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Lemma 34.2. Let \( X \) be a path-connected topological space on which a group \( G \) acts continuously and discretely. We consider the corresponding covering \( p : X \rightarrow X/G \) . Then \( p : X \rightarrow X/G \) is regular and the map\n\n\[ \Phi : G \rightarrow D\left( {p : X \rightarrow X/G}\right) \]\n\n\[... | Proof. It is straightforward to see that the maps \( x \mapsto {gx} \) are indeed deck transformations. It follows almost immediately that the covering \( p : X \rightarrow X/G \) is regular. Also it follows easily from the axioms of a group action that \( \Phi \) is a homomorphism of groups. Furthermore it is clear th... | Yes |
Lemma 34.3. Any 2-fold covering of path-connected topological spaces is regular and the deck transformation group contains precisely two elements. | Proof. Let \( p : X \rightarrow B \) be a 2-fold covering of path-connected topological spaces. By Lemma 16.4 we know that\n\n\[ d : X \rightarrow X \]\n\n\[ x \mapsto \text{the unique other element of}{p}^{-1}\left( {p\left( x\right) }\right) \]\n\nis a deck transformation. Evidently it is regular. It follows easily f... | Yes |
Lemma 34.4. Let \( G \) be a group and let \( H \subset G \) be a subgroup.\n\n(1) The subgroup \( H \) of \( G \) is normal if and only if \( N\left( H\right) = G \) . | Proof. Both statements are easily verified. Alternatively see [Isa94, p. 26] for the elementary proof. | No |
Proposition 34.5. Let \( p : \left( {X,{x}_{0}}\right) \rightarrow \left( {B,{b}_{0}}\right) \) be a covering of pointed topological spaces. We suppose that \( X \) and \( B \) are both path-connected and locally path-connected. We write \( \pi \mathrel{\text{:=}} {\pi }_{1}\left( {B,{b}_{0}}\right) \) and \( \Gamma \m... | Proof. Let \( p : \left( {X,{x}_{0}}\right) \rightarrow \left( {B,{b}_{0}}\right) \) be a covering of pointed path-connected topological spaces. We write \( \pi \mathrel{\text{:=}} {\pi }_{1}\left( {B,{b}_{0}}\right) ,\Gamma \mathrel{\text{:=}} {p}_{ * }\left( {{\pi }_{1}\left( {X,{x}_{0}}\right) }\right) \) and \( D \... | Yes |
This shows that \( \Psi \circ \Phi \) is the identity. | Now let \( g \in N\left( \Gamma \right) /\Gamma \) . We pick a representative loop \( s \) in \( {b}_{0} \) and we denote by \( \widetilde{s} \) the lift of \( s \) to the starting point \( {x}_{0} \) . We denote by \( {x}_{1} \) the endpoint of \( \widetilde{s} \) . Then\n\n\[ \Phi \left( {\Psi \left( g\right) }\right... | Yes |
Proposition 34.6. Let \( \\left( {X,{x}_{0}}\\right) \) be a pointed topological space that is path-connected, locally path-connected and semi-locally simply connected. Furthermore let \( \\gamma : {\\pi }_{1}\\left( {X,{x}_{0}}\\right) \\rightarrow G \) be an epimorphism onto a group \( G \) . We denote by \( p : \\le... | Proof. First note that \( {p}_{ * }\\left( {{\\pi }_{1}\\left( {\\widetilde{X},{\\widetilde{x}}_{0}}\\right) }\\right) = \\ker \\left( \\gamma \\right) \) is a normal subgroup of \( {\\pi }_{1}\\left( {X,{x}_{0}}\\right) \) . Therefore the normalizer of \( \\Gamma \\mathrel{\\text{:=}} {p}_{ * }\\left( {{\\pi }_{1}\\le... | Yes |
Corollary 34.7. Let \( p : \left( {X,{x}_{0}}\right) \rightarrow \left( {B,{b}_{0}}\right) \) be a covering of pointed topological spaces that are path-connected and locally path-connected. Then the following statements are equivalent:\n\n(1) The covering is regular.\n\n(2) The deck transformation group \( D\left( {p :... | Proof. Let \( p : \left( {X,{x}_{0}}\right) \rightarrow \left( {B,{b}_{0}}\right) \) be a covering of pointed path-connected topological spaces. We write \( \pi \mathrel{\text{:=}} {\pi }_{1}\left( {B,{b}_{0}}\right) ,\Gamma \mathrel{\text{:=}} {p}_{ * }\left( {{\pi }_{1}\left( {X,{x}_{0}}\right) }\right) \) and \( D =... | Yes |
Proposition 34.8. Let \( p : \left( {X,{x}_{0}}\right) \rightarrow \left( {B,{b}_{0}}\right) \) be a covering of pointed topological spaces that are path-connected, locally path-connected and semi-locally simply connected. If p is a finite-degree covering, then there exists a covering \( q : \left( {Y,{y}_{0}}\right) \... | Proof. Let \( p : \left( {X,{x}_{0}}\right) \rightarrow \left( {B,{b}_{0}}\right) \) be a finite-degree covering of pointed topological spaces that are path-connected, that are locally path-connected and that are semi-locally simply connected. In the following we write \( G \mathrel{\text{:=}} {\pi }_{1}\left( {X,{x}_{... | Yes |
Proposition 34.9. Let \( g \in {\mathbb{N}}_{ \geq 2} \) . We denote by \( {\sum }_{g} \) the surface of genus \( g \) . We pick a base point \( {x}_{0} \in {\sum }_{g} \) . There exists a monomorphism \( {\pi }_{1}\left( {{\sum }_{g},{x}_{0}}\right) \rightarrow \mathrm{{SL}}\left( {2,\mathbb{R}}\right) \) . | Proof. Throughout the proof we will make use of the notion of a Möbius structure, of a Möbius manifold and of a Möbius map between Möbius manifolds. We refer to pages 887 and 897 for the definitions, in case the reader needs a refresher.\n\nNow let \( g \geq 2 \) . As in the proof of Proposition 33.2 we can and will ma... | Yes |
Proposition 34.10. Let \( \sum \) be a closed connected 2-dimensional smooth manifold. We pick a base point \( {x}_{0} \in \sum \). The following two statements hold:\n\n(1) The group \( {\pi }_{1}\left( {\sum ,{x}_{0}}\right) \) is linear over \( \mathbb{R} \).\n\n(2) The group \( {\pi }_{1}\left( {\sum ,{x}_{0}}\righ... | Proof. Let \( \sum \) be a closed connected 2-dimensional smooth manifold. By Proposition 17.3 we know that there exists a finite covering \( p : \left( {\widetilde{\sum },{\widetilde{x}}_{0}}\right) \rightarrow \left( {\sum ,{x}_{0}}\right) \) such that \( \widetilde{\sum } \) is a closed orientable connected 2-dimens... | Yes |
Lemma 35.1. Let \( \mathbb{K} \) be a field. The Zariski open subsets form a topology on \( {\mathbb{K}}^{n} \) . | Proof. Let \( \mathbb{K} \) be a field.\n\n(1) The vanishing set of the empty set, i.e. \( V\left( \varnothing \right) \) is all of \( {\mathbb{K}}^{n} \), hence the empty set \( \varnothing = {\mathbb{K}}^{n} \smallsetminus V\left( \varnothing \right) \) is open.\n\n(2) The vanishing set of the constant polynomial \( ... | Yes |
Lemma 36.1. Let \( n \in {\mathbb{N}}_{0} \) .\n\n(1) We can view \( {\mathbb{{RP}}}^{n} \) as a CW-complex with the following properties:\n\n(a) the CW-structure has exactly one cell in the dimensions \( 0,1,\ldots, n \) and no other cells,\n\n(b) for each \( k \leq n \) the \( k \) -skeleton of \( {\mathbb{{RP}}}^{n}... | Proof.\n\n(1) We prove the statement by induction on \( n \) . For \( n = 0 \) the topological space \( {\mathbb{{RP}}}^{0} \) consists of a single point which means that it is a CW-complex with a single 0-cell. Now suppose that we have equipped \( {\mathbb{{RP}}}^{n - 1} \) with a CW-structure with the properties stat... | Yes |
Lemma 36.2. Let \( C = {\left( {c}_{ij}\right) }_{i, j \in \mathbb{N}} \) be an infinite diagonal matrix. Then the map \( {}^{584} \n\n\[ \n\gamma : {\mathbb{R}}^{\infty } \rightarrow {\mathbb{R}}^{\infty } \n\] \n\n\[ \nx \mapsto C \cdot x \n\] \n\nis continuous with respect to the topology \( \mathcal{T} \) . | Proof. Let \( U \in \mathcal{T} \) . We need to show that \( {\gamma }^{-1}\left( U\right) \) lies in \( \mathcal{T} \) . So given any \( n \in \mathbb{N} \) we need to show that \( {\gamma }^{-1}\left( U\right) \cap {\mathbb{R}}^{n} \) is an open subset of \( {\mathbb{R}}^{n} \) . So let \( n \in \mathbb{N} \) . Given... | Yes |
Lemma 36.4. The infinite-dimensional sphere \( {S}^{\infty } \) is contractible. | Proof. We want to show that \( {S}^{\infty } \) is contractible. We recall that by Lemma 18.13 it suffices to show that there exists a homotopy \( H \) from \( {\mathrm{{id}}}_{{S}^{\infty }} \) to a constant map. Our goal is to find such \( H \) . We set\n\n\[ \nX \mathrel{\text{:=}} \left\{ {\left( {0,{p}_{2},{p}_{3}... | No |
(1) For each \( n \in \mathbb{N} \) the obvious map \( {\mathbb{{RP}}}^{n} \rightarrow {\mathbb{{RP}}}^{\infty } \) is an embedding. Furthermore these maps define a homeomorphism \( \underline{\lim }{\mathbb{{RP}}}^{n} \triangleq {\mathbb{{RP}}}^{\infty } \) . | (1) We leave it as a very mildly interesting exercise to the reader to show that for each \( n \in \mathbb{N} \) the obvious map \( {\mathbb{{RP}}}^{n} \rightarrow {\mathbb{{RP}}}^{\infty } \) is an embedding. Next we note that we have the following equalities:\n\n\[ \mathop{\lim }\limits_{ \rightarrow }{\mathbb{{RP}}}... | No |
(1) For each \( n \in \mathbb{N} \) the obvious map \( {\mathbb{{CP}}}^{n} \rightarrow {\mathbb{{CP}}}^{\infty } \) is an embedding. Furthermore the maps \( {\mathbb{{CP}}}^{n} \rightarrow {\mathbb{{CP}}}^{\infty } \) define a homeomorphism \( \underline{\lim }{\mathbb{{CP}}}^{n}\overset{ \cong }{ \rightarrow }{\mathbb... | The proof of this lemma is basically verbatim the same as the proof of the corresponding parts of Lemma 36.5. | No |
Lemma 36.8. Let \( X \) be a CW-complex. The following three topologies on \( X \times \left\lbrack {0,1}\right\rbrack \) agree:\n\n(1) The product topology on the product \( X \times \left\lbrack {0,1}\right\rbrack \) .\n\n(2) The topology on \( X \times \left\lbrack {0,1}\right\rbrack \) viewed as the direct limit \(... | Proof. Let \( X \) be a CW-complex. First note that it follows immediately from the fact that \( X = \underline{\lim }{X}_{k} \) together with Lemma 3.8 (1c) that any subset that is open with respect to (1) is also open with respect to (2). Furthermore it follows from Lemma 36.7 (1) together with Lemma 3.8 (2b) that an... | Yes |
Corollary 36.9. Let \( X \) be a CW-complex and let \( \Phi : X \times \left\lbrack {0,1}\right\rbrack \rightarrow Y \) be a map to a topological space \( Y \) . The following statements are equivalent:\n\n(1) The map \( f \) is continuous.\n\n(2) For every \( k \in {\mathbb{N}}_{0} \) the restriction of \( f \) to \( ... | Proof. The corollary follows immediately from the equivalence of the three topologies in Lemma 36.8. | Yes |
Proposition 36.11. A CW-complex is regionally compact if and only if it is locally finite. | Proof. We will not make use of Proposition 36.11, except that in the negative sense, that now we know that many CW-complexes we are interested in are unfortunately not regionally compact. The proposition is proved in Geo08, Proposition 10.1.8]. | No |
Lemma 36.12. Let \( f : X \rightarrow Y \) be a map between CW-complexes. Suppose the following three conditions are satisfied:\n\n(1) \( f \) is a bijection,\n\n(2) \( f \) is continuous,\n\n(3) the images of the cells of \( X \) are precisely the cells of \( Y \) .\n\nThen \( f : X \rightarrow Y \) is a homeomorphism... | Proof (*). We denote by \( g = {f}^{-1} : Y \rightarrow X \) the inverse of \( f \) . It remains to show that \( g \) is continuous. Let \( {\left\{ {\Phi }_{i} : {\bar{B}}^{{m}_{i}} \rightarrow X\right\} }_{i \in I} \) be the characteristic maps of the CW-complex \( X \) and let \( {\left\{ {\Phi }_{j} : {\bar{B}}^{{n... | Yes |
Proposition 36.13. Every CW-complex is paracompact. | Proof. As always, when it comes to CW-complexes, the proof is slightly delicate. Since we alternatively to [Hat2, Proposition 1.20], [Lee00, Theorem 5.22], [LW69, Theorem II.4.2] and [Miy52], for a proof. | No |
Corollary 36.15. A CW-complex is compact if and only if it is finite. | Proof. In Proposition 36.10 (3) we showed that every finite CW-complex is compact. Now we want to prove the converse. Thus let \( X \) be a compact CW-complex. It follows from Theorem 36.14 that \( X \) is contained in a finite subcomplex \( Y \) of \( X \). But from \( X \subset Y \subset X \) it follows that \( X = Y... | Yes |
The maps\n\n\[ \text{CW-complex}X \mapsto \left| X\right| \]\n\n\[ \left( {f : X \rightarrow Y}\right) \mapsto \left( {f : \left| X\right| \rightarrow \left| Y\right| }\right) \]\n\ndefine a covariant functor from the category \( \mathcal{C}\mathcal{W} \) of CW-complexes to the category of topological spaces. The funct... | The statements are basically obvious. The fact that we just found a continuous map which is not cellular gives us that the functor \( \mathcal{C}\mathcal{W} \rightarrow \mathcal{T} \) op is not fully faithful. | Yes |
Lemma 36.18. Let \( X \) be a CW-complex and let \( A \) be a subcomplex of \( X \) .\n\n(1) The subset \( A \) admits a CW-structure such that for each \( n \in {\mathbb{N}}_{0} \) we have the equality \( \left| {A}^{n}\right| = A \cap \left| {X}^{n}\right| \) and such that the characteristic maps of \( A \) correspon... | Proof. We start out with the proof of Statement (1). We set \( {Y}^{-1} = \left| {Y}^{-1}\right| \mathrel{\text{:=}} \varnothing \) . For \( n = 0,1,2,3,\ldots \) we iteratively do the following:\n\n(1) Let \( {\left\{ {\varphi }_{i} : {S}^{n - 1} \rightarrow \left| {Y}^{n - 1}\right| \right\} }_{i \in I} \) be the att... | Yes |
Lemma 36.19. (*) Let \( f : A \rightarrow X \) be a cellular embedding between two CW-complexes.\n\nThe following statements hold:\n\n(1) The image \( f\left( A\right) \) is a subcomplex of \( A \) and the map \( f : A \rightarrow f\left( A\right) \) is a cellular isomorphism.\n\n(2) The map \( f : A \rightarrow X \) i... | Proof (*). It follows immediately from the above definition of a subcomplex and of a cellular embedding that the image of a cellular embedding \( f : A \rightarrow X \) is a subcomplex. Next we consider the map \( {f}^{-1} : f\left( A\right) \rightarrow A \) . We deduce from Lemma 36.7 together with Proposition 2.43 (3... | Yes |
Lemma 36.20. Let \( X \) be a CW-complex and let \( {\left\{ {X}_{i}\right\} }_{i \in I} \) be a family of subcomplexes such that \( \mathop{\bigcup }\limits_{{i \in I}}{X}_{i} = X \) . The following three statements hold:\n\n(1) A subset \( U \subset X \) is open if and only if each \( {X}_{i} \cap U \) is an open sub... | Proof.\n\n(1) This statement is an easy consequence of Lemma 36.7 (2).\n\n(2) The \ | No |
Lemma 36.21. Let \( X \) be a CW-complex.\n\n(1) For any \( n \in {\mathbb{N}}_{0} \) the \( n \) -skeleton \( {X}^{n} \) is a subcomplex of \( X \) . | Proof.\n\n(1) This statement is clear. | No |
Proposition 36.22. Let \( X \) be a CW-complex and let \( Y = {Y}_{0} \subset {Y}_{1} \subset {Y}_{2} \subset \ldots \) be a sequence of subcomplexes such that the following two conditions are satisfied: (1) we have \( X = \mathop{\bigcup }\limits_{{i \in \mathbb{N}}}{Y}_{i} \), (2) each \( {Y}_{i} \) is a deformation ... | Proof \( \left( *\right) \) . By hypothesis there exists for each \( k \in {\mathbb{N}}_{0} \) a deformation retraction \[ R\left( k\right) : {Y}_{k} \times \left\lbrack {0,1}\right\rbrack \rightarrow {Y}_{k - 1} \] We denote by \( {r}_{k} \mathrel{\text{:=}} R{\left( k\right) }_{1} : {Y}_{k} \rightarrow {Y}_{k - 1} \)... | Yes |
Proposition 36.23. Let \( X \) and \( Y \) be two CW-complexes. We use the notation from the above definition.\n\n(1) The above data turns \( Z \) into a CW-complex.\n\n(2) The characteristic maps \( {\Theta }_{\left( i, j\right) } \) define a continuous bijection \( \Theta : Z \rightarrow X \times Y \).\n\n(3) Suppose... | Proof (*). Let \( X \) and \( Y \) be two CW-complexes.\n\n(1) This statement follows basically immediately from the definitions.\n\n(2) It follows from Lemma 36.7 (4) that the map \( \Theta : Z \rightarrow X \times Y \), which is defined via the characteristic maps \( \overline{{\Theta }_{\left( i, j\right) }} \), is ... | Yes |
Proposition 36.25. Let \( X \) and \( Y \) be two CW-complexes. If at least one of the two CW-complexes has only countably many cells, then the map \( \Theta : Z \rightarrow X \times Y \) from Proposition 36.23 (2) is a homeomorphism, in particular \( X \times Y \) has a natural CW-structure. | Proof. This statement is proved in [LW69, Theorem II.5.2]. If one of \( X \) or \( Y \) is a finite CW-complex, then the statement is also proved in Hat02, Theorem A.6]. If both CW-complexes have countably many cells, then the statement is also proved in FrPi90a, Proposition 2.2.3] or [Miln56b, Lemma 2.1]. | Yes |
Lemma 36.27. (*) Let \( \Theta : A \rightarrow B \) be a continuous bijection between two topological spaces which has the property that a subset \( K \subset A \) is compact if and only if \( \Theta \left( K\right) \subset B \) is compact. If \( A \) is Hausdorff, then for any \( {a}_{0} \in A \) the map \( {\Theta }_... | Proof \( \left( *\right) \) . We write \( {b}_{0} \mathrel{\text{:=}} f\left( {a}_{0}\right) \) . The proof of the lemma naturally breaks up into showing that \( {\Theta }_{ * } : {\pi }_{1}\left( {A,{a}_{0}}\right) \rightarrow {\pi }_{1}\left( {B,{b}_{0}}\right) \) is an epimorphism and that it is a monomorphism.\n\nL... | No |
Theorem 36.28. There exists a full subcategory \( C \) of the category of topological spaces with the following properties:\n\n(1) Every regionally compact Hausdorff space is an object of \( C \) .\n\n(2) Every CW-complex is an object of \( C \) .\n\n(3) Every direct system in \( \mathcal{C} \) has a direct limit and e... | We will outline the proof of Theorem 36.28 by giving an explicit example of a convenient category. We will need the following definitions.\n\nDefinition.\n\n(1) A subset \( A \) of a topological space \( X \) is called compactly closed if for every map \( \varphi : K \rightarrow X \), where \( K \) is a compact Hausdor... | No |
Proposition 36.29. Every CW-complex is a CGWH-space. | Proof. Let \( X \) be a CW-complex.\n\n(1) Let \( A \) be a compactly closed subset of \( X \) . It follows easily from Lemma 36.7 (3) that \( A \) is in fact a closed subset of \( X \) .\n\n(2) By Proposition 36.10 (2) we know that \( X \) is Hausdorff. It follows from Lemma 2.17 (2) that \( X \) is in fact weakly Hau... | Yes |
Theorem 36.30. The category CGWH of CGWH-spaces has the following properties:\n\n\( \\left( 0\\right) \) The category CGWH is a full subcategory of the category of topological spaces.\n\n(1) Every regionally compact Hausdorff space is an object of CGWH.\n\n(2) Every CW-complex is an object of CGWH.\n\n(3) Every direct ... | Sketch of Proof. Most of the statements below were first proved by Michael McCord McCor69, building on work by Norman Steenrod Stee67. Arguably the best account of the proof is given in [Stri]. In the following we show that our theorem, as formulated, can be deduced from the statements in [Stri].\n\n(0) This statement ... | Yes |
(1) Let \( f : X \rightarrow Y \) be a bijection between two topological spaces. If \( f \) is proper, i.e. if the preimage of every compact subset of \( Y \) is a compact subset of \( X \), and if \( X \) and \( Y \) are CGWH-spaces, then \( f \) is actually a homeomorphism. | Proof. This proposition is proved in [Stri, Propositions 2.20, 2.40 and 3.17]. | No |
(1) Let \( {\left\{ {X}_{i}\right\} }_{i \in I} \) be a family of CW-complexes. The disjoint union \( \mathop{\bigsqcup }\limits_{{i \in I}}{X}_{i} \) admits a natural CW-structure such that each \( {X}_{i} \) is a subcomplex. | Proof \( \left( *\right) \) .\n\n(1) The first statement is basically obvious. | No |
Lemma 36.33. Let \( X \) be a CW-complex and let \( n \in \mathbb{N} \) . Let \( {\left\{ {\varphi }_{i} : {\bar{B}}^{n + 1} \rightarrow {X}^{n}\right\} }_{i \in I} \) be the characteristic maps of the \( \left( {n + 1}\right) \) -cells of \( X \) . For each \( i \in I \) let \( {\bar{B}}_{i}^{n + 1} \) be a copy of \(... | Proof \( \left( *\right) \) .\n\n(1) First note that it is basically clear that the given map \( f \) is a bijection. Furthermore it follows immediately from Lemma 18.25 that \( f \) is continuous. We use Lemma 36.32 to view both sides of Lemma 36.33 as CW-complexes. The map \( f \) gives a bijection between the cells ... | Yes |
Lemma 36.34. Let \( {f}_{i} : {X}_{i} \rightarrow {X}_{i + 1}, i \in \mathbb{N} \) be a sequence of cellular inclusion maps between CW-complexes. Then the direct limit \( {}^{602}\mathop{\lim }\limits_{ \rightarrow }{X}_{i} = \mathop{\bigcup }\limits_{{i \in I}}{X}_{i} \) admits a CW-structure such that each \( {X}_{i}... | Proof. We leave it to the reader to write down the, notationally messy but mathematically elementary, proof. | No |
Corollary 36.35. Let \( X \) be a CW-complex\n\n(1) The cone\n\n\[ \operatorname{Cone}\left( X\right) = \left( {X \times \left\lbrack {0,1}\right\rbrack }\right) / \sim \;\text{ where }\left( {x,0}\right) \sim \left( {y,0}\right) \text{ for every }x, y \in X \] | Proof. All four statements can be proved fairly easily using Proposition 36.23 (3b) and Lemma 36.32 We only point out that the CW-structure on the mapping torus Tor \( \left( {X, f}\right) \) is obtained in a 2-stage process, first we equip the mapping cylinder \( \operatorname{Cyl}\left( {f : X \rightarrow X}\right) \... | No |
Proposition 36.36. There exists a finite 3-dimensional CW-complex that admits an open subset that does not admit a CW-structure. | Proof. This statement is proved in Cau92, Exemple 2]. | No |
Proposition 36.37. Let \( p : \widetilde{X} \rightarrow X \) be a covering of a connected CW-complex \( X \) . We can equip \( \widetilde{X} \) with a natural structure of a CW-complex such that the following statements hold:\n\n(1) The map \( p : \widetilde{X} \rightarrow X \) is a cellular map.\n\n(2) For any \( k \i... | Proof. Let \( p : \widetilde{X} \rightarrow X \) be a covering of a connected CW-complex \( X \) of degree \( n \mathrel{\text{:=}} \left\lbrack {\widetilde{X} : X}\right\rbrack \) . Let \( \psi : {\bar{B}}^{k} \rightarrow X \) be a characteristic map of a \( k \) -cell of \( X \) . We write \( x = \psi \left( 0\right)... | No |
Theorem 37.1. (Seifert-van Kampen Theorem for CW-complexes) Let \( X \) be a CW-complex and let \( X = A \cup B \) be a decomposition of \( X \) in two subcomplexes such that \( A \cap B \) is path-connected. Let \( {x}_{0} \in A \cap B \) . Then\n\n\[ \n{\pi }_{1}\left( {X,{x}_{0}}\right) \cong {\pi }_{1}\left( {A,{x}... | Proof of Theorem \( {37.1}\left( *\right) \) . \( {}^{608} \) Let \( X \) be a CW-complex and let \( X = A \cup B \) be a decomposition of \( X \) in two subcomplexes such that \( A \cap B \) is path-connected. Note that \( A \cap B \) is also a subcomplex. Let \( {x}_{0} \in A \cap B \) .\n\nWe use the notation of Pro... | Yes |
Theorem 37.2. (HNN-Seifert-van Kampen Theorem for CW-complexes) Suppose we are given a path-connected CW-complex \( X \) and two path-connected disjoint sub-complexes \( A \) and \( B \) . Let \( f : A \rightarrow B \) be a cellular isomorphism. We pick a base point \( {x}_{0} \in A \) . Let \( \gamma : \left\lbrack {0... | Sketch of A proof of Theorem 37.2. Let \( X \) be a path-connected CW-complex \( X \) and let \( f : A \rightarrow B \) be a cellular isomorphism between two path-connected disjoint subcom-plexes of \( X \) . Let \( {x}_{0} \in A \) and let \( \gamma : \left\lbrack {0,1}\right\rbrack \rightarrow X \) be an embedding wi... | No |
Proposition 37.4. Let \( p : \widetilde{X} \rightarrow X \) be a finite covering of a finite CW-complex \( X \) . We equip \( \widetilde{X} \) with the structure of a CW-complex given by Proposition 36.37. Then\n\n\[ \chi \left( \widetilde{X}\right) = \left\lbrack {\widetilde{X} : X}\right\rbrack \cdot \chi \left( X\ri... | Proof. Let \( d \mathrel{\text{:=}} \left\lbrack {\widetilde{X} : X}\right\rbrack \) . As we mentioned in Proposition 36.37 (2), it follows immediately from the construction of the CW-structure on \( \widetilde{X} \) that for each \( k \) we have\n\n\[ \# k\text{-cells of}\widetilde{X} = \left\lbrack {\widetilde{X} : X... | Yes |
Lemma 37.5. Let \( X \) be a finite CW-complex and let \( A \) be a subcomplex. We equip the quotient \( X/A \) with the CW-structure given by Lemma 36.32 (3). Then \[ \chi \left( {X/A}\right) = \chi \left( X\right) - \chi \left( A\right) + 1. \] | Proof. It follows immediately from the construction of the CW-structure on \( X/A \) given in Lemma 36.32 (3) that for each \( k \in {\mathbb{N}}_{0} \) we have the following equalities: \[ \# k\text{-cells of}X/A = \left\{ \begin{array}{ll} \# k\text{-cells of}X\text{not in}A, & \text{ if }k > 0, \\ \# k\text{-cells o... | Yes |
Proposition 37.6. Let \( X \) be a non-empty finite connected 1-dimensional CW-complex. Then \( \chi \left( X\right) = 0 \) if and only if \( X \) is contractible. | Proof. This statement is an immediate consequence of Proposition 20.5 and the fact that we can view any 1-dimensional CW-complex as a topological graph of the same Euler characteristic. | No |
Theorem 37.9. Let \( X \) be a non-empty connected 1-dimensional CW-complex. Furthermore let \( {x}_{0} \in X \) . Then the following holds:\n\n(1) The group \( {\pi }_{1}\left( {X,{x}_{0}}\right) \) is isomorphic to a free group. In fact, if \( T \) is a spanning tree for \( X \), then the cardinality of the generatin... | Proof. Let \( X \) be a non-empty connected 1-dimensional CW-complex. By Proposition 37.7 we know that \( X \) admits a spanning tree \( T \) . We denote by \( E \) the set of 1-cells of \( X \) that are not contained in \( T \) . Let \( {x}_{0} \) be a point in \( T \) . We denote by \( p : X \rightarrow X/T \) the pr... | Yes |
Every subgroup of a free group (of any rank) is again a free group. | Proof. Let \( F = \langle S\rangle \) be a free group on a set \( S \) and let \( G \) be a subgroup of \( F \) .\n\n(1) By attaching one 1-cell for each \( s \in S \) to a one-point set \( \{ P\} \) we obtain a 1-dimensional CW-complex \( X \) with one 0-cell \( P \) and with one 1-cell for each \( s \in S \) . By The... | Yes |
Corollary 37.12. Let \( \left\langle {{g}_{1},\ldots ,{g}_{k} \mid {r}_{1},\ldots ,{r}_{m}}\right\rangle \) be a finite presentation for a group \( \pi \) . If \( Y \) is the associated 2-dimensional CW-complex, then \( {\pi }_{1}\left( Y\right) \cong \pi \) . | Proof. Let \( \pi = \left\langle {{g}_{1},\ldots ,{g}_{k} \mid {r}_{1},\ldots ,{r}_{m}}\right\rangle \) be a finitely presented group. We use the notation introduced in the above construction of the CW-complex associated to \( \pi \) . We denote by \( X \) the wedge of the \( k \) circles and we denote by \( {x}_{0} \)... | No |
Proposition 37.13. Let \( X \) be a connected CW-complex and let \( {x}_{0} \) be a point in the 0- skeleton of \( X \) .\n\n(1) If \( X \) has finitely many cells in each dimension, then the following statements hold:\n\n(a) The inclusion induced map \( {}^{617}{\pi }_{1}\left( {{X}^{1},{x}_{0}}\right) \rightarrow {\p... | Proof.\n\n(1) We first suppose that \( X \) is a finite CW-complex. Then the two statements follow immediately from iteratively applying Proposition 37.11 finitely many times. If \( X \) is an infinite CW-complex, but with finitely many cells in each dimension, then one can reduce the two statements to the finite case ... | No |
Proposition 37.15. Every finite-index subgroup of a finitely presented group is finitely presented. | Proof. Let \( \pi = \left\langle {{g}_{1},\ldots ,{g}_{k} \mid {r}_{1},\ldots ,{r}_{m}}\right\rangle \) be a finitely presented group. Let \( Y \) be the finite 2-complex associated to this presentation that we constructed on page 995 . By Corollary 37.12 we know that \( {\pi }_{1}\left( Y\right) \cong \pi \) . Now let... | Yes |
Corollary 38.2. Let \( X \) be connected CW-complex, let \( P \) be a point in the 0 -skeleton of \( X \) and let \( Q \) be some other point on \( Y \) .\n\n(1) Given any path \( \gamma : \left\lbrack {0,1}\right\rbrack \rightarrow X \) with \( \gamma \left( 0\right) = P \) and \( \gamma \left( 1\right) = Q \) there e... | Proof. Let \( X \) be connected CW-complex, let \( P \) be a point in the 0 -skeleton of \( X \) and let \( Q \) be some other point on \( Y \) .\n\n(1) Let \( \gamma : \left\lbrack {0,1}\right\rbrack \rightarrow X \) be a path with \( \gamma \left( 0\right) = P \) and \( \gamma \left( 1\right) = Q \) . We write \( A \... | Yes |
Lemma 38.4. If \( X \) is a CW-complex and if \( A \) is a subcomplex of \( X \), then there exists a deformation retraction from \( X \times \left\lbrack {0,1}\right\rbrack \) to the subcomplex \( \left( {X\times \{ 0\} }\right) \cup \left( {A \times \left\lbrack {0,1}\right\rbrack }\right) \) . | Proof. Let \( X \) be a CW-complex and let \( A \) be a subcomplex. For each \( n \in {\mathbb{N}}_{0} \) we set\n\n\[ \n{Z}_{n} \mathrel{\text{:=}} X \times \{ 0\} \cup \left( {{X}^{n - 1} \cup A}\right) \times \left\lbrack {0,1}\right\rbrack .\n\]\n\nNote that each \( {Z}_{n} \) is a subcomplex of the product CW-comp... | Yes |
Proposition 38.5. Let \( f : A \rightarrow B \) be a cellular map between CW-complexes that has the following property:\n\n(*) For each \( n \) -cell \( \sigma \) of \( A \) we have either \( f\left( \sigma \right) \subset {B}^{n - 1} \) or \( {\left. f\right| }_{\sigma } \) is a homeomorphism. We consider the correspo... | Proof. First note that by Lemma 36.19 we can and will view \( A \sqcup B \) as a subcomplex of \( Y \) . In the following let \( R : \operatorname{Cyl}\left( {f : A \rightarrow B}\right) \times \left\lbrack {0,1}\right\rbrack \rightarrow B \) be the natural deformation retraction given by Lemma 24.8. Furthermore note t... | Yes |
Proposition 38.6. Let \( \\left( {Y,{y}_{0}}\\right) \) be a path-connected topological space. If \( {\\pi }_{1}\\left( {Y,{y}_{0}}\\right) \) is abelian, then the map\n\n\[ \n\\left\\langle {\\left( {{S}^{1}, * }\\right) ,\\left( {Y,{y}_{0}}\\right) }\\right\\rangle \\rightarrow \\left\\lbrack {{S}^{1}, Y}\\right\\rbr... | Proof. It follows from the fact that \( Y \) is path-connected together with Proposition 18.33 (1) that the map is a surjection. Furthermore it follows from the hypothesis that \( {\\pi }_{1}\\left( \\overline{Y,{y}_{0}}\\right) \) is abelian together with Proposition 18.33 (2) that the map is an injection. | Yes |
Corollary 38.8. (1) Let \( \\left( {X,{x}_{0}}\\right) \) be a path-connected pointed topological space and let \( n \\in \\mathbb{N} \) . If \( n = 1 \) , then we assume that \( {\\pi }_{1}\\left( {X,{x}_{0}}\\right) \) is abelian and if \( n \\geq 2 \) we assume that \( {\\pi }_{1}\\left( {X,{x}_{0}}\\right) \) is tr... | Proof of Corollary 38.8. (1) (a) The case \( n = 1 \) is precisely the content of Proposition 38.6. (b) Now let \( n \\in {\\mathbb{N}}_{ \\geq 2} \) . We can and will equip \( {S}^{n} \) with a CW-structure such that \( {x}_{0} \) is a 0-cell. The desired statement now follows from Proposition 38.7 (2). | Yes |
Lemma 38.11. Let \( X \) be a CW-complex, let \( A \) be a subset of \( X \) and let \( Y \) be a topological space. Finally let \( {W}_{0} \subset {W}_{1} \subset {W}_{2} \subset \ldots \) be a sequence of subcomplexes of \( X \) such that for each \( k \in {\mathbb{N}}_{0} \) there exists an \( l \in {\mathbb{N}}_{0}... | \[ g : X \rightarrow Y \] \[ x \mapsto {f}_{k}\left( x\right) \;\text{where}k \in {\mathbb{N}}_{0}\text{such that}x \in {W}_{k}\text{.} \] For \( k = 0,1,2,\ldots \) we write \( {s}_{k} = \mathop{\sum }\limits_{{j = 1}}^{k}\frac{1}{{2}^{j}} \) . Then the map \[ H : X \times \left\lbrack {0,1}\right\rbrack \rightarrow Y... | Yes |
Theorem 38.14. (Cellular Approximation Theorem for Pairs) Suppose we are given a map \( g : \left( {X, A}\right) \rightarrow \left( {Y, B}\right) \) between pairs of CW-complexes. Let \( Z \) be a subcom-plex of \( A \) such that the restriction of \( g \) to \( Z \) is a cellular map. Then \( g \) is homotopic rel \( ... | Proof assuming the Cellular Approximation Theorem [38.13] The desired homotopy is constructed as follows:\n\n(1) By the Cellular Approximation Theorem 38.13 applied to \( {\left. g\right| }_{A} : A \rightarrow B \) there exists a homotopy \( F \) rel \( Z \) from \( {\left. g\right| }_{A} : A \rightarrow B \) to a cell... | Yes |
Proposition 38.15. Let \( V \) be a CW-complex and \( W \) be a topological space. Furthermore let \( \psi : {S}^{n - 1} \rightarrow {V}^{n - 1} \) be a map to the \( \left( {n - 1}\right) \) -skeleton of \( V \) and let \( \varphi : {S}^{k - 1} \rightarrow W \) be another map. Finally let\n\n\[ h : V{ \cup }_{\psi }{\... | The statement of Proposition 38.15 is fairly obvious, if there exists a point \( P \in {B}^{k} \) that does not get hit by \( h\left( {B}^{n}\right) \) . Indeed, if such a point \( P \) exists, then we can use the deformation retraction \( {\bar{B}}^{n} \smallsetminus \{ p\} \rightarrow {S}^{n - 1} \) to push \( h \) o... | No |
Let \( Z \) be a finite \( n \) -dimensional CW-complex.\n\n(1) Let \( g : Z \rightarrow Y \) be a map to a \( k \) -dimensional CW-complex \( Y \) with \( g\left( {Z}^{n - 1}\right) \subset {Y}^{n - 1} \) . Let \( B \) be a subcomplex of \( Z \) with \( g\left( B\right) \subset {Y}^{n} \) . If \( n < k \), then \( g \... | Proof. Let \( Z \) be a finite \( n \) -dimensional CW-complex. By Proposition 36.10 (3) the CW-complex \( Z \) is compact. It follows from Theorem 36.14 that the image of the compact set \( Z \) in a CW-complex is contained in a finite subcomplex. Therefore we can assume in both (1) and (2) that \( Y \) is in fact a f... | Yes |
Lemma 38.17. Let \( W \) be a topological space and let \( \varphi : {S}^{k - 1} \rightarrow W \) be map. Furthermore let \( g : {\left\lbrack 0,1\right\rbrack }^{n} \rightarrow Z \mathrel{\text{:=}} W{ \cup }_{\varphi }{\bar{B}}^{k} \) be a map. Then there exists a (possibly empty) polyhedron \( K \subset {\left\lbrac... | Proof. Let \( W \) be a topological space and let \( \varphi : {S}^{k - 1} \rightarrow W \) be map. Furthermore let \( g : {\left\lbrack 0,1\right\rbrack }^{n} \rightarrow Z \mathrel{\text{:=}} W{ \cup }_{\varphi }{\bar{B}}^{k} \) be a map. We write \( {D}_{1} \mathrel{\text{:=}} {\bar{B}}_{\frac{1}{4}}\left( 0\right) ... | Yes |
Lemma 38.18. Let \( W \) be a topological space and let \( \varphi : {S}^{k - 1} \rightarrow W \) be the attaching map of a \( k \) -cell. Furthermore let \( h : {\bar{B}}^{n} \rightarrow Z \mathrel{\text{:=}} W{ \cup }_{\varphi }{\bar{B}}^{k} \) be a map with \( h\left( {S}^{n - 1}\right) \subset W \) . If \( n < k \)... | Proof. Let \( W \) be a topological space and let \( \varphi : {S}^{k - 1} \rightarrow W \) be the attaching map of a \( k \) -cell. Furthermore let \( h : {\bar{B}}^{n} \rightarrow Z \mathrel{\text{:=}} W{ \cup }_{\varphi }{\bar{B}}^{k} \) be a map with \( h\left( {S}^{n - 1}\right) \subset W \) . We suppose that \( n... | Yes |
Theorem 39.1. Let \( X \) be a CW-complex. Given any subcomplex \( A \) the corresponding inclusion map \( i : A \rightarrow X \) is a closed cofibration. | Proof. The statement of the Homotopy Extension Theorem 38.1 is precisely that the inclusion \( A \rightarrow X \) is a cofibration. By Lemma 36.18 (2) we know that the cofibration is in fact a closed cofibration. | Yes |
Lemma 39.3. If \( i : A \rightarrow X \) is a map between two topological spaces, then the following two statements are equivalent:\n\n(1) The map \( i : A \rightarrow X \) is a cofibration.\n\n(2) The map \( {}^{643} \)\n\n\[ s : \overline{\operatorname{Cyl}}\left( i\right) \rightarrow X \times \left\lbrack {0,1}\righ... | Proof (*). First we prove the \ | No |
Lemma 39.5. There exists a retraction\n\n\\[ \n\\rho : \\left\\lbrack {0,1}\\right\\rbrack \\times \\left\\lbrack {0,1}\\right\\rbrack \\rightarrow \\left( {\\{ 0\\} \\times \\left\\lbrack {0,1}\\right\\rbrack }\\right) \\cup \\left( {\\left\\lbrack {0,1}\\right\\rbrack \\times \\{ 0\\} }\\right) \n\\]\n\nthat has the ... | Proof. In Figure 664 we show how such a retraction \\( \\rho \\) can be constructed.\n\n\n\nFIGURE 664. Illustration for the proof of Lemma 39.5 | Yes |
Proposition 39.7. Let \( f : X \rightarrow Y \) be a map between topological spaces. We denote by \( i : X \rightarrow \operatorname{Cyl}\left( f\right) \) the natural inclusion. The natural retraction \( r : \operatorname{Cyl}\left( f\right) \rightarrow Y \) is a homotopy equivalence and the following diagram commutes... | Proof. We showed in Lemma 24.8 (2a) that the natural retraction \( r : \operatorname{Cyl}\left( f\right) \rightarrow Y \) is a homotopy equivalence. It is clear that the given diagram commutes. Thus we are done by Proposition 39.6 (1). That was quick. | Yes |
Lemma 39.9. Let \( X \) be a topological space and let \( A \subset X \) be a non-empty subset. We denote by \( i : A \rightarrow X \) the inclusion map. If \( i \) is a closed cofibration, then the following three statements hold:\n\n(1) The map \n\n\[ \varphi : \operatorname{Cone}\left( {i : A \rightarrow X}\right) \... | Proof. Let \( X \) be a topological space and let \( A \subset X \) be a closed subset such that the inclusion \( i : A \rightarrow X \) is a cofibration.\n\n(1) Similar to the proof of Lemma 39.8 (1) we consider the following two maps:\n\n(a) First of all we consider the obvious inclusion map \( j : X \times \{ 0\} \r... | Yes |
Corollary 39.10. Let \( X \) be a CW-complex and let \( A \subset X \) be a subcomplex.\n\n(1) If \( A \) is contractible, then the projection map \( X \rightarrow X/A \) is a homotopy equivalence.\n\n(2) If \( {a}_{0} \in A \) is a point such that \( \left\{ {a}_{0}\right\} \) is a deformation retract of \( A \), then... | Proof. By Theorem 39.1 we know that the inclusion \( A \rightarrow X \) is a closed cofibration. The two statements now follow immediately from the corresponding statements in Lemma 39.9. | No |
Proposition 39.11. Let \( X \) be a 0-connected CW-complex.\n\n(1) There exists a contractible 1-dimensional subcomplex \( T \) which contains all 0-cells \( {}^{649} \)\n\n(2) Given any \( T \) as in (1) the quotient \( X/T \) is a CW-complex with precisely one 0-cell. Furthermore the projection \( X \rightarrow X/T \... | Proof. Let \( X \) be a 0 -connected CW-complex.\n\n(1) In Proposition 37.7 (1) we showed that the 1-skeleton \( {X}^{1} \) admits a subcomplex \( T \) that contains all 0-cells of \( X \) and that admits a deformation retraction to \( {x}_{0} \).\n\n(2) By Lemma 36.32 the quotient \( X/T \) admits a CW-structure with ... | Yes |
(1) If \( f : A \rightarrow X \) and \( g : B \rightarrow Y \) are two (closed) cofibrations, then the product map \( f \times g : A \times B \rightarrow X \times Y \) is also a (closed) cofibration. | It is clear that Statement (2) is an immediate consequence of Statement (1). Thus it remains to prove Statement (1). Now let \( f : A \rightarrow X \) and \( g : B \rightarrow Y \) be two cofibrations,\n\nBy Lemma 39.8 we can assume that \( f : A \rightarrow X \) and \( g : B \rightarrow Y \) are both inclusions. Note ... | Yes |
Theorem 39.13. Let \( X \) be a topological space and let \( A \) be a closed subset. The following two statements are equivalent:\n\n(1) The inclusion \( i : A \rightarrow X \) is a cofibration.\n\n(2) There exists a map \( \phi : X \rightarrow \left\lbrack {0,1}\right\rbrack \) such that \( A = {\phi }^{-1}\left( {\{... | Proof (*). First we deal with the \ | No |
Proposition 39.14. Let \( A \subset X \) and \( B \subset Y \) be subsets of topological spaces. If the inclusion maps \( i : A \rightarrow X \) and \( j : B \rightarrow Y \) are closed cofibrations, then the inclusion map\n\n\[ \left( {A \times Y}\right) \cup \left( {X \times B}\right) \rightarrow X \times Y \]\n\nis ... | Proof. Since \( i \) and \( j \) are closed cofibrations we can pick maps \( \phi : X \rightarrow \left\lbrack {0,1}\right\rbrack ,\psi : Y \rightarrow \left\lbrack {0,1}\right\rbrack \) and homotopies \( G : X \times \left\lbrack {0,1}\right\rbrack \rightarrow X \) and \( H : Y \times \left\lbrack {0,1}\right\rbrack \... | Yes |
Let \( X \) be a topological space and let \( A \subset X \) be a subset. If the inclusion \( i : A \rightarrow X \) is a closed cofibration, then the inclusion \( \left( {X\times \{ 0,1\} }\right) \cup \left( {A \times \left\lbrack {0,1}\right\rbrack }\right) \rightarrow X \times \left\lbrack {0,1}\right\rbrack \) is ... | On page 1024 we saw that the inclusion map \( \{ 0,1\} \rightarrow \left\lbrack {0,1}\right\rbrack \) is a cofibration. It follows from this observation together with Proposition 39.14 that the given inclusion map is a closed cofibration. | No |
Proposition 39.19. Let \( \varphi : Y \rightarrow {Y}^{\prime } \) be a homotopy equivalence between two topological spaces. Furthermore let \( A \) be a topological space and let \( f : A \rightarrow Y \) and \( {f}^{\prime } : A \rightarrow {Y}^{\prime } \) be two maps such that \( \varphi \circ f \) is homotopic to ... | Proof. Let \( F : A \times \left\lbrack {0,1}\right\rbrack \rightarrow {Y}^{\prime } \) be a homotopy between \( \varphi \circ f \) and \( {f}^{\prime } \). We consider the following commutative diagram\n\n\n\nThe ... | Yes |
Lemma 40.2. Let \( \left( {X,{x}_{0}}\right) \) be a pointed topological space and let \( f : \left( {{I}^{n},\partial {I}^{n}}\right) \rightarrow \left( {X,{x}_{0}}\right) \) be a map. Furthermore let \( A \subset {I}^{n} \) be a boundary-trivial cuboid and let \( C \) be a non-degenerate cuboid that is contained in \... | Proof (*). Let \( f : \left( {{I}^{n},\partial {I}^{n}}\right) \rightarrow \left( {X,{x}_{0}}\right) \) be a map, let \( A = \left\lbrack {{a}_{1},{b}_{1}}\right\rbrack \times \cdots \times \left\lbrack {{a}_{n},{b}_{n}}\right\rbrack \subset {I}^{n} \) be a boundary-trivial cuboid and let \( C = \left\lbrack {{c}_{1},{... | Yes |
Proposition 40.3. For any pointed topological space \( \left( {X,{x}_{0}}\right) \) and any \( n \geq 2 \) the homotopy group \( {\pi }_{n}\left( {X,{x}_{0}}\right) \) is abelian. | Proof. Suppose we are given two maps \( f, g : \left( {{I}^{n},\partial {I}^{n}}\right) \rightarrow \left( {X,{x}_{0}}\right) \) . We have to show that \( f * g \) is homotopic to \( g * f \) . The homotopy is described in Figure 684 for \( n = 2 \) . It is straightforward to write down an explicit homotopy between \( ... | No |
Lemma 40.4. Let \( X \) be a topological space, let \( {x}_{0} \in X \) and let \( n \geq 1 \) . Furthermore let \( f : \left( {{S}^{n}, * }\right) \rightarrow \left( {X,{x}_{0}}\right) \) be a map. Then the following statements are equivalent:\n\n(1) the map \( f : {S}^{n} \rightarrow X \) is homotopic to a constant m... | Proof (*). The equivalence of (2) and (3) is proved in a similar way as Lemma 14.1. We leave it to the reader to make the fairly straightforward modifications.\n\nWe turn to the proof that (3) implies (2). This is basically obvious. Indeed, the fact that \( f \) represents the trivial element in \( {\pi }_{n}\left( {X,... | No |
Proposition 40.5. Let \( X \) be a topological space, let \( {x}_{0} \) and \( {x}_{1} \) be two points in \( X \) and let \( \gamma : \left\lbrack {0,1}\right\rbrack \rightarrow X \) be a path from \( {x}_{0} \) to \( {x}_{1} \) . Given a map \( f : \left( {{I}^{n},\partial {I}^{n}}\right) \rightarrow \left( {X,{x}_{1... | Proof. Let \( X \) be a topological space, let \( {x}_{0} \) and \( {x}_{1} \) be two points in \( X \) and let \( \gamma : \left\lbrack {0,1}\right\rbrack \rightarrow X \) be a path from \( {x}_{0} \) to \( {x}_{1} \) . We need to prove the following claim. Claim.\n\n(1) The map \( \Phi : {\pi }_{n}\left( {X,{x}_{1}}\... | No |
Proposition 40.6. Let \( n \geq 1 \) . Recall that we denote by PTop the category of pointed topological spaces and that we denote by \( \mathcal{G}r \) be the category of groups. Then\n\n\[ \mathrm{{Ob}}\left( {P\text{ Top }}\right) \rightarrow \mathrm{{Ob}}\left( {\mathcal{G}r}\right) \]\n\n\[ \left( {X,{x}_{0}}\righ... | Proof. It follows immediately from the definitions that\n\n\[ {\mathrm{{id}}}_{ * } = \mathrm{{id}}\;\text{ for all pointed pairs }\left( {X,{x}_{0}}\right) ,\]\n\nand that\n\n\[ {\left( g \circ f\right) }_{ * } = {g}_{ * } \circ {f}_{ * }\;\text{ for all maps }f : \left( {X,{x}_{0}}\right) \rightarrow \left( {Y,{y}_{0... | Yes |
Proposition 40.8. Let \( A \) and \( B \) be two topological spaces and let \( {a}_{0} \in A \) and \( {b}_{0} \in B \) . We consider the inclusion maps\n\n\[ \n\begin{aligned} i : A & \rightarrow A \times B \\ a & \mapsto \left( {a,{b}_{0}}\right) \end{aligned}\;\text{ and }\;\begin{aligned} j : B & \rightarrow A \tim... | Proof. The proof of this proposition is almost verbatim the same as the proof of Proposition 16.20. We leave it to the reader to make the minute modifications. | No |
Proposition 40.10. Let \( n \in \mathbb{N} \) . For any \( k \in \{ 1,\ldots, n - 1\} \) we have\n\n\[{\pi }_{k}\left( {S}^{n}\right) = 0.\] | Proof.\n\n(1) Again, as on page \( \left| \overline{935}\right| \) we consider the \( n \) -sphere \( {S}^{n} \) as a CW-complex with one 0-cell and one \( n \) -cell. With this CW-structure it follows immediately from Proposition 40.9 that \( {\pi }_{k}\left( {S}^{n}\right) = 0 \) for \( k = 1,\ldots, n - 1 \) .\n\n(2... | No |
Corollary 40.12. For every \( n \in \mathbb{N} \) we have \( {\pi }_{n}\left( {S}^{\infty }\right) = 0 \) . | Proof. We equip \( {S}^{\infty } \) and all spheres \( {S}^{k}, k \in \mathbb{N} \) with the CW-complex structure that we defined on page 942. We have\n\n\[ \text{by Proposition 40.11 and by definition by Proposition 40.10 we have} \]\n\n\[ \text{of the topology of}{S}^{\infty }\;{\pi }_{n}\left( {S}^{k}\right) = 0\tex... | Yes |
Corollary 40.14. Let \( X \) be a topological space that is path-connected, locally path-connected and semi-locally simply connected. Given any \( n \geq 2 \) the universal covering \( p : \widetilde{X} \rightarrow \) \( X \) induces an isomorphism\n\n\[ \n{p}_{ * } : {\pi }_{n}\left( {\text{ universal covering }\widet... | Proof of Proposition 40.13. Let \( p : \left( {X,{x}_{0}}\right) \rightarrow \left( {B,{b}_{0}}\right) \) be a covering of pointed topological spaces and furthermore let \( n \geq 2 \) . Our goal is to construct an inverse to the map \( {p}_{ * } : {\pi }_{n}\left( {X,{x}_{0}}\right) \rightarrow {\pi }_{n}\left( {B,{b}... | No |
Lemma 40.15. Let \( X \) be a 1-dimensional CW-complex and let \( {x}_{0} \in X \) be base point. Then \( {\pi }_{n}\left( {X,{x}_{0}}\right) = 0 \) for every \( n \geq 2 \) . | Sketch of proof. Let \( X \) be a 1-dimensional CW-complex, let \( {x}_{0} \in X \) and let \( n \geq 2 \) . We first assume that \( X \) is finite and connected. We write \( k = 1 - \chi \left( X\right) \) . We have\n\n\( {\pi }_{n}\left( X\right) \cong {\pi }_{n}\left( {\mathop{\bigvee }\limits_{{i = 1}}^{k}{S}^{1}}\... | No |
Lemma 40.19. Let \( k \in {\mathbb{N}}_{0} \) .\n\n(1) Let \( \left( {X,{x}_{0}}\right) \) be a pointed topological space. We denote by \( * \) the North Pole of \( \sum \left( X\right) \) . The map\n\n\[ \n{\pi }_{k}\left( {X,{x}_{0}}\right) \rightarrow {\pi }_{k + 1}\left( {\sum \left( X\right) , * }\right)\n\]\n\n\[... | (1) It follows from Lemma 24.4 (4) that this map is well-defined \( {}^{665} \) We leave it the task of verifying that the map is in fact a group homomorphism as a somewhat challenging exercise to the reader. Alternatively we refer to [Hilt53, Theorem 2.1] for a proof.\n\n(2) This statement follows immediately from (1)... | No |
Lemma 41.1. Let \( n \in \mathbb{N} \) .\n\n(1) Every n-simplex, in particular the standard n-simplex \( {\Delta }^{n} \), is compact. | Proof.\n\n(1) We had proved this statement in Exercise 2.36 (b). Of course it is not difficult to prove the statement \ | No |
Proposition 41.2. Let \( X \) be a topological space. For any \( n \in {\mathbb{N}}_{0} \) the composition\n\n\[ \n{\partial }_{n - 1} \circ {\partial }_{n} : {\mathrm{C}}_{n}\left( X\right) \rightarrow {\mathrm{C}}_{n - 2}\left( X\right)\n\]\n\nis the zero map. | Proof. It suffices to show that \( {\partial }_{n - 1}\left( {{\partial }_{n}\left( \sigma \right) }\right) = 0 \) for every singular \( n \) -simplex \( \sigma : {\Delta }^{n} \rightarrow X \) . Thus let \( \sigma : {\Delta }^{n} \rightarrow X \) be a singular \( n \) -simplex. Then it follows immediately from the def... | Yes |
Lemma 41.4. Let \( X \) be a topological space.\n\n(1) The map \( {\epsilon }_{X} \circ {\partial }_{1} : {\mathrm{C}}_{1}\left( X\right) \rightarrow \mathbb{Z} \) is the zero map. | Proof. Let \( X \) be a topological space.\n\n(1) Let \( c = \mathop{\sum }\limits_{{i = 1}}^{n}{b}_{i}{\sigma }_{i} \in {\mathrm{C}}_{1}\left( X\right) \) . We calculate that\n\n\[ \epsilon \left( {{\partial }_{1}\left( c\right) }\right) = \epsilon \left( {\mathop{\sum }\limits_{{i = 1}}^{n}{b}_{i} \cdot {\partial }_{... | Yes |
Proposition 41.5. Let \( X \) be a path-connected non-empty topological space and let \( P \) be a point in \( X \) . Then the augmentation map\n\n\[ \n\epsilon = {\epsilon }_{X} : {\mathrm{H}}_{0}\left( X\right) \rightarrow \mathbb{Z} \n\]\n\nis a natural isomorphism where the inverse is given by the map\n\n\[ \n\iota... | Proof. Let \( X \) be a non-empty path-connected topological space. Let \( P \) be a point in \( X \) . It is clear that \( \epsilon \circ \iota = {\operatorname{id}}_{\mathbb{Z}} \), in particular the homomorphism \( \epsilon \) is surjective. It remains to show that \( \epsilon \) is injective.\n\nThus let \( c \in {... | Yes |
Lemma 41.7. Let \( f : {C}_{ * } \rightarrow {D}_{ * } \) be a chain map between chain complexes \( {C}_{ * } \) and \( {D}_{ * } \). Then the map\n\n\[ \n{f}_{ * } : {\mathrm{H}}_{n}\left( {C}_{ * }\right) \rightarrow {\mathrm{H}}_{n}\left( {D}_{ * }\right)\n\]\n\n\[ \n\left\lbrack c\right\rbrack \mapsto \left\lbrack ... | Proof. Let \( c \in {C}_{n} \) be a cycle. First we have to show that \( {f}_{n}\left( c\right) \) is again a cycle. Indeed we have\n\n\[ \n\begin{array}{l} {\partial }_{n}\left( {{f}_{n}\left( c\right) }\right) = {f}_{n - 1}\left( {{\partial }_{n}c}\right) = {f}_{n}\left( 0\right) = 0. \\ \end{array}\n\]\n\ndefinition... | Yes |
(1) Let \( f : X \rightarrow Y \) be a map between topological spaces and let \( n \in {\mathbb{N}}_{0} \) . For every \( c \in {\mathrm{C}}_{n}\left( X\right) \) we have\n\n\[ \n{f}_{ * }\left( {{\partial }_{n}c}\right) = {\partial }_{n}\left( {{f}_{ * }\left( c\right) }\right) \in {\mathrm{C}}_{n}\left( Y\right) .\n\... | (1) Let \( f : X \rightarrow Y \) be a map between topological spaces and let \( \sigma : {\Delta }^{n} \rightarrow X \) be a singular \( n \) -simplex. We have\n\n\[ \n{f}_{ * }\left( {\partial \sigma }\right) = {f}_{ * }\left( {\mathop{\sum }\limits_{{j = 0}}^{n}{\left( -1\right) }^{j} \cdot \sigma \circ {i}_{j}^{n}}... | Yes |
For each \( n \in {\mathbb{N}}_{0} \) the map\n\n\[ \nX \mapsto {\mathrm{H}}_{n}\left( X\right) \n\]\n\ntogether with the map\n\n\[ \n\left( {f : X \rightarrow Y}\right) \mapsto \left( \begin{matrix} {f}_{ * } : {\mathrm{H}}_{n}\left( X\right) & \rightarrow & {\mathrm{H}}_{n}\left( Y\right) \\ \left\lbrack {\mathop{\su... | Example. Let \( X \) and \( Y \) be non-empty topological spaces. Let \( f : X \rightarrow Y \) be a map. We suppose that \( f \) is constant, i.e. we suppose that there exists a \( y \in Y \) such that \( f\left( X\right) = \{ y\} \) . We denote by \( i : \{ y\} \rightarrow Y \) the inclusion. Let \( n \in \mathbb{N} ... | Yes |
Lemma 41.13. If \( {\left\{ {\mathrm{C}}_{a}\right\} }_{a \in A} \) is a family of chain complexes, then the obvious inclusion maps \( {\mathrm{C}}_{a} \rightarrow \mathop{\bigoplus }\limits_{{a \in A}}{\mathrm{C}}_{a} \) and \( {\mathrm{C}}_{a} \rightarrow \mathop{\prod }\limits_{{a \in A}}{\mathrm{C}}_{a} \) induce i... | \[ \mathop{\bigoplus }\limits_{{a \in A}}{\mathrm{H}}_{n}\left( {\mathrm{C}}_{a}\right) \overset{ \cong }{ \rightarrow }{\mathrm{H}}_{n}\left( {\mathop{\bigoplus }\limits_{{a \in A}}{\mathrm{C}}_{a}}\right) \;\text{ and }\;\mathop{\prod }\limits_{{a \in A}}{\mathrm{H}}_{n}\left( {\mathrm{C}}_{a}\right) \overset{ \cong ... | Yes |
Lemma 41.14. Let \( X \) be a topological space with path-components \( {\left\{ {X}_{a}\right\} }_{a \in A} \). Given \( a \in A \) we denote by \( {i}_{a} : {X}_{a} \rightarrow X \) the inclusion map. Then the map\n\n\[ \mathop{\bigoplus }\limits_{{a \in A}}{i}_{a} : \;\mathop{\bigoplus }\limits_{{a \in A}}{\mathrm{C... | Proof. Evidently the standard simplex \( {\Delta }^{n} \) is path-connected. Thus it follows from Lemma 2.70 that the image of a map \( \sigma : {\Delta }^{n} \rightarrow X \) is contained in a path-component \( {X}_{a} \). This shows that\n\n\[ \mathop{\bigsqcup }\limits_{{a \in A}}\text{singular}n\text{-simplices in}... | Yes |
Lemma 42.2. Let \( f, g : {C}_{ * } \rightarrow {D}_{ * } \) be two chain maps between chain complexes. If \( f \) and \( g \) are chain homotopic, then \( {f}_{ * } = {g}_{ * } : {\mathrm{H}}_{n}\left( C\right) \rightarrow {\mathrm{H}}_{n}\left( D\right) \) for all \( n \in {\mathbb{N}}_{0} \) . | Proof. Let \( P = {\left\{ {P}_{n}\right\} }_{n \in {\mathbb{N}}_{0}} \) be a chain homotopy between \( f \) and \( g \) . Then for any cycle \( z \in {C}_{n} \) we get the equalities\n\n\[ f\left( z\right) - g\left( z\right) = \left( {f - g}\right) \left( z\right) = \left( {{\partial }_{n + 1}{P}_{n} + {P}_{n - 1}{\pa... | Yes |
Proposition 42.5. Let \( X \) and \( Y \) be two topological spaces and let \( f, g : X \rightarrow Y \) be two maps. If \( f \) and \( g \) are homotopic, then the following hold:\n\n(1) the induced maps \( {f}_{ * } \) and \( {g}_{ * } \) from \( {\mathrm{C}}_{ * }\left( X\right) \) to \( {\mathrm{C}}_{ * }\left( Y\r... | We start out with a discussion of the key idea behind the proof of Proposition 42.5. Afterwards we will fill in the details. So let \( f, g : X \rightarrow Y \) be two maps between topological spaces and let \( F : X \times \left\lbrack {0,1}\right\rbrack \rightarrow Y \) be a homotopy between the maps \( f \) and \( g... | No |
Lemma 42.6. Let \( n \in {\mathbb{N}}_{0} \) . Given \( t \in \{ 0,1\} \) we write\n\n\[ \n{\eta }_{t} : {\Delta }^{n} \rightarrow {\Delta }^{n} \times \left\lbrack {0,1}\right\rbrack \n\]\n\n\[ \nx \mapsto \left( {x, t}\right) .\n\]\n\nWith this notation we have\n\n\[ \n{\partial }_{n + 1}\left( {\Omega }_{n}\right) +... | Proof. The proof of the lemma is basically an elementary, albeit slightly confusing calculation. First note that\n\n\[ \n\partial {\Omega }_{n} = \partial \left( {\mathop{\sum }\limits_{{j = 0}}^{n}{\left( -1\right) }^{j} \cdot \left\lbrack {{v}_{0},\ldots ,{v}_{j},{w}_{j},\ldots ,{w}_{n}}\right\rbrack }\right)\n\]\n\n... | Yes |
Lemma 42.7. Let \( X \) be a topological space and let \( F : X \times \left\lbrack {0,1}\right\rbrack \rightarrow X \) be a homotopy from a map \( f \) to a map \( g \) . Given any \( n \in {\mathbb{N}}_{0} \) we define\n\n\[ \n{P}_{n} : {\mathrm{C}}_{n}\left( X\right) \rightarrow {\mathrm{C}}_{n + 1}\left( Y\right) \... | Proof. Let \( X \) be a topological space and let \( F : X \times \left\lbrack {0,1}\right\rbrack \rightarrow X \) be a homotopy from a map \( f \) to a map \( g \) . Furthermore let \( \sigma : {\Delta }^{n} \rightarrow X \) be a singular \( k \) -simplex. Then\n\n\[ \n\left( {\partial {P}_{n} + {P}_{n - 1}\partial }\... | Yes |
Corollary 42.8. Let \( X \) and be \( Y \) topological spaces and let \( n \in {\mathbb{N}}_{0} \). (1) If \( f : X \rightarrow Y \) is a homotopy equivalence (e.g. a homeomorphism), then the induced map \( {f}_{ * } : {\mathrm{H}}_{n}\left( X\right) \rightarrow {\mathrm{H}}_{n}\left( Y\right) \) is an isomorphism. | (1) Let \( f : X \rightarrow Y \) be a homotopy equivalence and let \( g : Y \rightarrow X \) be a homotopy inverse of \( f \), i.e. \( g \) is a map such that \( f \circ g \simeq {\operatorname{id}}_{Y} \) and \( g \circ f \simeq {\operatorname{id}}_{X} \). It follows from Proposition 42.5 and the functoriality of hom... | Yes |
Lemma 43.2. If \( X \) is a non-empty topological space, then the natural sequence\n\n\[ 0 \rightarrow {\widetilde{\mathrm{H}}}_{0}\left( X\right) \overset{\text{ x }}{ \rightarrow }{\mathrm{H}}_{0}\left( X\right) \xrightarrow[\text{ from Lemma }]{\text{ augmentation map }{\epsilon }_{X}}\mathbb{Z} \rightarrow 0 \]\n\n... | Proof. Let \( P \in X \) be a point. We consider the following diagram:\n\n\n\nWe make the following observations:\n\n(1) It follows immediately from the definitions that the diagram commutes.\n\n(2) By Lemma 43.1 ... | No |
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