proposition 70.21. Let \( M \) and \( \widetilde{M} \) be two compact oriented \( n \) -dimensional smooth manifolds. Suppose that \( M \) is decomposed into two submanifolds \( A \) and \( B \) and that \( \widetilde{M} \) is decomposed into two submanifolds \( \widetilde{A} \) and \( \widetilde{B} \) . We equip \( A ... | Proof. We pick a fundamental cycle \( \\mu \\in {\\mathrm{C}}_{n - 1}\\left( {A \\cap B}\\right) \) for the closed oriented \( \\left( {n - 1}\\right) \) - dimensional smooth manifold \( A \\cap B \) . We consider the map , let \( M \) be an \( n \) -dimensional smooth manifold and let \( A \) be a union of components of \( \partial M \) . Furthermore let \( N \) be a compact oriented \( k \) -dimensional smooth manifold and let \( \varphi : \left( {N,\partial N}\right) \rightarrow \left( {M,... | Proof. Let \( n \in \mathbb{N} \), let \( M \) be an \( n \) -dimensional smooth manifold and let \( A \) be a union of components of \( \partial M \) . Furthermore let \( N \) be a compact oriented \( k \) -dimensional smooth manifold and let \( \varphi : \left( {N,\partial N}\right) \rightarrow \left( {M, A}\right) \... | Yes |
Lemma 70.23. We continue with the above notation. The internal connected sum \( {N}_{0}1{\# }_{\gamma }{N}_{1} \) , viewed as a smooth manifold in its own right, is diffeomorphic to the connected sum \( {N}_{0}\# {N}_{1} \) . | Proof. The lemma can be proved fairly easily using the Collar Neighborhood Theorem 8.12 applied to \( {N}_{0} \smallsetminus \Phi \left( {D \times \left\{ {P}_{0}\right\} }\right) \) and \( {N}_{1} \smallsetminus \Phi \left( {D \times \left\{ {P}_{1}\right\} }\right) \) . We leave it to the reader to fill in the detail... | No |
Proposition 70.16. Let \( n \in \mathbb{N} \), let \( M \) be a connected \( n \) -dimensional smooth manifold and let \( A \) be a union of components of \( \partial M \) . Furthermore let \( N \) be a compact oriented \( k \) -dimensional submanifold with \( \partial N \subset N \) . We suppose that \( n - k \geq 2 \... | Sketch of PROOF. To simplify the notation we assume that \( N \) has precisely two components \( {N}_{0} \) and \( {N}_{1} \) . As explained in Construction 70.22, since \( n - k \geq 2 \) we can perform an internal connected sum \( {N}_{0}{\# }_{\gamma }{N}_{1} \) . As in the proof of Proposition 70.13 one can deduce,... | No |
Lemma 71.1. Let \( K \) be an abstract simplicial complex. As always we denote by \( \operatorname{sd}\left( K\right) \) its barycentric subdivision. The following two statements are equivalent:\n\n(1) \( K \) is a closed \( n \) -dimensional simplicial homology manifold.\n\n(2) For every vertex \( {v}^{1052} \) of the... | Proof. The lemma follows immediately from the second part of the following claim.\n\nClaim. Let \( K \) be an abstract simplicial complex and let \( s \) be a \( k \) -simplex of \( K \) .\n\n(a) There is a simplicial isomorphism \( \operatorname{Lk}\left( {K, s}\right) * \partial s \cong \operatorname{Lk}\left( {\oper... | No |
Lemma 71.3. Let \( K \) be an abstract simplicial complex. If \( \left| K\right| \) is an \( n \) -dimensional topological manifold with \( \partial \left( \left| K\right| \right) = \varnothing \), then \( K \) is an \( n \) -dimensional simplicial homology manifold. | Proof. Let \( s \) be a \( k \) -simplex of \( K \) . We need to show that \( \operatorname{Lk}\left( {K, s}\right) \) is a simplicial homology \( \left( {n - k - 1}\right) \) -sphere. We pick a point \( P \) in the corresponding open simplex \( \langle s\rangle \) . We perform the following calculation:\n\n\[\n\begin{... | Yes |
Theorem 71.5. (Poincaré Duality Theorem) Let \( M \) be a closed \( n \) -dimensional smooth manifold and let \( k \in {\mathbb{N}}_{0} \) . If \( M \) is orientable, then for every choice of orientation and every commutative ring \( R \) the map\n\n\[ \n{\mathrm{H}}^{k}\left( {M;R}\right) \rightarrow {\mathrm{H}}_{n -... | Proof. It follows easily from Lemma 8.9 that we can assume that \( M \) is actually connected. Recall that by Theorem [64.2] we know that \( M \) admits a smooth simplicial structure \( \left( {K = \left( {V, S}\right), f : \left| K\right| \rightarrow M}\right) \) . Without loss of generality we can assume that \( \lef... | No |
Lemma 71.6. Let \( K = \left( {V, S}\right) \) be a closed \( n \) -dimensional pseudomanifold.\n\n(1) The barycentric subdivision \( \operatorname{sd}\left( K\right) \) is a closed \( n \) -dimensional pseudomanifold.\n\n(2) Suppose that \( K \) is equipped with an orientation. Let \( t \) be an \( n \) -simplex of \(... | SKETCH OF PROOF. Statements (1) and (2a) follow almost immediately from the definitions, the proof is outsourced to Exercise 66.5. Furthermore Statement (2b) can be deduced easily from the definition of the fundamental class on page 1653 and the definition of the subdivision map \( {u}_{n} : {\mathrm{C}}_{n}^{\operator... | No |
Lemma 67.4. Let \( K = \left( {V, S}\right) \) be an abstract simplicial complex. Given any ordered \( k \) -simplex \( s = \left( {{v}_{0},\ldots ,{v}_{k}}\right) \) we have the following equality: | \[ {\delta }_{k}\left( {\left\lbrack {v}_{0},\ldots ,{v}_{k}\right\rbrack }^{ * }\right) = {\left( -1\right) }^{k + 1}.\;\sum \;{\left\lbrack {v}_{0},\ldots ,{v}_{k}, w\right\rbrack }^{ * } \in {\mathrm{C}}_{\operatorname{simp}}^{k + 1}\left( {K;\mathbb{Z}}\right) . \] \[ w \in V\text{such that} \] \[ \left\{ {{v}_{0},... | No |
Proposition 71.9. Let \( K = \left( {V, S}\right) \) be a closed oriented \( n \) -dimensional pseudomanifold.\n\n(1) For each \( k \in {\mathbb{N}}_{0} \) the map\n\n\[ \Lambda : \operatorname{Hom}\left( {{\mathrm{C}}_{k}^{\text{simp }}\left( K\right) ,\mathbb{Z}}\right) \rightarrow {\mathrm{D}}_{n - k}^{\text{simp }}... | ## Proof.\n\n(1) First note that \( K \) is by definition a finite abstract simplicial complex. Thus it follows immediately from Lemma 67.2 and Lemma 71.7 (2) that the maps \( \Lambda \) are isomorphisms. Basically by definition the maps \( \Lambda \) are natural. | No |
Lemma 71.10. Given any closed oriented n-dimensional pseudomanifold \( K \) the following diagram commutes:  | Proof. Let \( K = \left( {V, S}\right) \) be a closed oriented \( n \) -dimensional pseudomanifold. Given two \( k \) -simplices \( s \) and \( t \) of \( K \) we write \( \underline{s} \leq \underline{t} \) if \( s \subset t \) . As we pointed out in Lemma 62.2, this order on the vertex set of \( \operatorname{sd}\lef... | No |
Lemma 71.12. Let \( K = \left( {V, S}\right) \) be a closed \( n \) -dimensional simplicial homology manifold and let \( k \in {\mathbb{N}}_{0} \) . (1) For \( i \neq k \) we have \( {\mathrm{H}}_{i}^{\mathrm{{simp}}}\left( {{X}^{k},{X}^{k - 1}}\right) = 0 \) . (2) The map \[ \Theta : {\mathrm{D}}_{k}^{\text{simp }}\le... | Proof. Let \( K = \left( {V, S}\right) \) be a closed \( n \) -dimensional simplicial homology manifold. Given \( l \in {\mathbb{N}}_{0} \) we denote by \( {S}_{l} \) the set of \( l \) -simplices of \( K \) . Given an \( \left( {n - k}\right) \) -simplex \( s \in {S}_{n - k} \) we define the following: \( Z\left( s\ri... | Yes |
Lemma 71.13. Let \( K = \left( {V, S}\right) \) be a closed \( n \) -dimensional simplicial homology manifold and let \( k \in {\mathbb{N}}_{0} \) .\n\n(1) For every \( i > k \) we have \( {\mathrm{H}}_{i}^{\operatorname{simp}}\left( {X}^{k}\right) = 0 \) .\n\n(2) For every \( i < k \) the inclusion induced map \( {\ma... | (1) Note that \( {X}^{k} \) is an abstract simplicial complex of dimension \( k \) . It follows immediately from the definition of simplicial homology that \( {\mathrm{H}}_{i}^{\text{simp }}\left( K\right) = 0 \) for \( i > k \) .\n\n(2) We fix \( i \in {\mathbb{N}}_{0} \) . Let \( m > i \) . We consider the following ... | Yes |
Proposition 71.14. Let \( M \) be a closed \( n \) -dimensional smooth manifold. Furthermore let \( \left( {K = \left( {V, S}\right) ,\Theta : \left| K\right| \rightarrow M}\right) \) be a smooth simplicial structure.\n\n(1) For each \( \left( {n - k}\right) \) -simplex \( s \) we have a homeomorphism \( \left| {Z\left... | Sketch of Proof. Let \( M \) be a closed \( n \) -dimensional smooth manifold. Furthermore let \( \left( {K = \left( {V, S}\right) ,\Theta : \left| K\right| \rightarrow M}\right) \) be a smooth simplicial structure. Note that we know by Theorem 64.14 that \( \left( {K = \left( {V, S}\right) ,\Theta : \left| K\right| \r... | Yes |
Lemma 72.1. If \( H \) and \( G \) are two hyperplanes in \( {\mathbb{R}}^{3} \) through the origin, then there exists a matrix \( A \in \mathrm{{SO}}\left( 3\right) \) such that for all \( v \in {\mathbb{R}}^{3} \) we have\n\n\[ \n{\rho }_{G}\left( v\right) = A \cdot {\rho }_{H}\left( v\right) \n\] | SKETCH OF PROOF. With respect to the standard basis of \( {\mathbb{R}}^{3} \) the maps \( {\rho }_{G} \) and \( {\rho }_{H} \) are represented by orthogonal matrices \( P \) and \( Q \) with determinant -1 . The matrix \( A = P{Q}^{-1} \) then lies in \( \mathrm{{SO}}\left( 3\right) \) and for any \( v \in {\mathbb{R}}... | No |
Lemma 72.3. Let \( f : M \rightarrow N \) be a map between compact oriented connected \( n \) -dimensional topological manifolds such that \( f\left( {\partial M}\right) \subset \partial N \) . Let \( x \in M \smallsetminus \partial M \) and let \( U \) be an open neighborhood of \( x \) with the property that \( f \) ... | Proof. Let \( M \) and \( N \) be compact connected \( n \) -dimensional topological manifolds that are equipped with orientations \( {\left\{ {\mu }_{x}\right\} }_{x \in M \smallsetminus \partial M} \) and \( {\left\{ {\nu }_{x}\right\} }_{x \in N \smallsetminus \partial N} \) . Furthermore let \( f, x \in M \smallset... | Yes |
Proposition 72.4. Let \( K \subset {\mathbb{R}}^{3} \) be a knot. If \( K \) is amphichiral, then the knot exterior \( {X}_{K} \) is also amphichiral. | Proof. Let \( K \subset {\mathbb{R}}^{3} \) be an amphichiral knot. By the Tubular Neighborhood Theorem 8.24 we can pick a tubular neighborhood \( {\nu K} = {\bar{B}}^{2} \times K \) . As on page 1728 we consider the knot exterior \( {X}_{K} \mathrel{\text{:=}} {S}^{3} \smallsetminus \left( {{B}^{2} \times K}\right) \)... | Yes |
Lemma 72.6. Let \( p \in \mathbb{N} \) and let \( q, r \in \mathbb{Z} \) with \( \gcd \left( {p, q}\right) = \gcd \left( {p, r}\right) = 1 \) .\n\n(1) In the case that \( q \equiv {r}^{\pm 1}{\;\operatorname{mod}\;p} \) there exists an orientation-preserving diffeomorphism \( L\left( {p, q}\right) \rightarrow L\left( {... | Proof. Let \( p \in \mathbb{N} \) and let \( q, r \in \mathbb{Z} \) with \( \gcd \left( {p, q}\right) = \gcd \left( {p, r}\right) = 1 \) . We suppose that \( q \equiv \pm {r}^{\pm 1}{\;\operatorname{mod}\;p} \) . As in the proof of Lemma 16.6 we define a diffeomorphism\n\n\[ \Phi : {S}^{3} \rightarrow {S}^{3} \]\n\n as... | Yes |
Lemma 73.3. Let \( \\left\\{ {{\\left\\{ {A}_{i}\\right\\} }_{i \\in \\mathbb{N}},{\\left\\{ {\\varphi }_{ij} : {A}_{i} \\rightarrow {A}_{j}\\right\\} }_{i \\leq j}}\\right\\} \) be a direct system of groups and let \( G \) be an abelian group. Then\n\n\[ \n\\left\\{ {\\{ \\operatorname{Hom}\\left( {{A}_{i}, G}\\right)... | Proof. As so often for arguments regarding limits, the proof follows mostly from unraveling the definitions. We refer to Rot09, Proposition 5.26] for more details. | No |
Lemma 73.4. 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 and let \( G \) be an abelian group. If \( C \) is free abelian, then the dual sequence\n\n\[ 0 \rightarrow \operatorname{Hom}\left( {C, G}\right) \overset{{p}^{ * ... | Proof. It follows from the hypothesis that \( C \) is a free abelian and from Lemma 46.1 that the initial short exact sequence splits. But this means that, according to Splitting Lemma 46.2, that there exists an isomorphism \( \Phi : B \rightarrow A \oplus C \) such that the following diagram commutes:\n\n![448f61af-e5... | Yes |
Lemma 73.7. If \( {\left\{ {C}_{a}\right\} }_{a \in A} \) is a family of cochain complexes, then the family of inclusion maps \( {C}_{a} \rightarrow \mathop{\bigoplus }\limits_{{a \in A}}{C}_{a}, a \in A \), and \( {C}_{a} \rightarrow \mathop{\prod }\limits_{{a \in A}}{C}_{a}, a \in A \), induce isomorphisms | \[ \mathop{\bigoplus }\limits_{{a \in A}}{\mathrm{H}}^{n}\left( {C}_{a}\right) \overset{ \cong }{ \rightarrow }{\mathrm{H}}^{n}\left( {\mathop{\bigoplus }\limits_{{a \in A}}{C}_{a}}\right) \;\text{ and }\;\mathop{\prod }\limits_{{a \in A}}{\mathrm{H}}^{n}\left( {C}_{a}\right) \overset{ \cong }{ \rightarrow }{\mathrm{H}... | Yes |
Lemma 73.8. Let \( G \) be an abelian group.\n\n(1) Let \( f = {\left\{ {f}_{n} : {C}_{n} \rightarrow {D}_{n}\right\} }_{n \in {\mathbb{N}}_{0}} \) be a chain map between chain complexes \( {C}_{ * } \) and \( {D}_{ * } \), then for each \( n \in {\mathbb{N}}_{0} \) the map\n\n\[ {f}^{ * } : \operatorname{Hom}\left( {{... | (1) Let \( f = {\left\{ {f}_{n} : {C}_{n} \rightarrow {D}_{n}\right\} }_{n \in {\mathbb{N}}_{0}} \) be a chain map. This means that for each \( n \in {\mathbb{N}}_{0} \) we have\n\n\[ {f}_{n - 1} \circ {\partial }_{n} = {\partial }_{n} \circ {f}_{n} \]\n\nwhich implies by the fact that \( A \mapsto \operatorname{Hom}\l... | Yes |
(1) The singular 1-cochains \( {\theta }_{\mathbb{R}} \) and \( {\theta }_{\mathbb{Z}} \) are cocycles, in particular \( {\theta }_{\mathbb{R}} \) defines an element in \( {\mathrm{H}}^{1}\left( {{S}^{1};\mathbb{R}}\right) \) and \( {\theta }_{\mathbb{Z}} \) defines an element in \( {\mathrm{H}}^{1}\left( {{S}^{1};\mat... | (1) We will first prove that the singular 1-cochain \( {\theta }_{\mathbb{R}} \) is a cocycle. Thus we have to show that \( {\delta }_{1}{\theta }_{\mathbb{R}} = 0 \in {\mathrm{C}}^{2}\left( {{S}^{1},\mathbb{R}}\right) = \operatorname{Hom}\left( {{\mathrm{C}}_{2}\left( {S}^{1}\right) ,\mathbb{R}}\right) \) . So let \( ... | Yes |
Proposition 73.11. Let \( X \) be a non-empty path-connected topological space and let \( G \) be an abelian group. Then the map \[ G \rightarrow {\mathrm{H}}^{0}\left( {X;G}\right) \] \[ g \mapsto \left\lbrack {0\text{-cochain that assigns to each}x \in X\text{the value}g}\right\rbrack \] is a well-defined map that is... | Proof. We consider the \ | No |
Lemma 73.13. Let \( G \) be an abelian group and let \( n \in {\mathbb{N}}_{0} \). (1) The maps \[ \left( {X, A}\right) \mapsto {\mathrm{H}}^{n}\left( {X, A;G}\right) \] \[ \left( {f : \left( {X, A}\right) \rightarrow \left( {Y, B}\right) }\right) \; \mapsto \;\left( {{f}^{ * } : \;{\mathrm{H}}^{n}\left( {Y, B;G}\right... | (1) This statement follows immediately from Lemma 73.8 (2) together with the fact that the map \( \left( {X, A}\right) \mapsto {\mathrm{C}}_{ * }\left( {X, A}\right) \) defines a covariant functor from the category PairTop of pairs of topological spaces to the category \( \mathcal{{ChC}}p\mathcal{{lx}} \) of chain comp... | Yes |
Lemma 73.14. Let \( X \) be a topological space and let \( G \) be an abelian group.\n\n(1) We denote by \( {\left\{ {X}_{j}\right\} }_{j \in J} \) the path-components of \( X \) . The inclusion maps \( {\iota }_{j} : {X}_{j} \rightarrow X \) , \( j \in J \), induce an isomorphism\n\n\[ \mathop{\prod }\limits_{{j \in J... | Proof. Let \( X \) be a topological space with path-components \( {\left\{ {X}_{j}\right\} }_{j \in J} \) and let \( G \) be an abelian group. We have the following isomorphisms of cochain complexes: \n\nThe first ... | Yes |
Lemma 73.17. Let \( X \) be a topological space and let \( B \subset X \) be a non-empty subset. Then given any abelian group \( G \) there exists a natural long exact sequence\n\n\[ \cdots \xrightarrow[]{{\delta }_{n - 1}}\;{\mathrm{H}}^{n}\left( {X, B;G}\right) \;\xrightarrow[]{{p}^{ * }}\;{\widetilde{\mathrm{H}}}^{n... | Proof. The lemma follows immediately from Lemma 73.15 applied to the triple \( \left( {X, B,\{ P\} }\right) \) where \( P \) is a point in \( B \), together with Lemma 73.16 (5). | Yes |
Proposition 73.18. Suppose \( \left( {{\mathrm{C}}_{ * },{\partial }_{ * }}\right) \) and \( \left( {{C}_{ * }^{\prime },{\partial }_{ * }^{\prime }}\right) \) are two chain complexes of free abelian groups. Suppose that for each \( n \in {\mathbb{N}}_{0} \) the homology groups \( {\mathrm{H}}_{n}\left( {C}_{ * }\right... | Proof. Suppose \( \left( {{\mathrm{C}}_{ * },{\partial }_{ * }}\right) \) and \( \left( {{C}_{ * }^{\prime },{\partial }_{ * }^{\prime }}\right) \) are two chain complexes of free abelian groups. By our hypothesis there exist isomorphisms \( {\gamma }_{n} : {\mathrm{H}}_{n}\left( {C}_{ * }\right) \rightarrow {\mathrm{H... | Yes |
Let \( \\left( {X, A}\\right) \) and \( \\left( {Y, B}\\right) \) be topological spaces and let \( G \) be an abelian group. Then \( {\\mathrm{H}}_{n}\\left( {X, A}\\right) \) and \( {\\mathrm{H}}_{n}\\left( {Y, B}\\right) \) are isomorphic for all \( n \\in {\\mathbb{N}}_{0} \n\\[ \n\\text{isomorphic for all}n \\in {\... | This corollary follows immediately from Proposition 73.18 and the observation  | Yes |
Corollary 73.20. Let \( f : \left( {X, A}\right) \rightarrow \left( {Y, B}\right) \) be a map between pairs of topological spaces and let \( G \) be an abelian group. Then the induced map \[ {f}_{ * } : {\mathrm{H}}_{n}\left( {X, A}\right) \rightarrow {\mathrm{H}}_{n}\left( {Y, B}\right) \text{is} \] \[ \Rightarrow \;{... | Proof. Let \( f : \left( {X, A}\right) \rightarrow \left( {Y, B}\right) \) be a map between pairs of topological spaces and let \( G \) be an abelian group. If \( {f}_{ * } : {\mathrm{H}}_{n}\left( {X, A}\right) \rightarrow {\mathrm{H}}_{n}\left( {Y, B}\right) \) is an isomorphism for all \( n \in {\mathbb{N}}_{0} \), ... | Yes |
Theorem 74.1. (Excision Theorem for Cohomology Groups) Let \( X \) be a topological space and let \( Z \subset A \subset X \) be subsets such that the closure of \( Z \) is contained in the interior of \( A \) . Then the inclusion\n\n\[ i : \left( {X \smallsetminus Z, A \smallsetminus Z}\right) \rightarrow \left( {X, A... | Proof. In the original Excision Theorem 43.19 we showed that the inclusion induced map\n\n\[ {i}_{ * } : {\mathrm{H}}_{n}\left( {X \smallsetminus Z, A \smallsetminus Z}\right) \rightarrow {\mathrm{H}}_{n}\left( {X, A}\right) \]\n\nis an isomorphism for every \( n \in {\mathbb{N}}_{0} \) . But then it follows immediatel... | Yes |
Proposition 74.2. Let \( G \) be an abelian group. Let \( X \) be a topological space and let \( A \subset X \) be a non-empty subset. We denote by \( p : \left( {X, A}\right) \rightarrow \left( {X/A, A/A}\right) \) the obvious projection map. We consider the following map:\n\n\[ \n{\widetilde{\mathrm{H}}}^{n}\left( {X... | Proof. At this stage we have all the tools to rewrite the proof of Proposition 43.22. In particular we can use the above Excision Theorem 74.1 for Cohomology Groups to replace the Excision Theorem 43.19. We leave it to the reader to verify that nothing goes wrong. | No |
Lemma 74.3. (*) Let \( G \) be an abelian group and let \( n \in {\mathbb{N}}_{0} \) . The maps \[ X \mapsto {\mathrm{H}}_{\mathrm{{CW}}}^{n}\left( X\right) \;\text{ and }\;\left( {f : X \rightarrow Y}\right) \mapsto \left( {{f}^{ * } : {\mathrm{H}}_{\mathrm{{CW}}}^{n}\left( {Y;G}\right) \rightarrow {\mathrm{H}}_{\math... | Proof. This lemma follows immediately from Lemmas 48.3 and 73.8 (2). | Yes |
Proposition 74.4. For every CW-complex \( X \), any abelian group \( G \) and any \( n \in {\mathbb{N}}_{0} \) there exists a natural isomorphism\n\n\[ \n{\mathrm{H}}^{n}\left( {X;G}\right) \overset{ \cong }{ \rightarrow }{\mathrm{H}}_{\mathrm{{CW}}}^{n}\left( {X;G}\right) \n\] | First Proof of Proposition 74.4. Let \( X \) be a CW-complex. Recall that on page 1290 we introduced the intermediate cellular chain complex that is given by\n\n\[ \n{\mathrm{C}}_{n}^{\text{int }}\left( X\right) \mathrel{\text{:=}} \ker \left( {{\mathrm{C}}_{n}\left( {X}^{n}\right) \overset{\partial }{ \rightarrow }{\m... | Yes |
Lemma 73.13 Proposition 74.4 since the coboundary map is zero |  is homotopy equivalent to  | No |
Corollary 74.5. Let \( X \) be a topological space. Suppose we are in one of the following two situations:\n\n(1) \( X \) is homotopy equivalent to a retract of a finite CW-complex,\n\n(2) \( X \) is a compact topological manifold.\n\nThen for every \( n \in {\mathbb{N}}_{0} \) the cohomology group \( {\mathrm{H}}^{n}\... | Proof. Statement (2) is an immediate consequence of Statement (1) and Theorem 85.12, We will prove Statement (1) in Exercise 74.1 | No |
Lemma 74.6. Let \( {C}_{ * } \) be a chain complex.\n\n(1) Let \( G \) be an abelian group. The map\n\n\[ \langle \text{,} \rangle : {\mathrm{H}}^{n}\left( {C;G}\right) \times {\mathrm{H}}_{n}\left( C\right) \rightarrow G \]\n\n\[ \left( {\left\lbrack \varphi \right\rbrack ,\left\lbrack c\right\rbrack }\right) \mapsto ... | Proof. Let \( {C}_{ * } \) be a chain complex.\n\n(1) Let \( G \) be an abelian group. Let \( \varphi \in \operatorname{Hom}\left( {{C}_{n}, G}\right) \) be a cocycle and let \( c \in {C}_{n} \) be a cycle. Furthermore let \( \alpha \in \operatorname{Hom}\left( {{C}_{n - 1}, G}\right) \) and let \( d \in {\mathrm{C}}_{... | Yes |
Lemma 74.8. Let \( A \) and \( B \) be two free abelian groups of rank \( k \), let \( {a}_{1},\ldots ,{a}_{k} \in A \) and let \( {b}_{1},\ldots ,{b}_{k} \in B \) . Furthermore let \( \langle \rangle ,\rangle : A \times B \rightarrow \mathbb{Z} \) be a bilinear pairing. If\n\n\[ \det \left( {\left\langle {a}_{i},{b}_{... | Proof. We denote by \( S \) the free abelian group generated by the set \( \left\{ {{a}_{1},\ldots ,{a}_{k}}\right\} \) and we denote by \( T \) the free abelian group generated by the set \( \left\{ {{b}_{1},\ldots ,{b}_{k}}\right\} \) . We denote by \( i : S \rightarrow A \) and by \( j : T \rightarrow B \) the obvio... | Yes |
Lemma 74.11. We denote by \( \alpha ,\beta \in {\mathrm{H}}^{1}\left( {{S}^{1} \times {S}^{1};\mathbb{Z}}\right) \) the basis given by Lemma 74.9. A basis for \( {\mathrm{H}}^{1}\left( {{\sum }_{g};\mathbb{Z}}\right) \cong {\mathbb{Z}}^{2g} \) is given by \( {p}_{i}^{ * }\left( \alpha \right) ,{p}_{i}^{ * }\left( \beta... | Proof. As we will see, the proof of the lemma is very similar to the proof of Lemma 74.9. For \( i = 1,\ldots, g \) we write \( {\varphi }_{i} = {p}_{i}^{ * }\left( \alpha \right) \) and \( {\psi }_{i} \mathrel{\text{:=}} {p}_{i}^{ * }\left( \beta \right) \) . We consider the singular 1-cycles \( {x}_{1},\ldots ,{x}_{g... | Yes |
Lemma 74.13. Let \( \left( {X,{A}_{1},{A}_{2}}\right) \) be a triad of topological spaces. The following statements are equivalent:\n\n(1) the triad is excisive,\n\n(2) for every \( n \in {\mathbb{N}}_{0} \) the inclusion induced map\n\n\[ \n{\mathrm{H}}_{n}^{\left\{ {A}_{1},{A}_{2}\right\} }\left( {{A}_{1} \cup {A}_{2... | Proof. Let \( \left( {X,{A}_{1},{A}_{2}}\right) \) be a triad of topological spaces. The equivalence of (1) and (2) is an immediate consequence of Corollary 42.3 and Proposition 49.2\n\nNow we turn to the proof that (2) and (3) are equivalent. We write \( Y = {A}_{1} \cup {A}_{2} \) . We consider the following sequence... | Yes |
Theorem 74.14. (Mayer-Vietoris Theorem) Given any excisive triad \( \left( {X, A, B}\right) \) with \( X = A \cup B \) there exists a natural \( {}^{1085} \) long exact sequence\n\n\[ \cdots \rightarrow {\mathrm{H}}_{n}\left( {A \cap B}\right) \xrightarrow[]{\frac{{i}_{A \cap B} \oplus - {i}_{A \cap B}}{\varnothing } \... | Proof. Let \( \left( {X, A, B}\right) \) be a triad with \( X = A \cup B \) . In the proof of the original Mayer-Vietoris Theorem 46.5 we showed that the sequence\n\n\[ 0 \rightarrow {\mathrm{C}}_{ * }\left( {A \cap B}\right) \xrightarrow[]{\;\frac{{i}_{A \cap B} \oplus - {i}_{A \cap B}}{\hslash }\;}{\mathrm{C}}_{ * }\... | No |
Theorem 74.15. (Mayer-Vietoris Theorem for Cohomology Groups) Let \( \\left( {X, A, B}\\right) \) be an excisive triad with \( X = A \\cup B \) and let \( G \) be an abelian group. Then there exists a natural\n\n\[ \n\\cdots \\rightarrow {\\mathrm{H}}^{n}\\left( {X;G}\\right) \\xrightarrow[]{{i}^{ * } \\oplus {i}^{ * }... | Proof. Let \( \\left( {X, A, B}\\right) \) be an excisive triad with \( X = A \\cup B \) and let \( G \) be an abelian group. We start out with the following claim.\n\nClaim. The inclusion map\n\n\[ \n{\\mathrm{C}}_{ * }^{\\{ A, B\\} }\\left( X\\right) \\rightarrow {\\mathrm{C}}_{ * }\\left( X\\right) \n\] \n\ninduces ... | No |
Lemma 74.16. (*) Let \( X \) and \( Y \) be topological spaces and let \( x \in X \) and \( y \in Y \) be good points \( {}^{1087} \) We use these points to define the wedge \( X \vee Y \) . Then \( \left( {X \vee Y, X, Y}\right) \) is an excisive triad. | Proof (*). We denote by \( z \) the point in \( X \vee Y \) that is given by identifying \( x \) with \( y \) . For each \( k \in {\mathbb{N}}_{0} \) we consider the following diagram\n\n\[ \begin{array}{l} 0 \rightarrow {\mathrm{H}}_{k}\left( {Y,\{ y\} ;\mathbb{Z}}\right) \rightarrow {\mathrm{H}}_{k}\left( {X,\{ x\} ;... | Yes |
Proposition 74.17. (*) 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 \( {a}_{k} \in {A}_{k} \) . Let \( G \) be an abelian group. Given \( j \in K \) we denote by\n\n\[ \n{i}_{j} : {A}_{j} \rightarrow \mathop{\bigvee }\lim... | Proof (*).\n\n(1) First we consider the case we are given just two topological spaces. By Lemma 74.16 we can use the Mayer-Vietoris Theorem 74.15 for Cohomology Groups to prove, completely analogously to Proposition 47.8, the desired statement.\n\n(2) The case that we are dealing with finitely many topological spaces f... | No |
Lemma 74.18. (*) Let \( k \in \mathbb{Z} \) and let \( G \) be an abelian group. Given any topological space \( X \) we have a natural isomorphism\n\n\[ \n{\sum }_{X} : {\widetilde{\mathrm{H}}}^{k}\left( {\sum \left( X\right) ;G}\right) \overset{ \cong }{ \rightarrow }{\widetilde{\mathrm{H}}}^{k + 1}\left( {X;G}\right)... | Proof \( \left( *\right) \) . The construction of the isomorphism \( {\sum }_{X} \) is basically the same as the construction of the corresponding isomorphism in cohomology. In the Lemma 46.8 we just need to replace the Mayer-Vietoris Theorem 46.5 by the Mayer-Vietoris Theorem 74.15 for Cohomology Groups. We leave it t... | No |
Lemma 75.1. For any abelian group \( G \) the contravariant functor \( \operatorname{Hom}\left( {-, G}\right) \) is left-exact. | Proof (*). Let \( G \) be an abelian group and let\n\n\[ A\;\overset{\alpha }{ \rightarrow }\;B\;\overset{\beta }{ \rightarrow }\;C\; \rightarrow \;0 \]\n\nbe an exact sequence of abelian groups. We need to show that the sequence\n\n\[ 0 \rightarrow \operatorname{Hom}\left( {C, G}\right) \overset{{\beta }^{ * }}{ \righ... | Yes |
Corollary 75.3. For every choice of abelian groups \( G \) and \( H \) we have \( {\operatorname{Ext}}_{n}\left( {H, G}\right) = 0 \) for \( n \geq 2 \) . | Proof. As we pointed out in the proof of Lemma 57.16 any abelian group \( H \) admits a free resolution of length 2 :\n\n\[ 0 \rightarrow {F}_{1}\overset{{f}_{1}}{ \rightarrow }{F}_{0}\overset{{f}_{0}}{ \rightarrow }H \rightarrow 0. \]\n\nThe corollary now follows immediately from Proposition 75.2. | No |
Lemma 75.4. Let \( G \) and \( H \) be abelian groups, then there exists a natura 1091 isomorphism\n\n\[{\operatorname{Ext}}_{0}\left( {H, G}\right) \overset{ \cong }{ \rightarrow }\operatorname{Hom}\left( {H, G}\right)\] | Proof (*). The proof of Lemma 75.4 is, not surprisingly, rather similar to the proof of Lemma 57.15. Let \( G \) and \( H \) be abelian groups. Let\n\n\[ \cdots \rightarrow {F}_{2}\overset{{f}_{2}}{ \rightarrow }{F}_{1}\overset{{f}_{1}}{ \rightarrow }{F}_{0}\overset{{f}_{0}}{ \rightarrow }H \rightarrow 0 \]\n\nbe the c... | Yes |
Lemma 75.5. Let \( G, H \) and \( {G}_{1},\ldots ,{G}_{k},{\mathrm{H}}_{1},\ldots ,{\mathrm{H}}_{k} \) be abelian groups. The following statements hold:\n\n(1) There exists a natural isomorphism\n\n\[ \operatorname{Ext}\left( {{\bigoplus }_{i = 1}^{k}{\mathrm{H}}_{i}, G}\right) \cong {\bigoplus }_{i = 1}^{k}\operatorna... | Proof. The first three statements are proved almost the same way as we proved Lemma 57.17. We leave it to the reader to fill in the details. | No |
Lemma 75.7. Let \( G \) and \( H \) be abelian groups. If \( G \) is injective, then \( \operatorname{Ext}\left( {H, G}\right) = 0 \) . | Proof. Let \( H \) be an abelian group. As we pointed out in the proof of Lemma 57.16 the abelian group \( H \) admits a free resolution of length 2 :\n\n\[ 0 \rightarrow {F}_{1}\overset{{\varphi }_{1}}{ \rightarrow }{F}_{0}\overset{{\varphi }_{0}}{ \rightarrow }H \rightarrow 0. \]\n\nBy Proposition 75.2 the group \( \... | Yes |
Proposition 75.10. The group \( \operatorname{Ext}\left( {\mathbb{Q},\mathbb{Z}}\right) \) is isomorphic to the additive group \( \left( {\mathbb{R}, + }\right) \) . | Proof. This proposition is proved in [Wie69]. Alternatively the proposition is almost proved in [Bre97, Lemma 14.8]. We also refer to [Boar10] for the calculation of Ext \( \left( {\mathbb{Q},\mathbb{Z}}\right) \) . | No |
Proposition 75.11. Let \( G \) be an abelian group. If \( G \) is not a finitely generated group, then \( \operatorname{Hom}\left( {G,\mathbb{Z}}\right) \) is uncountable or \( \operatorname{Ext}\left( {G,\mathbb{Z}}\right) \) is uncountable. | Proof. We will barely make use of this proposition, thus we refer to Hat02, Proposition 3.F.12] instead for a proof. | No |
Theorem 75.13. (Universal Coefficient Theorem for Cohomology Groups) Let \( \left( {X, A}\right) \) be a pair of topological spaces and let \( G \) be an abelian group. Then for each \( n \in {\mathbb{N}}_{0} \) there exists a short exact sequence\n\n\[ 0 \rightarrow \operatorname{Ext}\left( {{\mathrm{H}}_{n - 1}\left(... | Proof of Theorem 7.5.13 Assuming Theorem 7.5.12. As the reader will have noticed there is really not much to say, except that we recall that we pointed out on page 1120 that the chain groups \( {\mathrm{C}}_{n}\left( {X, A}\right), n \in {\mathbb{N}}_{0} \) are free abelian groups. | No |
(1) Let \( {C}_{ * } \) be a chain complex. The following two statements are equivalent:\n\n(a) \( {\mathrm{H}}_{k}\left( {C}_{ * }\right) = 0 \) for all \( k \in {\mathbb{N}}_{0} \).\n\n(b) \( {\mathrm{H}}^{k}\left( {{C}_{ * };\mathbb{Z}}\right) = 0 \) for all \( k \in {\mathbb{N}}_{0} \). | (1) We start out with the following claim.\n\nClaim. Let \( A \) be an abelian group. If \( A \) is non-trivial, then \( \operatorname{Hom}\left( {A,\mathbb{Z}}\right) \) or \( \operatorname{Ext}\left( {A,\mathbb{Z}}\right) \) is non-trivial.\n\nLet \( A \) be a non-trivial abelian group. If \( A \) is finitely generat... | Yes |
Proposition 75.15. Let \( M \) be a connected \( n \) -dimensional topological manifold. Let \( G \) be an abelian group.\n\n(1) Let \( x \in M \smallsetminus \partial M \) . For \( i \neq n \) we have \( {\mathrm{H}}_{i}\left( {M, M\smallsetminus \{ x\} ;G}\right) = 0 \) . Furthermore the map\n\n\[ \text{ev:}{\mathrm{... | Proof. Let \( M \) be a compact connected \( n \) -dimensional topological manifold and let \( G \) be an abelian group.\n\n(1) This statement follows immediately from Lemma 86.4 and the Universal Coefficient Theorem 75.13. | Yes |
Lemma 75.16. Let \( M \) be a compact oriented connected non-empty \( n \) -dimensional topological manifold. Then we have \( {\mathrm{H}}^{n}\left( {M,\partial M;\mathbb{Z}}\right) \cong \mathbb{Z} \). Furthermore there exists a unique generator \( {\left\lbrack M\right\rbrack }^{ * } \in {\mathrm{H}}^{n}\left( {M,\pa... | Proof (*). Let \( M \) be a compact oriented connected \( n \) -dimensional non-empty topological manifold.\n\nIt follows from the Universal Coefficient Theorem 75.13 for Cohomology Groups together with Proposition 87.22 that the map\n\n\[ \n{\mathrm{H}}^{n}\left( {M,\partial M;\mathbb{Z}}\right) \overset{\text{ ev }}{... | Yes |
Lemma 75.17. Let \( f : M \rightarrow N \) be a map between closed oriented connected non-empty \( n \) -dimensional topological manifolds. Then\n\n\[ \n{f}^{ * }\left( {\left\lbrack N\right\rbrack }^{ * }\right) = \deg \left( f\right) \cdot {\left\lbrack M\right\rbrack }^{ * }.\n\] | Proof. Let \( f : M \rightarrow N \) be a map between closed, oriented connected \( n \) -dimensional nonempty topological manifolds. It follows from Lemma 75.16 that any class in \( \varphi = {\mathrm{H}}^{n}\left( {M;\mathbb{Z}}\right) \)\n\n\( {}^{1097} \) Here \( \left\lbrack M\right\rbrack \in {\mathrm{H}}_{n}\lef... | Yes |
Proposition 75.18. Let \( X \) be a path-connected topological space, let \( {x}_{0} \in X \) and let \( G \) be an abelian group. Then the Hurewicz homomorphism \( {\pi }_{1}\left( {X,{x}_{0}}\right) \rightarrow {\mathrm{H}}_{1}\left( X\right) \) from page 1314 and the evaluation homomorphism ev: \( {\mathrm{H}}^{1}\l... | Proof. As we pointed out on page 1867 and Proposition 52.2 (4), both maps are natural. So it remains to show that both maps are isomorphisms.\n\nWe insert one extra map into the above sequence of maps to see that the maps are indeed isomorphisms | No |
Corollary 75.20. Let \( \left( {X, A}\right) \) be a pair of topological space and let \( k \in {\mathbb{N}}_{0} \) such that \( {\mathrm{H}}_{k}\left( {X, A;\mathbb{Z}}\right) \) and \( {\mathrm{H}}_{k - 1}\left( {X, A;\mathbb{Z}}\right) \) are finitely generated. (By Proposition 85.13 (4) we know if \( X \) and \( A ... | Proof. First note that Statement (2) is an immediate consequence of Statement (1) and the discussion on page 1422. So let turn to the proof of Statement (1). To simplify the notation we only consider the case \( A = \varnothing \) . Let \( X \) be a topological space and let \( k \in {\mathbb{N}}_{0} \) such that \( {\... | Yes |
Lemma 57.3 (3) | \[ \begin{aligned} {\mathrm{H}}_{k}\left( {X;\mathbb{F}}\right) & \cong {\mathrm{H}}_{k}\left( {X;\mathbb{Z}}\right) \otimes \mathbb{F} \oplus \operatorname{Tor}\left( {{\mathrm{H}}_{k - 1}\left( {X;\mathbb{Z}}\right) ,\mathbb{F}}\right) \cong {\mathbb{F}}^{s} \oplus {\mathbb{F}}^{\gamma } \oplus {\mathbb{F}}^{\alpha }... | Yes |
Theorem 75.21. Let \( \\mathbb{F} \) be a field and let \( \\left( {{V}_{ * },{v}_{ * }}\\right) \) be a chain complex over \( \\mathbb{F} \) . Then the map \[ {\\operatorname{ev}}_{\\mathbb{F}} : {\\mathrm{H}}^{k}\\left( {V;\\mathbb{F}}\\right) \\mapsto {\\operatorname{Hom}}_{\\mathbb{F}}\\left( {{\\mathrm{H}}_{k}\\le... | A proof of that theorem is provided by any self-respecting book on homological algebra, see e.g. DaK01, Corollary 2.31]. | No |
Lemma 75.22. Let \( C \) be an abelian group and let \( \mathbb{F} \) be a field. Then the map 1099\n\n\( {\Phi }_{C} : {\operatorname{Hom}}_{\mathbb{Z}}\left( {C,\mathbb{F}}\right) \rightarrow {\operatorname{Hom}}_{\mathbb{F}}\left( {C \otimes \mathbb{F},\mathbb{F}}\right) \)\n\n\[ \n\left( {\varphi : C \rightarrow \m... | Proof. It is straightforward to verify that an inverse map to \( {\Phi }_{C} \) is given by the following homomorphism of \( \mathbb{F} \) -vector spaces:\n\n\[ \n{\Psi }_{C} : {\operatorname{Hom}}_{\mathbb{F}}\left( {C \otimes \mathbb{F},\mathbb{F}}\right) \rightarrow {\operatorname{Hom}}_{\mathbb{Z}}\left( {C,\mathbb... | Yes |
Corollary 75.23. Let \( {C}_{ * } \) be a chain complex and let \( \mathbb{F} \) be a field. The maps \[ {\Phi }_{{\mathrm{C}}_{k}} : {\operatorname{Hom}}_{\mathbb{Z}}\left( {{C}_{k},\mathbb{F}}\right) \rightarrow {\operatorname{Hom}}_{\mathbb{F}}\left( {{\mathrm{C}}_{k} \otimes \mathbb{F},\mathbb{F}}\right) \;\text{ w... | Proof. By Lemma 75.22 the maps \( {\Phi }_{{\mathrm{C}}_{k}}, k \in {\mathbb{N}}_{0} \), are isomorphisms of the cochain vector spaces. It is straightforward to see that these maps are also cochain maps. | Yes |
Lemma 75.25. Let \( G \) be an abelian group.\n\n(1) For any abelian group \( H \) the map\n\n\[ \mu : \operatorname{Hom}\left( {H,\mathbb{Z}}\right) \otimes G \rightarrow \operatorname{Hom}\left( {H, G}\right) \]\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{m}{\varphi }_{i} \otimes {g}_{i} \mapsto \left( \begin{aligned} H &... | Proof. The proof is elementary and is left to the reader. | No |
Lemma 75.27. Let \( G \) be an abelian group. If \( H \) is a free abelian group of finite rank, then the map\n\n\[ \n\mu : \operatorname{Hom}\left( {H,\mathbb{Z}}\right) \otimes G \rightarrow \operatorname{Hom}\left( {H, G}\right)\n\]\n\nis an isomorphism. | Proof. Let \( {v}_{1},\ldots ,{v}_{n} \) be a basis for the finitely generated free abelian group \( H \) . We denote by \( {\varphi }_{1},\ldots ,{\varphi }_{n} \) the dual basis of \( \operatorname{Hom}\left( {H,\mathbb{Z}}\right) \) . In other words, these are the homomorphisms that are uniquely determined by \( {\v... | No |
Lemma 75.28. Let \( \left( {{\mathrm{C}}_{ * },{\partial }_{ * }}\right) \) be a chain complex of free abelian groups. If all cohomology groups \( {\mathrm{H}}^{n}\left( {{\mathrm{C}}_{ * };\mathbb{Z}}\right) \) are finitely generated, then all homology groups \( {\mathrm{H}}_{n}\left( {C}_{ * }\right) \) are also fini... | Proof of Lemma 75.28. Let \( \left( {{\mathrm{C}}_{ * },{\partial }_{ * }}\right) \) be a chain complex of free abelian groups such that all cohomology groups \( {\mathrm{H}}^{n}\left( {{\mathrm{C}}_{ * };\mathbb{Z}}\right) \) are finitely generated. By the Universal Coefficient Theorem 75.12 for Cohomology Groups ther... | Yes |
Theorem 75.29. (Universal Coefficient Theorem) Let \( X \) be a topological space and let \( G \) be an abelian group. If all homology groups \( {\mathrm{H}}_{n}\left( X\right) \) of \( X \) are finitely generated (e.g. \( X \) could be a compact topological manifold or \( X \) could be a CW-complex that has only finit... | Proof. This theorem follows immediately from applying the Universal Coefficient Theorem 75.26 to the singular chain complex of \( X \) . | No |
Lemma 75.30. Let \( X \) be a topological space, let \( G \) be an abelian group and let \( f : \mathbb{Z} \rightarrow G \) be a homomorphism. The following diagram commutes: | Proof. The statement follows immediately from explicitly writing down the homomorphisms. | No |
Lemma 75.31. (*) We denote by \( \varphi : \mathbb{Z} \rightarrow \mathbb{Q} \) the obvious inclusion. Let \( n \in {\mathbb{N}}_{0} \) .\n\n(1) Let \( X \) be a topological space such that all homology groups are finitely generated. We suppose that \( {\mathrm{H}}^{n}\left( {X;\mathbb{Z}}\right) \) is a finitely gener... | Proof \( \left( *\right) \) .\n\n(1) This statement follows easily from the Universal Coefficient Theorem 75.29 together with Lemma 57.17 (4) and Lemma 75.30. | No |
Proposition 76.1. Let \( \left( {I, \leq }\right) \) be a directed set and let \( \left( {{\left\{ {X}_{i}\right\} }_{i \in I},{\left\{ {f}_{ij}\right\} }_{i \leq j}}\right) \) be a direct system over one of the following categories:\n\n(1) the category of abelian groups,\n\n(2) the category of \( R \) -modules, where ... | Proof. This explicit construction of the direct limit was already given on page 733. The proof of the proposition is a not overly exciting exercise in going through all the definitions. We leave this task to the reader. | No |
Lemma 76.2. Let \( \\left( {I, \\leq }\\right) \) be a directed set and let \( \\left( {{\\left\\{ {X}_{i}\\right\\} }_{i \\in I},{\\left\\{ {f}_{ij} : {X}_{i} \\rightarrow {X}_{j}\\right\\} }_{i \\leq j}}\\right) \) be a direct system over I in one of the four categories that we had considered in Proposition 76.1. The... | (1) The first statement follows immediately from the explicit description of the direct limit given in Proposition 76.1. | No |
Lemma 76.3. Let \( \\left( {I, \\leq }\\right) \) be a directed set and let \( \\left( {{\\left\\{ {X}_{i}\\right\\} }_{i \\in I},{\\left\\{ {f}_{ij} : {X}_{i} \\rightarrow {X}_{j}\\right\\} }_{i \\leq j}}\\right) \) be a direct system over I in one of the four categories that we had considered in Proposition 76.1. Let... | The first part of the lemma is just an immediate consequence of the definition of the direct limit over the directed set \( J \) . As we will see, the second statement is a fairly straightforward consequence of the next lemma. But it is also a good exercise to prove the above second statement \ | No |
Lemma 76.4. Let \( {\left\{ {f}_{i} : {K}_{i} \rightarrow {L}_{\varphi \left( i\right) }\right\} }_{i \in I} \) be a homomorphism between two direct systems \( \left( {{\left\{ {K}_{i}\right\} }_{i \in I},{\left\{ {\chi }_{{i}_{1}{i}_{2}}\right\} }_{{i}_{1} \leq {i}_{2}}}\right) \) and \( \left( {{\left\{ {L}_{j}\right... | Sketch of Proof. Let \( j \in J \) . Since \( \varphi : I \rightarrow J \) is cofinal we can pick \( i \in I \) with \( f\left( i\right) \geq j \) .\n\nWe define\n\n\[ {g}_{j} : {L}_{j} \rightarrow \mathop{\lim }\limits_{ \rightarrow }{K}_{i} \]\n\n\[ x \mapsto \left\lbrack {{f}_{i}^{-1}\left( {{\lambda }_{j, f\left( i... | No |
Proposition 76.5. Let \( \left( {{\left\{ {X}_{i}\right\} }_{i \in \mathbb{N}},{\left\{ {g}_{i} : {X}_{i + 1} \rightarrow {X}_{i}\right\} }_{i \in \mathbb{N}}}\right) \) be an inverse system over one of the following categories:\n\n(1) the category of abelian groups,\n\n(2) the category of \( R \) -modules, where \( R ... | Proof. This proposition is implicitly proved on page 750. | No |
Lemma 76.6. Let \( {f}_{n} : {X}_{n} \rightarrow {X}_{n + 1} \) be a sequence of morphisms in any category. Suppose there exists an \( N \in \mathbb{N} \) such that all \( {f}_{n} \) for \( n \geq N \) are isomorphisms. Then \( \lim {X}_{n} \) exists and the natural map \( \mathop{\lim }\limits_{ \leftarrow }{X}_{n} \r... | Proof. We leave it to the reader to prove this elementary lemma. | No |
Lemma 47.6. Let \( \\left( {{\\left\\{ {\\mathrm{C}}_{i}\\right\\} }_{i \\in \\mathbb{N}},{\\left\\{ {g}_{i} : {\\mathrm{C}}_{i} \\rightarrow {\\mathrm{C}}_{i + 1}\\right\\} }_{i \\in \\mathbb{N}}}\\right) \) be a direct system of chain complexes. Then for each \( n \\in {\\mathbb{N}}_{0} \) we obtain an induced direct... | \[ \\mathop{\\lim }\\limits_{ \\rightarrow }{\\mathrm{H}}_{n}\\left( {\\mathrm{C}}_{i}\\right) \\overset{ \\cong }{ \\rightarrow }{\\mathrm{H}}_{n}\\left( {\\mathop{\\lim }\\limits_{ \\rightarrow }{\\mathrm{C}}_{i}}\\right) \] | Yes |
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