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Proposition 110.14. There is no topological space \( X \) such that \( {\mathrm{H}}^{ * }\left( {X;{\mathbb{Z}}_{3}}\right) \) is isomorphic to \( {\mathbb{Z}}_{3}\left\lbrack \varphi \right\rbrack /\left( {\varphi }^{4}\right) \) with \( \varphi \in {\mathrm{H}}^{8}\left( {X;{\mathbb{Z}}_{3}}\right) \) . | Proof. We provide a proof by contradiction. So suppose there exists a topological space \( X \) such that \( {\mathrm{H}}^{ * }\left( {X;{\mathbb{Z}}_{3}}\right) \) is isomorphic to \( {\mathbb{Z}}_{3}\left\lbrack \varphi \right\rbrack /\left( {\varphi }^{4}\right) \) with \( \varphi \in {\mathrm{H}}^{8}\left( {X;{\mat... | Yes |
Let \( n \geq 2 \) . The map \[ {\pi }_{n}\left( {X, A,{x}_{0}}\right) \times {\pi }_{n}\left( {X, A,{x}_{0}}\right) \rightarrow {\pi }_{n}\left( {X, A,{x}_{0}}\right) \] \[ \left( {\left\lbrack f\right\rbrack ,\left\lbrack g\right\rbrack }\right) \mapsto \left\lbrack {f * g}\right\rbrack \] is well-defined and it defi... | The proof of the first statement is quite similar to the proof of Proposition 40.1. We leave it to the reader to fill in the details. | No |
Lemma 111.6. Let \( n \in \mathbb{N} \) . There exists an explicit \( {}^{1515} \) map \( \varphi : {I}^{n} \rightarrow {\bar{B}}^{n} \) with the following three properties:\n\n(1) we have \( \varphi \left( {J}^{n - 1}\right) = * \mathrel{\text{:=}} \left( {0,\ldots ,0,1}\right) \) and \( \varphi \left( {I}^{n - 1}\rig... | \n\nFigure 1554. Illustration for Lemma 111.6.\n\n any point \( x \in {I}^{n} \smallsetminus \{ Q\} \) we de... | Yes |
Lemma 111.7. Let \( X \) be a topological space, let \( A \subset X \) and let \( k \in \mathbb{N} \) . The following two statements are equivalent:\n\n(1) The pair \( \left( {X, A}\right) \) is \( k \) -connected.\n\n(2) Every path-component of \( X \) contains a point of \( A \), furthermore for every \( a \in A \) a... | Proof. We denote by \( i : A \rightarrow X \) the inclusion map. Let \( a \in A \) . By Proposition 11.4 we have the following long exact sequence of homotopy groups:\n\n\[ \ldots \rightarrow {\pi }_{k}\left( {A, a}\right) \overset{{i}_{ * }}{ \rightarrow }{\pi }_{k}\left( {X, a}\right) \rightarrow {\pi }_{k}\left( {X,... | Yes |
Proposition 111.8. Let \( X \) be a topological space and let \( A \subset X \) . For any \( n \in \mathbb{N} \) the following two statements are equivalent:\n\n(1) Every map \( \left( {{\bar{B}}^{n},{S}^{n - 1}}\right) \rightarrow \left( {X, A}\right) \) is homotopic rel \( {S}^{n - 1} \) to a map \( {\bar{B}}^{n} \ri... | Proof of Proposition 11.8. First note that it is elementary to see that for \( n = 0 \) the statements (1) and (2’) are equivalent. Now let \( n \geq 1 \) . The implication (1) \( \Rightarrow \) (2) is an immediate consequence of Lemma 111.3 and the by now well-known fact that there exists a homeomorphism \( \left( {{I... | Yes |
Proposition 111.9. Let \( X \) be a CW-complex and let \( k \in {\mathbb{N}}_{0} \). (1) If \( A \) is a subcomplex such that all cells in \( X \smallsetminus A \) have dimension greater than \( k \), then the pair \( \left( {X, A}\right) \) is \( k \) -connected. | Proof. (1) Let \( X \) be a CW-complex, let \( A \) be a subcomplex and let \( k \in {\mathbb{N}}_{0} \). Suppose that all cells in \( X \smallsetminus A \) have dimension greater than \( k \). This implies in particular that every path-component of \( X \) contains at least one point in \( A \). The remainder of the p... | No |
(1) Let \( \\left( {X, A,{x}_{0}}\\right) \) be a pointed pair of topological spaces and let \( n \\in {\\mathbb{N}}_{ \\geq 2} \) . The map\n\n\[ \n{\\Phi }_{\\left( X, A,{x}_{0}\\right) } : {\\pi }_{n}\\left( {X, A,{x}_{0}}\\right) \\rightarrow {\\mathrm{H}}_{n}\\left( {X, A;\\mathbb{Z}}\\right) \n\]\n\n\[ \n\\left\\... | (1) The proof of the first statement is evidently just a straightforward variation on the proof of Lemma 53.1. We refer to [Bre93, Lemma VII.10.2] or [Hat02, Proposition 4.36 for details. | No |
Lemma 111.12. Let \( \\left( {X, A,{x}_{0}}\\right) \) be a pointed pair of topological spaces and let \( n \\in {\\mathbb{N}}_{ \\geq 2} \) . Furthermore let \( \\sigma \\in {\\pi }_{n}\\left( {X, A,{x}_{0}}\\right) \) . For any \( g \\in {\\pi }_{1}\\left( {A,{x}_{0}}\\right) \) we have\n\n\[ \n{\\Phi }_{\\left( X, A... | Proof. Let \( \\left( {X, A,{x}_{0}}\\right) \) be a pointed pair of topological spaces and let \( n \\in {\\mathbb{N}}_{ \\geq 2} \) . Furthermore let \( \\left\\lbrack f\\right\\rbrack \\in {\\pi }_{n}\\left( {X, A,{x}_{0}}\\right) \) and let \( \\left\\lbrack \\gamma \\right\\rbrack \\in {\\pi }_{1}\\left( {A,{x}_{0... | Yes |
Theorem 111.13. (Relative Hurewicz Theorem) Let \( \left( {X, A,{x}_{0}}\right) \) be a pointed pair of topological spaces such that \( X \) and \( A \) are path-connected. Let \( n \geq 2 \) . If the pair \( \left( {X, A}\right) \) is \( \left( {n - 1}\right) \) -connected, then the following two statements hold:\n\n(... | Proof. We will not provide a proof of the Relative Hurewicz Theorem. A full proof is given in [Bre93, Theorem VII.10.7] or alternatively in [Hat02, Theorem 4.37] and [Spa95, Proposition 7.5.1]. If \( A \) is simply connected, then it is a great exercise to figure out to what degree the proof of the Hurewicz Theorem 53.... | No |
Theorem 111.14. Let \( X \) and \( Y \) be two topological spaces and furthermore let \( f : X \rightarrow Y \) be a map. We pick a base point \( {x}_{0} \in X \) and we write \( {y}_{0} = f\left( {x}_{0}\right) \). (1) If for every \( i \in {\mathbb{N}}_{ \geq 2} \) the induced map \[ {f}_{ * } : {\pi }_{i}\left( {X,{... | Proof. The idea is to \ | No |
Corollary 111.15. Let \( n \in {\mathbb{N}}_{ \geq 2} \) . If \( \sum \) is a homotopy \( n \) -sphere, then for every \( i \in {\mathbb{N}}_{0} \) we have \( {\pi }_{i}\left( \sum \right) \cong {\pi }_{i}\left( {S}^{n}\right) \) . | Proof. Let \( \sum \) be a homotopy \( n \) -sphere. Note that by Theorem 11.14(2) it suffices to find a map \( f : {S}^{n} \rightarrow \sum \) such that for every \( i \in {\mathbb{N}}_{i \geq 2} \) the induced map \( {f}_{ * } : {\mathrm{H}}_{i}\left( {{S}^{n};\mathbb{Z}}\right) \rightarrow {\mathrm{H}}_{i}\left( {\s... | No |
Lemma 111.16. We consider the 2-dimensional sphere \( X = {S}^{2} \) together with the subset \( A \mathrel{\text{:=}} {S}_{ \leq 0}^{2} = \left\{ {\left( {x, y, z}\right) \in {S}^{2} \mid z \leq 0}\right\} \) given by the lower hemisphere and together with the subset \( \bar{Z} = \{ \left( {0,0, - 1}\right) \} \) that... | Proof. We start out with the following observations: (1) On page 116 we saw that \( A = {S}_{ \leq 0}^{2} \) is homeomorphic to \( {\bar{B}}^{2} \) . In particular \( A \) is contractible which implies by Proposition 40.7 (2) that \( {\pi }_{2}\left( A\right) = {\pi }_{3}\left( A\right) = 0 \) . \( {}^{1524} \) Here ev... | Yes |
Theorem 111.17. (Blakers-Massey Theorem)\n\n(1) Let \( X \) be a topological space and let \( A \) and \( B \) be subsets such that \( X = \mathring{A} \cup \mathring{B} \) . If \( \left( {A, A \cap B}\right) \) is \( n \) -connected and if \( \left( {B, A \cap B}\right) \) is \( m \) -connected, then the inclusion ind... | Proof. The first statement was originally proved in 1952 by Albert Blakers and William Massey [BlM52]. Proofs can also be found in Gra75, Corollary 16.27], [tD08, Theorem 6.4.1], [tDKP70, p. 211] or [WhdG78, Chapter VII.7]. The second statement can presumably be deduced from the first statement. Alternatively see Hat02... | No |
Lemma 112.1. The map \( p : {S}^{1} \rightarrow \left\lbrack {-1,1}\right\rbrack \) that is given by the projection onto the \( x \) -coordinate is not a Serre fibration. | Proof. We consider the topological space \( X = {\bar{B}}^{1} = \left\lbrack {-1,1}\right\rbrack \) together with the maps \( \widetilde{f} : X \times \{ 0\} \rightarrow {S}^{1} \) and \( F : \left\lbrack {-1,1}\right\rbrack \times \left\lbrack {0,1}\right\rbrack \rightarrow \left\lbrack {-1,1}\right\rbrack \) that are... | Yes |
(1) For each \( n \in \mathbb{N} \) there exists an orientation-preserving homeomorphism \[ \left( {{\bar{B}}^{n} \times \left\lbrack {0,1}\right\rbrack ,{\bar{B}}^{n}\times \{ 0\} \cup {S}^{n - 1} \times \left\lbrack {0,1}\right\rbrack }\right) \overset{ \cong }{ \rightarrow }\left( {{\bar{B}}^{n} \times \left\lbrack ... | SKETCH OF PROOF (*). (1) For \( n = 1 \) such a homeomorphism is illustrated in Figure 1565. The general case is given by \ | No |
Lemma 112.3. Let \( p : Y \rightarrow B \) be a Serre fibration. Let \( {y}_{0} \in Y \). We write \( {b}_{0} = p\left( {y}_{0}\right) \). Furthermore let \( n \in \mathbb{N} \). Given any map \( \varphi : \left( {{I}^{n},\partial {I}^{n}}\right) \rightarrow \left( {B,\left\{ {b}_{0}\right\} }\right) \) there exists a ... | Proof. Let \( \Psi : \left( {{I}^{n},{J}^{n - 1}}\right) \overset{ \cong }{ \rightarrow }\left( {{I}^{n},{I}^{n - 1}\times \{ 0\} }\right) \) be a homeomorphism as in Lemma 112.2 (2). We consider the diagram Note that the diagram without the red dashed arrow commutes. Since \( p \) is a Serre fibration and since \( {I}... | Yes |
Proposition 112.5. Let \( p : Y \rightarrow B \) be a fibration. Given \( b \in B \) we write \( {F}_{b} \mathrel{\text{:=}} {p}^{-1}\left( b\right) \) . Let \( \alpha : \left\lbrack {0,1}\right\rbrack \rightarrow B \) be a path from a point \( {b}_{0} \) to a point \( {b}_{1} \) . We consider the following diagram\n\n... | Proof (*). Let \( p : Y \rightarrow B \) be a fibration over a topological space \( B \) . Given \( b \in B \) we write \( {F}_{b} \mathrel{\text{:=}} {p}^{-1}\left( b\right) \) . Let \( \alpha : \left\lbrack {0,1}\right\rbrack \rightarrow B \) be a path from a point \( {b}_{0} \) to a point \( {b}_{1} \) . We consider... | Yes |
Proposition 112.6. Let \( p : Y \rightarrow B \) be a map between topological spaces. We assume that \( B \) is path-connected.\n\n(1) If \( p : Y \rightarrow B \) is a fibration, then for any two points \( {b}_{0},{b}_{1} \in B \) the fibers \( {p}^{-1}\left( {b}_{0}\right) \) and \( {p}^{-1}\left( {b}_{1}\right) \) a... | Proof.\n\n(1) This statement follows immediately from Proposition 112.5 (3) and the hypothesis that \( B \) is path-connected. | Yes |
Proposition 112.6 (1) allows us to make the following definition.\n\nDefinition. Let \( p : Y \rightarrow B \) be a fibration over a path-connected topological space \( B \) . We refer to the homotopy type of the fibers as the homotopy fiber of the fibration. | Remark. A more precise statement of Proposition 112.6 (1) holds: any path \( \gamma : \left\lbrack {0,1}\right\rbrack \rightarrow B \) from \( {b}_{0} \) to \( {b}_{1} \) defines naturally a homotopy equivalence between the fibers \( {p}^{-1}\left( {b}_{0}\right) \) and \( {p}^{-1}\left( {b}_{1}\right) \) . We refer to... | Yes |
Proposition 40.8. Let \( X \) and \( Y \) be two topological spaces and let \( {x}_{0} \in X \) and \( {y}_{0} \in Y \) . We consider the maps\n\n\[ \n\begin{aligned} i : X & \rightarrow X \times Y \\ x & \mapsto \left( {x,{y}_{0}}\right) \end{aligned}\;\text{ and }\;\begin{aligned} j : Y & \rightarrow X \times Y \\ y ... | Proof. We denote by \( p : X \times Y \rightarrow X \) and \( q : X \times Y \rightarrow Y \) the two obvious projections. On page [2643] we saw that both projections are fibrations. We concentrate on the first projection map \( p : X \times Y \rightarrow X \) . It is a fibration with fiber \( {p}^{-1}\left( {x}_{0}\ri... | Yes |
Proposition 112.8. Let \( p : Y \rightarrow B \) be a Serre fibration and let \( \left( {K, L}\right) \) be a pair of CW-complexes. We denote by \( i : K \times \{ 0\} \cup L \times \left\lbrack {0,1}\right\rbrack \rightarrow K \times \left\lbrack {0,1}\right\rbrack \) the inclusion map. Given any map \( F : K \times \... | Proof. We start out with a reformulation of our task: Given \( n = 0,1,2,3,\ldots \) we need to define maps\n\n\[ \n{\widetilde{F}}^{n} : \left( {K\times \{ 0\} }\right) \cup \left( {{K}^{n} \cup L}\right) \times \left\lbrack {0,1}\right\rbrack \rightarrow Y \n\]\n\nthat make the above diagram commute and which have th... | Yes |
Proposition 112.9. Let \( p : Y \rightarrow B \) be a Serre fibration and let \( \left( {K, L}\right) \) be a pair of CW-complexes such that \( L \) is a deformation retract of \( K \) . We denote by \( i : L \rightarrow K \) the inclusion map. Given any map \( f : K \rightarrow B \) and given any map \( g : L \rightar... | Proof. We make the following preparations:\n\n(1) Since \( L \) is a deformation retract of \( K \) there exists by definition of a deformation retraction, see page 548, a homotopy \( H : K \times \left\lbrack {0,1}\right\rbrack \rightarrow K \) rel \( L \) such that \( {H}_{0} = {\operatorname{id}}_{K} \) and \( {H}_{... | Yes |
Lemma 112.11. Let \( n \in \mathbb{N} \) . If \( x \in {S}^{n - 1} \), then \( \left( {{\bar{B}}^{n} \times \partial \left\lbrack {0,1}\right\rbrack }\right) \cup \left( {\{ x\} \times \left\lbrack {0,1}\right\rbrack }\right) \) is a deformation retract of \( {\bar{B}}^{n} \times \left\lbrack {0,1}\right\rbrack \) . | Proof of Lemma 112.11 (*). As usual we write \( I = \left\lbrack {0,1}\right\rbrack \) . Given \( k \in \{ 1,\ldots, n\} \) we consider\n\n\[ \n{X}_{k} \mathrel{\text{:=}} \left( {{I}^{n - 1} \times \partial I}\right) \cup \left( {\left\{ \underset{ \in {\mathbb{R}}^{n - k}}{\underbrace{\left( 1,1,\ldots ,1\right) }}\r... | No |
Lemma 112.12. The maps\n\n\[ \n\\left( {X,{x}_{0}}\\right) \\mapsto \\left( {\\Omega \\left( {X,{x}_{0}}\\right) ,{c}_{{x}_{0}}}\\right) \n\]\n\n\[ \n\\left( {f : \\left( {X,{x}_{0}}\\right) \\rightarrow \\left( {Y,{y}_{0}}\\right) }\\right) \\mapsto \\left( \\begin{matrix} \\Omega \\left( {X,{x}_{0}}\\right) & \\right... | Proof. Let \( f : \\left( {X,{x}_{0}}\\right) \\rightarrow \\left( {Y,{y}_{0}}\\right) \) be a map between pointed topological spaces. It follows from Lemma 5.5 (1) that the map\n\n\[ \n\\Omega \\left( {X,{x}_{0}}\\right) \\rightarrow \\Omega \\left( {Y,{y}_{0}}\\right) \n\]\n\n\[ \n\\left( {\\gamma : \\left\\lbrack {0... | Yes |
(2) If \( \left( {X,{x}_{0}}\right) \) is a pointed topological space, then \( \left\{ {c}_{{x}_{0}}\right\} \) is a deformation retract of the path space \( P\left( {X,{x}_{0}}\right) \), in particular the path space \( P\left( {X,{x}_{0}}\right) \) is contractible. | Proof. We prove statement (2). The proof of statement (1) is basically identical. Thus let \( \left( {X,{x}_{0}}\right) \) be a pointed topological space. We consider the \ | No |
(2) Let \( \\left( {X,{x}_{0}}\\right) \) be a pointed topological space. The evaluation map\n\n\[ p : P\\left( {X,{x}_{0}}\\right) \\rightarrow X \]\n\n\[ \\left( {f : \\left\\lbrack {0,1}\\right\\rbrack \\rightarrow X}\\right) \\mapsto f\\left( 1\\right) \]\n\n is a fibration with fiber \( {p}^{-1}\\left( {x}_{0}\\ri... | Proof. We prove statement (2). The proof of statement (1) is basically identical and is thus left to the reader. Thus let \( \\left( {X,{x}_{0}}\\right) \) be a pointed topological space. First note that the map \( p : P\\left( {X,{x}_{0}}\\right) \\rightarrow X \) is continuous by Proposition 5.4 (2). Next recall that... | No |
Corollary 112.15. Given any pointed topological space \( \left( {X,{x}_{0}}\right) \) and any \( n \geq 1 \) the map \( {}^{1541} \n\n\[ \n{\partial }_{n} : {\pi }_{n}\left( {X,{x}_{0}}\right) \overset{ \cong }{ \rightarrow }{\pi }_{n - 1}\left( {\Omega \left( {X,{x}_{0}}\right) ,{c}_{{x}_{0}}}\right) \n\] \n\n\[ \n\le... | Proof. There are two approaches to proving the corollary, namely one can prove it by \ | No |
Lemma 112.16. Let \( f : A \rightarrow B \) be a map between topological spaces and let \( q : Z \rightarrow B \) be a fibration. As in Lemma 25.16 (1) we consider the pullback\n\n\[ \n{f}^{ * }Z \mathrel{\text{:=}} \left\{ {\left( {a, z}\right) \mid a \in A}\right. \text{ and }\overset{\text{i.e. }q\left( z\right) = f... | Proof.\n\n(1) Let \( X \) be any topological space. We consider the following diagram where initially we ignore the diagonal arrows:\n\n\n\nHere the maps \( G : X \times \left\lbrack {0,1}\right\rbrack \rightarrow ... | No |
Proposition 112.17. Let \( f : X \rightarrow Y \) be a map between topological spaces. We denote by \( p : {P}_{f} \rightarrow Y \) the corresponding mapping path fibration.\n\n(1) The map\n\n\[ h : X \rightarrow {P}_{f} \]\n\n\[ x \mapsto \left( {x\text{, constant path}{c}_{f\left( x\right) }}\right) \]\n\nis a homoto... | Proof of Proposition 112.17. Let \( f : X \rightarrow Y \) be a map between topological spaces. We consider the map\n\n\[ p : {P}_{f} = {f}^{ * }\left( {Y}^{\left\lbrack 0,1\right\rbrack }\right) = \left\{ {\left( {x,\gamma : \left\lbrack {0,1}\right\rbrack \rightarrow Y}\right) \in X \times {Y}^{\left\lbrack 0,1\right... | Yes |
Proposition 113.1. Let \( p : Y \rightarrow B \) be a map between topological spaces. If for every \( b \in B \) there exists an open neighborhood \( U \) of \( b \) such that \( p : {p}^{-1}\left( U\right) \rightarrow U \) is a Serre fibration, then \( p : Y \rightarrow B \) itself is a Serre fibration. | Proof. Let \( p : Y \rightarrow B \) be a map between topological spaces. In this proof we say \( U \subset B \) is small if \( p : {p}^{-1}\left( U\right) \rightarrow U \) is a Serre fibration.\n\nNow assume that \( p : Y \rightarrow B \) has the property that \( B \) can be covered by small sets. We want to show that... | No |
Proposition 113.2. Let \( p : Y \rightarrow B \) be a fiber bundle with fiber \( F \). (1) The map \( p : Y \rightarrow B \) is a Serre fibration. | (1) This part of the proposition is an immediate consequence of the definitions, Proposition 113.1 and the discussion on page 2643. | No |
(1) Let \( X \) be a topological space and let \( f : X \rightarrow X \) be a homeomorphism. The natural projection \[ \begin{array}{l} \overset{\text{basically the mapping torus of }\left( {X, f}\right) }{\overbrace{\operatorname{Tor}\left( {X, f}\right) = \left( {\left\lbrack {0,1}\right\rbrack \times X}\right) /\lef... | Proof. (1) We denote by \( \pi : \left\lbrack {0,1}\right\rbrack \rightarrow \left\lbrack {0,1}\right\rbrack /0 \sim 1 \) the obvious projection map. We consider the two open sets \( {U}_{1} \mathrel{\text{:=}} \pi \left( \left( {\frac{1}{4},\frac{3}{4}}\right) \right) \) and \( {U}_{2} \mathrel{\text{:=}} \pi \left( {... | Yes |
Lemma 113.5. Let \( g : C \rightarrow B \) be a map between topological spaces and let \( p : Y \rightarrow B \) be a fiber bundle with fiber \( F \) . As in Lemma 25.16 (1) we consider the pullback\n\n\[ \n{g}^{ * }Y \mathrel{\text{:=}} \left\{ {\left( {c, f}\right) \mid c \in C}\right. \text{and}\left. {f \in {p}^{-1... | Proof. The proof is, not surprisingly, also almost identical to the proof of Lemma ?? | No |
(1) Every bundle over a compact interval \( \left\lbrack {a, b}\right\rbrack \) is trivial. | (1) This statement will be proved in Exercise 113.2. | No |
Proposition 113.7. Let \( X \) be a topological space. If \( X \) is paracompact and contractible, then every bundle over \( X \) is trivial. | Proof. The statement of the proposition is almost the same as the statement of Proposition ?? (3) for vector bundles. The reference we gave for vector bundles, namely Hat2, Theorem 1.6], also applies to general bundles. | No |
Proposition 113.8. (*) Let \( n \in \mathbb{N} \) and let \( K \) be a closed \( k \) -dimensional submanifold of the smooth manifold \( {S}^{n} \) . For \( i = 1,\ldots, n - k - 2 \) we have \( {\pi }_{i}\left( {{S}^{n} \smallsetminus K}\right) = 0 \) . | Proof (*). Let \( n \in \mathbb{N} \) and \( K \) be a closed \( k \) -dimensional submanifold of the smooth manifold \( {S}^{n} \) . If \( n - k - 2 \leq 0 \), then there is nothing to prove. Thus we assume that \( n - k - 2 \geq 1 \) , i.e. we assume that \( n \geq k + 3 \) .\n\nWe want to show that \( {\pi }_{i}\lef... | Yes |
Proposition 113.9. Let \( n \in \mathbb{N} \). (2) The Hopf map \( q : {S}^{{4n} + 3} \rightarrow {\mathbb{{HP}}}^{n} \) is an \( {S}^{3} \) -bundle. | Proof. We only provide the proof of (2). The proof of (1) is a straightforward modification of the proof of (2). Given \( j \in \{ 0,\ldots, n\} \) we define\n\n\[ \n{V}_{j} \mathrel{\text{:=}} \left\{ {\left\lbrack {{h}_{0} : \cdots : {h}_{n}}\right\rbrack \in {\mathbb{{HP}}}^{n} \mid {h}_{j} \neq 0}\right\} .\n\]\n\n... | Yes |
(1) For all \( n \geq 2 \) we have \( {\pi }_{n}\left( {S}^{4}\right) \cong {\pi }_{n}\left( {S}^{7}\right) \oplus {\pi }_{n - 1}\left( {S}^{3}\right) \) . | (1) By Lemma 60.9 (2) we can identify the quaternionic projective space \( {\mathbb{{HP}}}^{1} \) with \( {S}^{4} \) . Therefore if we apply Proposition 113.9 (2) with \( n = 1 \) we obtain an \( {S}^{3} \) -bundle \( q : {S}^{7} \rightarrow {\mathbb{{HP}}}^{1} = {S}^{4} \) . We pick \( {y}_{0} \in {S}^{7} \) and we wr... | Yes |
Let \( n \in \mathbb{N} \) . We denote by \( p : {S}^{{2n} + 1} \rightarrow {\mathbb{{CP}}}^{n} \) the Hopf map from page 2688. We have\n\n\[ \n{\pi }_{k}\left( {\mathbb{{CP}}}^{n}\right) = \left\{ \begin{array}{ll} \mathbb{Z} \cdot \left\lbrack {\mathrm{{id}}}_{{\mathbb{{CP}}}^{1}}\right\rbrack , & \text{ if }k = 2, \... | (1) By Proposition 113.9 (1) we know that for every \( n \in \mathbb{N} \) the Hopf map \( p : {S}^{{2n} + 1} \rightarrow {\mathbb{{CP}}}^{n} \) is an \( {S}^{1} \) -bundle. Thus we obtain from Proposition 113.2 the following long exact sequence of homotopy groups:\n\n\[ \n\ldots \rightarrow {\pi }_{k}\left( {S}^{1}\ri... | Yes |
Proposition 113.13. There exists a fiber bundle \( p : {S}^{15} \rightarrow {S}^{8} \) with fiber \( {S}^{7} \) . | The proof of Proposition 113.13 is rather longish, thus we will first point out that it allows us to prove the following theorem. | No |
(1) For all \( n \geq 2 \) we have \( {\pi }_{n}\left( {S}^{8}\right) \cong {\pi }_{n}\left( {S}^{15}\right) \oplus {\pi }_{n - 1}\left( {S}^{7}\right) \) . | The proof is basically the same as the proof of Theorem 113.11. We just need to replace Proposition 113.9 (2) by Proposition 113.13 | No |
Proposition 113.15. Let \( X \) be a topological space that is compact and Hausdorff. Furthermore let \( \varphi : X \times X \rightarrow X \) be a map. Given \( x \in X \) we consider the maps 1568\n\n\[\\begin{aligned} {L}_{x} : X & \\rightarrow X \\\\\na & \\mapsto \\varphi \\left( {x, a}\\right) \\end{aligned}\\;\\... | Proof. We consider the following two subsets of \( \\sum \\left( X\\right) \) :\n\n\[C \\mathrel{\\text{:=}} \\left( {\\left\\lbrack {-1,\\frac{1}{2}}\\right\\rbrack \\times X}\\right) /\\{ - 1\\} \\times X\\;\\text{ and }\\;D \\mathrel{\\text{:=}} \\left( {\\left\\lbrack {-\\frac{1}{2},1}\\right\\rbrack \\times X}\\ri... | No |
Lemma 113.16. Given any \( z \in {S}^{7} \) the maps \( {L}_{z} : {S}^{7} \rightarrow {S}^{7} \) and \( {R}_{z} : {S}^{7} \rightarrow {S}^{7} \), as defined in Proposition 113.15, are homeomorphisms. | Proof. First note that we can not recycle the argument from page 2693 since there we used that the quaternions are associative.\n\nBy Corollary 50.8 (2) it suffices to show that the maps \( {L}_{z} \) and \( {R}_{z} \) are injective. First we show that given any \( z \in {S}^{7} \) the map \( {L}_{z} : {S}^{7} \rightar... | Yes |
(1) We have \( \operatorname{Sp}\left( 1\right) = {S}^{3} \) . | (1) In Lemma 60.1 (5) we saw that for any quaternion \( h \in \mathbb{H} \) we have \( h \cdot \bar{h} = \parallel h{\parallel }^{2} \) . It follows immediately that \( \operatorname{Sp}\left( 1\right) = {S}^{3} \) . | No |
(1) Let \( n \in \mathbb{N} \). (a) The set \( \mathrm{O}\left( n\right) \) is a submanifold of \( \mathrm{M}\left( {n \times n,\mathbb{R}}\right) = {\mathbb{R}}^{{n}^{2}} \) of dimension \( \frac{1}{2}n\left( {n - 1}\right) \). (b) The set \( \mathrm{U}\left( n\right) \) is a submanifold of \( \mathrm{M}\left( {n \tim... | Proof. In Lemma 6.55 we had basically proved statements (1) and (2) for the groups \( \mathrm{O}\left( n\right) \) and \( \mathrm{U}\left( n\right) \). The only aspect that we did not mention explicitly so far is that \( \mathrm{O}\left( n\right) \) is a submanifold of \( \mathrm{O}\left( {n + 1}\right) \). We leave it... | No |
Lemma 114.4. Every Lie group is parallelizable 1579 | Sketch of Proof. Let \( G \) be an \( n \) -dimensional Lie group with trivial element \( e \) . We denote by \( {R}_{g} : G \rightarrow G \) the map that is given by right-multiplication by \( g \) . Since \( G \) is a Lie group this map is in fact a diffeomorphism. The map \[ G \times {T}_{e}G \rightarrow {TG} = \lef... | No |
Lemma 114.5. Let \( G \) be a Lie group and let \( K \) be a finite normal subgroup. The quotient \( G/K \) admits a unique smooth manifold structure such that the projection \( G \rightarrow G/K \) is a local diffeomorphism. Furthermore the group structure on the quotient group \( G/K \) defines a Lie group structure ... | Proof. In Lemma 114.1 we already saw that the action of \( K \) on \( G \) is free and continuous. Since \( G \) is a Lie group one also sees that the action is smooth. Finally, since \( G \) is finite we get for free that the action is proper. Thus we obtain from Proposition 6.32 that \( G/K \) admits a unique smooth ... | No |
Theorem 114.6. Any compact Lie group is isomorphic (in the obvious sense) to a Lie subgroup of \( \mathrm{{GL}}\left( {n,\mathbb{R}}\right) \) for some \( n \in \mathbb{N} \) . | The theorem is a consequence of the so-called Peter-Weyl Theorem. We only stated  Theorem III.4.1] or alternatively to [Sep07, Theorem 3.28] for a proof. | No |
Lemma 114.7. Let \( n \in \mathbb{N} \) . The maps\n\n\[ \begin{matrix} \mathrm{O}\left( {n + 1}\right) /\mathrm{O}\left( n\right) \overset{p}{ \rightarrow }{S}^{n} & \mathrm{U}\left( {n + 1}\right) /\mathrm{U}\left( n\right) \overset{q}{ \rightarrow }{S}^{{2n} + 1} & \text{ and } & \mathrm{{Sp}}\left( {n + 1}\right) /... | Proof.\n\n(1) First we show that the given map \( p : \mathrm{O}\left( {n + 1}\right) /\mathrm{O}\left( n\right) \rightarrow {S}^{n} \) is well-defined. To do so we consider the map\n\n\[ \mathrm{O}\left( {n + 1}\right) \rightarrow {S}^{n} \]\n\n\[ V \mathrel{\text{:=}} \left( {{v}_{1}\ldots {v}_{n + 1}}\right) \mapsto... | Yes |
(1) Let \( G \) be a Lie group. If \( H \) is a closed subgroup of \( G \), then the projection \( G \rightarrow G/H \) is a bundle projection with fiber \( H \) . | (1) This statement is proved in Kiri08, Theorem 2.11. Alternatively, if \( G \) is a closed subgroup of some \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \), then the proposition is proved in Bre93, p. 110]. | No |
(1) For any \( n \in \mathbb{N} \) the inclusions \( i : \mathrm{{SO}}\left( n\right) \rightarrow \mathrm{{SL}}\left( {n,\mathbb{R}}\right) \) and \( j : \mathrm{{SL}}\left( {n,\mathbb{R}}\right) \rightarrow {\mathrm{{GL}}}_{ + }{\left( n,\mathbb{R}\right) }^{1585} \) are homotopy equivalences.\n\n(2) Let \( n \in {\ma... | Proof.\n\n(1) We proved this statement in Lemma 8.8.\n\n(2) First we consider the case \( n = 2 \) . We have isomorphisms\n\n\[ \n\begin{aligned} \mathbb{Z} & \cong {\pi }_{1}\left( {{S}^{1},\mathrm{{id}}}\right) \; = {\pi }_{1}\left( {\mathrm{{SO}}\left( 2\right) ,\mathrm{{id}}}\right) \overset{ \cong }{ \Rightarrow }... | Yes |
Proposition 114.13. Let \( G \) be a connected Lie group and let \( p : \widetilde{G} \rightarrow G \) be some covering (e.g. it could be the universal covering). We denote by \( {1}_{G} \) the neutral element in \( G \) . We pick a point \( {1}_{\widetilde{G}} \in {p}^{-1}\left( {1}_{G}\right) \) . Then \( \widetilde{... | Sketch of THE PROOF. First note that we saw in Proposition 17.1 that \( \widetilde{G} \) is a topological manifold that admits a unique smooth structure such that \( p : \widetilde{G} \rightarrow G \) is a smooth map. Now we turn towards defining the Lie group structure on the smooth manifold \( \widetilde{G} \) . In f... | No |
Lemma 114.15. The above group structures on \( \mathrm{O},\mathrm{U} \) and \( \mathrm{{Sp}} \) are continuous. | Proof. This lemma can surely be proved, with a little effort, \ | No |
Lemma 114.16. (*) The topological spaces \( \mathrm{O} \), U and \( \mathrm{{Sp}} \) admit \( \mathrm{{CW}} \) -structures such that each \( \mathrm{O}\left( n\right) \subset \mathrm{O},\mathrm{U}\left( n\right) \subset \mathrm{U} \) and \( \mathrm{{Sp}}\left( n\right) \subset \mathrm{{Sp}} \) is a subcomplex. | Proof. In Whd \( \mathbf{{J44}} \) it is shown that we can equip the \( \mathrm{O}\left( n\right) \) with CW-structures such that for each \( n \) the subset \( \mathrm{O}\left( n\right) \subset \mathrm{O}\left( {n + 1}\right) \) is actually a subcomplex. Analogous statements for \( \mathrm{U}\left( n\right) \) and \( ... | Yes |
Lemma 114.17. Given any \( k \in \mathbb{N} \) we have\n\n(1) \( \;{\pi }_{k}\left( \mathrm{O}\right) = \mathop{\lim }\limits_{ \rightarrow }{\pi }_{k}\left( {\mathrm{O}\left( n\right) }\right) \rightleftarrows {\pi }_{k}\left( {\mathrm{O}\left( N\right) }\right) \; \) for any \( N \in \mathbb{N} \) with \( N > k + 1 \... | Proof (*). We prove (1). The proofs of (2) and (3) are almost identical. We start out with the following claim.\n\nClaim. Given any compact subset \( K \) of \( \mathrm{O} \) there exists an \( n \in \mathbb{N} \) with \( K \subset \mathrm{O}\left( n\right) \) .\n\nLet \( K \) be a compact subset of \( \mathrm{O} \) . ... | Yes |
Theorem 114.19. (Bott Periodicity Theorem) For any \( i \in {\mathbb{N}}_{0} \) we have \( {}^{1593} \)\n\n(1)\n\[{\pi }_{i + 2}\left( \mathrm{U}\right) \cong {\pi }_{i}\left( \mathrm{U}\right)\]\n\n(2)\n\[{\pi }_{i + 4}\left( \mathrm{{Sp}}\right) \cong {\pi }_{i}\left( \mathrm{O}\right)\]\n\n(3)\n\[{\pi }_{i + 4}\left... | Proof of the Bott Periodicity Theorems 114.18 and 114.19. First note that Theorem 114.19 follows immediately from Theorem 114.18 together with Corollary 112.15 All the known proofs are non-trivial and go well-beyond what we can do in these modest notes. | No |
Proposition 114.20. The homotopy groups of \( \mathrm{O},\mathrm{U} \) and \( \mathrm{{Sp}} \) are as follows:\n\n<table><thead><tr><th>\( i{\;\operatorname{mod}\;8} \)</th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th></tr></thead><tr><td>\( {\pi }_{i}\left( \mathrm{U}\right) \)</td><... | Proof. The homotopy groups of \( \mathrm{U} \) are straightforward to determine. More precisely, let \( i \in {\mathbb{N}}_{0} \) . We have\n\n\[ \n{\pi }_{i}\left( \mathrm{U}\right) = {\pi }_{i{\;\operatorname{mod}\;2}}\left( \mathrm{U}\right) \in {\pi }_{i{\;\operatorname{mod}\;2}}\left( {\mathrm{U}\left( 1\right) }\... | Yes |
Theorem 114.21. Let \( n \in \mathbb{N} \) . If \( {S}^{n} \) admits the structure of an H-space, then \( n \in \{ 1,3,7\} \) . | Proof. Let \( \mu : {S}^{n} \times {S}^{n} \rightarrow {S}^{n} \) be the multiplication map of an H-space structure and let \( e \in X \) be the \ | No |
Lemma 114.22. Let \( \left( {X,{x}_{0}}\right) \) be a pointed topological space. We denote by \( \Omega \left( {X,{x}_{0}}\right) \) the loop space as defined on page 2661.\n\n(1) The map\n\n\[ \mu : \Omega \left( {X,{x}_{0}}\right) \times \Omega \left( {X,{x}_{0}}\right) \rightarrow \Omega \left( {X,{x}_{0}}\right) \... | Proof (*). Let \( \left( {X,{x}_{0}}\right) \) be a pointed topological space. To simplify the notation we write \( {\Omega X} \) instead of \( \Omega \left( {X,{x}_{0}}\right) \).\n\n(1) We need to show that the map \( \mu : {\Omega X} \times {\Omega X} \rightarrow {\Omega X} \) is in fact continuous. By the \ | No |
Lemma 114.23. If \( X \) is a path-connected \( H \) -space, then \( {\pi }_{1}\left( X\right) \) is abelian. | First PROOF OF LEMMA 114.23 (*). We denote by \( \mu : X \times X \rightarrow X \) and \( e \in X \) the objects in the definition of an H-space. Let \( f, g : \left\lbrack {0,1}\right\rbrack \rightarrow X \) be loops in \( e \) . We need to show that the path \( f * g \) is path-homotopic to the path \( g * f \) .\n\n... | Yes |
Proposition 114.26. Let \( G \) be a Lie group. There exists a compact Lie subgroup \( {K}^{1601} \) such that \( G \) is homeomorphic to \( K \times {\mathbb{R}}^{n} \) for some \( n \in {\mathbb{N}}_{0} \) . | Proof of Proposition 114.26. Like many other foundational results on Lie groups this proposition has its origins in the work of Henri Cartan Carta36. A full proof is given in [Mal45, Theorem 11] or [Mos49, Theorem 2]. A sketch of the proof is provided in [Sam52, Chapter 7]. | No |
Proposition 40.7 (2) it suffices to prove the desired statement for \( K \) . Since \( K \) is in particular a compact smooth manifold we obtain from Proposition 64.6 that \( {\bigoplus }_{i = 0}^{\infty }{\mathrm{H}}_{i}\left( {X;\mathbb{Z}}\right) \) is a finitely generated abelian group. | Thus we can conclude the proof by appealing to Theorem 114.24. | Yes |
Lemma 114.28. Let \( X \) be a path-connected \( H \) -space and let \( {x}_{0} \in X \) . In the following we denote by \( \mu : X \times X \rightarrow X \) the multiplication map of the \( H \) -space. We consider the following two binary operations on \( {\pi }_{1}\left( {X,{x}_{0}}\right) \) :\n\n\[ \n\\begin{array... | Proof. Let \( X \) be a path-connected H-space and let \( {x}_{0} \in X \) . We denote by \( \mu : X \times X \rightarrow X \) the multiplication map of the H-space. Since \( \mu \) is an \( H \) -space structure one sees easily that \ | No |
Lemma 115.2. Let \( m \in \mathbb{N} \) . Given any real number \( r \in \left( {0,1}\right) \) there exists a homotopy \( F : {\bar{B}}^{m}/{S}^{m - 1} \times \left\lbrack {0,1}\right\rbrack \rightarrow {\bar{B}}^{m}/{S}^{m - 1} \) rel \( * \) with the following three properties:\n\n(1) \( {F}_{0} = \mathrm{{id}} \) ,... | Proof. Let \( r \in \left( {0,1}\right) \) . We consider the map\n\n\[ \left( {{\bar{B}}^{m}/{S}^{m - 1}}\right) \times \left\lbrack {0,1}\right\rbrack \rightarrow {\bar{B}}^{m}/{S}^{m - 1} \]\n\n\[ \left( {\left\lbrack x\right\rbrack, t}\right) \mapsto \left\{ \begin{array}{ll} \left\lbrack {x \cdot \left( {1 - t + t ... | No |
Lemma 115.4. Let \( K \) be the trivial knot and let \( g : {\bar{B}}^{2} \times K \rightarrow {\mathbb{R}}^{3} \) be a thickening.\n\n(1) If \( g \) is a thickening with self-linking number zero, then \( \left\lbrack {\rho }_{\left( K, g\right) }\right\rbrack = 0 \in {\pi }_{3}\left( {{S}^{2}, * }\right) \) . | (1) Let \( g \) be the thickening of the trivial knot \( K \) with self-linking number zero. We want to show that \( \left\lbrack {\rho }_{\left( K, g\right) }\right\rbrack = 0 \in {\pi }_{3}\left( {{S}^{2}, * }\right) \) . By Proposition 54.9 it suffices to show that the map \( {\rho }_{\left( K, g\right) } : {S}^{3} ... | No |
Lemma 115.5. Let \( n, k \in {\mathbb{N}}_{0} \) with \( n + k \geq 2 \) . Let \( \left( {{N}_{i},{g}_{i}}\right), i = 1,\ldots, m \) be thickened \( k \) - dimensional submanifolds of \( {\mathbb{R}}^{n + k} \) . If the images \( {g}_{i}\left( {{\bar{B}}^{n} \times {N}_{i}}\right) \) are contained in \( m \) disjoint ... | Proof. This lemma follows easily from Proposition 53.11. We leave it to the reader to fill in the details. | No |
Proposition 115.6. Let \( k, n \in {\mathbb{N}}_{0} \) . Given any \( \varphi \in {\pi }_{n + k}\left( {{S}^{n}, * }\right) \) there exists a thickened \( k \) -dimensional submanifold \( \left( {N, g}\right) \) of \( {\mathbb{R}}^{n + k} \) with \( \left\lbrack {\rho }_{\left( N, g\right) }\right\rbrack = \varphi \) . | Proof. Throughout this proof we find it convenient to view each \( m \) -dimensional sphere as the quotient \( {\bar{B}}^{m}/{S}^{m - 1} \) . So suppose we are given a map \( \varphi : {\bar{B}}^{n + k}/{S}^{n + k - 1} \rightarrow {\bar{B}}^{n}/{S}^{n - 1} \) with \( \varphi \left( *\right) = * \) . As we pointed out i... | No |
Lemma 115.7. Let \( \left( {M, g}\right) \) be a thickened submanifold of \( {\mathbb{R}}^{m} \) and let \( \varphi \) be a diffeomorphism of \( {\mathbb{R}}^{m} \) . If \( \varphi \) is diffeotopic to the identity, then \( \left( {\varphi \left( M\right) ,{\varphi }_{ * }\left( g\right) }\right) \) is cobordant to \( ... | Proof. Let \( \left( {M, g : {\bar{B}}^{m - k} \times M \rightarrow {\mathbb{R}}^{m}}\right) \) be a thickened \( k \) -dimensional submanifold of \( {\mathbb{R}}^{m} \) and furthermore let \( F : {\mathbb{R}}^{m} \times \left\lbrack {0,1}\right\rbrack \rightarrow {\mathbb{R}}^{m} \) be a diffeotopy from the identity t... | Yes |
Theorem 115.11. (Thom-Pontryagin Theorem) Let \( n, k \in {\mathbb{N}}_{0} \) with \( n + k \geq 2 \) . The map\n\n\[ \Psi : {\Omega }_{k}^{\mathrm{{th}}}\left( {\mathbb{R}}^{n + k}\right) \rightarrow {\pi }_{n + k}\left( {{S}^{n}, * }\right) \]\n\n\[ \left\lbrack \left( {N, g}\right) \right\rbrack \mapsto \left\lbrack... | SKETCH OF PROOF. Fortunately we have already done basically all the work:\n\n(1) First we argue that \( \Psi \) is well-defined. Thus suppose that we are given thickened \( k \) -dimensional submanifolds \( \left( {M, g}\right) \) and \( \left( {N, h}\right) \) of \( {\mathbb{R}}^{n + k} \) that are cobordant. We pick ... | No |
Corollary 115.12. For any \( m \geq 2 \) the map\n\n\[ \n\mathbb{Z} \rightarrow {\pi }_{m}\left( {{S}^{m}, * }\right)\n\]\n\n\[ \nn \mapsto n \cdot \left\lbrack {\operatorname{id}}_{{S}^{m}}\right\rbrack\n\]\n\nis an isomorphism. | Proof. Let \( P \in {\mathbb{R}}^{m} \) and let \( g \) be a thickening for the submanifold \( \{ P\} \) . We consider the two maps\n\n\[ \n\begin{aligned} \mathbb{Z} & \rightarrow {\Omega }_{0}^{\mathrm{{th}}}\left( {\mathbb{R}}^{m}\right) \\ n & \mapsto n \cdot \left\lbrack \left( {\{ 0\}, g\text{ with }\operatorname... | Yes |
Lemma 115.13. If \( \left( {L, g}\right) = \left( {{L}_{1} \sqcup \cdots \sqcup {L}_{m},{g}_{1} \sqcup \cdots \sqcup {g}_{m}}\right) \) is a thickened link, then\n\n\[ \operatorname{slk}\left( {L, g}\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{i \neq j}}\operatorname{lk}\left( {{L}_{i},{L}_{j}}\right) + \mathop{... | Proof. We will prove the lemma in Exercise 115.2, | No |
Theorem 115.14. The map\n\n\[ \Psi : {\Omega }_{1}^{\mathrm{{th}}}\left( {\mathbb{R}}^{3}\right) \rightarrow \mathbb{Z} \]\n\n\[ \left\lbrack \left( {L, g}\right) \right\rbrack \mapsto \text{self-linking number}\operatorname{slk}\left( {L, g}\right) \]\n\n\nis well-defined and it is an isomorphism. In particular we hav... | (1) We consider the trivial knot \( K \) with a thickening \( g \) of self-linking number +1 . By Theorem 115.14 the thickened knot \( \left( {K, g}\right) \) defines a generator of \( {\Omega }_{1}^{\text{th }}\left( {\mathbb{R}}^{3}\right) \) . It follows from the isomorphism provided by the Thom-Pontryagin Theorem 1... | Yes |
Lemma 115.15. Given any oriented link \( L \subset {S}^{3} \) there exists a compact oriented connected surface \( F \subset {S}^{3} \) such that \( \partial F = L. \) | Proof of Lemma 115.15. Let \( L \subset {S}^{3} \) be an oriented link. A modest generalization of Proposition 99.6 shows that there exists a compact oriented surface \( G \subset {S}^{3} \) with \( \partial G = L \) . In Exercise 1 \( F \) with \( \partial F = L \) . | No |
Lemma 115.16. If \( K = {K}_{1} \sqcup \cdots \sqcup {K}_{m} \) and \( L = {L}_{1} \sqcup \cdots \sqcup {L}_{n} \) are two oriented links, then the following statements hold:\n\n(1) There exist compact oriented proper 2-dimensional submanifolds \( S \) and \( T \) of \( {\bar{B}}^{4} \) with \( \partial S = K \) and \(... | ## Proof of Lemma 115.16.\n\n(1) The proof of this statement is virtually the same as the proof of Lemma 99.14, we just need to replace Proposition 99.6 by Lemma 115.15.\n\n(2) We leave it to the reader to verify that the proof of Lemma 99.14 (2) can be modified to deal not only with knots but also with links. | No |
Lemma 115.17. Let \( K \) be an oriented knot in \( {\mathbb{R}}^{3} \) and let \( g \) and \( h \) be two thickenings for \( K \) in \( {\mathbb{R}}^{3} \) . If \( \operatorname{slk}\left( {K, g}\right) = \operatorname{slk}\left( {K, h}\right) \), then \( \left\lbrack \left( {K, g}\right) \right\rbrack = \left\lbrack ... | Proof of Lemma \( {115.17}\left( *\right) \) . Since \( \operatorname{slk}\left( {K, g}\right) = \operatorname{slk}\left( {K, h}\right) \) it follows from Lemma 99.1 that there exists a smooth isotopy \( F : \left( {{\bar{B}}^{2} \times K}\right) \times \left\lbrack {0,1}\right\rbrack \rightarrow {\mathbb{R}}^{3} \) be... | Yes |
Lemma 115.18. Let \( W \) be a compact oriented proper connected 2-dimensional submanifold of \( {\mathbb{R}}^{3} \times \lbrack 0,1) \) with boundary components \( {L}_{1},\ldots ,{L}_{m} \) . Given \( {s}_{1},\ldots ,{s}_{m - 1} \in \mathbb{Z} \) there exists a thickening \( h \) for \( W \) such that for \( i = 1,\l... | Proof. Evidently we can assume that \( m \geq 1 \) . In this case \( W \) is a compact oriented proper connected 2-dimensional submanifold with non-empty boundary. Thus by Proposition 10.6 (3) there exists a thickening \( f : {\bar{B}}^{2} \times W \rightarrow {\mathbb{R}}^{3} \times \lbrack 0,1) \) for \( W \) . For \... | Yes |
Proposition 115.19. Let \( n \in \mathbb{N} \) and let \( * \in {\bar{B}}^{2n} \) . The map\n\n\[ \Psi : {\Omega }_{{2n} - 1}^{\text{th }}\left( {\mathbb{R}}^{{4n} - 1}\right) \rightarrow \mathbb{Z} \]\n\n\[ \left\lbrack \left( {N, g : {\bar{B}}^{2n} \times N \rightarrow {\mathbb{R}}^{{4n} - 1}}\right) \right\rbrack \m... | Proof. In Exercise 115.7 we will verify that \( \Psi \) is a homomorphism. Now let us actually show that \( \Psi \) is well-defined. Thus suppose that we are given thickened \( \left( {{2n} - 1}\right) \) -dimensional submanifolds \( \left( {{N}_{0},{g}_{0}}\right) \) and \( \left( {{N}_{1},{g}_{1}}\right) \) of \( {\m... | No |
Corollary 87.28 (2) that \( {\left( {i}_{0}\right) }_{ * }\left( \left\lbrack {{g}_{0}\left( {* \times {N}_{0}}\right) }\right\rbrack \right) = {\left( {i}_{1}\right) }_{ * }\left( \left\lbrack {{g}_{1}\left( {* \times {N}_{1}}\right) }\right\rbrack \right) \) | \[ = \operatorname{lk}\left( {{N}_{1},{g}_{1}\left( {* \times {N}_{1}}\right) }\right) \text{.} \] | No |
Proposition 115.20. For any \( n \in \mathbb{N} \) there exists a thickened \( \left( {{2n} - 1}\right) \) -dimensional sub-manifold \( \left( {N, g}\right) \) in \( {\mathbb{R}}^{{4n} - 1} \) with \( \Psi \left( \left\lbrack \left( {N, g}\right) \right\rbrack \right) = 2 \) . | Sketch of Proof. We consider the \ | No |
Proposition 116.4. Let \( m \in {\mathbb{N}}_{ \geq 2} \) and let \( k \in \{ 0,\ldots, m\} \) . The map\n\n\[ \Phi : {\Omega }_{k}^{\mathrm{{th}}}\left( {\mathbb{R}}^{m}\right) \rightarrow {\Omega }_{k}^{\mathrm{{fr}}}\left( {\mathbb{R}}^{m}\right) \]\n\n\[ \left\lbrack \left( {N, g : {\bar{B}}^{m - k} \times N}\right... | SKETCH OF PROOF.\n\n(1) It follows basically from the definitions that the framing associated to a thickened submanifold is indeed a framing. Applying the same procedure to a thickened cobordism between thickened submanifolds we obtain a framed cobordism between framed submanifolds. This shows that \( \Phi \) is well-d... | No |
Lemma 116.5. Let \( m \in {\mathbb{N}}_{ \geq 2} \) and let \( k \in \{ 0,\ldots, m\} \) .\n\n(1) The map\n\n\[ \n{\Omega }_{k}^{\mathrm{{fr}}}\left( {\mathbb{R}}^{m}\right) \rightarrow {\Omega }_{k}^{\mathrm{{fr}}}\left( {\mathbb{R}}^{m + 1}\right) \n\]\n\n\[ \n\left\lbrack \left( {N, v}\right) \right\rbrack \mapsto \... | Proof.\n\n(1) The first statement is almost obvious.\n\n(2) We leave it to the reader to go through all the definitions to verify this statement. | No |
Theorem 116.7. Let \( n, k \in \mathbb{N} \). (1) The suspension homomorphism \( {\Omega }_{k}^{\mathrm{{fr}}}\left( {\mathbb{R}}^{n + k}\right) \rightarrow {\Omega }_{k}^{\mathrm{{fr}}}\left( {\mathbb{R}}^{n + k + 1}\right) \) is an epimorphism whenever \( n \geq k + 1 \). | The basic behind the proof of Theorem 116.7 (1) is quite simple: given a framed \( k \) - dimensional submanifold of \( {\mathbb{R}}^{n + k + 1} \) we want to use the \ | No |
Lemma 116.9. Let \( \\left( {M, v}\\right) \) be a framed \( k \) -dimensional submanifold in \( {\\mathbb{R}}^{n + k} \\times \\{ 0\\} \\subset \) \( {\\mathbb{R}}^{n + k + 1} \) . If \( n \\geq k + 1 \), then there exists a framed cobordism to \( \\left( {M, w = \\left( {{w}_{1},\\ldots ,{w}_{n + 1}}\\right) }\\right... | Proof. Let \( \\left( {M, u = \\left( {{u}_{1},\\ldots ,{u}_{n + 1}}\\right) : M \\rightarrow \\mathrm{{GL}}\\left( {n + k + 1, n + 1}\\right) }\\right) \) be a framed \( k \) -dimensional submanifold that is contained in \( {\\mathbb{R}}^{n + k} \\times \\{ 0\\} \\subset {\\mathbb{R}}^{n + k + 1} \) . By Lemma 116.3 (... | No |
(1) Given any \( g \in {\mathbb{N}}_{0} \) there exists a proper smooth embedding \( \psi : {\sum }_{g,1} \rightarrow \left\lbrack {0,1}\right\rbrack \times {\mathbb{R}}^{2} \) and a framing \( w \) for \( \psi \left( {\sum }_{g,1}\right) \) such that \( \partial \psi \left( {\sum }_{g,1}\right) = {\mathrm{C}}_{2} \) a... | (1) By now the seasoned reader should be fairly convinced that there exists a proper smooth embedding \( \psi : {\sum }_{g,1} \rightarrow \left\lbrack {0,1}\right\rbrack \times {\mathbb{R}}^{2} \) with \( \psi \left( {\partial {\sum }_{g,1}}\right) = {\mathrm{C}}_{2} \) and such that the smooth embedding is a \ | No |
(1) Under the isomorphism \( {\Omega }_{1}^{\mathrm{{fr}}}\left( {\mathbb{R}}^{3}\right) \cong {\pi }_{3}\left( {{S}^{2}, * }\right) \) the element \( \left\lbrack \left( {{\mathrm{C}}_{3},{h}_{3}}\right) \right\rbrack \) corresponds, up to a sign, to the element represented by the Hopf map \( {S}^{3} \rightarrow {S}^{... | Proof. The first statement follows easily from Lemma 115.4. | No |
Proposition 117.4. Let \( n \in {\mathbb{N}}_{ \geq 4} \) . The map \[ {\mathbb{Z}}_{2} \rightarrow {\Omega }_{1}^{\mathrm{{fr}}}\left( {\mathbb{R}}^{n}\right) \] \[ \left\lbrack k\right\rbrack \mapsto k \cdot \left\lbrack \left( {{C}_{n},{h}_{n}}\right) \right\rbrack \] is well-defined and it is an epimorphism. | Our proof of Proposition 117.4 requires one definition and two lemmas. | No |
Lemma 117.5. Let \( n \in {\mathbb{N}}_{ \geq 3} \) and let \( \gamma : {S}^{1} \rightarrow {C}_{n} \) be the obvious diffeomorphism. We consider the loop\n\n\[ \alpha : {S}^{1} \rightarrow {\mathrm{{GL}}}_{ + }\left( {n - 1,\mathbb{R}}\right) \]\n\n\[ z = {e}^{\mathrm{i}\varphi } \mapsto \left( \begin{matrix} \cos \le... | Proof (*). The first statement follows immediately from the definitions. We turn to the proof of the second statement. As we will see, the proof of the second statement is given by a slightly tedious calculation. To simplify the notation we only deal with the case \( n = 3 \) .\n\n\( {}^{1647} \) Here \( \bar{\alpha } ... | No |
Lemma 117.6. Let \( n \in {\mathbb{N}}_{ \geq 2} \) and let \( \gamma : {S}^{1} \rightarrow {\mathbb{R}}^{n} \) be a smooth embedding. Furthermore let \( f : \gamma \left( {S}^{1}\right) \rightarrow \mathrm{{GL}}\left( {n, n - 1}\right) \) be a framing for \( \gamma \left( {S}^{1}\right) \) . Finally let \( \alpha ,\be... | Proof of Lemma 117.6 (*). We write \( M = \gamma \left( {S}^{1}\right) \) . Since \( \alpha ,\beta : {S}^{1} \rightarrow {\mathrm{{GL}}}_{ + }\left( {n - 1,\mathbb{R}}\right) \) are smoothly homotopic we can pick a smooth homotopy \( H : {S}^{1} \times \left\lbrack {0,1}\right\rbrack \rightarrow {\mathrm{{GL}}}_{ + }\l... | Yes |
Lemma 117.5 \( \left( 1\right) \; \) since \( n \geq 4 \) we know by Proposition 114.11 \( \left( 2\right) \) that \( {\pi }_{1}\left( {{\mathrm{{GL}}}_{ + }\left( {n - 1,\mathbb{R}}\right) }\right) \cong {\mathbb{Z}}_{2} \) | therefore we see that \( \left\lbrack \alpha \right\rbrack \cdot \left\lbrack \alpha \right\rbrack = 0 \in \overline{{\pi }_{1}\left( {{\mathrm{{GL}}}_{ + }\left( {n - 1,\mathbb{R}}\right) }\right) } \) , this implies by Proposition 14.7 that \( \alpha \) is path-homotopic to \( \bar{\alpha } \) , in fact, by the Whitn... | Yes |
Let \( n \geq 4 \) . Note that by Proposition 117.4 we now know that there exists an epimorphism \( {\mathbb{Z}}_{2} \rightarrow {\Omega }_{1}^{\mathrm{{fr}}}\left( {\mathbb{R}}^{n}\right) \cong {\pi }_{n}\left( {S}^{n - 1}\right) \) . In Theorem 109.13 we showed that \( {\pi }_{n + 1}\left( {S}^{n}\right) \) is non-tr... | Therefore it follows that \( {\pi }_{n + 1}\left( {S}^{n}\right) \cong {\mathbb{Z}}_{2} \), which means we have proved Theorem 117.1. | Yes |
Proposition 117.8. \( {}^{1649} \) This sentence contains a slight technical nuisance which we cleverly hide in a footnote. First of all, it should be pretty clear by now that \( \sum \) admits an orientation-reversing self-diffeomorphism. Thus, after possibly precomposing \( {\varphi }_{n} \) with such a diffeomorphis... | Using Corollary 30.2 and using the Collar Neighborhood Theorem 8.12 we can then arrange, after possibly precomposing \( \overline{{\varphi }_{n}\mathrm{\;b}}\mathrm{y} \) another diffeomorphism, that \( {\left. {\psi }_{n}\right| }_{\partial \sum } = {\left. {\varphi }_{n}\right| }_{\partial \sum } \). | Yes |
Lemma 117.9. Let \( C \) and \( D \) be two closed oriented 1-dimensional submanifolds of a closed oriented framed 2-dimensional submanifold \( \left( {F, v}\right) \) of \( {\mathbb{R}}^{n + 2} \) . If \( \left\lbrack C\right\rbrack = \left\lbrack D\right\rbrack \in {\mathrm{H}}_{1}\left( {F;\mathbb{Z}}\right) \), the... | Proof. Note that it follows from our hypothesis that \( \left\lbrack C\right\rbrack = \left\lbrack D\right\rbrack \in {\mathrm{H}}_{1}\left( {F;\mathbb{Z}}\right) \) together with Corollary 98.7 that there exists a compact oriented proper 2-dimensional submanifold \( W \subset \left\lbrack {0,1}\right\rbrack \times F \... | Yes |
Proposition 117.10. The map \( {q}_{\left( F, v\right) } : {\mathrm{H}}_{1}\left( {F;{\mathbb{Z}}_{2}}\right) \rightarrow {\mathbb{Z}}_{2} \) has the property, that for any \( x, y \in {\mathrm{H}}_{1}\left( {F;{\mathbb{Z}}_{2}}\right) \) we have\n\n\[ q\left( {x + y}\right) = q\left( x\right) + q\left( y\right) + {Q}_... | Proof. This requires some thought. Note that it suffices to deal with the case that \( F \) is standard in \( {\mathbb{R}}^{3} \) . | No |
Proposition 117.11. Let \( V \) be a finite-dimensional vector space over \( {\mathbb{F}}_{2} \). (1) Every non-singular +++++ even +++++ symmetric form \( \langle \) , \( \rangle : V \times V \rightarrow {\mathbb{F}}_{2} \) admits a symplectic basis, i.e. there exists a basis \( {a}_{1},\ldots ,{a}_{k},{b}_{1},\ldots ... | (1) The proposition follows immediately from Proposition 101.22. | No |
(1) The quadratic forms \( {\sigma }_{0} \oplus {\sigma }_{0} \) and \( {\sigma }_{1} \oplus {\sigma }_{1} \) on \( \mathcal{W} \oplus \mathcal{W} \) are isometric. | (1) Let \( {x}_{1},{y}_{1},{x}_{2},{y}_{2} \) be the obvious basis of \( \mathcal{W} \oplus \mathcal{W} = {\mathbb{F}}_{2}^{2} \oplus {\mathbb{F}}_{2}^{2} \) . We consider the isomorphism \( \varphi : \mathcal{W} \oplus \mathcal{W} \rightarrow \mathcal{W} \oplus \mathcal{W} \) that is determined by\n\n\[ \n{x}_{1} \map... | Yes |
Lemma 117.13. Given any \( m \in \mathbb{N} \) we have\n\n\[ \n\# N\left( {m \cdot {\sigma }_{0}}\right) > \# P\left( {m \cdot {\sigma }_{0}}\right) \;\text{ and } \n\]\n\n\[ \n\# N\left( {\left( {m - 1}\right) \cdot {\sigma }_{0} \oplus {\sigma }_{1}}\right) < \# P\left( {\left( {m - 1}\right) \cdot {\sigma }_{0} \opl... | Proof. Given a quadratic form \( \left( {V, q}\right) \) over \( {\mathbb{F}}_{2} \) we set\n\n\[ \nr\left( q\right) \mathrel{\text{:=}} \# \{ v \in V \mid q\left( v\right) = 0\} - \# \{ v \in V \mid q\left( v\right) = 1\} .\n\]\n\nThe following claim contains the key calculation that we need to complete the proof of t... | Yes |
(1) Let \( \left( {V, q}\right) \) be a non-degenerate quadratic form. We set \( m \mathrel{\text{:=}} \dim \left( V\right) \) .\n\n(a) \( \left( {V, q}\right) \) is isometric to \( m \cdot {\sigma }_{0} \) if and only if \( \operatorname{Arf}\left( q\right) = 0 \) .\n\n(b) \( \left( {V, q}\right) \) is isometric to \(... | (1) This statement \( f \) is an immediate consequence of Lemmas 117.12 and 117.13. | No |
Proposition 117.15. Let \( \\left( {V, q}\\right) \) be a non-degenerate quadratic form over \( {\\mathbb{F}}_{2} \). Given any symplectic basis \( {a}_{1},\\ldots ,{a}_{n},{b}_{1},\\ldots ,{b}_{n} \) of the associated symmetric form we have the following equality:\n\n\[ \n\\operatorname{Arf}\\left( q\\right) = \\matho... | Proof. Let \( \\left( {V, q}\\right) \) be a non-degenerate quadratic form over \( {\\mathbb{F}}_{2} \) and let \( {a}_{1},\\ldots ,{a}_{n},{b}_{1},\\ldots ,{b}_{n} \) be a symplectic basis of the associated symmetric form. As in the proof of Lemma 117.12 we denote by \( {V}_{i} \) the span of \( \\left\{ {{a}_{i},{b}_... | Yes |
Proposition 117.16. Let \( \left( {F, v}\right) \) be a closed oriented framed 2-dimensional submanifold of \( {\mathbb{R}}^{n} \) . If \( \left\lbrack \left( {F, v}\right) \right\rbrack = 0 \in {\Omega }_{2}^{\mathrm{{fr}}}\left( {\mathbb{R}}^{n + 2}\right) \), then \( \left. {\operatorname{Arf}\left( {\lbrack F, v)}\... | Proof. Let \( \left( {F, v}\right) \) be a closed oriented framed 2-dimensional submanifold of \( {\mathbb{R}}^{n + 2} \) which represents the trivial element in \( {\Omega }_{2}^{\mathrm{{fr}}}\left( {\mathbb{R}}^{n + 2}\right) \) . By definition this means that there exists a compact oriented proper framed 3-dimensio... | No |
Proposition 117.17. (1) Let \( n \geq 2 \) . The map \[ \text{Arf} : {\Omega }_{2}^{\mathrm{{fr}}}\left( {\mathbb{R}}^{n + 2}\right) \rightarrow {\mathbb{Z}}_{2} \] \[ \left\lbrack \left( {F, v}\right) \right\rbrack \mapsto \operatorname{Arf}\left( {F, v}\right) \] is a well-defined homomorphism. | Proof. (1) By Proposition 117.16 we know that the map \( \left\lbrack \left( {F, v}\right) \right\rbrack \mapsto \operatorname{Arf}\left( {F, v}\right) \) is well-defined. It follows easily from Theorem 117.14 that the map is actually a homomorphism. | Yes |
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