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Lemma 19.32. Let \( K \) be a compact space, let \( \mathcal{S} \) be a commutative semigroup of self-homeomorphisms of \( K \) acting minimally on \( K \), let \( {\varphi }_{1},\ldots ,{\varphi }_{k} \in \mathcal{S} \) and let \( I \subseteq \mathbb{N} \) be an IP set. Suppose that for every \( r \in \mathbb{N} \), f... | Proof. Let \( \varnothing \neq V \subseteq K \) be an open set. Since \( K \smallsetminus \mathop{\bigcup }\limits_{{\psi \in \mathcal{S}}}{\psi }^{-1}V \neq K \) is closed and invariant under the action of \( \mathcal{S} \), it must be empty by minimality of the action. By compactness there are \( {\psi }_{1},\ldots ,... | Yes |
Theorem 19.36 (Gallai, Witt, Finitary Version). Let \( r, m \in \mathbb{N} \) and let \( F \subseteq {\mathbb{N}}^{m} \) be a finite nonempty set. Then there exists \( N = N\left( {r, m, F}\right) \in \mathbb{N} \) such that whenever\n\n\[ \n\{ 1,2,\ldots, N{\} }^{m} = {A}_{1} \cup {A}_{2} \cup \cdots \cup {A}_{r} \n\]... | Proof. Fix \( k \in \mathbb{N} \) with \( F \subseteq \{ 1,\ldots, k{\} }^{m} \) . Let \( K \mathrel{\text{:=}} \{ 1,\ldots, r{\} }^{{\mathbb{N}}^{m}} \) be the space of all \( r \) -colorings of \( {\mathbb{N}}^{m} \) and define the mapping\n\n\[ \n\lambda : K \rightarrow \mathbb{N},\;\lambda \left( x\right) \mathrel{... | Yes |
Theorem 20.5 (Szemerédi, Finitary Version). For every \( \varepsilon > 0 \) and \( k \in \mathbb{N}, k \geq 2 \) there is \( N = N\left( {\varepsilon, k}\right) \) such that whenever \( A \subseteq \mathbb{N} \) is contained in an interval of length \( \ell \geq N \) and \( \operatorname{card}\left( A\right) \geq \vare... | Proof. Suppose by contradiction that the statement is false, i.e., that there is \( \varepsilon > 0 \) and \( k \in \mathbb{N} \) such that for every \( N \) there is \( {A}_{N} \subseteq \mathbb{N} \) without \( k \) -term arithmetic progressions but contained in an interval of length \( \ell \geq N \) and having card... | Yes |
Proposition 20.6. Let \( \left( {X,\sum ,\mu }\right) \) be a probability space, let \( \varepsilon > 0 \) and \( k \in \mathbb{N}, k \geq 2 \) be given, and \( N \mathrel{\text{:=}} N\left( {\varepsilon /2, k}\right) \) obtained from the finitary Szemerédi Theorem 20.5. Suppose that \( {A}_{1},\ldots ,{A}_{N} \in \sum... | Proof. Define\n\n\[ A\left( x\right) \mathrel{\text{:=}} \left\{ {i : i \in \{ 1,2,\ldots, N\}, x \in {A}_{i}}\right\} \subseteq \mathbb{N}. \]\n\nFor the function \( f \) defined by \( f\left( x\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{i = 1}}^{N}{\mathbf{1}}_{{A}_{i}}\left( x\right) = \operatorname{card}\le... | Yes |
Theorem 20.7 (Furstenberg, Multiple Recurrence). Let \( \left( {\mathrm{X};\varphi }\right) ,\mathrm{X} = \left( {X,\sum ,\mu }\right) \) , be a measure-preserving system, and let \( A \in \sum \) with \( \mu \left( A\right) > 0 \) . Then for every \( k \in \mathbb{N} \) there is \( n \in \mathbb{N} \) such that \[ \mu... | Proof. Take \( {A}_{j} \mathrel{\text{:=}} {\varphi }^{-j}\left( A\right) ,\varepsilon \mathrel{\text{:=}} \mu \left( A\right) \), and let \( n, a \in \{ 1,2,\ldots, N\} \) be as in Proposition 20.6. Since \( \varphi \) is measure-preserving, the assertion follows. | No |
Corollary 20.8. Let \( \\left( {\\mathrm{X};\\varphi }\\right) ,\\mathrm{X} = \\left( {X,\\sum ,\\mu }\\right) \\), be a measure-preserving system with Koopman operator \( T \\mathrel{\\text{:=}} {T}_{\\varphi } \\), and let \( 0 < f \\in {\\mathrm{L}}^{\\infty }\\left( \\mathrm{X}\\right) \) . For every \( k \\in \\ma... | Proof. Apply Theorem 20.7 to the set \( A \\mathrel{\\text{:=}} \\left\\lbrack {f > \\frac{1}{2}\\parallel f{\\parallel }_{\\infty }}\\right\\rbrack \) and use that \( {T}^{j}f \\geq \) \( \\frac{1}{2}\\parallel f{\\parallel }_{\\infty }{T}^{j}{\\mathbf{1}}_{A} \) for each \( j \\in \\mathbb{N}. \) | Yes |
Lemma 20.9. Let \( \\left( {K;\\varphi }\\right) \) be a topological system with \( K \) metrizable, \( T \\mathrel{\\text{:=}} {T}_{\\varphi } \) be the Koopman operator on \( \\mathrm{C}\\left( K\\right) \) and \( 0 < g \\in \\mathrm{C}\\left( K\\right) \) . For every \( k \\in \\mathbb{N} \) and \( \\varepsilon > 0 ... | Proof. Suppose by contradiction that there are \( k,{n}_{0} \\in \\mathbb{N} \) and \( \\varepsilon > 0 \) such that for every \( j \\in \\mathbb{N} \) and for every \( {N}_{0} \\in \\mathbb{N} \) with \( {N}_{0} \\geq {n}_{0} \) there is an invariant probability measure \( {\\mu }_{j,{N}_{0}} \) with \( \\left\\langle... | Yes |
Lemma 20.10. Let \( \left( {\mathrm{X};\varphi }\right) \) and \( \left( {\mathrm{Y};\psi }\right) \) be measure-preserving systems and let \( S \in \mathrm{M}\left( {\mathrm{Y};\mathrm{X}}\right) \) be a Markov embedding that intertwines the Koopman operators \( {T}_{\varphi } \) and \( {T}_{\psi } \) . Then for all \... | Proof. By Theorem 13.9 a Markov embedding is multiplicative, the assertion hence follows from the identity \( {T}_{\varphi }S = S{T}_{\psi } \) . | No |
For every \( k \in \mathbb{N} \) and \( \varepsilon > 0 \) there is a constant \( {c}_{1}\left( {k,\varepsilon }\right) > 0 \) and there is \( {N}_{1} \mathrel{\text{:=}} {N}_{1}\left( {k,\varepsilon }\right) \in \mathbb{N} \) such that for every measure-preserving system \( \left( {\mathrm{X};\varphi }\right) \) with ... | Consider the shift system \( \left( {{\mathcal{W}}_{2}^{ + };\tau }\right) \), and let \( g \in \mathrm{C}\left( {\mathcal{W}}_{2}^{ + }\right) \) be the \( {0}^{\text{th }} \) coordinate projection, i.e., \( g\left( {\left( {x}_{n}\right) }_{n \in {\mathbb{N}}_{0}}\right) = {x}_{0} \) . For \( k \) and \( \varepsilon ... | Yes |
Proposition 20.15. Let \( \left( {\mathrm{X};\varphi }\right) \) be a weakly mixing measure-preserving system and let \( T \mathrel{\text{:=}} {T}_{\varphi } \) be the Koopman operator on \( {\mathrm{L}}^{2}\left( \mathrm{X}\right) \) . Then\n\n\[ \mathop{\lim }\limits_{{N \rightarrow \infty }}\frac{1}{N}\mathop{\sum }... | Recall that the van der Corput Lemma 9.28 played the central role in the proof. | No |
Proposition 20.16. Let \( \left( {\mathrm{X};\varphi }\right) \) be an ergodic measure-preserving system and let \( T \mathrel{\text{:=}} {T}_{\varphi } \) be the Koopman operator on \( {\mathrm{L}}^{2}\left( \mathrm{X}\right) \) . If \( T \) has discrete spectrum on \( {\mathrm{L}}^{2}\left( \mathrm{X}\right) \) , the... | Proof. Let \( {f}_{1},\ldots ,{f}_{k - 1} \in {\mathrm{L}}^{\infty }\left( \mathrm{X}\right) \) be eigenvectors of \( T \) corresponding to unimodular eigenvalues \( {\lambda }_{1},\ldots ,{\lambda }_{k - 1} \) . Then we have\n\n\[ \frac{1}{N}\mathop{\sum }\limits_{{n = 1}}^{N}\left( {{T}^{n}{f}_{1}}\right) \cdot \left... | Yes |
Lemma 20.18. Let \( \left( {\mathrm{X};\varphi }\right) \) be an ergodic measure-preserving system. Let \( f, g \in \) \( {\mathrm{L}}^{\infty }\left( \mathrm{X}\right) \) such that \( f \in {E}_{\text{aws }} \) or \( g \in {E}_{\text{aws }} \) . Then\n\n\[ \mathop{\lim }\limits_{{N \rightarrow \infty }}\frac{1}{N}\mat... | Proof. The proof is an application of the van der Corput lemma. We let \( {u}_{n} \mathrel{\text{:=}} \) \( \left( {{T}^{n}f}\right) \cdot \left( {{T}^{2n}g}\right) \) and write\n\n\[ \left( {{u}_{n} \mid {u}_{n + m}}\right) = {\int }_{\mathrm{X}}\left( {{T}^{n}f}\right) \cdot \left( {{T}^{2n}g}\right) \cdot \left( {{T... | Yes |
Theorem 20.22 (Furstenberg Correspondence Principle for Squares). If for every ergodic measure-preserving system \( \left( {\mathrm{X};\varphi }\right) \), its Koopman operator \( T \mathrel{\text{:=}} {T}_{\varphi } \) on \( {\mathrm{L}}^{2}\left( \mathrm{X}\right) \) and every \( 0 < f \in {\mathrm{L}}^{\infty }\left... | The proof of this correspondence principle is analogous to the one of Theorem 20.4. Then, to prove Theorem 20.21 one first shows that for a measure-preserving system \( \left( {\mathrm{X};\varphi }\right) \) and its Koopman operator \( T \mathrel{\text{:=}} {T}_{\varphi } \) the limit\n\n\[ \n\mathop{\lim }\limits_{{N ... | No |
Proposition 20.23. Let \( \left( {\mathrm{X};\varphi }\right) \) be an ergodic measure-preserving system. Then the limit (20.19) is strictly positive for every \( 0 < f \in {\mathrm{L}}^{\infty }\left( \mathrm{X}\right) \) . | The proof is analogous to that of Theorem 20.19 for \( k = 3 \) and again uses the decomposition \( {\mathrm{L}}^{2}\left( \mathrm{X}\right) = {E}_{\text{aws }} \oplus {E}_{\text{rev }} \), the vanishing of the above limit on \( {E}_{\text{aws }} \) and a relative denseness argument on \( {E}_{\mathrm{{rev}}} \), see E... | No |
Theorem 21.2. For a bounded sequence \( {\left( {a}_{n}\right) }_{n \in \mathbb{N}} \) in \( \mathbb{C} \) the following assertions are equivalent:\n\n(i) The sequence \( {\left( {a}_{n}\right) }_{n \in \mathbb{N}} \) is a good weight for the mean ergodic theorem.\n\n(ii) For every isometry \( T \) on a Hilbert space \... | Proof. First of all we note that the sum on the right-hand side of (21.3) is strongly convergent for every bounded function \( c : \mathbb{T} \rightarrow \mathbb{C} \), because by Exercise 16.19 eigenvectors of a contraction to different unimodular eigenfunctions are orthogonal.\n\nThe implications (i) \( \Rightarrow \... | Yes |
Corollary 21.6. Let \( \left( {\mathrm{X};\varphi }\right) \) be a measure-preserving system and \( f \in {\mathrm{L}}^{\infty }\left( \mathrm{X}\right) \) . Then for almost every \( x \) the sequence of weights \( {\left( f\left( {\varphi }^{n}\left( x\right) \right) \right) }_{n \in \mathbb{N}} \) is good for the mea... | The almost everywhere convergence of (21.4) for a fixed \( \lambda \in \mathbb{T} \) follows from Birkhoff’s Theorem 11.1 applied to the product system \( \left( {Y, v;\psi }\right) \) and \( g \in {\mathrm{L}}^{1}\left( {Y, v}\right) \) for \( Y \mathrel{\text{:=}} \mathbb{T} \times X \) with product measure \( v,\psi... | Yes |
Lemma 21.7. Take a bounded sequence \( {\left( {a}_{n}\right) }_{n \in \mathbb{N}} \) in \( \mathbb{C} \), an ergodic measure-preserving system \( \left( {\mathrm{X};\varphi }\right) \) and \( f,{f}_{1},{f}_{2},\ldots \) integrable functions on \( X \) with \( \mathop{\lim }\limits_{{j \rightarrow \infty }}{\begin{Vmat... | Proof. Since \( x \) is generic for \( \left| {f}_{j}\right| \) and the sequence \( {\left( {\begin{Vmatrix}{f}_{j}\end{Vmatrix}}_{1}\right) }_{j \in \mathbb{N}} \) is bounded, the sequence \( {\left( {b}_{j}\right) }_{j \in \mathbb{N}} \) is bounded and hence has a convergent subsequence, say \( {b}_{{j}_{m}} \rightar... | Yes |
Theorem 21.8 (Uniform Wiener-Wintner Theorem). Let \( T \) be a Koopman operator as above and let \( f \in {\mathrm{L}}^{1}\left( \mathrm{X}\right) \) be orthogonal to all eigenfunctions of \( T \) . Then\n\n\[ \mathop{\lim }\limits_{{N \rightarrow \infty }}\mathop{\sup }\limits_{{\lambda \in \mathbb{T}}}\left| {\frac{... | The proof is analogous to the one of Theorem 21.5 using a finitary version of the van der Corput lemma, see, e.g., Assani (2003, Ch. 2) and Schreiber (2013a). | No |
Theorem 21.10 (Bourgain Return Time Theorem). Let \( \left( {\mathrm{X};\varphi }\right) \) be a measure-preserving system and \( f \in {\mathrm{L}}^{\infty }\left( \mathrm{X}\right) \) . Then for almost every \( x \in X \), the sequence \( {\left( f\left( {\varphi }^{n}\left( x\right) \right) \right) }_{n \in \mathbb{... | For the proof we refer to the original paper of Bourgain (1989) or to a more detailed version in Assani (2003). Further proofs and generalizations are in Rudolph (1994, 1998), Ornstein and Weiss (1992) and Zorin-Kranich (2014a, 2014b), see also Demeter et al. (2008) and the survey article by Assani and Presser (2013). | No |
Theorem 21.13. Let \( \left( {\mathrm{X};\varphi }\right) ,\mathrm{X} = \left( {X,\sum ,\mu }\right) \), be a measure-preserving system and \( f \) be an integrable function on \( X \) . Then there exists a set \( {X}^{\prime } \in \sum \) with \( \mu \left( {X}^{\prime }\right) = 1 \) such that the weighted averages\n... | Proof. By the approximation argument based on Lemma 21.7, cf. the proof of Theorem 21.5, it suffices to take \( f \in {\mathrm{L}}^{\infty }\left( \mathrm{X}\right) \) . Let \( S \) be an operator on a Banach space \( E \) with relatively weakly compact orbits (hence power-bounded), \( y \in E,{y}^{\prime } \in {E}^{\p... | Yes |
Theorem 21.14. Let \( {\left( {k}_{n}\right) }_{n \in \mathbb{N}} \) be a subsequence of \( \mathbb{N} \) . Then the following assertions are equivalent:\n\n(i) For every contraction \( T \) on a Hilbert space \( H \), the averages\n\n\[ \n\frac{1}{N}\mathop{\sum }\limits_{{n = 1}}^{N}{T}^{{k}_{n}}f \n\]\n\n(21.6)\n\nc... | The proof of the implication (iv) \( \Rightarrow \) (ii) is Exercise 4, and (ii) \( \Rightarrow \) (i) follows as in the proof of Theorem 21.2. | No |
Theorem 21.17. Let \( q : \mathbb{N} \rightarrow \mathbb{N} \) be a polynomial. Then for every contraction \( T \) on a Hilbert space \( H \), the averages\n\n\[ \n\frac{1}{N}\mathop{\sum }\limits_{{n = 1}}^{N}{T}^{q\left( n\right) }f \n\]\n\n(21.7)\n\nconverge strongly for every \( f \in H \) as \( N \rightarrow \inft... | Proof. By Exercise 16.19 the eigenspaces corresponding to different unimodular eigenvalues are orthogonal. Since \( c\left( \lambda \right) \) exists-as will be shown in a moment-and is bounded by 1 for \( \lambda \in {\sigma }_{\mathrm{p}}\left( T\right) \cap \mathbb{T} \), the sum \( \mathop{\sum }\limits_{{\lambda \... | No |
Corollary 1. Let \( \left( {X, A}\right) \) be a \( {CW} \) pair. If \( A \) is contractible, then \( X/A \sim X \) . More precisely: The projection \( X \rightarrow X/A \) is a homotopy equivalence. | Proof. Let \( p \) be the projection \( X \rightarrow X/A \) . Since \( A \) is contractible, there is a homotopy \( {f}_{t} : A \rightarrow A \) such that \( {f}_{0} = {\mathrm{{id}}}_{A} \) and \( {f}_{1} = \) const. By Borsuk’s theorem, there exists a homotopy \( {F}_{t} : X \rightarrow X \) such that \( {F}_{0} = {... | No |
Corollary 2. If \( \left( {X, A}\right) \) is a CW pair, then \( X/A \sim X \cup {CA} \), where \( {CA} \) is a cone over \( A \) . | Proof. \( X/A = \left( {X \cup {CA}}\right) /{CA} \sim X \cup {CA} \) . The latter follows from Corollary 1 applied to the CW complex \( X \cup {CA} \) and its contractible CW subcomplex \( {CA} \) . | Yes |
Proposition 1. Let \( y \in T \) . A deck transformation \( f : T \rightarrow T \) is fully determined by the image \( f\left( y\right) \) of \( y \) . In particular, if a deck transformation \( f \) has a fixed point, then \( f = \mathrm{{id}} \) . | Proof. Let \( z \in T \) . Choose a path \( s : I \rightarrow T \) joining \( y \) with \( z \) . Let \( {s}^{\prime } : I \rightarrow T \) be a path obtained by lifting \( p \circ s : I \rightarrow X \) with the beginning \( f\left( y\right) \) . The paths \( f \circ s \) and \( {s}^{\prime } \) have a common beginnin... | Yes |
Proposition 2. Let \( y,{y}^{\prime } \in T \) and \( p\left( y\right) = p\left( {y}^{\prime }\right) \) . A deck transformation \( f : T \rightarrow T \) such that \( f\left( y\right) = {y}^{\prime } \) exists if and only if \( {p}_{ * }{\pi }_{1}\left( {T, y}\right) = {p}_{ * }{\pi }_{1}\left( {T,{y}^{\prime }}\right... | Proof. Let there be a deck transformation \( f : T \rightarrow T \) with \( f\left( y\right) = {y}^{\prime } \) . Let \( s : I \rightarrow T \) be a loop with the beginning \( y \) representing an arbitrarily chosen element \( \alpha \) of \( {\pi }_{1}\left( {T, y}\right) \) . Consider the lifting \( {s}^{\prime } : I... | Yes |
If \( \Gamma \) is a discrete subgroup of a topological group \( G \), then there arises a regular covering \( G \rightarrow G/\Gamma \). | For example, there are many known discrete subgroups in the group \( {SO}\left( 3\right) \) : the dihedral groups, the groups of symmetries of Platonic solids, and so forth. For each of these groups \( \Gamma \) there arises a regular covering \( {SO}\left( 3\right) \rightarrow \) \( {SO}\left( 3\right) /\Gamma \) ; si... | No |
Theorem. Let \( p : T \rightarrow X \) be a covering, and let \( Z \) be a path connected space. Let \( {\widetilde{x}}_{0} \in T,{x}_{0} = p\left( {\widetilde{x}}_{0}\right) \in X \), and \( {z}_{0} \in Z \) be base points and let \( f : Z \rightarrow X \) be a continuous map such that \( f\left( {z}_{0}\right) = {x}_... | Proof. Let \( {F}^{\prime },{F}^{\prime \prime } \) be two such maps. Let \( z \in Z \) be an arbitrary point, and let \( s : I \rightarrow Z \) be a path joining \( {z}_{0} \) with \( z \) . The paths \( {F}^{\prime } \circ s,{F}^{\prime \prime } \circ s : I \rightarrow T \) both begin at \( {\widetilde{x}}_{0} \) and... | Yes |
Every element of \( {\pi }_{1}\left( {X,{x}_{0}}\right) \) may be presented as a product\n\n\[ \n{i}_{{k}_{1}}\left( {\alpha }_{1}\right) \ldots {i}_{{k}_{N}}\left( {\alpha }_{N}\right) \]\n\n(*) \n\nwhere \( {k}_{s} = 1 \) or 2 and \( {\alpha }_{s} \in {\pi }_{1}\left( {{U}_{{k}_{s}},{x}_{0}}\right) \) . | Proof. Let \( \sigma : I \rightarrow X \) be a loop representing the chosen element of \( {\pi }_{1}\left( {X,{x}_{0}}\right) \) . A simple fact from analysis states that there exists an \( n \) such that for each \( r = \) \( 1,\ldots, n,\sigma \left( \left\lbrack {\frac{r - 1}{n},\frac{r}{n}}\right\rbrack \right) \) ... | Yes |
Proposition 2. The word (1) is equal to \( 1 \in {\pi }_{1}\left( {X,{x}_{0}}\right) \) if and only if it can be reduced by admissible transformations to the trivial word \( \left( {{i}_{1}\left( 1\right) }\right. \) or \( \left. {{i}_{2}\left( 1\right) }\right) \) . | Proof. The if part of this proposition is obvious [admissible transformations do not change the product (1)]. Prove the only if part.\n\nSuppose that a product \( \left( *\right) \) is equal to \( 1 \in {\pi }_{1}\left( {X,{x}_{0}}\right) \) . Let \( {\sigma }_{r} : I \rightarrow {U}_{{k}_{r}} \subset X \) be a loop (o... | No |
Theorem 3. The fundamental group of the complement to the trefoil is a group with two generators, \( a \) and \( b \), and one relation: \( {aba} = {bab} \) . | One can take for the generators \( u = {ab} \) and \( v = {bab} \) ; then the relation takes the form \( {u}^{3} = {v}^{2} \). | Yes |
Theorem 4. The fundamental group of the complement to the trefoil is not commutative. | Indeed, the formulas \( f\left( a\right) = \left( {213}\right), f\left( b\right) = \left( {132}\right) \) define a homomorphism of the group \( {\pi }_{1}\left( {{\mathbb{R}}^{3} - K}\right) \) onto \( S\left( 3\right) \) [because (213) and (132) satisfy the above relation and generate the group \( S\left( 3\right) \rb... | Yes |
Theorem 1. Let \( p : T \rightarrow X \) be a covering, let \( {\widetilde{x}}_{0} \in T \), and let \( {x}_{0} = p\left( {\widetilde{x}}_{0}\right) \in X \) . If \( n \geq 2 \), then \( {p}_{ * } : {\pi }_{n}\left( {T,{\widetilde{x}}_{0}}\right) \rightarrow {\pi }_{n}\left( {X,{x}_{0}}\right) \) is an isomorphism. | This follows from results of Sect. 6.10 (see the corollary in this section) and from the simply connectedness of the sphere \( {S}^{n} \) for \( n \geq 2 \) . | No |
Theorem 2.\n\n\[ \n{\pi }_{n}\left( {S}^{1}\right) = \left\{ \begin{array}{l} \mathbb{Z},\text{ if }n = 1 \\ 0,\text{ if }n \geq 2 \end{array}\right. \n\] | The first is already known (Sect. 6.3), the second follows from the fact that there is a covering \( \mathbb{R} \rightarrow {S}^{1} \), and the line \( \mathbb{R} \) is contractible. | No |
Proposition 1. If\n\n\n\n\nis a commutative diagram with exact rows, \( {\varphi }_{1} \) is an epimorphism, and \( {\varphi }_{2},{\varphi }_{4} \) are monomorphisms, then \( {\varphi }_{3} \) is also a monomorphism... | Proof of Proposition 1. Let \( {a}_{3} \in {A}_{3} \) and \( {\varphi }_{3}\left( {a}_{3}\right) = 0 \) . Then \( {g}_{3} \circ {\varphi }_{3}\left( {a}_{3}\right) = \) \( 0 \Rightarrow {\varphi }_{4} \circ {f}_{3}\left( {a}_{3}\right) = 0 \) (commutativity of the third square) \( \Rightarrow {f}_{3}\left( {a}_{3}\righ... | Yes |
Proposition 1. The definition of a Serre fibration is equivalent to the definition which states CHP only for the case when \( \left( {X, Y}\right) = \left( {{D}^{n},{S}^{n - 1}}\right) \) for all \( n \) . | Proof. Repeat cases 3 and 4 of the proof in Sect. 9.3. | No |
Proposition 2. The definition of a Serre fibration is equivalent to the definition which states CHP only in the absolute form. | Proof. By Proposition 1, it is sufficient to deduce the relative CHP for \( \left( {X, Y}\right) = \) \( \left( {{D}^{n},{S}^{n - 1}}\right) \) from the absolute CHP for \( X = {D}^{n} \) . But the two statements are essentially the same, as Fig. 47 shows. | Yes |
Example 2 (Path fibration). Let \( W \) be an arbitrary topological space with a base point \( {w}_{0} \) . Put \( E = E\left( {W,{w}_{0}}\right) \) (the space of paths of \( W \) beginning at \( {w}_{0} \) ), \( B = W \) , and define \( p : E \rightarrow B \) by the formula \( p\left( s\right) = s\left( 1\right) \) . ... | Indeed, let \( \widetilde{\varphi } : X \rightarrow E \) be a continuous map, and let \( \Phi : X \times I \rightarrow B = W \) be a homotopy such that \( \Phi \left( {x,0}\right) = \left( {\widetilde{\varphi }\left( x\right) }\right) \left( 1\right) \) for every \( x \in X \) (see Fig. 48). The covering homotopy \( \w... | Yes |
Proposition 1. The homology \( {\widetilde{H}}_{n}\left( X\right) \) of the reduced singular complex (called the reduced homology of \( X \) ) is related to the usual homology as follows. If \( X \) is not empty, then\n\n\[ \n{H}_{n}\left( X\right) = \left\{ \begin{array}{l} {\widetilde{H}}_{n}\left( X\right) ,\;\text{... | Proof. Obvious. | No |
Proposition 2. If chain maps \( \varphi ,\psi : \mathcal{C} \rightarrow {\mathcal{C}}^{\prime } \) are homotopic, then the induced homology maps \( {\varphi }_{ * },{\psi }_{ * } : {H}_{n}\left( \mathcal{C}\right) \rightarrow {H}_{n}\left( {\mathcal{C}}^{\prime }\right) \) are equal. | Proof. Let \( D = \left\{ {D}_{n}\right\} \) be a homotopy between \( \varphi \) and \( \psi \) . If \( c \in \operatorname{Ker}{\partial }_{n} \subset {C}_{n} \), then\n\n\[ \n{\psi }_{n}\left( c\right) - {\varphi }_{n}\left( c\right) = {D}_{n - 1} \circ {\partial }_{n}\left( c\right) + {\partial }_{n + 1}^{\prime } \... | Yes |
Example 1 (Barycentric Transformator). The barycentric subdivision of the standard simplex \( {\Delta }^{n} \) (see Fig. 21 in Sect. 5.8) consists of \( \left( {n + 1}\right) \) ! \( n \) -dimensional Euclidean simplices corresponding to chains \( {\delta }^{0} \subset {\delta }^{1} \subset \cdots \subset {\delta }^{n}... | \[ {u}_{k}^{\sigma } = \frac{{v}_{\sigma \left( 0\right) } + {v}_{\sigma \left( 1\right) } + \cdots + {v}_{\sigma \left( k\right) }}{k + 1}, k = 0,1,\ldots, n, \] where \( {v}_{0},{v}_{1},\ldots ,{v}_{n} \) are the vertices of \( {\Delta }^{n} \) in their natural order. The correspondence \( {v}_{i} \mapsto {u}_{i}^{\s... | Yes |
Theorem 1. For all \( n \) ,\n\n\[ \n{\widetilde{H}}_{m}\left( {S}^{n}\right) = \left\{ \begin{array}{l} \mathbb{Z},\text{ if }m = n, \\ 0,\text{ if }m \neq n. \end{array}\right.\n\] | Proof of Theorem 1 Consider a portion of the reduced homology sequence of the pair \( \left( {{D}^{n},{S}^{n - 1}}\right) \) :\n\n\[ \n\begin{matrix} {\widetilde{H}}_{m}\left( {D}^{n}\right) \rightarrow {H}_{m}\left( {{D}^{n},{S}^{n - 1}}\right) \rightarrow {\widetilde{H}}_{m - 1}\left( {S}^{n - 1}\right) \rightarrow {... | No |
Theorem 2. For any topological space \( X \) and any \( n \) , \[ {\widetilde{H}}_{n}\left( {\sum X}\right) = {\widetilde{H}}_{n - 1}\left( X\right) \] | Proof. It follows from the reduced homology sequence of the pair \( \left( {{CX}, X}\right) \), the contractibility of \( {CX} \), the equality \( {\sum X} = {CX}/X \), and the (obvious) fact that \( \left( {{CX}, X}\right) \) is a Borsuk pair. | No |
Theorem 1. Let \( A \) be an arbitrary set and let \( {S}_{\alpha }^{n},\alpha \in A \), be copies of the standard \( n \) -dimensional sphere. Then\n\n\[ \n{\widetilde{H}}_{n}\left( {\mathop{\bigvee }\limits_{{\alpha \in A}}{S}_{\alpha }^{n}}\right) = \left\{ \begin{array}{ll} {\bigoplus }_{\alpha \in A}\mathbb{Z}\alp... | Proof. This follows from Theorem 2 of Sect. 13.1, since \( \mathop{\bigvee }\limits_{{\alpha \in A}}{S}_{\alpha }^{n} \) is homotopy equivalent to the suspension of \( \mathop{\bigvee }\limits_{{\alpha \in A}}{S}_{\alpha }^{n - 1} \) (and even is homeomorphic to this suspension if the latter is understood in the base p... | Yes |
Theorem 2. If \( \left( {{X}_{\alpha },{x}_{\alpha }}\right) \) are base point spaces which are Borsuk pairs, then for any \( m \) , \[ {\widetilde{H}}_{m}\left( {\mathop{\bigvee }\limits_{{\alpha \in A}}{X}_{\alpha }}\right) = {\bigoplus }_{\alpha \in A}{\widetilde{H}}_{m}\left( {X}_{\alpha }\right) \] | Proof. A bouquet is the quotient space of a disjoint union under the union of the base points. | No |
Theorem 1. If the number of n-dimensional cells of a CW complex \( X \) is \( N \), then the group \( {H}_{n}\left( X\right) \) is generated by at most \( N \) generators; in particular, the nth Betti number \( {B}_{n}\left( X\right) \) does not exceed \( N \) . For example, if \( X \) does not have \( n \) -dimensiona... | It follows directly from previous results. | No |
Theorem 2 (Excision Theorem). Let \( X \) be a \( {CW} \) complex and let \( A, B \) be \( {CW} \) subcomplexes of \( X \) such that \( A \cup B = X \) . Then (for every \( n \) )\n\n\[{H}_{n}\left( {X, A}\right) = {H}_{n}\left( {B, A \cap B}\right) .\] | Indeed, \( X/A \) and \( B/\left( {A \cap B}\right) \) are the same as CW complexes. | No |
Theorem 3 (Mayer-Vietoris Sequence). Let \( X \) be a CW complex and let \( A, B \) be \( {CW} \) subcomplexes of \( X \) such that \( A \cup B = X \) . Then there exists an exact sequence\n\n\[ \cdots \rightarrow {H}_{n}\left( {A \cap B}\right) \rightarrow {H}_{n}\left( A\right) \oplus {H}_{n}\left( B\right) \rightarr... | Proof. Let \( Y = \left( {A \times 0}\right) \cup \left( {\left( {A \cap B}\right) \times I}\right) \cup \left( {B \times 1}\right) \subset X \times I \) and let \( C \subset Y \) be \( \left( {A \cap B}\right) \times I \) . Then \( Y/C \) and \( \sum \left( {A \cap B}\right) \) (actually with the vertices merged; this... | Yes |
Lemma 2. The kernel \( \operatorname{Ker}\left( {A \otimes {F}_{2} \rightarrow A \otimes {F}_{1}}\right) \) does not depend on the choice of presentation \( B = {F}_{2}/{F}_{1} \) . | Proof The proof consists in constructing a canonical isomorphism\n\n\[ \operatorname{Ker}\left( {A \otimes {F}_{2}^{\prime } \rightarrow A \otimes {F}_{1}^{\prime }}\right) \cong \operatorname{Ker}\left( {A \otimes {F}_{2} \rightarrow A \otimes {F}_{1}}\right) \]\n\nfor an arbitrary other presentation \( B = {F}_{1}^{\... | Yes |
Corollary 1. For every homomorphism \( f : {H}_{n}\left( X\right) \rightarrow G \), there exists a cohomology class \( \gamma \in {H}^{n}\left( {X;G}\right) \) such that \( f\left( \alpha \right) = \langle \gamma ,\alpha \rangle \) for every \( \alpha \in {H}_{n}\left( G\right) \) . | Remark also that this \( \gamma \) is defined up to an element of \( \operatorname{Ext}\left( {{H}_{n}\left( X\right), G}\right) \) ; in particular, if \( {H}_{n}\left( X\right) \) and \( G \) are finitely generated, then this Ext group is finite, so \( \gamma \) is defined by \( f \) up to adding an element of finite ... | No |
Theorem 1. Let \( {X}_{1},{X}_{2} \) be topological spaces. Then for any \( n \) ,\n\n(1) There is a (noncanonical) isomorphism\n\n\[ \n{H}_{n}\left( {{X}_{1} \times {X}_{2}}\right) \cong \n\]\n\n\[ \n\mathop{\bigoplus }\limits_{{i + j = n}}\left( {{H}_{i}\left( {X}_{1}\right) \otimes {H}_{j}\left( {X}_{2}\right) }\rig... | We will deduce Theorem 1 from an algebraic result related to the tensor product of complexes.\n\nDefinition. Let\n\n\[ \n\begin{array}{l} \left( \mathcal{C}\right) \;\ldots \xrightarrow[]{{\partial }_{n + 1}}{C}_{n}\xrightarrow[]{{\partial }_{n}}{C}_{n - 1}\xrightarrow[]{{\partial }_{n}}\ldots \\ \left( {\mathcal{C}}^{... | Yes |
Theorem 2. If the complexes \( \mathcal{C} \), and \( {\mathcal{C}}^{\prime } \) are free (that is, all \( {C}_{n},{C}_{n}^{\prime } \) are free Abelian groups), then, for every \( n \) ,\n\n(1) There is a (noncanonical) isomorphism\n\n\[ \n{H}_{n}\left( {\mathcal{C} \otimes {\mathcal{C}}^{\prime }}\right) \cong \n\]\n... | Proof. Begin with part (2). Let \( {Z}_{n} = \operatorname{Ker}{\partial }_{n},{B}_{n - 1} = \operatorname{Im}{\partial }_{n} \) . Consider the diagram \n\nThe rows of this diagram are complexes, the columns are exac... | Yes |
Theorem 2. Every compact smooth manifold is homeomorphic to a triangulated subset of an Euclidean space, and the homeomorphism can be made smooth on every simplex of the triangulation. | We do not prove these theorems. Theorem 1 is proved in many textbooks in differential topology. Its proof is not hard. The situation with Theorem 2 is worse. Since the 1920s, the topologist regarded this fact as obvious. There are many geometric approaches to this result which look promising. For example, take a compac... | No |
Proposition 1. (1) A triangulated space \( X \) is an \( n \) -dimensional homology manifold if and only if for every vertex \( v \) of \( X \), the link \( \operatorname{Lk}\left( v\right) \) is a homological \( \left( {n - 1}\right) \) - dimensional sphere (that is, has the same homology groups as \( {S}^{n - 1} \) )... | Proof. Open stars of vertices, \( \operatorname{st}\left( v\right) = \operatorname{St}\left( v\right) - \operatorname{Lk}\left( v\right) \), for an open cover of \( X \) . Also, \( \operatorname{St}\left( v\right) \) is a cone over \( \operatorname{Lk}\left( v\right) \) with the vertex \( v \) . Thus, if \( {x}_{0} \in... | Yes |
Proposition 2. Every connected n-dimensional homology manifold is an \( n \) -dimensional pseudomanifold. | Proof. Let \( X \) be an \( n \) -dimensional homology manifold. Since the link of every vertex of \( X \) is an \( \left( {n - 1}\right) \) -dimensional homological sphere, this link contains simplices of dimension \( \geq n - 1 \) ; hence, every vertex is a vertex of an \( n \) -dimensional simplex. There cannot be s... | Yes |
Theorem 1. Let \( X \) be a smooth closed oriented \( n \) -dimensional manifold, and let \( {\alpha }_{1} \in {H}_{m}\left( X\right) ,{\alpha }_{2} \in {H}_{n - m}\left( X\right) \) . Let \( {Y}_{1} \) and \( {Y}_{2} \) be closed oriented submanifolds of \( X \) of dimensions \( m \) and \( n - m \) which realize \( {... | As usual (see the warning in the beginning of this lecture), we do not give a rigorous proof of these statements; but from the point of view of common sense they are obvious. We can make the simplices of a triangulation of \( X \) much smaller than the distances between the intersection points of \( {Y}_{1} \) and \( {... | No |
Theorem 3. Let \( X \) be a connected closed orientable manifold of even dimension \( {2k} \), and let \( {H}_{k}^{0}\left( X\right) \) be the free part of \( {H}_{k}\left( X\right) \) . Then the integral bilinear form \( \phi \) (the intersection index) on \( {H}_{k}^{0}\left( X\right) \) is unimodular [that is, the m... | Proof of Theorem 3. Consider the homomorphism \( {\omega }_{i} : {H}_{k}^{0}\left( X\right) \rightarrow \mathbb{Z},{\omega }_{i}\left( {\alpha }_{j}\right) = {\delta }_{ij} \) . By part (2) of Theorem 2, there exists a \( {\beta }_{i} \in {H}^{k}\left( {X;\mathbb{Z}}\right) \) such that \( \left\langle {{\beta }_{i},\a... | Yes |
Theorem 4. Let \( {X}_{1},{X}_{2} \) be a compact oriented homology manifold of dimensions \( {n}_{1},{n}_{2} \), and let \( {\gamma }_{1} \in {H}^{{q}_{1}}\left( {{X}_{1};G}\right) ,{\gamma }_{2} \in {H}^{{q}_{2}}\left( {{X}_{2};G}\right) \) . Then\n\n\[ \n{D}_{{X}_{1} \times {X}_{2}}\left( {{\gamma }_{1} \times {\gam... | Proof. We use the obvious relation \( \left( {{\alpha }_{1} \times {\alpha }_{2}}\right) \frown {p}_{1}^{ * }\gamma = \left( {a \frown \gamma }\right) \times \beta \) ), where \( {\alpha }_{1} \in {H}_{{q}_{1}}\left( {X}_{1}\right) ,{\alpha }_{2} \in {H}_{{q}_{2}}\left( {X}_{2}\right) ,\gamma \in {H}^{r}\left( {{X}_{1}... | Yes |
Theorem 1. Let \( X \) be a finitely triangulated space, and let \( f : X \rightarrow X \) be a continuous map. If \( f \) has no fixed points, then \( \mathcal{L}\left( f\right) = 0 \) . | Proof. We assume that \( X \) is furnished with a metric in which every simplex is isometric to the standard simplex. Then there is a positive \( \delta \) such that \( \operatorname{dist}\left( {x, f\left( x\right) }\right) > \delta \) for every \( x \in X \) . By applying to \( X \) the barycentric subdivision suffic... | Yes |
Theorem 2. Let \( X \) be a compact smooth manifold (not necessarily orientable, and maybe with a nonempty boundary), and let \( \xi \) be a vector field on \( X \) . Suppose that \( \xi \) has no zeroes and that on the boundary \( \partial X \) it is directed inside \( X \) . Then \( \chi \left( X\right) = 0 \) . | Proof of Theorem 2. A vector field \( \xi \) on \( X \) (with or without zeroes) determines a \ | No |
Lemma 1. \( \phi \left( {{f}_{ * }{\alpha }_{1},{\alpha }_{2}}\right) = {\left( -1\right) }^{\dim {\alpha }_{1}}\phi \left( {{F}_{ * }\left\lbrack X\right\rbrack ,{\alpha }_{1} \times {\alpha }_{2}}\right) \) . | Proof of Lemma 1. Let \( {\alpha }_{1} = D{\gamma }_{1},{\alpha }_{2} = D{\gamma }_{2} \) . Then\n\n\[ \phi \left( {{F}_{ * }\left\lbrack X\right\rbrack ,{\alpha }_{1} \times {\alpha }_{2}}\right) = \phi \left( {{\left( \operatorname{id} \times f\right) }_{ * } \circ {\Delta }_{ * }\left\lbrack X\right\rbrack ,{\alpha ... | Yes |
Lemma 2. Let \( {\alpha }_{1},\ldots ,{\alpha }_{N} \) be a basis in the free part of the full homology group of a compact oriented homology manifold \( X \) [first, the basis in \( {H}_{0}\left( X\right) \), then \( {H}_{1}\left( X\right) \), and so on], and let \( {\alpha }_{1}^{ * },\ldots ,{\alpha }_{N}^{ * } \) be... | Proof. By part (2) of Theorem 2 in Sect. 17.5, it is sufficient to prove that\n\n\[ \phi \left( {{\Delta }_{ * }\left\lbrack X\right\rbrack ,{\alpha }_{p} \times {\alpha }_{q}}\right) = \phi \left( {\mathop{\sum }\limits_{i}\left( {{\alpha }_{i}^{ * } \times {\alpha }_{i}}\right) ,{\alpha }_{p} \times {\alpha }_{q}}\ri... | No |
Proposition 1. Let \( X, Y \), and \( f \) be as above, and let \( \alpha \in {H}_{q}\left( Y\right) ,\beta \in {H}_{m - q}\left( X\right) \) . Then\n\n\[{\phi }_{X}\left( {{f}^{!}\alpha ,\beta }\right) = {\phi }_{Y}\left( {\alpha ,{f}_{ * }\beta }\right)\]\n\n( \( {\phi }_{X} \) and \( {\phi }_{Y} \) denote the inters... | Proof.\n\n\[{\phi }_{X}\left( {{f}^{!}\alpha ,\beta }\right) = {\phi }_{X}\left( {D{f}^{ * }{D}^{-1}\alpha ,\beta }\right) = \left\langle {{f}^{ * }{D}^{-1}\alpha ,\beta }\right\rangle\]\n\n\[= \left\langle {{D}^{-1}\alpha ,{f}_{ * }\beta }\right\rangle = {\phi }_{Y}\left( {\alpha ,{f}_{ * }\beta }\right) .\]\n\nBy par... | Yes |
Proposition 2. Let \( X, Y \) be compact oriented homological manifolds, and let \( p : X \times \) \( Y \rightarrow Y \) be the projection. Then, for any \( \alpha \in {H}_{m}\left( Y\right) \) , \[ {p}^{!}\alpha = \left\lbrack X\right\rbrack \times \alpha \] | Proof. Let \( \alpha = {D\gamma },\gamma \in {H}^{n - m}\left( {Y;\mathbb{Z}}\right) \) . Then \[ {p}^{!}\alpha = {D}_{X \times Y}{p}^{ * }\gamma = {D}_{X \times Y}\left( {1 \times \gamma }\right) = {D}_{X}1 \times {D}_{Y}\gamma = \left\lbrack X\right\rbrack \times \alpha . \] | Yes |
Proposition 3. Let \( X, Y \) be connected compact oriented manifolds of the same dimension \( n \), and let \( f : X \rightarrow Y \) be a continuous map of degree \( d \) . Then the compositions\n\n\[ \n\\begin{matrix} {H}_{m}\left( Y\right) \\xrightarrow[]{{f}^{!}}{H}_{m}\left( X\right) \\xrightarrow[]{{f}_{ * }}{H}... | Here is a proof of the first statement. Let \( \\alpha \\in {H}_{m}\left( Y\\right) ,\\alpha = {D}_{Y}\\gamma ,\\gamma \\in {H}^{n - m}\left( {Y;\\mathbb{Z}}\\right) \) . Then \( {f}_{ * }{f}^{!}\\alpha = {f}_{ * }{D}_{X}{f}^{ * }\\gamma = {f}_{ * }\\left( {\\left\\lbrack X\\right\\rbrack \\frown {f}^{ * }\\gamma }\\ri... | Yes |
Theorem 1. Let \( {Y}_{1},{Y}_{2} \) be closed oriented submanifolds of a smooth closed oriented manifold \( X \) transverse to each other; the latter means that the inclusion map \( {i}_{1} \) of \( {Y}_{1} \) in \( X \) is transversely regular to \( {Y}_{2} \) . Then the intersection \( Z = {Y}_{1} \cap {Y}_{2} = {i}... | Proof of Theorem 1. \[ D\left( {{\alpha }_{1} \smile {\alpha }_{2}}\right) = \left\lbrack X\right\rbrack \frown \left( {{\alpha }_{1} \smile {\alpha }_{2}}\right) = \left( {\left\lbrack X\right\rbrack \frown {\alpha }_{1}}\right) \frown {\alpha }_{2} = \left( {D{\alpha }_{1}}\right) \frown {\alpha }_{2} \] \[ = {i}_{1}... | Yes |
Proposition 2. Poincaré isomorphisms described above, together with Poincaré's isomorphisms for the manifold \( \partial X \), form an isomorphism between homology and cohomology sequences of the pair \( \left( {X,\partial X}\right) \) ; more precisely, there arises a plus-minus commutative diagram\n\n\[ \cdots \;{H}_{... | Proof. We will prove the plus-minus commutativity of the first square; for the third square the proof is more or less the same, while the commutativity of the second square is obvious.\n\nTake a \( c \in {C}^{n - m}\left( {\partial X;G}\right) \) and extend it to \( \widetilde{c} \in {C}^{n - m}\left( {X;G}\right) \) .... | Yes |
Theorem 1. The obstruction cochain is a cocycle: \( \delta {c}_{f} = 0 \) . | Proof. The statement may be regarded as a variation on the theme of \( \partial \partial = 0 \) [we need to prove that \( {c}_{f}\left( {\partial a}\right) = 0 \), but the cochain \( {c}_{f} \) itself is defined by means of boundaries], but the accurate proof requires some work. For example, it can be deduced from the ... | Yes |
Lemma 1. For any continuous map \( f : {X}^{n} \rightarrow Y \) and any cochain \( d \in {\mathcal{C}}^{n}\left( {X;{\pi }_{n}\left( Y\right) }\right) \) , there exists a continuous map \( g : {X}^{n} \rightarrow Y \) which agrees with \( f \) on \( {X}^{n - 1} \) and is such that \( {d}_{f, g} = d \) . | Proof. Consider an \( n \) -dimensional cell \( e \) of \( X \) and distinguish a small ball in \( e \) . Than change the map \( f \) on this ball in such a way that the two maps of the ball, the old one and the new one, compose a spheroid of the class \( d\left( e\right) \) (see Fig. 74). Having such a change made on ... | No |
Lemma 2. \( \delta {d}_{f, g} = {c}_{g} - {c}_{f} \) . | Proof. Consider, for simplicity’s sake, the case when \( f \) and \( g \) are different on only one \( n \) -dimensional cell \( e \subset X \) (the general case, essentially, is not different from this case). Let \( \sigma \) be an \( \left( {n + 1}\right) \) -dimensional cell of \( X \) ; we want to show that\n\n\[ \... | Yes |
Corollary 1. A CW complex of the type \( K\left( {\pi, n}\right) \) is homotopically unique. Hence, a topological space of the type \( K\left( {\pi, n}\right) \) is weakly homotopically unique. | Proof. Let \( X,{X}^{\prime } \) be CW complexes of the type \( K\left( {\pi, n}\right) \), and let \( {F}_{\pi }\; \in \) \( {H}^{n}\left( {X;\pi }\right) ,{F}_{\pi }^{\prime } \in {H}^{n}\left( {{X}^{\prime };\pi }\right) \) be the fundamental classes. According to the theorem, there exist continuous maps \( f : X \r... | Yes |
Theorem 1 (Hopf). For every \( n \) -dimensional CW complex \( X \), there is a bijection\n\n\[ \n{H}^{n}\left( {X;\mathbb{Z}}\right) \leftrightarrow \pi \left( {X,{S}^{n}}\right) ,\left\lbrack f\right\rbrack \mapsto {f}^{ * }\left( s\right) ,\n\]\n\nwhere \( s = 1 \in \mathbb{Z} = {H}^{n}\left( {{S}^{n};\mathbb{Z}}\ri... | Proof. This classical theorem (proved, actually, before the appearance of not only the obstruction theory, but also cohomology) is, from a modern point of view, a corollary of the theorem in Sect. 18.3. Indeed, the construction of the space\n\n. Let an n-dimensional CW complex \( X \) contain as a CW sub-complex a sphere \( {S}^{n - 1} \) . This sphere is a retract of \( X \) if and only if the inclusion homomorphism \( {H}^{n - 1}\left( {X;\mathbb{Z}}\right) \rightarrow {H}^{n - 1}\left( {{S}^{n - 1};\mathbb{Z}}\right) \) is an epimorphism. | Proof. The only if part is obvious: If \( r : X \rightarrow {S}^{n - 1} \) is a retraction, then the composition\n\n\[ \n{H}^{n - 1}\left( {{S}^{n - 1};\mathbb{Z}}\right) \overset{{r}^{ * }}{ \rightarrow }{H}^{n - 1}\left( {X;\mathbb{Z}}\right) \overset{{j}^{ * }}{ \rightarrow }{H}^{n - 1}\left( {{S}^{n - 1};\mathbb{Z}... | Yes |
Proposition 1. Let \( {\pi }_{0}\left( F\right) = {\pi }_{1}\left( F\right) = \cdots = {\pi }_{n - 1}\left( F\right) = 0 \), and let \( s,{s}^{\prime } : {B}^{n} \rightarrow E \) be two sections. Then \( {C}_{s} = {C}_{{s}^{\prime }} \in {H}^{n + 1}\left( {B;{\pi }_{n}\left( F\right) }\right) \) . | Proof of Proposition 1. It is clear that a homotopy of a section \( s : {B}^{k} \rightarrow E \) will not affect either \( {c}_{s} \) or \( {C}_{s} \) . Suppose that the given sections \( s,{s}^{\prime } \) are homotopic over \( {B}^{k} \) for some \( k,0 \leq k < n - 1 \) (since the fiber \( F \) is connected, this is... | Yes |
Proposition 2. \( \left\langle {C\left( {\tau }_{X}\right) ,\left\lbrack X\right\rbrack }\right\rangle = \chi \left( X\right) \) . | Proof. A section of the fibration \( {\tau }_{X} \) is the same as a nowhere vanishing vector field on \( X \) . It is easy to understand that a generic vector field on \( X \) has only isolated zeroes. Take a local coordinate system with the origin at the isolated zero \( {x}_{0} \) of a vector field \( \xi \), take a... | Yes |
Theorem 3. Let \( \\xi \) be an n-dimensional real vector bundle with a CW base. Then\n\n\[ \n{w}_{i}\\left( {{S}^{r}\\xi }\\right) = {G}_{n;r;i}\\left( {{w}_{1}\\left( \\xi \\right) ,{w}_{2}\\left( \\xi \\right) ,\\ldots }\\right) ; \n\]\n\nin particular,\n\n\[ \n{w}_{1}\\left( {{S}^{r}\\xi }\\right) = \\left( \\begin... | The proof of Theorem 3 is so close to the proof of Theorem 2 that we do not feel any necessity in detailing it [just mention that it is based on the relation \( {S}^{r}\\xi = \)\n\n\( \\bigoplus \\;\\left( {{\\zeta }_{{j}_{1}} \\otimes \\cdots \\otimes {\\zeta }_{{j}_{r}}}\\right) \\rbrack . \)\n\n\( 1 \\leq {j}_{1} \\... | No |
Theorem (Cartan-Serre). Let \( X \) be a simply connected space with finitely generated homology groups. Suppose that the rational cohomology of \( X \) is a free skew-commutative algebra,\n\n\[ \n{H}^{ * }\left( {X;\mathbb{Q}}\right) = {\Lambda }_{\mathbb{Q}}\left( {{x}_{1},\ldots ,{x}_{m}}\right) \otimes \mathbb{Q}\l... | Proof of Theorem. Since \( {H}^{ * }\left( {X;\mathbb{Q}}\right) = {H}^{ * }\left( {X;\mathbb{Z}}\right) \otimes \mathbb{Q} \) and the homology groups of \( X \) are finitely generated, there exist nonzero integers \( {a}_{1},\ldots ,{a}_{m},{b}_{1},\ldots ,{b}_{\ell } \) such that the classes \( {a}_{1}{x}_{1},\ldots ... | Yes |
Corollary 2 (The Stiefel Theorem). Every closed orientable three-dimensional manifold is parallelizable. | Proof. To prove that a closed orientable three-dimensional manifold \( X \) is parallelizable, it is sufficient to construct two linearly independent (at every point) vector fields on \( X \), that is, to construct a section of the fibration \( E\xrightarrow[]{\;V\left( {3,2}\right) }X \) associated with the tangent bu... | Yes |
Lemma 1. The group \( {\operatorname{comp}}_{\bar{p}}{\pi }_{N + q}\left( {X\left( s\right) }\right) \) does not depend on \( s \) . | Proof. According to Sect. 34.1, there is a fibration \( X\left( s\right) \overset{X\left( {s + 1}\right) }{ \rightarrow }{Y}_{s} \) . Since the base of this fibration is a product of spaces of the type \( K\left( {{\mathbb{Z}}_{p}, m}\right) \), the homotopy groups of the base have no nontrivial non- \( p \) -component... | Yes |
For \( q < n \), there exists an \( {s}_{0} \) such that\n\n\[{\operatorname{comp}}_{p}{\pi }_{N + q}\left( {X\left( s\right) }\right) = 0\]\n\nfor \( s > {s}_{0} \) . | Proof. Let \( m \) be a smallest integer \( < n \) for which\n\n\[{\operatorname{comp}}_{p}{H}_{N + m}\left( {X\left( s\right) }\right) \neq 0.\]\n\nThen \( m \) is also the smallest integer \( < n \) for which\n\n\[{\operatorname{comp}}_{p}{\pi }_{N + m}\left( {X\left( s\right) }\right) \neq 0\]\n\n(strictly speaking,... | Yes |
Lemma 3. Let \( q < n \) . (1). If \( 0 \leq s < {s}^{\prime } \), then \( {\pi }_{N + q}\left( {X\left( s\right), X\left( {s}^{\prime }\right) }\right) \) is a p-group. (2). If \( {s}^{\prime } \) is sufficiently large, then \( {\pi }_{N + q}\left( {X\left( s\right), X\left( {s}^{\prime }\right) }\right) = {\operatorn... | Proof. The statement follows from Lemmas 1 and 2 and the exactness of the homotopy sequence\n\n\[ \n{\pi }_{N + q}\left( {X\left( {s}^{\prime }\right) }\right) \rightarrow {\pi }_{N + q}\left( {X\left( s\right) }\right) \rightarrow {\pi }_{N + q}\left( {X\left( s\right), X\left( {s}^{\prime }\right) }\right) \n\]\n\n\[... | Yes |
Proposition 1. The two definitions of \( \circ \) are equivalent. | Proof. The map \( {\left( -1\right) }^{Nk}\left( {a\# b}\right) = \left( {{\left( -1\right) }^{Nk}a}\right) \# b : {S}^{N + k}\# {S}^{N + \ell } \rightarrow {S}^{N}\# {S}^{N} \) can be presented as a composition of two maps,  \( {\left( -1\right) }^{k\ell }\alpha \circ \beta \) . | Proof. Let \( a : {S}^{N + k} \rightarrow {S}^{N} \) and \( b : {S}^{N + \ell } \rightarrow {S}^{N} \) be spheroids of classes \( \alpha \) and \( \beta \) . There is an obvious homotopy commutative diagram\n\n\n\nBy... | Yes |
Proposition 3. The composition product is distributive:\n\n\[ \left( {\beta + \gamma }\right) \circ \alpha = \beta \circ \alpha + \gamma \circ \alpha ;\alpha \circ \left( {\beta + \gamma }\right) = \alpha \circ \beta + \alpha \circ \gamma . \] | Proof. Since we have already proven that the composition product is skew-commutative, it is sufficient to prove any one of these formulas, and we will prove the second one. Actually, we will prove a stronger statement, namely, that for any \( \beta ,\gamma \in {\pi }_{m}\left( {S}^{n}\right) \) and \( \alpha \in {\pi }... | Yes |
Proposition 1. (i) For any virtual vector bundle \( \alpha \in K\\left( X\\right) \) there exist a usual vector bundle a and an integer \( N \) such that \( \alpha = \\{ a - N\\} \) . (Recall that in the theory of vector bundles \( N \) denotes the standard trivial vector bundle of dimension \( N, X \times {\\mathbb{C}... | Proof of Proposition. (i) Let \( \\alpha = \\{ c - d\\} \), and let \( \\bar{d} \) be a bundle such that \( d \oplus \\bar{d} = N \) . Then the virtual bundle \( \\{ c - d\\} \) is equivalent to \( \\{ \\left( {c \oplus \\bar{d}}\\right) - \\left( {d \oplus \\bar{d}}\\right) \\} = \\{ a - N\\} \), where \( a = c \oplus... | Yes |
This is an epimorphism whose kernel is the group \( {S}^{1} \subset {\operatorname{Spin}}^{\mathbb{C}}\left( {2n}\right) \) of multiplication by complex numbers of absolute value 1 . The group \( {\operatorname{Spin}}^{\mathbb{C}}\left( {2n}\right) \) is precisely \( {\tau }^{-1}\left( {{SO}\left( {2n}\right) }\right) ... | Obviously, \( {\varphi }_{v} \in {\operatorname{Pin}}^{\mathbb{C}}\left( {2n}\right) - {\operatorname{Spin}}^{\mathbb{C}}\left( {2n}\right) \) for any unit vector \( v \in {\mathbb{C}}^{n} \) . Furthermore, \( \tau \left( {\varphi }_{v}\right) \) is in this case the reflection of \( {\mathbb{C}}^{n} \) in the real hype... | No |
Proposition 2. \( {\Lambda }^{\text{even }}{\mathbb{C}}^{n} \) and \( {\Lambda }^{\text{odd }}{\mathbb{C}}^{n} \) are isomorphic as representations of the group \( {\operatorname{Spin}}^{\mathbb{C}}\left( {{2n} - 2}\right) \subset {\operatorname{Spin}}^{\mathbb{C}}\left( {2n}\right) \) . | Proof. The embedding \( {\operatorname{Spin}}^{\mathbb{C}}\left( {{2n} - 2}\right) \rightarrow {\operatorname{Spin}}^{\mathbb{C}}\left( {2n}\right) \) is induced by the embedding \( {\mathbb{C}}^{n - 1} \rightarrow {\mathbb{C}}^{n} \) . Let \( v \in {\mathbb{C}}^{n} \) be a unit vector orthogonal to \( {\mathbb{C}}^{n ... | Yes |
Proposition 3. If \( n > 1 \), then\n\n\[ \n{H}^{1}\left( {{\operatorname{Spin}}^{\mathbb{C}}\left( {2n}\right) ;\mathbb{Z}}\right) = \mathbb{Z};{H}^{2}\left( {{\operatorname{Spin}}^{\mathbb{C}}\left( {2n}\right) ;\mathbb{Z}}\right) = 0.\n\] | Proof of Proposition 3. The fibration \( {\operatorname{Spin}}^{\mathbb{C}}\left( {2n}\right) \rightarrow {SO}\left( {2n}\right) \) is simple since its fibers have canonical orientations as cosets of the group \( {S}^{1} \) . The \( {E}_{2} \) -term of the spectral sequence of this fibration looks like\n\n<table><tr><t... | Yes |
Proposition 5. \( {H}^{ * }\left( {B{\operatorname{Spin}}^{\mathbb{C}}\left( {2n}\right) }\right) \) is the ring of polynomials of the following variables: \( c\left( {\dim c = 2}\right) ,{p}_{1},\ldots ,{p}_{n}\left( {\dim {p}_{i} = {4i}}\right) ,\chi \left( {\dim \chi = {2n}}\right) \) . Thus, rational characteristic... | The first statement follows from the spectral sequence of the fibration\n\n\[ \nB{\operatorname{Spin}}^{\mathbb{C}}\xrightarrow[]{K\left( {\mathbb{Z},2}\right) }{BSO}\left( {2n}\right) \n\]\n\nand the computation of the rational cohomology of the space \( {BSO}\left( {2n}\right) \) from Lecture 19. The rest of Proposit... | No |
Theorem 2. Let \( \varphi : {H}^{n}\left( {{BO}\left( N\right) ;{\mathbb{Z}}_{2}}\right) \rightarrow {\mathbb{Z}}_{2} \) be a homomorphism. There exists a smooth closed \( n \) -dimensional manifold \( M \) such that for every \( \alpha \in {H}^{n}\left( {{BO}\left( N\right) ;{\mathbb{Z}}_{2}}\right) \) , \n\n\[ \n\lef... | (The reader can prove this theorem for an exercise or find the proof in Stong's book.) | No |
Theorem 3. \( {\Omega }_{ * }^{SO} \otimes \mathbb{Q} \) is the ring of polynomials (over \( \mathbb{Q} \) ) of variables of degree \( {4i} \) . More precisely: The homomorphism \( {\Omega }^{SO} \rightarrow \mathbb{Z}\left\lbrack {{t}_{1},{t}_{2},\ldots }\right\rbrack ,\deg \left( {t}_{i}\right) = {4i} \) determined b... | A proof of Theorem 3 is relatively simple: It is based on Cartan-Serre's theorem (Sect. 26.4). First of all, we already know that\n\n\[ \operatorname{rank}{H}_{s}\left( {{BSO}\left( N\right) }\right) = \left\{ \begin{array}{l} 0,\;\text{ if }s\text{ is not divisible by }4, \\ \text{ the number of partitions of }t \\ \t... | Yes |
Theorem 4 (Milnor, Averbuch). The ring \( {\Omega }_{ * }^{SO} \) does not have elements of an odd order. | The most elegant proof of this result uses the Adams spectral sequence modulo an odd prime. | No |
Theorem 10 (Milnor, Novikov). The ring \( {\Omega }_{ * }^{U} \) is isomorphic to the ring of integral polynomials of generators of dimension \( {2i}, i = 1,2,\ldots \) The generators are represented by some complex projective algebraic manifolds. | Notice that after a tensor multiplication by \( \mathbb{Q} \), this theorem becomes a corollary of the Cartan-Serre theorem (compare with Theorem 3). Moreover, Theorem 10 shows that the Chern numbers fully determine the class of a stably almost complex manifold in \( {\Omega }_{ * }^{U} \). | No |
Theorem 1.2 (Hölder’s inequality) Let \( \Omega \) be a measurable set in \( {\mathbb{R}}^{n} \), either bounded or unbounded. If \( u \in {L}^{p}\left( \Omega \right) \) and \( v \in {L}^{q}\left( \Omega \right) \), where\n\n\[ \frac{1}{p} + \frac{1}{q} = 1,\;1 \leq p, q \leq \infty ,\]\n\nthen \( {uv} \in {L}^{1}\lef... | The Hölder inequality can be used in any domain since its proof is just a simple application of Young's inequality\n\n\[ {ab} \leq \frac{{a}^{p}}{p} + \frac{{b}^{q}}{q}\;\text{ for }\;\frac{1}{p} + \frac{1}{q} = 1,\;1 \leq p, q \leq \infty \]\n\n(1.5)\n\n(see Exercise 1.1 for a proof of (1.5)). | No |
Theorem 1.16 Let \( \Omega \subset {\mathbb{R}}^{3} \) be a smooth bounded domain. For each \( k \in \mathbb{N} \) there exists a linear operator \( {E}_{k} : {H}^{k}\left( \Omega \right) \rightarrow {H}^{k}\left( {\mathbb{R}}^{3}\right) \) and a constant \( {C}_{k} \) such that\n\n(i) \( {\left. {E}_{k}u\right| }_{\Om... | Proof We prove only the remark about the uniformity for domains \( {R\Omega } \), where \( R \geq 1 \) ; for the construction of an extension operator see Adams &Fournier (2003), Constantin & Foias (1988), Evans (1998), Robinson (2001), or Stein (1970), among others. Suppose that we have an extension operator for \( {H... | Yes |
Lemma 1.17 (Sobolev interpolation inequality on \( \Omega \) ) Let \( s,{s}_{1},{s}_{2} \in \mathbb{N} \) with \( 0 \leq {s}_{1} \leq s \leq {s}_{2} \), and choose \( \theta \in \left\lbrack {0,1}\right\rbrack \) such that \( s = \theta {s}_{1} + \left( {1 - \theta }\right) {s}_{2} \) . Then for any \( u \in {H}^{{s}_{... | Proof Use Theorem 1.16 to extend \( u \) to a function \( {E}_{{s}_{2}}\left\lbrack u\right\rbrack \in {H}^{{s}_{2}}\left( {\mathbb{R}}^{n}\right) \) such that\n\n\[ {\begin{Vmatrix}{E}_{{s}_{2}}\left\lbrack u\right\rbrack \end{Vmatrix}}_{{H}^{{s}_{2}}\left( {\mathbb{R}}^{n}\right) } \leq {C}_{{s}_{2}}\parallel u{\para... | Yes |
Lemma 1.27 If \( X \) is a reflexive Banach space and \( u \in {L}^{1}\left( {0, T;X}\right) \) then there exists a unique \( g \in X \) such that\n\n\[ \langle f, g\rangle = {\int }_{0}^{T}\langle f, u\left( s\right) \rangle \mathrm{d}s \] \n\nfor any \( f \in {X}^{ * } \) . In this case we define\n\n\[ {\int }_{0}^{T... | Proof Define a map \( \Gamma : {X}^{ * } \rightarrow \mathbb{R} \) by setting\n\n\[ \Gamma \left( f\right) = {\int }_{0}^{T}\langle f, u\left( s\right) \rangle \mathrm{d}s \] \n\nfor each \( f \in {X}^{ * } \) . This mapping \( \Gamma \) is clearly linear, and it is bounded since\n\n\[ \left| {\Gamma \left( f\right) }\... | Yes |
Lemma 1.28 If \( B \) is a countable orthonormal basis of a Hilbert space \( H \) then the set\n\n\[ \nX = \\left\\{ {u : u = \\mathop{\\sum }\\limits_{{j = 1}}^{N}{c}_{j}\\left( t\\right) {v}_{j}, N \\in \\mathbb{N},{c}_{j} \\in {C}^{\\infty }\\left( \\left\\lbrack {0, T}\\right\\rbrack \\right) ,{v}_{j} \\in B}\\righ... | Proof Take any \( u \\in {L}^{2}\\left( {0, T;H}\\right) \) and choose \( \\delta > 0 \) . Define\n\n\[ \n{c}_{k}\\left( t\\right) = {\\left\\langle u\\left( t\\right) ,{v}_{k}\\right\\rangle }_{H}\\;\\text{ and }\\;{u}_{N} = \\mathop{\\sum }\\limits_{{j = 1}}^{N}{c}_{j}\\left( t\\right) {v}_{j}.\n\]\n\nThen\n\n\[ \n\\... | Yes |
Corollary 1.32 Suppose that \( u, g \in {L}^{1}\left( {0, T;X}\right) \) and that for every \( f \in {X}^{ * } \)\n\n\[ \left\langle {f, u\left( {t}_{2}\right) }\right\rangle - \left\langle {f, u\left( {t}_{1}\right) }\right\rangle = {\int }_{{t}_{1}}^{{t}_{2}}\langle f, g\left( s\right) \rangle \mathrm{d}s \]\n\n(1.29... | Proof It follows from (1.29) that \( \langle f, u\left( t\right) \rangle \) is absolutely continuous and almost everywhere differentiable with derivative \( \langle f, g\left( t\right) \rangle \) . By assumption \( \langle f, u\left( t\right) \rangle \) and \( \langle f, g\left( t\right) \rangle \) are both elements of... | Yes |
Theorem 1.33 If \( u \in {L}^{2}\left( {0, T;{H}_{0}^{1}}\right) \) and \( {\partial }_{t}u \in {L}^{2}\left( {0, T;{H}^{-1}}\right) \) then we have \( u \in {C}^{0}\left( {\left\lbrack {0, T}\right\rbrack ;{L}^{2}}\right) \) with\n\n\[ \mathop{\sup }\limits_{{t \in \left\lbrack {0, T}\right\rbrack }}\parallel u\left( ... | For the proof of the result in this form see Lemma 10.4 in Renardy & Rogers (2004), for example. A more general version can be found in Chapter 7 of Roubíček (2013). | No |
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