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Proposition 12.3.1 (i) The Hofer distance is symmetric, i.e.\n\n\[ \rho \left( {{\phi }_{0},{\phi }_{1}}\right) = \rho \left( {{\phi }_{1},{\phi }_{0}}\right) \]\n\nfor all \( {\phi }_{0},{\phi }_{1} \in {\operatorname{Ham}}_{\mathrm{c}}\left( {M,\omega }\right) \) | Proof: Abbreviate \( {\phi \psi } \mathrel{\text{:=}} \phi \circ \psi \) for two diffeomorphism \( \phi ,\psi \) of \( M \) . Choose compactly supported Hamiltonians \( F, G : \left\lbrack {0,1}\right\rbrack \times M \rightarrow \mathbb{R} \) such that \( {\phi }_{F} = {\phi }_{1}{\phi }_{0}^{-1} \) and \( {\phi }_{G} ... | No |
Theorem 12.3.3 Let \( \phi ,\psi \in {\operatorname{Ham}}_{\mathrm{c}}\left( {M,\omega }\right) \) . Then\n\n\[ \rho \left( {\phi ,\psi }\right) = 0\; \Rightarrow \;\phi = \psi . \] | Proof: See page 475 for \( M = {\mathbb{R}}^{2n} \) . The proof assumes Theorem 12.1.1. | No |
Theorem 12.3.4 The displacement energy is a relative symplectic capacity for subsets of \( {\mathbb{R}}^{2n} \) and it satisfies\n\n\[ e\left( {{B}^{2n}\left( r\right) }\right) = e\left( {{Z}^{2n}\left( r\right) }\right) = \pi {r}^{2} \]\n\nfor \( r > 0 \) . In particular, \( {w}_{G}\left( A\right) \leq e\left( A\right... | Proof: See page 475. The proof assumes Theorem 12.1.1. | No |
Lemma 12.3.8 Let \( \mathbb{R} \times M \rightarrow \mathbb{R} : \left( {t, z}\right) \mapsto {H}_{t}\left( z\right) \) be a compactly supported Hamiltonian function and denote by \( {\left\{ {\phi }_{t}\right\} }_{t \in \mathbb{R}} \) the Hamiltonian isotopy determined by \( H \) via equation (12.3.1). Then the map\n\... | Proof: Exercise. | No |
Theorem 12.3.11. (Bialy-Polterovich) Let \( {V}_{1},{V}_{2} : {\mathbb{R}}^{2n} \rightarrow \mathbb{R} \) be compactly supported smooth functions with sufficiently small second derivatives. Then\n\n\[ \rho \left( {{\psi }_{{V}_{1}},{\psi }_{{V}_{2}}}\right) = \begin{Vmatrix}{{V}_{1} - {V}_{2}}\end{Vmatrix}. \] | Proof: See Bialy-Polterovich [64]. | No |
Theorem 12.3.13 Let \( \left( {M,\omega }\right) \) be either \( \left( {{\mathbb{R}}^{2n},{\omega }_{0}}\right) \) or a closed symplectic manifold, and let \( \left\lbrack {{t}_{0},{t}_{1}}\right\rbrack \times M \rightarrow M : \left( {t, z}\right) \mapsto {\phi }_{t}\left( z\right) \) be a compactly supported regular... | Proof: See Bialy-Polterovich [64] for the case \( M = {\mathbb{R}}^{2n} \), Lalonde-McDuff [391] for the proof of (i) for general closed \( M \), and McDuff [460] for (ii). | No |
Lemma 12.4.1 Let \( f : {\mathbb{R}}^{m} \rightarrow {\mathbb{R}}^{m} \) be continuously differentiable and assume \( x\left( t\right) = x\left( {t + T}\right) \in {\mathbb{R}}^{m} \) is a periodic solution of the differential equation \( \dot{x} = f\left( x\right) \) . If \( T \cdot \mathop{\sup }\limits_{x}\parallel ... | Proof: Since \( x\left( 0\right) = x\left( T\right) \), an easy calculation shows that\n\n\[ \dot{x}\left( t\right) = {\int }_{0}^{t}\frac{s}{T}\ddot{x}\left( s\right) {ds} + {\int }_{t}^{T}\frac{s - T}{T}\ddot{x}\left( s\right) {ds}. \]\n\nThis implies that\n\n\[ \left| {\dot{x}\left( t\right) }\right| \leq {\int }_{0... | Yes |
Theorem 12.4.2 The map \( \left( {M,\omega }\right) \mapsto {\mathrm{c}}_{\mathrm{{HZ}}}\left( {M,\omega }\right) \) satisfies the monotonicity, conformality, and normalization axioms of a symplectic capacity. Moreover, \[ {\mathrm{c}}_{\mathrm{{HZ}}}\left( {{B}^{2n}\left( r\right) }\right) = {\mathrm{c}}_{\mathrm{{HZ}... | Proof: See page 483. | No |
Theorem 12.4.3 Assume \( H \in \mathcal{H}\left( {{Z}^{2n}\left( 1\right) }\right) \) with \( \sup H > \pi \) . Then the Hamiltonian flow of \( H \) has a nonconstant periodic orbit of period 1 . | Proof: See Section 12.5, page 501. | No |
Lemma 12.4.4 Let \( Q \) be a compact connected hypersurface in \( {\mathbb{R}}^{m} \). Then \( Q \) is oriented and \( {\mathbb{R}}^{m} \smallsetminus Q \) has two connected components. One of these connected components is bounded. | Proof: The proof is by standard arguments in differential topology and we only sketch the main points. The idea is to define, for every point \( x \notin Q \), a map\n\n\[ \n{f}_{x} : Q \rightarrow {S}^{m - 1} \n\]\n\nby\n\n\[ \n{f}_{x}\left( \xi \right) \mathrel{\text{:=}} \frac{\xi - x}{\left| \xi - x\right| } \n\]\n... | Yes |
Theorem 12.4.5 Let \( Q \) be a compact \( \left( {{2n} - 1}\right) \) -dimensional manifold without boundary, and let \( \iota : Q \times \left\lbrack {0,1}\right\rbrack \rightarrow {\mathbb{R}}^{2n} \) be any embedding. Then for a dense set of parameters \( s \in \left\lbrack {0,1}\right\rbrack \) the Hamiltonian flo... | Proof, assuming Theorem 12.4.3: By Lemma 12.4.4 the set \( {\mathbb{R}}^{2n} \smallsetminus \operatorname{Im}\iota \) has two connected components \( {U}_{0} \) and \( {U}_{1} \) . Denote by \( {U}_{0} \) the unbounded component and by \( {U}_{1} \) the bounded component and assume without loss of generality that\n\n\[... | Yes |
Theorem 12.4.6 (Weinstein conjecture) If \( Q \subset {\mathbb{R}}^{2n} \) is a hypersurface of contact type then its characteristic foliation has a closed orbit. | Proof, assuming Theorem 12.4.3: By Proposition 3.5.31, there exists a vector field \( X : V \rightarrow {\mathbb{R}}^{2n} \) defined on a neighbourhood of \( Q \) such that \( X \) is transverse to \( Q \) and \( {\mathcal{L}}_{X}{\omega }_{0} = {\omega }_{0} \) . Denote by \( {\phi }_{s} : {V}_{s} \rightarrow V \) the... | Yes |
Lemma 12.5.3 Suppose that \( {H}_{0} : {Z}^{2n}\left( 1\right) \rightarrow \mathbb{R} \) satisfies (I),(II),(III). Then there exists a Hamiltonian function \( H : {\mathbb{R}}^{2n} \rightarrow \mathbb{R} \) which satisfies the following conditions.\n\n(i) There exists a constant \( R > 0 \) such that \( K \subset {B}^{... | Proof: First choose a number \( 0 < \varepsilon < \pi /2 \) such that \( M > \pi + \varepsilon \) . Then there exists a smooth function \( f : \lbrack 0,\infty ) \rightarrow \mathbb{R} \) such that\n\n\[ f\left( s\right) = M,\;\text{ for }0 \leq s \leq 1, \]\n\n\[ f\left( s\right) = \left( {\pi + \varepsilon }\right) s... | Yes |
Lemma 12.5.4 Suppose that \( H : {\mathbb{R}}^{2n} \rightarrow \mathbb{R} \) satisfies the quadratic growth condition (12.5.1). Then, for \( \tau > 0 \) sufficiently small, there exists a unique function \( {V}_{\tau } : {\mathbb{R}}^{2n} \rightarrow \mathbb{R} \) which satisfies the following conditions.\n\n(i) \( \le... | Proof: That the boundary value problem (12.5.3) has a unique solution for all \( {x}_{1} \) and \( {y}_{0} \) whenever \( \tau \) is sufficiently small follows from the implicit function theorem and Exercise 12.5.2. Hence equation (12.5.4) can be used to define \( {V}_{\tau } \) . We prove that \( {V}_{\tau } \) conver... | No |
Corollary 12.5.6 The map \( {C}^{\infty }\left( {\mathbb{R}/\mathbb{Z},{\mathbb{R}}^{2n}}\right) \rightarrow {\mathbb{R}}^{2nN} : z \mapsto \{ z\left( {j/N}\right) {\} }_{j \in \mathbb{Z}} \) identifies the critical points of the symplectic action \( {\mathcal{A}}_{H} \) with the critical points of the discrete symplec... | Proof: Lemma 12.5.4. | No |
Lemma 12.5.7 Assume that \( \Phi \) satisfies the Palais-Smale condition and let \( c \in \) \( \mathbb{R} \) be a regular value of \( \Phi \) . Then for every \( T > 0 \) there exists a number \( \delta > 0 \) such that \[ \Phi \left( x\right) \leq c + \delta \; \Rightarrow \;\Phi \left( {{\phi }_{T}\left( x\right) }\... | Proof: The proof is by contradiction. If the assertion were false then there would exist a sequence \( {x}_{\nu } \in X \) such that \( \Phi \left( {x}_{\nu }\right) \) and \( \Phi \left( {{\phi }_{T}\left( {x}_{\nu }\right) }\right) \) both converge to \( c \) . Since \[ \frac{d}{dt}\Phi \left( {{\phi }_{t}\left( {x}_... | Yes |
Theorem 13.2.3 (Gromov) Consider a Lagrangian embedding \( \iota : L \rightarrow {\mathbb{R}}^{2n} \) of a compact \( n \) -dimensional manifold \( L \) into \( \left( {{\mathbb{R}}^{2n},{\omega }_{0}}\right) \) . Then the closed 1-form\n\n\[ \n{\iota }^{ * }{\lambda }_{0} \in {\Omega }^{1}\left( L\right) ,\;{\lambda }... | Proof: See [287] and also [470, Theorem 9.2.1]. | No |
Let \( {S}^{n} \) denote the sphere \( \mathop{\sum }\limits_{{j = 0}}^{n}{\xi }_{j}^{2} = 1 \) in \( {\mathbb{R}}^{n + 1} \) . The immersion\n\n\[ \iota : {S}^{n} \rightarrow {\mathbb{R}}^{2n} : \left( {{\xi }_{0},\ldots ,{\xi }_{n}}\right) \mapsto \left( {{\xi }_{1},\ldots ,{\xi }_{n},{\xi }_{0}{\xi }_{1},\ldots ,{\x... | is Lagrangian since \( {\iota }^{ * }{\omega }_{0} = \mathop{\sum }\limits_{{j = 1}}^{n}{\xi }_{j}d{\xi }_{j} \land d{\xi }_{0} = 0 \) . Note that \( \iota : {S}^{n} \rightarrow {\mathbb{R}}^{2n} \) is not an embedding since the north and the south pole are both mapped to the origin (see Fig. 13.2). | Yes |
Consider the following 1-form on \( {\mathbb{R}}^{4} \) :\n\n\[ \sigma = \cos \left( {r}_{1}^{2}\right) \left( {{x}_{1}d{y}_{1} - {y}_{1}d{x}_{1}}\right) + \cos \left( {r}_{2}^{2}\right) \left( {{x}_{2}d{y}_{2} - {y}_{2}d{x}_{2}}\right) ,\n\nwhere \( {r}_{1}^{2} \mathrel{\text{:=}} {x}_{1}^{2} + {y}_{1}^{2} \) and \( {... | Furthermore, the 2-form\n\n\[ {d\sigma } = a\left( {r}_{1}\right) d{x}_{1} \land d{y}_{1} + a\left( {r}_{2}\right) d{x}_{2} \land d{y}_{2},\;a\left( r\right) = 2\cos {r}^{2} - 2{r}^{2}\sin {r}^{2} \]\n\nis nondegenerate except at points where one of \( a\left( {r}_{1}\right) \) or \( a\left( {r}_{2}\right) \) vanishes.... | Yes |
Proposition 13.2.7 A complete simply connected Kähler manifold of nonpositive sectional curvature is symplectomorphic to standard Euclidean space. | Proof: See McDuff [446]. | No |
Gromov [287] has proved that any symplectic structure on the connected sum \( M = {\mathbb{{CP}}}^{3}\# {\mathbb-T}}^{6} \) is nonstandard on the submanifold \( {\mathbb{{CP}}}^{2} \subset {\mathbb{{CP}}}^{3} \) . | Details of the proof may be found in [451]. (One can also prove this by using part (ix) of Remark 4.5.2 in Chapter 4. Namely, if there were a symplectic form \( \sigma \) on \( M = {\mathbb{{CP}}}^{3}\# {T}^{6} \) which had the standard \( {\mathbb{{CP}}}^{2} \) as a symplectic submanifold, then a neighbourhood of infi... | No |
Consider the manifold \( M \mathrel{\text{:=}} {\mathbb{T}}^{2} \times {S}^{2} \times {S}^{2} \). Identify the 2-torus with the product of two circles. For \( \theta \in {S}^{1} \) and \( z \in {S}^{2} \subset {\mathbb{R}}^{3} \), denote the rotation about the axis through \( z \) by the angle \( \theta \) by \( {\phi ... | The proof that \( \omega \) and \( {\psi }^{ * }\omega \) are not isotopic cannot be based on the Gromov-Witten invariants because these are invariant under deformation of symplectic forms (equivalence relation (B) on page 504). In other words, the evaluation maps from the relevant moduli spaces of holomorphic spheres ... | Yes |
A quite different example is provided by Ruan [546]. He finds two symplectic structures on a certain closed \( 6 \) -manifold \( Z \) which are not deformation equivalent (i.e. related by a diffeomorphism followed by a path of symplectic forms). | Observe first that a Kähler manifold carries a natural deformation class of symplectic forms: the set of Kähler forms on a complex manifold is convex, and so all such forms are homotopic (i.e. can be joined by a path of symplectic forms). Ruan considers pairs of homeomorphic but nondiffeomorphic Kähler surfaces \( \lef... | Yes |
In [346], Ionel and Parker computed the Gromov invariants, and hence the Seiberg-Witten invariants, of products of the circle with three-dimensional mapping tori. As a result they were able to construct, for each integer \( n \geq 2 \), infinitely many symplectic four-manifolds that are homeomorphic, but not diffeomorp... | The manifold \( E\left( n\right) \) can be obtained as the fibre connected sum of \( n \) copies of the nine point blowup\n\n\[ E\left( 1\right) \mathrel{\text{:=}} {\mathbb{{CP}}}^{2}\# 9{\overline{\mathbb{{CP}}}}^{2} \]\n\nof the projective plane which admits the structure of a Lefschetz fibration over \( {\mathbb{{C... | Yes |
A longstanding question in symplectic topology is whether two closed smooth manifolds of the same dimension are diffeomorphic if and only if their cotangent bundles with their canonical symplectic structures are symplectomorphic. | In [3], Abouzaid proved that if \( \sum \) is a homotopy \( \left( {{4k} + 1}\right) \) -sphere that does not bound a parallelizable manifold then \( {T}^{ * }\sum \) and \( {T}^{ * }{S}^{{4k} + 1} \) are not symplectomorphic. Such manifolds exist for many, but not all, \( k \geq 2 \) . For example there are six diffeo... | Yes |
Theorem 13.3.10 (Taubes) Let \( \left( {M,\omega }\right) \) be a closed symplectic four-manifold with \( {b}^{ + } \geq 2 \) and define \( a \mathrel{\text{:=}} \left\lbrack \omega \right\rbrack \in {H}^{2}\left( {M;\mathbb{R}}\right) \) . Then\n\n\[ \operatorname{SW}\left( {M,{\mathfrak{o}}_{\omega },{\Gamma }_{\omeg... | Proof: See Taubes \( \left\lbrack {{606},{607}}\right\rbrack \) . | No |
Proposition 13.3.11 Let \( M \) be a closed smooth four-manifold. Two cohomolo-gous symplectic forms on \( M \) have equivalent spin \( {}^{c} \) structures and hence have the same first Chern class. | Proof: Let \( \omega ,{\omega }^{\prime } \) be cohomologous symplectic forms on \( M \) and \( a \mathrel{\text{:=}} \left\lbrack \omega \right\rbrack = \left\lbrack {\omega }^{\prime }\right\rbrack \) . Choose the orientation such that \( {a}^{2} > 0 \) and choose \( e \) such that \( {\Gamma }_{{\omega }^{\prime }} ... | Yes |
Proposition 13.3.13 If a closed symplectic four-manifold \( \left( {M,\omega }\right) \) is diffeomorphic to an oriented connected sum \( X\# Y \) by an orientation-preserving diffeomorphism, then either \( {b}^{ + }\left( X\right) = 0 \) or \( {b}^{ + }\left( Y\right) = 0 \) . | Proof: If \( {b}^{ + }\left( X\right) \) and \( {b}^{ + }\left( Y\right) \) are both positive, then the Seiberg-Witten invariants of the oriented connected sum \( X\# Y \) vanish by Remark 13.3.9. Hence it follows from Theorem 13.3.10 that \( X\# Y \) does not carry a symplectic form that is compatible with the orienta... | Yes |
Let \( M \) be the 2-point blowup of \( {\mathbb{{CP}}}^{2} \) and let \( L,{E}_{1},{E}_{2} \) be the standard basis of \( {H}_{2}\left( {M;\mathbb{Z}}\right) \). Suppose that \( \omega \) is a symplectic form with\n\n\[ \n{c}_{1}\left( \omega \right) = \mathrm{{PD}}\left( {{3L} - {E}_{1} - {E}_{2}}\right) \n\]\n\nSinc... | Moreover, the homology class \( E \mathrel{\text{:=}} {2L} - 2{E}_{1} - {E}_{2} \) is the image of \( {E}_{2} \) under the reflection about the exceptional sphere in the class \( L - {E}_{1} - {E}_{2} \) and hence can be represented by a smoothly embedded sphere with self-intersection number minus one. It satisfies \( ... | Yes |
Theorem 13.3.17 Let \( M \) be a closed smooth four-manifold and let \( \sum \subset M \) be an oriented connected 2-dimensional submanifold of genus \( g \) representing a nontorsion homology class \( A \in {H}^{2}\left( {M;\mathbb{Z}}\right) \) . (i) Assume \( {b}^{ + } > 1 \) and \( A \cdot A \geq 0 \) and let \( c ... | Proof: Part (i) was proved by Kronheimer-Mrowka [381] and independently by Morgan-Szabó-Taubes [494]. | No |
Corollary 13.3.18 (Symplectic Thom conjecture) Every symplectic surface in a closed symplectic 4-manifold minimizes the genus in its homology class. | Proof: The genus of a connected symplectic surface in the homology class \( A \) is given by \( {2g} - 2 = A \cdot A - {c}_{1}\left( A\right) \) (see Example 4.4.5). Hence the assertion follows from part (ii) of Theorem 13.3.17. | Yes |
Consider the complex projective plane \( \left( {M,\omega }\right) = \left( {{\mathbb{{CP}}}^{2},{\omega }_{\mathrm{{FS}}}}\right) \) and let \( L \mathrel{\text{:=}} \left\lbrack {\mathbb{{CP}}}^{1}\right\rbrack \in {H}_{2}\left( {{\mathbb{{CP}}}^{2};\mathbb{Z}}\right) \) be the homology class of a line. Then \( {c}_{... | A similar calculation shows that \( k\left( {2L}\right) = 5 \) and, since there is a unique conic through any five generic points in \( {\mathbb{{CP}}}^{2} \), it follows that \( \operatorname{Gr}\left( {2L}\right) = 1 \) . More generally, one obtains \( k\left( {dL}\right) = \frac{{d}^{2} + {3d}}{2} \) and each holomo... | Yes |
Theorem 13.3.21 (Taubes) Let \( \left( {M,\omega }\right) \) be a closed symplectic four-manifold with \( {b}^{ + } > 1 \) . Then \( \mathrm{{SW}}\left( {M,{\mathfrak{o}}_{\omega },{\Gamma }_{\omega ,\mathrm{{PD}}\left( A\right) }}\right) = \operatorname{Gr}\left( {M,\omega, A}\right) \) for all \( A \in {H}_{2}\left( ... | Proof: See Taubes \( \left\lbrack {{608},{609},{611}}\right\rbrack \) . | No |
Theorem 13.3.22 (Li-Liu) Let \( \\left( {M,\\omega }\\right) \) be a closed symplectic 4-manifold with \( {b}^{ + } = 1 \) and choose a homology class \( A \\in {H}_{2}\\left( {M;\\mathbb{Z}}\\right) \) such that\n\n\[ A \\cdot E \\geq - 1\\;\\forall E \\in {\\mathcal{E}}_{\\omega }\n\]\n\nThen \( \\operatorname{SW}\\l... | Proof: See Li-Liu [423]. | No |
Corollary 13.3.23 (Taubes) Let \( \left( {M,\omega }\right) \) be a closed symplectic 4-manifold and let \( A \in {H}_{2}\left( {M;\mathbb{Z}}\right) \) . If \( {b}^{ + } > 1 \) assume \( \operatorname{SW}\left( {M,{\mathfrak{o}}_{\omega },{\Gamma }_{\omega ,\operatorname{PD}\left( A\right) }}\right) \neq 0 \) and if \... | Proof: By assumption and Theorems 13.3.21 and 13.3.22, \( \operatorname{Gr}\left( {M,\omega, A}\right) \neq 0 \) and hence the homology class \( A \) can be represented by a symplectic submanifold \( C = {C}_{1} \cup \cdots \cup {C}_{\ell } \) of \( M \), where the \( {C}_{i} \) denote the connected components of \( C ... | Yes |
Corollary 13.3.24 (Taubes) Let \( \left( {M,\omega }\right) \) be a closed symplectic four-manifold with \( {b}^{ + } > 1 \) . Then the following holds.\n\n(i) \( \left( {M,\omega }\right) \) has Seiberg-Witten simple type.\n\n(ii) The canonical class \( K \) is Poincaré dual to a symplectic submanifold of \( M \) .\n\... | Proof: Let \( c \in {H}^{2}\left( {M;\mathbb{Z}}\right) \) be a Seiberg-Witten basic class. Then there exists a \( {\operatorname{spin}}^{c} \) structure \( \Gamma \) with first Chern class \( {c}_{1}\left( \Gamma \right) = c \) such that \( \operatorname{SW}\left( {M,{\mathfrak{o}}_{\omega },\Gamma }\right) \neq 0 \) ... | Yes |
Corollary 13.3.25 (Taubes) Let \( \omega \) be any symplectic form on \( {\mathbb{CP}}^{2} \). Then there exists a symplectically embedded sphere \( C \subset {\mathbb{CP}}^{2} \) such that \( C \cdot C = 1 \). | Proof: Since \( {\mathbb{CP}}^{2} \) admits a diffeomorphism that acts as minus the identity on \( {H}_{2}\left( {M;\mathbb{Z}}\right) \), we may assume without loss of generality that \( {\int }_{{\mathbb{CP}}^{1}}\omega > 0 \). Then we have \( {\mathbf{p}}_{\omega } = {\mathbf{p}}_{{\omega }_{\mathrm{FS}}} \) by Rema... | Yes |
Corollary 13.3.26 (Taubes) Any two cohomologous symplectic forms on \( {\mathbb{{CP}}}^{2} \) are diffeomorphic. | Proof: Let \( {\omega }_{\mathrm{{FS}}} \) be the Fubini-Study form and \( \omega \) any other symplectic form in the same cohomology class. By Corollary \( {13.3.25} \) the homology class \( \left\lbrack {\mathrm{{CP}}}^{1}\right\rbrack \) can be represented by an \( \omega \) -symplectic sphere \( C \) . Hence a theo... | Yes |
Corollary 13.3.27 Let \( \left( {M,\omega }\right) \) be a closed symplectic four-manifold.\n\n(i) If \( E \in \mathcal{E} \) satisfies \( {c}_{1}\left( E\right) = 1 \) then \( E \) can be represented by a symplectically embedded sphere.\n\n(ii) If \( {b}^{ + } > 1 \) then every homology class \( E \in \mathcal{E} \) s... | Proof: Assume first that \( {b}^{ + } > 1 \) and denote \( m \mathrel{\text{:=}} {c}_{1}\left( E\right) \) . Then it follows from part (i) of Lemma 13.3.15 that the homology class \( A = {mE} \) satisfies the hypotheses of Corollary 13.3.23. Hence \( {mE} \) can be represented by a symplectic submanifold \( C = {C}_{1}... | Yes |
Lemma 13.3.31 (Inflation lemma) Let \( \\left( {M,\\omega }\\right) \) be a closed symplectic 4-manifold with \( {b}^{ + } = 1 \) and let \( A \\in {H}_{2}\\left( {M;\\mathbb{Z}}\\right) \) such that \( A \\cdot A \\geq 0 \) and \( \\operatorname{Gr}\\left( A\\right) \\neq 0 \) . If \( {A}^{2} = 0 \) assume also that \... | Proof: See Lalonde-McDuff [392]. | No |
Theorem 13.3.32 Let \( \\left( {M,\\omega }\\right) \) be a closed symplectic four-manifold with \( {b}^{ + } = 1 \) . Then any path of symplectic forms on \( M \) with cohomologous endpoints is homotopic with fixed endpoints to an isotopy. | Proof: See McDuff [458], Biran [65], and Li-Liu [424, Proposition 4.11]. | No |
Theorem 13.3.33 Let \( M \) be \( {\mathbb{{CP}}}^{2} \) or a ruled surface.\n\n(i) Any two cohomologous symplectic forms on \( M \) are diffeomorphic.\n\n(ii) Let \( \widetilde{M} \) be the blowup of \( M \) at \( k \) points, and suppose that \( {\widetilde{\omega }}_{0} \) and \( {\widetilde{\omega }}_{1} \) are two... | Proof: We prove (i). For \( M = {\mathbb{{CP}}}^{2} \) this is the assertion of Corollary 13.3.26. If \( M \) is ruled then, by part (i) of Remark 13.3.28, we can assume both forms are compatible with the same ruling. Therefore, they are homotopic by Exercise 6.2.7 and isotopic by Theorem 13.3.32. Hence they are diffeo... | Yes |
Example 13.4.1 (The projective plane) Let \( M = {\mathbb{{CP}}}^{2} \), fix a cohomology class \( a \in {H}^{2}\left( {M;\mathbb{R}}\right) \) with \( {a}^{2} \neq 0 \), and let \( \omega \) be the Fubini-Study form in the class \( a \) . By a theorem of Taubes [608] every symplectic form in the class \( a \) is diffe... | Thus \( {\mathcal{S}}_{a} \) is connected if and only if every diffeomorphism of \( {\mathbb{{CP}}}^{2} \) that acts as the identity on cohomology is isotopic to the identity, i.e. if and only if \( {\operatorname{Diff}}_{h}\left( {\mathbb{{CP}}}^{2}\right) \) is connected. (This is an open problem.) | No |
Example 13.4.2 (The product \( {S}^{2} \times {S}^{2} \) ) The discussion of Example 13.4.1 carries over to \( M \mathrel{\text{:=}} {S}^{2} \times {S}^{2} \) as follows. First, every class \( a \in {H}^{2}\left( {M;\mathbb{R}}\right) \) with \( {a}^{2} \neq 0 \) is represented by a symplectic form. Second, theorems of... | \[ {\mathcal{S}}_{a} \cong \operatorname{Diff}\left( {M, a}\right) /\operatorname{Symp}\left( {M,\omega }\right) \cong {\operatorname{Diff}}_{h}\left( M\right) /{\operatorname{Symp}}_{h}\left( {M,\omega }\right) \] and the group \( {\operatorname{Symp}}_{h}\left( {M,\omega }\right) \) is connected. This shows that \( {... | Yes |
Let \( M \) be an orientable smooth four-manifold that admits the structure of a fibration \( {S}^{2} \hookrightarrow M\overset{\pi }{ \rightarrow }\sum \) over a closed ori-entable surface \( \sum \) of positive genus \( g\left( \sum \right) \) with fibres diffeomorphic to the 2-sphere. Fix an orientation of \( M \) a... | (i) By Taubes Seiberg-Witten theory (Section 13.3) a class \( a \in {H}^{2}\left( {M;\mathbb{R}}\right) \) is represented by a symplectic form if and only if \( {a}^{2} \neq 0 \) and \( \langle a, F\rangle \neq 0 \) . Any such cohomology class is uniquely determined by the numbers \( {a}^{2} \) and \( \langle a, F\rang... | Yes |
Is \( \left( {\widetilde{M},\widetilde{\omega }}\right) \) symplectomorphic to a blowup of \( \left( {M,\omega }\right) \) with weight \( \pi {\lambda }^{2} \) ? | A positive answer to question (b) is necessary for addressing question (a), and question (c) is of course a separate issue that one can consider for any manifold. However, if questions (b) and (c) have positive answers, then so does question (a) by the blowup construction in Section 7.1. In general one could attempt to... | No |
Example 13.4.6 (Deformation equivalence) In [592, 593], Ivan Smith constructed a simply connected four-manifold \( X \) that admits two symplectic forms \( {\omega }_{0},{\omega }_{1} \) such that \( {c}_{1}\left( {\omega }_{0}\right) \) is divisible by three and \( {c}_{1}\left( {\omega }_{1}\right) \) is primitive. H... | In fact, for each integer \( N \) he constructed a simply connected four-manifold \( {X}_{N} \) with at least \( N \) different deformation equivalence classes of symplectic forms, distinguished by the divisibility properties of their first Chern classes. Taking the product \( {X}_{N} \times {\mathbb{T}}^{2n} \), one o... | No |
Let \( M \) be a smooth \( K \) 3-surface. Then every class \( a \in {H}^{2}\left( {M;\mathbb{R}}\right) \) with \( {a}^{2} > 0 \) can be represented by a symplectic form with an associated hyperKähler structure. | To see this, denote by \( {\mathcal{J}}_{\text{int }}\left( M\right) \) the space of integrable almost complex structures on \( M \) and, for \( J \in {\mathcal{J}}_{\text{int }}\left( M\right) \), denote by \( {\mathcal{K}}_{J} \subset {H}^{2}\left( {M;\mathbb{R}}\right) \) the Kähler cone, i.e. the set of all cohomol... | Yes |
Every cohomology class \( a \in {H}^{2}\left( {{\mathbb{T}}^{4};\mathbb{R}}\right) \) with \( {a}^{2} \neq 0 \) is represented by a translation invariant symplectic form and, by Taubes’ Theorem 13.3.10, every symplectic form on \( {\mathbb{T}}^{4} \) has first Chern class zero (see Exercise 13.3.30). There are infinite... | The Conolly-Lê-Ono argument in [126], reproduced in the proof of Proposition 13.3.12, removes only half of these classes as candidates for containing a symplectic form. | No |
Example 13.4.11 (Topology of \( {\mathcal{S}}_{a} \) ) It is also interesting to study the topology of the space \( {\mathcal{S}}_{a} \) of symplectic forms in a given cohomology class beyond the question of connectedness. Thus let us consider the connected component \( {\mathcal{S}}_{\omega } \) of \( {\mathcal{S}}_{a... | If there is a symplectomorphism \( \phi \) that is smoothly, but not symplectically, isotopic to the identity one obtains a loop\n\n\[ \n\left\lbrack {0,1}\right\rbrack \rightarrow {\mathcal{S}}_{\omega } : t \rightarrow {\omega }_{t} \mathrel{\text{:=}} {\phi }_{t}^{ * }\omega ,\;{\omega }_{0} = {\omega }_{1} = \omega... | Yes |
Let \( \left( {M,\omega }\right) \) be a closed symplectic four-manifold and let \( J \) be an \( \omega \) -tame almost complex structure on \( M \) . Does there exist a symplectic form that is compatible with \( J \) (and represents the same cohomology class as \( \omega \) in the case \( {b}^{ + } = 1 \) )? | This question has a positive answer for \( M = {\mathbb{{CP}}}^{2} \) by Taubes [618] and for \( M = {S}^{2} \times {S}^{2} \) by Li-Zhang [425] (see Remark 4.1.3). | No |
Let \( \\left( {M,\\omega }\\right) \) be the \( k \) -point blowup of the complex projective plane with \( k \\geq 5 \) and any symplectic form. Is \( {\\operatorname{Symp}}_{h}\\left( {M,\\omega }\\right) \\subset {\\operatorname{Diff}}_{0}\\left( M\\right) \) ? | Now consider the image of the homomorphism (14.2.1). The isotopy class of a diffeomorphism \( \\phi : M \\rightarrow M \) belongs to this image if and only if \( \\phi \) is isotopic to a symplectomorphism. Necessary conditions are the following.\n\n(S) \( {\\phi }^{ * }\\omega - \\omega \) is exact and \( \\omega ,{\\... | No |
Problem 22 (Autonomous Hamiltonian conjecture)\n\nThe number \( \operatorname{aut}\left( {M,\omega }\right) \) is infinite for every closed symplectic manifold \( \left( {M,\omega }\right) \) . | Polterovich-Shelukhin [533] verify this conjecture for Riemann surfaces of genus at least four and their products with symplectically aspherical manifolds. They also derive consequences of this result in Hamiltonian dynamics. | No |
Fix an integer \( n \geq 3 \), let \( {\iota }_{0} : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{2n} \) be the standard Lagrangian embedding given by\n\n\[ \n{\iota }_{0}\left( x\right) \mathrel{\text{:=}} \left( {x,0}\right) \;\text{ for }x \in {\mathbb{R}}^{n}, \n\]\n\nand let \( \iota : {\mathbb{R}}^{n} \rightarrow {... | In a similar vein, Dimitroglou Rizell-Evans [137] proved that if \( n \geq 5 \) is an odd integer, then any two embedded Lagrangian tori in \( \left( {{\mathbb{R}}^{2n},{\omega }_{0}}\right) \) are smoothly isotopic. | No |
The Reeb vector field of every contact form on a closed contact manifold has a closed orbit. | For contact hypersurfaces of Euclidean space the Weinstein conjecture was settled by Viterbo [643] in the late 1980s (see Theorem 12.4.6). It was confirmed by Hofer [317] for the 3-sphere and by Taubes [612] for all closed contact 3- manifolds. In [130], Cristofaro-Gardiner and Hutchings used embedded contact homology ... | No |
Which closed contact manifolds \( \left( {M,\xi }\right) \) admit (contractible) positive loops of contactomorphisms? | Positive loops of contactomorphisms obviously exist on contact manifolds with periodic Reeb flows, such as pre-quantization circle bundles over symplectic manifolds (Example 3.5.11) and unit cotangent bundles of closed Riemannian manifolds with periodic geodesic flows (Example 3.5.7). These loops are all non-contractib... | No |
Under what conditions on \( {a}_{i} \) and \( {b}_{i} \) does there exist a symplectic embedding of \( E\left( a\right) \) into \( E\left( b\right) \) (with respect to the standard symplectic structure \( {\omega }_{0} \) )? | For \( n = 1 \) the obvious answer is if and only if \( {a}_{1} \leq {b}_{1} \) . For \( n = 2 \) an answer to the symplectic ellipsoid embedding problem was given by McDuff [463] in 2009. This answer was later reformulated by Hutchings [338,339] and McDuff [467]. Define the ordered sequence\n\n\[ 0 < {c}_{1}\left( a\r... | Yes |
Let \( \left( {M,\omega }\right) \) be a connected, simply connected, symplectic \( {2n} \) -manifold and let \( J \) be an \( \omega \) -compatible almost complex structure such that the Riemannian metric \( \langle \cdot , \cdot \rangle \mathrel{\text{:=}} \omega \left( {\cdot, J \cdot }\right) \) is complete and has... | By Hadamard’s theorem the manifold \( M \) in Problem 51 is diffeomorphic to \( {\mathbb{R}}^{2n} \) . By a theorem of McDuff [446] the question has a positive answer whenever \( J \) is integrable. | No |
Lemma 51.1 (transitivity). | The argument is identical for \( \left( {h \circ f}\right) \circ \left( {g \circ k}\right) = h \circ \left( {f \circ g}\right) \circ k \) . | No |
Problem 13.1. Let \( X \) be a topological space; let \( A \) be a subset of \( X \) . Suppose that for each \( x \in A \) there is an open set \( U \) containing \( x \) such that \( U \subset A \) . Show that \( A \) is open in \( X \) . | Solution: Let \( {\mathcal{C}}_{A} \) the collection of open sets \( U \) where \( x \in U \subseteq A \) for some \( x \in A \) . Suppose \( {U}_{0} = \mathop{\bigcup }\limits_{{U \in {\mathcal{C}}_{A}}}U \) . Since \( X \) is a topological space, \( {U}_{0} \) is open in \( X \) . Clearly if \( x \in A \), then \( x ... | Yes |
Problem 13.2. Consider the nine topologies on the set \( X = \{ a, b, c\} \) indicated in Example 1 of \( § \) 12. Compare them; that is, for each pair of topologies, determine whether they are comparable, and if so, which is the finer. | Solution: I label the topologies in Example 1 as follows:\n\n\[ \begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} \]\n\nThe table below compares them. The number indicates the finer of the two topologies. \ | No |
Problem 13.2. (a) If \( \{ {\mathcal{T}}_{\alpha }\} \) is a family of topologies on \( X \) , show that \( \bigcap {\mathcal{T}}_{\alpha } \) is a topology on \( X \) . Is \( \bigcup {\mathcal{T}}_{\alpha } \) a topology on \( X \) ? | Part (a) Let \( \mathcal{T} = \mathop{\bigcap }\limits_{\alpha }{T}_{\alpha } \) . Obviously \( X \) and \( \varnothing \) are element sof \( \mathcal{T} \) since they are contained in each \( \alpha \) . Now suppose \( \left\{ {U}_{\beta }\right\} \) is an indexed family of non-empty open sets in \( \mathcal{T} \) . I... | Yes |
Problem 13.6. Show that the topologies of \( {\mathbb{R}}_{l} \) and \( {\mathbb{R}}_{K} \) are not comparable. | Solution: It suffices to show that neither of the topologies is finer than the other. Let \( {\mathcal{T}}_{l} \) and \( {\mathcal{T}}_{K} \) be the topologies of \( {\mathbb{R}}_{l} \) and \( {\mathbb{R}}_{K} \) , respectively. A given \( x \in \mathbb{R} \) is contained in the basis element \( \lbrack x, b) \) of \( ... | Yes |
Problem 13.8. (a) Apply Lemma 13.2 to show that the countable collection \( \mathcal{B} = \{ \left( {a, b}\right) \mid a < b, \) a and b rational\} is a basis that generates the standard topology on \( \mathbb{R} \) . | Solution: Part (a) Let \( \mathcal{T} \) be the topology generated by \( \mathcal{B} \) and \( {\mathcal{T}}_{\mathbb{R}} \) be the standard topology on \( \mathbb{R} \) . Let \( {U}_{0} \) be an open set in \( \mathcal{T} \) . It follows that \( {U}_{0} \) is the union of some subcollection \( \left\{ {B}_{k}\right\} ... | Yes |
Problem 16.2. If \( \mathcal{T} \) and \( {\mathcal{T}}^{\prime } \) are topologies on \( X \) and \( {\mathcal{T}}^{\prime } \) is strictly finer than \( \mathcal{T} \) , what can you say about the corresponding subspace topologies on the subset \( Y \) of \( X \) ? | Solution: Let \( {U}_{0} \) be open in the subspace topology \( {\mathcal{T}}_{Y} \) of \( \mathcal{T} \) . There is a \( {W}_{0} \in \mathcal{T} \) where \( {U}_{0} = {W}_{0} \cap Y \) . But since \( {\mathcal{T}}^{\prime } \) is finer than \( \mathcal{T} \), it follows that \( {W}_{0} \) is open in \( {\mathcal{T}}^{... | Yes |
Problem 17.1. Let \( \mathcal{C} \) be a collection of subsets of the set \( X \) . Suppose that \( \varnothing \) and \( X \) are in \( \mathcal{C} \) , and that finite unions and arbitrary intersections of elements of \( \mathcal{C} \) are in \( \mathcal{C} \) . Show that the collection \( \mathcal{T} = \{ X - C|C{in... | Solution: Clearly \( X \smallsetminus X = \varnothing \) and \( x \smallsetminus \varnothing = X \) are open in \( \mathcal{T} \) . Suppose \( U = \mathop{\bigcup }\limits_{\alpha }{U}_{\alpha } \) for some indexed collection \( \left\{ {U}_{\alpha }\right\} \) of open sets in \( \mathcal{T} \) . Since each \( {U}_{\al... | Yes |
Show that if \( A \) is closed in \( Y \) and \( Y \) is closed in \( X \), then \( A \) is closed in \( X \) . | This question requires us to prove Theorem 17.3. By hypothesis, \( Y \smallsetminus A \) is open in \( Y \) , and there is some \( U \) open in \( X \) where \( U \cap Y = Y \smallsetminus A \) . Therefore \( X \smallsetminus U \) is closed in \( X \) . Consequently, since \( A \subseteq Y \) :\n\n\[ \nY \cap \left( {X... | Yes |
Problem 17.4. Show that if \( U \) is open in \( X \) and \( A \) is closed in \( X \) , then \( U - A \) is open in \( X \) , and \( A - U \) is closed in \( X \) . | Solution: The set \( X \smallsetminus A \) is open in \( X \) . We have:\n\n\[ U \cap \left( {X \smallsetminus A}\right) = \left( {U \cap X}\right) \smallsetminus \left( {U \cap A}\right) = U \smallsetminus \left( {U \cap A}\right) = \left( {U \smallsetminus U}\right) \cup \left( {U \smallsetminus A}\right) = U \smalls... | Yes |
Problem 17.6. Let \( A \) , \( B \) , and \( {A}_{\alpha } \) be subsets of space \( X \) . Prove the following: (a) If \( A \subset B \) , then \( \overline{A} \subset \overline{B} \) . | Solution: Part (a) This is an important result. We have \( A \subseteq B \subseteq \overline{B} \) . Since \( \overline{B} \) is a closed set containing \( A \), it contains the closure of \( A \) . Consequently, \( \bar{A} \subseteq \bar{B} \) . | Yes |
Let \( A \) , \( B \) , and \( {A}_{\alpha } \) denote subsets of space \( X \) . Determine whether the following equations hold; if an equality fails, determine whether one of the inclusions \( \supset \) or \( \subset \) holds. (a) \( \overline{A \cap B} \) . (b) \( \overline{\bigcap {A}_{\alpha }} = \bigcap {\overli... | Solution: Part (a) See part (b).\n\nPart (b) Since \( \bigcap {A}_{\alpha } \subseteq {A}_{\beta } \) for all \( {A}_{\beta } \) in the collection \( \left\{ {A}_{\alpha }\right\} \), it follows from exercise 17.6(a) that \( \overline{\bigcap {A}_{\alpha }} \subseteq \overline{{A}_{\beta }} \) . Intersecting all \( \ov... | No |
Problem 17.9. Let \( A \subset X \) and \( B \subset Y \) . Show that in the space \( X \times Y,\overline{A \times B} = \bar{A} \times \bar{B} \) . | Solution: If \( \left( {x, y}\right) \in \overline{A} \times \overline{B} \) , then \( x \in \overline{A} \) and \( y \in \overline{B} \) . Applying Theorem 17.5, every basis element \( {B}_{x} \) of \( X \) containing \( x \) intersects \( A \) . Similarly, every basis element \( {B}_{y} \) containing \( y \) intersec... | Yes |
Problem 17.10. Show that every order topology is Hausdroff. | Solution: Suppose \( \mathcal{T} \) is an order topology on a given set \( X \) . Let \( {x}_{1},{x}_{2} \) be distinct points in \( X \) where \( {x}_{1} < {x}_{2} \) . If \( {x}_{2} \) is not the immediate successor of \( {x}_{1} \), there is some \( c \in \left( {{x}_{1},{x}_{2}}\right) \) . If \( {x}_{1} \) and \( ... | Yes |
Problem 17.11. Show that the product of two Hausdorff spaces is Hausdroff. | Solution: Suppose \( X \) and \( Y \) are Hausdorff spaces. Given distinct \( \left( {{x}_{0},{y}_{0}}\right) ,\left( {{x}_{1},{y}_{1}}\right) \) of \( X \times Y \), if \( {x}_{0} \neq {x}_{1} \) and \( {y}_{0} \neq {y}_{1} \) , there are neighborhoods \( {A}_{0} \) of \( {x}_{0} \) and \( {A}_{1} \) of \( {x}_{1} \) ... | Yes |
Problem 17.12. Show that the subspace of a Hausdorff space is Hausdorff. | Solution: Let \( X \) be a Hausdorff space and \( Y \) a subset of \( X \) . Given any distinct \( {x}_{0},{x}_{1} \) in \( Y \), there are neighborhoods \( U \) of \( {x}_{0} \) and \( V \) of \( {x}_{1} \) in \( X \) that are disjoint. By definition, \( {U}^{\prime } = U \cap Y \) and \( {V}^{\prime } = V \cap V \) a... | Yes |
Show that \( X \) is Hausdorff if and only if the \( \mathbf{diagonal}\Delta = \{ x \times x|x \in X\} \) is closed in \( X \times X \) . | Solution: Suppose \( X \) is a Hausdorff space. Given \( x \in X \), from Theorem 17.8 the singleton \( \{ x\} \) is closed in \( X \), so \( \overline{\{ x\} } = \{ x\} \) . Accordingly, for every \( y \neq x \) , there is some neighborhood of \( y \) in \( X \) that does not intersect \( \{ x\} \) . Next we will show... | Yes |
Problem 17.14. In the finite complement topology on \( \mathbb{R} \) , to what point or points does the sequence \( {x}_{n} = 1/n \) converge? | Solution: On page 99 of the text, Munkres states without proof that \( \mathbb{R} \) in the finite complement toplogy is not Hausdorff. We’ll quickly prove this result here. Given space \( X \) with the finite complement topology, assume \( X \) is Hausdorff. It follows that for distinct \( u, v \in X \), there are nei... | Yes |
Problem 17.15. Show the \( {T}_{1} \) axiom is equivalent to the condition that for each pair of points of \( X \) , each has a neighborhood not containing the other. | Solution: Suppose a space \( X \) satisfies the \( {T}_{1} \) axiom. Given \( A = \{ a\} \) and \( B = \{ b\} \) where \( a \neq b \) , each singleton is closed. Therefore \( A = \bar{A} \) and \( B = \bar{B} \) . Since \( b \notin \bar{A} \), we infer from Theorem 17.5 that there is a neighborhood of \( b \) that does... | Yes |
Theorem 1. Suppose \( T \) is a Hausdorff space. If \( U \) is finer than \( T \), then \( U \) is Hausdorff. | Proof. For any distinct points \( x \) and \( y \) , there are neighborhoods \( A \) of \( x \) and \( B \) of \( y \) in \( T \) that are disjoint. Since \( T \subseteq U \), it follows that \( A \) and \( B \) are open in \( U \) . Hence, \( U \) is Hausdorff. | Yes |
Problem 18.2. Suppose that \( f : X \rightarrow Y \) is continuous. If \( f \) is a limit point of the subset \( A \) of \( X \) , is it necessarily true that \( f\left( x\right) \) is a limit point of \( f\left( A\right) \) ? | Solution: No, this isn’t necessarily true. Suppose \( f \) is a constant function where \( f\left( X\right) = \{ \gamma \} \) for some \( \gamma \in Y \) . It follows that \( f\left( x\right) = \gamma \), which is not a limit point of \( f\left( A\right) \) because any neighborhood of \( f\left( x\right) \) cannot inte... | Yes |
Problem 18.3. Let \( X \) and \( {X}^{\prime } \) denote a single set in two topologies \( \mathcal{T} \) and \( {\mathcal{T}}^{\prime } \) , respectively. Let \( i : {X}^{\prime } \rightarrow X \) be the identity function. (a) Show that \( i \) is continuous \( \Leftrightarrow \bar{{\mathcal{T}}^{\prime }} \) is finer... | Solution: Part(a) Suppose \( i \) is continuous. Given a basis element \( B \) of \( X \) , the set \( {f}^{-1}\left( B\right) = {V}^{\prime } \) is open in \( {X}^{\prime } \) . Since \( i \) is the identity function, \( x \in B \) if and only if \( x \in {V}^{\prime } \), so \( B = {V}^{\prime } \) . Further, if \( x... | Yes |
Problem 18.4. Given \( {x}_{0} \in X \) and \( {y}_{0} \in Y \), show that the maps \( f : X \rightarrow X \times Y \) and \( g : Y \rightarrow X \times Y \) defined by \( f\left( x\right) = x \times {y}_{0} \) and \( g\left( y\right) = {x}_{0} \times y \) are imbeddings. | Solution: The function \( f \) is an injection. Given \( a, b \in X \), we have \( f\left( a\right) = \left( {a,{y}_{0}}\right) = \left( {b,{y}_{0}}\right) = f\left( b\right) \), so \( a = b \) . The set \( X \times {y}_{0} \) is a subspace of \( X \times Y \) . Let \( {f}^{\prime } : X \rightarrow X \times {y}_{0} \) ... | Yes |
Problem 18.6. Find a function \( f : \mathbb{R} \rightarrow \mathbb{R} \) that is continuous at precisely one point. | Solution: We can use a variation of the Dirichlet function:\n\n\[ f\left( x\right) = \left\{ \begin{array}{lll} x & \text{ if } & x \in \mathbb{Q}, \\ 0 & \text{ if } & x \in \mathbb{R} \smallsetminus \mathbb{Q}. \end{array}\right. \]\n\nThis function is continuous only at \( x = 0 \) . Let \( U \) be a neighborhood of... | Yes |
Problem 18.9. Let \( \\left\\{ {A}_{\\alpha }\\right\\} \) be a collection of subsets of \( X \) ; let \( X = \\mathop{\\bigcup }\\limits_{\\alpha }{A}_{\\alpha } \). Let \( f : X \\rightarrow Y \) ; suppose that \( f \\mid {A}_{\\alpha } \) is continous for each \( \\alpha \). (a) Show that if the collection \( \\{ {A... | Solution: Part (a) Since \( f|{A}_{\\alpha } \) is continuous, by Theorem 18.1 for every closed set \( B \) of \( Y \), the set \( {\\left( f|{A}_{\\alpha }\\right) }^{-1}\\left( B\\right) \) is closed. It follows that:\n\n\[ \n{f}^{-1}\\left( B\\right) = \\mathop{\\bigcup }\\limits_{\\alpha }{\\left( f \\mid {A}_{\\al... | Yes |
Let \( f : A \rightarrow B \) and \( g : C \rightarrow D \) be continuous functions. Let us define a map \( f \times g : A \times C \rightarrow B \times D \) by the equation \( \left( {f \times g}\right) \left( {a \times c}\right) = f\left( a\right) \times g\left( c\right) \) . Show that \( f \times g \) is continuous. | Solution: A basis for \( B \times D \) is the collection \( \{ U \times V : U \) is a basis element of \( B \) and \( V \) is a basis element of \( D\} \) . Given a basis element \( U \times V \) , it follows that \( {f}^{-1}\left( U\right) \) are \( {g}^{-1}\left( V\right) \) are open. By the definition of a product t... | Yes |
Problem 18.11. Let \( F : X \times Y \rightarrow Z \) . We say that \( F \) is continuous in each variable separately if for each \( {y}_{0} \) in \( Y \) , the map \( h : X \rightarrow Z \) defined by \( h\left( x\right) = F\left( {x \times {y}_{0}}\right) \) is continuous, and for each \( {x}_{0} \) in \( X \) , the ... | Solution: Suppose \( F \) is continuous. Given \( {y}_{0} \in Y \), for each \( {x}_{0} \in \), let \( U \) be a neighborhood of \( h\left( {x}_{0}\right) = F\left( {{x}_{0},{y}_{0}}\right) \) . It follows that \( {F}^{-1}\left( U\right) = V \times W \) is open in \( X \times Y \), and \( \left( {{x}_{0},{y}_{0}}\right... | Yes |
Problem 18.12. Let \( F : \mathbb{R} \times \mathbb{R} \) be defined by the equation:\n\n\[ F\left( {x \times y}\right) = \left\{ \begin{array}{lll} {xy}/\left( {{x}^{2} + {y}^{2}}\right) & \text{ if } & x \times y \neq 0 \times 0. \\ 0 & \text{ if } & x \times y = 0 \times 0. \end{array}\right. \]\n\n(a) Show that \( ... | Solution: Part (a) To make things easy, I’ll use basic analysis to prove continuity. For each \( {y}_{0} \in Y \) , define:\n\n\[ {h}_{{y}_{0}} = F\left( {x \times {y}_{0}}\right) = \left\{ \begin{array}{lll} x{y}_{0}/\left( {{x}^{2} + {y}_{0}^{2}}\right) & \text{ if } & {x}_{0} \neq 0\text{ or }{y}_{0} \neq 0, \\ 0 & ... | Yes |
Prove Theorem 19.4. | Given distinct \( \mathbf{x},\mathbf{y} \in \mathop{\prod }\limits_{\alpha }{X}_{\alpha } \), there is some \( \delta \) such that \( {\mathbf{x}}_{\delta } \neq {\mathbf{y}}_{\delta } \) . Because \( {X}_{\delta } \) is Hausdorff, there is some neighborhood \( {U}_{\delta }^{\prime } \) of \( {\mathbf{x}}_{\delta } \)... | Yes |
Show that \( \left( {{X}_{1} \times \cdots \times {X}_{n - 1}}\right) \times {X}_{n} \) is homeomorphic with \( {X}_{1} \times \cdots \times {X}_{n} \) . | Designate \( Y = \mathop{\prod }\limits_{{j = 1}}^{{n - 1}}{X}_{n} \) and \( Z = \mathop{\prod }\limits_{{j = 1}}^{n}{X}_{n} \) . Define \( f : Y \times {X}_{n} \rightarrow Z \) as \( f\left( {\left( {{a}_{1},\ldots ,{a}_{n - 1}}\right) ,{a}_{n}}\right) = \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) . A simple verificati... | Yes |
Problem 19.5. One of the implications stated in Theorem 19.6 holds for the box topology. Which one? | Solution: Since the box topology is finer than the product topology, any open set in the latter is open in the former. Therefore if \( f \) is continuous in the box topology, it must be continuous in the product topology. The topologies of each \( {X}_{\alpha } \) are unchanged. Applying Theorem 19.6, each \( {f}_{\alp... | Yes |
Problem 19.7. Let \( {\mathbb{R}}^{\infty } \) be the subset of \( {\mathbb{R}}^{\omega } \) consisting of all sequences that are ”eventually zero,” that is, all sequences \( \left( {{x}_{1},{x}_{2},\ldots }\right) \) such that \( {x}_{i} \neq 0 \) for only finitely many values of \( i \) . What is the closure of \( {\... | Solution: In the product topology, any sequence \( \left( {x}_{n}\right) \) is in the closure of \( {\mathbb{R}}^{\infty } \) . If \( {U}^{\prime } \) is a neighborhood of \( \left( {x}_{n}\right) \), a basis element of \( U \) must be of the form \( B = \mathop{\bigcup }\limits_{\alpha }{B}_{\alpha } \) where only a f... | Yes |
Problem 19.8. Given the sequence \( \left( {{a}_{1},{a}_{2},\ldots }\right) \) and \( \left( {{b}_{1},{b}_{2},\ldots }\right) \) of real numbers with \( {a}_{i} > 0 \) for all \( i \) , define \( h : {\mathbb{R}}^{\omega } \rightarrow {\mathbb{R}}^{\omega } \) by the equation \( h\left( {{x}_{1},{x}_{2},\ldots }\right)... | Solution: Let \( {\phi }_{k} : \mathbb{R} \rightarrow \mathbb{R} \) where, for a given real \( {a}_{k},{b}_{k} \) where \( {a}_{k} > 0 \), we define \( {\phi }_{k}\left( {x}_{k}\right) = {a}_{k}{x}_{k} + {b}_{k} \) . This function is a bijection. If \( {\phi }_{k}\left( {x}_{k}\right) = {a}_{k}{x}_{k} + {b}_{k} = {a}_{... | Yes |
Problem 19.9. [See problem.] | Solution: Part(a) For each \( \alpha \), let \( {C}_{\alpha } = \left\{ {D \subseteq A : {f}_{\alpha }^{-1}\left( {B}_{\alpha }\right) = D}\right. \) for some basis element \( \left. {{B}_{\alpha }\text{of}{X}_{\alpha }}\right\} \) . Let \( \mathcal{U} = \) \( \mathop{\bigcup }\limits_{{\alpha \in J}}{C}_{\alpha } \cup... | Yes |
Problem 20.2. Show that \( \mathbb{R} \times \mathbb{R} \) in the dictionary order topology is metrizable. | Solution: Define the function \( d : \left( {\mathbb{R} \times \mathbb{R}}\right) \times \left( {\mathbb{R} \times \mathbb{R}}\right) \rightarrow \mathbb{R} \) as:\n\n\[ d\left( {{x}_{1} \times {x}_{2},{y}_{1} \times {y}_{2}}\right) = \left\{ \begin{array}{lll} 1 & \text{ for } & {x}_{1} \neq {y}_{1}, \\ \inf \left\{ {... | Yes |
Problem 20.3. Let \( X \) be a metric space with metric \( d \) . (a) Show that \( d : X \times X \rightarrow \mathbb{R} \) is continuous. (b) Let \( {X}^{\prime } \) denote a space having the same underlying set as \( X \) . Show that if \( d : {X}^{\prime } \times {X}^{\prime } \rightarrow \mathbb{R} \) is continuous... | Solution: Part (a) Let \( U \) be a basis element of \( \mathbb{R} \), so \( U = \left( {a, b}\right) \) for some \( a < b \) . It follows that \( {d}^{-1}\left( U\right) = \{ x \times y \in \) \( X \times X : d\left( {x, y}\right) \in \left( {a, b}\right) \} \) (in other word, each pair of points in \( X \times X \) t... | Yes |
Problem 20.4. [See problem.] | Solution: Part (b) Skipped. | No |
Let \( A \subset X \) . If \( d \) is a metric for the topology of \( X \) , show that \( d|A \times A \) is a metric for the subspace topology on \( A \) . | Solution: Designate \( f : A \times A \rightarrow \mathbb{R} \) where \( f = d \mid A \times A \) and the topology generated in \( A \) by it \( \mathcal{T} \) . Clearly if \( \mathbf{a},\mathbf{b} \in A \) , then \( f\left( {\mathbf{a},\mathbf{b}}\right) = d\left( {\mathbf{a},\mathbf{b}}\right) \) . Suppose \( B = {B}... | Yes |
Problem 21.2. Let \( X \) and \( Y \) be metric spaces with metrics \( {d}_{X} \) and \( {d}_{Y} \) , respectively. Let \( f : X \rightarrow Y \) have the property that for every pair of points \( {x}_{1},{x}_{2} \) of \( X \) , \( {d}_{Y}\left( {f\left( {x}_{1}\right), f\left( {x}_{2}\right) }\right) = {d}_{X}\left( {... | Solution: Let \( g : X \rightarrow f\left( X\right) \) be a restriction of \( f \) where \( g\left( x\right) = f\left( x\right) \) for all \( x \in X \) . Obviously \( g \) is surjective. For any \( {x}_{1},{x}_{2} \in X \), if \( g\left( {x}_{1}\right) = g\left( {x}_{2}\right) = y \), then \( {d}_{Y}\left( {g\left( {x... | Yes |
Problem 21.7. Let \( X \) be a set, and let \( {f}_{n} : X \rightarrow \mathbb{R} \) be a sequence of functions. Let \( \bar{\rho } \) be the uniform metric on the space \( {\mathbb{R}}^{X} \) . Show that the sequence \( \left( {f}_{n}\right) \) converges uniformly to the function \( f : X \rightarrow \mathbb{R} \) if ... | Solution: Suppose \( \left( {f}_{n}\right) \) converges uniformly to \( f \) in \( \mathbb{R} \) . Define \( \mathbf{y},{\mathbf{z}}_{n} \in {\mathbb{R}}^{X} \) such that \( \mathbf{y}\left( x\right) = f\left( x\right) \) and \( {\mathbf{z}}_{n}\left( x\right) = {f}_{n}\left( x\right) \) for all \( n \in \mathbb{N} \) ... | Yes |
Problem 24.3. | Solution: Define \( g : X \rightarrow \mathbb{R} \) where \( g\left( x\right) = f\left( x\right) - {i}_{\mathbb{R}}\left( x\right) = f\left( x\right) - x \) where \( {i}_{\mathbb{R}} \) is the identity function. Since \( f \) and \( {i}_{\mathbb{R}} \) are continuous, \( g \) is continuous by Theorems 18.2(e) and 21.5.... | Yes |
Problem 24.10. | Solution: Suppose \( U \) is an open connected subspace of \( {\mathbb{R}}^{2} \) . Let \( {x}_{0} \in U \) be given and \( V \) be the set of points in \( U \) to which there is a path from \( {x}_{0} \) in \( U \) . Since \( U \) is open in \( {\mathbb{R}}^{2} \), there is some non-empty basis element \( {B}_{0} \sub... | No |
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