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Proposition 4.1.15 (Serre duality) Let \( X \) be a compact complex manifold. For any holomorphic vector bundle \( E \) on \( X \) the natural pairing\n\n\[ \n{H}^{p, q}\left( {X, E}\right) \times {H}^{n - p, n - q}\left( {X,{E}^{ * }}\right) \rightarrow \mathbb{C} \n\]\n\nis non-degenerate. | Proof. Fix hermitian structures \( h \) and \( g \) on \( E \) and \( X \), respectively. Then consider the pairing \( {\mathcal{H}}^{p, q}\left( {X, E}\right) \times {\mathcal{H}}^{n - p, n - q}\left( {X,{E}^{ * }}\right) \rightarrow \mathbb{C} \) . In order to show that this pairing is non-degenerate, we have to show... | Yes |
Corollary 4.1.16 For any holomorphic vector bundle \( E \) over a compact complex manifold \( X \) there exist natural \( \mathbb{C} \) -linear isomorphisms (Serre duality): | Remark 4.1.17 The isomorphism \( {\bar{ * }}_{E} : {\mathcal{H}}^{p, q}\left( {X, E}\right) \cong {\mathcal{H}}^{n - p, n - q}\left( {X,{E}^{ * }}\right) \) induces an isomorphism \( {H}^{p, q}\left( {X, E}\right) \cong {H}^{n - p, n - q}\left( {X,{E}^{ * }}\right) \) . But this isomorphism is only \( \mathbb{C} \) -an... | No |
Proposition 4.2.3 If \( \nabla \) and \( {\nabla }^{\prime } \) are two connections on a vector bundle \( E \) , then \( \nabla - {\nabla }^{\prime } \) is \( {\mathcal{A}}_{M}^{0} \) -linear and can, therefore, be considered as an element in \( {\mathcal{A}}^{1}\left( {M,\operatorname{End}\left( E\right) }\right) \) .... | Proof. We have to show that \( \left( {\nabla - {\nabla }^{\prime }}\right) \left( {f \cdot s}\right) = f \cdot \left( {\nabla - {\nabla }^{\prime }}\right) \left( s\right) \), which is an immediate consequence of the Leibniz rule (4.2). | Yes |
Corollary 4.2.4 The set of all connections on a vector bundle \( E \) is in a natural way an affine space over the (infinite-dimensional) complex vector space \( {\mathcal{A}}^{1}\left( {M,\operatorname{End}\left( E\right) }\right) \) . | Remark 4.2.5 Often, local calculations are performed by using the following statement: Any connection \( \nabla \) on a vector bundle \( E \) can locally be written as \( d + A \), where \( A \) is a matrix valued one-form.\n\nIndeed, if \( E \) is the trivial vector bundle \( E = M \times {\mathbb{C}}^{r} \), then \( ... | No |
Proposition 4.2.14 Let \( \left( {E, h}\right) \) be a holomorphic vector bundle together with an hermitian structure. Then there exists a unique hermitian connection \( \nabla \) that is compatible with the holomorphic structure. This connection is called the Chern connection on \( \left( {E, h}\right) \) . | Proof. Let us first show the uniqueness. This is is a purely local problem. Thus, we may assume that \( E \) is the trivial holomorphic vector bundle, i.e. \( E = X \times {\mathbb{C}}^{r} \) . According to Remark 4.2.5 the connection \( \nabla \) is of the form \( \nabla = d + A \), where \( A = \left( {a}_{ij}\right)... | Yes |
Proposition 4.2.19 A holomorphic vector bundle \( E \) admits a holomorphic connection if and only if its Atiyah class \( A\left( E\right) \in {H}^{1}\left( {X,{\Omega }_{X} \otimes \operatorname{End}\left( E\right) }\right) \) is trivial. | Proof. First note that \( d{\psi }_{ij} = \partial {\psi }_{ij} \), as the \( {\psi }_{ij} \) are holomorphic.\n\nLocal holomorphic connections on \( {U}_{i} \times {\mathbb{C}}^{r} \) are of the form \( \partial + {A}_{i} \) . Those can be glued to a connection on the bundle \( E \) if and only if\n\n\[{\psi }_{i}^{-1... | Yes |
Lemma 4.3.2 The curvature homomorphism \( {F}_{\nabla } : {\mathcal{A}}^{0}\left( E\right) \rightarrow {\mathcal{A}}^{2}\left( E\right) \) is \( {\mathcal{A}}^{0} \) - linear. | Proof. For a local section \( s \) of \( E \) and a local function \( f \) on \( M \) one computes\n\n\[ \n{F}_{\nabla }\left( {f \cdot s}\right) = \nabla \left( {\nabla \left( {f \cdot s}\right) }\right) = \nabla \left( {{df} \otimes s + f \cdot \nabla \left( s\right) }\right) \n\]\n\n\[ \n= \underset{0}{\underbrace{{... | Yes |
Lemma 4.3.5 (Bianchi identity) If \( {F}_{\nabla } \in {\mathcal{A}}^{2}\left( {M,\operatorname{End}\left( E\right) }\right) \) is the curvature of a connection \( \nabla \) on a vector bundle \( E \), then\n\n\[ 0 = \nabla \left( {F}_{\nabla }\right) \in {\mathcal{A}}^{3}\left( {M,\operatorname{End}\left( E\right) }\r... | Proof. This follows from \( \nabla \left( {F}_{\nabla }\right) \left( s\right) = \nabla \left( {{F}_{\nabla }\left( s\right) }\right) - {F}_{\nabla }\left( {\nabla \left( s\right) }\right) = \nabla \left( {{\nabla }^{2}\left( s\right) }\right) - \) \( {\nabla }^{2}\left( {\nabla \left( s\right) }\right) = 0 \) . Here w... | No |
ii) On the tensor product \( {E}_{1} \otimes {E}_{2} \) the curvature is given by\n\n\[ {F}_{{\nabla }_{1}} \otimes 1 + 1 \otimes {F}_{{\nabla }_{2}} \] | Proof. Let us prove ii). This is the following straightforward calculation\n\n\[ {F}_{\nabla }\left( {{s}_{1} \otimes {s}_{2}}\right) \]\n\n\[ = \nabla \left( {\nabla \left( {{s}_{1} \otimes {s}_{2}}\right) }\right) = \nabla \left( {{\nabla }_{1}\left( {s}_{1}\right) \otimes {s}_{2} + {s}_{1} \otimes {\nabla }_{2}\left... | No |
We claim that the curvature \( F \) of the Chern connection on the holomorphic line bundle \( \mathcal{O}\left( 1\right) \) endowed with this hermitian metric is\n\n\[ \frac{i}{2\pi }F = {\omega }_{\mathrm{{FS}}} \]\n\nwhere \( {\omega }_{\mathrm{{FS}}} \) is the Fubini-Study Kähler form (see Example 3.1.9, i)). | This can be verified locally. On a standard open subset \( {U}_{i} \subset {\mathbb{P}}^{n} \) one has \( {\omega }_{\mathrm{{FS}}} = \frac{i}{2\pi }\partial \bar{\partial }\log \left( {1+\sum {\left| {w}_{i}\right| }^{2}}\right) . \)\n\nThe hermitian structure of \( {\left. \mathcal{O}\left( 1\right) \right| }_{{U}_{i... | Yes |
Lemma 4.3.17 The curvature of the induced connection \( {\nabla }_{1} \) on \( {E}_{1} \) is given by \( {F}_{{\nabla }_{1}} = {\operatorname{pr}}_{1} \circ {F}_{\nabla } - {b}_{2} \circ {b}_{1} \) . | Now let \( {E}_{1} \) be a holomorphic subbundle of the trivial holomorphic vector bundle \( E = {\mathcal{O}}^{\oplus r} \) endowed with the trivial constant hermitian structure. The curvature of the Chern connection \( \nabla \) on \( E \) is trivial, as \( \nabla \) is just the exterior differential. Hence, \( {F}_{... | No |
Proposition 4.3.18 The curvature \( {F}_{{\nabla }_{1}} \) of the Chern connection \( {\nabla }_{1} \) of a subbundle \( {E}_{1} \subset E = {\mathcal{O}}^{\oplus r} \) (with the induced hermitian structure) is semi-negative. | Proof. By the previous lemma we have \( {F}_{{\nabla }_{1}} = - {b}_{2} \circ {b}_{1} \) . Thus, if \( {h}_{2} \) is the induced hermitian structure on the quotient \( E/{E}_{1} \) then\n\n\[ \n{h}_{1}\left( {{F}_{{\nabla }_{1}}\left( s\right), s}\right) = - {h}_{1}\left( {{b}_{2}\left( {{b}_{1}\left( s\right) }\right)... | Yes |
Example 4.3.20 The Euler sequence on \( {\mathbb{P}}^{n} \) twisted by \( \mathcal{O}\left( {-1}\right) \) is of the form\n\n\[ 0 \rightarrow \mathcal{O}\left( {-1}\right) \rightarrow {\mathcal{O}}^{\oplus n + 1} \rightarrow {\mathcal{T}}_{{\mathbb{P}}^{n}}\left( {-1}\right) \rightarrow 0. \] | Hence, \( {\mathcal{T}}_{{\mathbb{P}}^{n}}\left( {-1}\right) \) admits an hermitian structure such that the curvature of the Chern connection is semi-positive. Twisting by \( \mathcal{O}\left( 1\right) \) yields a connection on \( {\mathcal{T}}_{{\mathbb{P}}^{n}} \) with positive curvature. In fact, this is the curvatu... | No |
Lemma 4.4.2 The k-multilinear symmetric map \( P \) is invariant if and only if for all \( B,{B}_{1},\ldots ,{B}_{k} \in \mathfrak{{gl}}\left( {r,\mathbb{C}}\right) \) one has\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{k}P\left( {{B}_{1},\ldots ,{B}_{j - 1},\left\lbrack {B,{B}_{j}}\right\rbrack ,{B}_{j + 1},\ldots ,{B}_{k}... | Proof. Use the invertible matrix \( C = {e}^{tB} \) and differentiate the invariance equation (4.5) at \( t = 0 \) . The converse is left as an exercise (cf. Exercise 4.4.8). \( ▱ \) | No |
Proposition 4.4.3 Let \( P \) be an invariant \( k \) -multilinear symmetric form on \( \mathfrak{{gl}}\left( {r,\mathbb{C}}\right) \) . Then for any vector bundle \( E \) of rank \( r \) and any partition \( m = \) \( {i}_{1} + \ldots + {i}_{k} \) there exists a naturally induced \( k \) -linear map:\n\n\[ P : \left( ... | Proof. Once a trivialization \( E\left( x\right) \cong {\mathbb{C}}^{r} \) is fixed, the above definition makes sense. Since \( P \) is invariant, it is independent of the chosen trivialization.\n\nClearly, the \( k \) -linear map defined in this way induces also a \( k \) -multilinear map on the level of global sectio... | Yes |
Lemma 4.4.4 For any forms \( {\gamma }_{j} \in {\mathcal{A}}^{{i}_{j}}\left( {M,\operatorname{End}\left( E\right) }\right) \) one has\n\n\[ \n{dP}\left( {{\gamma }_{1},\ldots ,{\gamma }_{k}}\right) = \mathop{\sum }\limits_{{j = 1}}^{k}{\left( -1\right) }^{\mathop{\sum }\limits_{{\ell = 1}}^{{j - 1}}{i}_{\ell }}P\left( ... | Proof. This can be seen by a local calculation. We write \( \nabla = d + A \), where \( A \) is the local connection matrix of \( \nabla \) . The induced connection on \( \operatorname{End}\left( E\right) \) is of the form \( \nabla = d + A \) with \( A \) acting as \( \gamma \mapsto \left\lbrack {A,\gamma }\right\rbra... | Yes |
Corollary 4.4.5 Let \( {F}_{\nabla } \) be the curvature of an arbitrary connection \( \nabla \) on a vector bundle \( E \) of rank \( r \) . Then for any invariant \( k \) -multilinear symmetric polynomial \( P \) on \( \mathfrak{{gl}}\left( {r,\mathbb{C}}\right) \) the induced \( {2k} \) -form \( \widetilde{P}\left( ... | Proof. This is an immediate consequence of the Bianchi identity (Lemma 4.3.5), which says \( \nabla \left( {F}_{\nabla }\right) = 0 \), and the previous lemma. | Yes |
Lemma 4.4.6 If \( \nabla \) and \( {\nabla }^{\prime } \) are two connections on the same bundle \( E \), then \( \left\lbrack {\widetilde{P}\left( {F}_{\nabla }\right) }\right\rbrack = \left\lbrack {\widetilde{P}\left( {F}_{{\nabla }^{\prime }}\right) }\right\rbrack \) | Proof. The space of all connections is an affine space over \( {\mathcal{A}}^{1}\left( {M,\operatorname{End}\left( E\right) }\right) \) , i.e. if \( \nabla \) is given, then any other connection is of the form \( {\nabla }^{\prime } = \nabla + A \) for some \( A \in {\mathcal{A}}^{1}\left( {\operatorname{End}\left( E\r... | Yes |
Let us compute the characteristic classes of a hypersurface \( Y \subset X \) . The normal bundle sequence in this case takes the form\n\n\[ \n{\left. 0 \rightarrow {\mathcal{T}}_{Y} \rightarrow {\mathcal{T}}_{X}\right| }_{Y} \rightarrow {\mathcal{O}}_{Y}\left( Y\right) \rightarrow 0.\n\] | Since any short exact sequence of holomorphic vector bundles splits as a sequence of complex vector bundles, the Whitney product formula yields \( {i}^{ * }\mathrm{c}\left( X\right) = \mathrm{c}\left( Y\right) \cdot {i}^{ * }\mathrm{c}\left( {\mathcal{O}\left( Y\right) }\right) \) . Therefore,\n\n\[ \n\mathrm{c}\left( ... | Yes |
Proposition 4.4.12 Let \( L \) be a complex line bundle over a differentiable manifold \( M \) . Then the image of \( \delta \left( L\right) \in {H}^{2}\left( {M,\mathbb{Z}}\right) \) under the natural map \( {H}^{2}\left( {M,\mathbb{Z}}\right) \rightarrow {H}^{2}\left( {M,\mathbb{C}}\right) \) equals \( - {\mathrm{c}}... | Proof. In order to prove this, we have to consider the two resolutions of the constant sheaf \( \mathbb{C} \) on \( M \) given by the de Rham complex and the Čech complex, respectively. They are compared as follows:\n\n denote the formal Chern roots of \( {\mathcal{T}}_{X} \) . Then \[ {\chi }_{y} = {\int }_{X}\mathop{\prod }\limits_{{i = 1}}^{n}\left( {1 + y{e}^{-{\gamma }_{i}}}\right) \frac{{\gamma }_{i}}{1 - {e}^{-{\gamma }_{i}}}. \] | Proof. This follows immediately from the definition of the Todd classes, Theorem 5.1.1 and the equality \[ \operatorname{ch}\left( {{\bigoplus }_{p = 0}^{n}{\Omega }_{X}^{p}{y}^{p}}\right) = \mathop{\prod }\limits_{{i = 1}}^{n}\left( {1 + y{e}^{-{y}_{i}}}\right) \] the proof of which is left to the reader. | No |
Lemma 5.2.3 (Nakano identity) Let \( \nabla \) be the Chern connection on \( E \) . Then \[ \left\lbrack {\Lambda ,{\bar{\partial }}_{E}}\right\rbrack = - i{\left( {\nabla }^{1,0}\right) }^{ * } = i\left( {{\bar{ * }}_{{E}^{ * }} \circ {\nabla }_{{E}^{ * }}^{1,0} \circ {\bar{ * }}_{E}}\right) . \] | Proof. The second equality is the definition of the adjoint operator of \( {\nabla }^{1,0} \) . The first equality is local and we may, therefore, use an orthonormal trivialization \( \psi : {\left. E\right| }_{U} \cong U \times {\mathbb{C}}^{r} \) . With respect to such a trivialization the Hodge operator \( {\bar{ * ... | Yes |
Consider \( \mathcal{O}\left( 1\right) \) on \( {\mathbb{P}}^{n} \), which is positive due to Example 4.3.12 (see also page 197). Thus, \( {H}^{q}\left( {{\mathbb{P}}^{n},{\Omega }^{p} \otimes \mathcal{O}\left( m\right) }\right) = 0 \) for \( p + q > n \) and \( m > 0 \). In particular, \( {H}^{q}\left( {{\mathbb{P}}^{... | Using Serre duality this yields\n\n\[ \n{H}^{q}\left( {{\mathbb{P}}^{n},\mathcal{O}\left( m\right) }\right) = \left\{ \begin{array}{ll} 0 & \text{ if }0 < q < n \\ 0 & \text{ if }q = 0, m < 0 \\ 0 & \text{ if }q = n, m > - n - 1. \end{array}\right. \n\] | No |
Proposition 5.2.7 Let \( L \) be a positive line bundle on a compact Kähler manifold \( X \) . For any holomorphic vector bundle \( E \) on \( X \) there exists a constant \( {m}_{0} \) such that\n\n\[ \n{H}^{q}\left( {X, E \otimes {L}^{m}}\right) = 0 \n\]\n\nfor \( m \geq {m}_{0} \) and \( q > 0 \) . | Proof. Choose hermitian structures on \( E \) and \( L \) and denote the associated Chern connections by \( {\nabla }_{E} \) and \( {\nabla }_{L} \), respectively. By assumption we may suppose that \( \left( {i/{2\pi }}\right) {F}_{{\nabla }_{L}} \) is a Kähler form \( \omega \) . We endow \( X \) with the correspondin... | Yes |
Lemma 5.3.2 For any line bundle \( M \) on \( X \) and integers \( {n}_{1},\ldots ,{n}_{\ell } > 0 \) the line bundle \( {\sigma }^{ * }\left( {{L}^{k} \otimes M}\right) \otimes \mathcal{O}\left( {-\sum {n}_{j}{E}_{j}}\right) \) on \( \widehat{X} \) is positive for \( k \gg 0 \) . | Proof. In a neighbourhood \( {x}_{j} \in {U}_{j} \subset X \) of each point \( {x}_{j} \) the blow-up can be seen as the incidence variety \( {\widehat{U}}_{j} = \mathcal{O}\left( {-1}\right) \subset {U}_{j} \times {\mathbb{P}}^{n - 1} \) . Moreover, \( \mathcal{O}\left( {E}_{j}\right) \) is isomorphic to to \( {p}_{j}... | Yes |
Corollary 5.3.3 A compact Kähler manifold \( X \) is projective if and only if \( {\mathcal{K}}_{X} \cap {H}^{2}\left( {X,\mathbb{Z}}\right) \neq \varnothing \) | Proof. By abuse of notation we write \( {\mathcal{K}}_{X} \cap {H}^{2}\left( {X,\mathbb{Z}}\right) \) instead of \( {\mathcal{K}}_{X} \cap \) \( \operatorname{Im}\left( {{H}^{2}\left( {X,\mathbb{Z}}\right) \rightarrow {H}^{2}\left( {X,\mathbb{R}}\right) }\right) \) . The Kähler cone \( {\mathcal{K}}_{X} \) is by defini... | Yes |
Corollary 5.3.5 Let \( X = V/\Gamma \) be a complex torus. Then \( X \) is projective if and only if there exists a Riemann form, i.e. an alternating bilinear form \( \omega : V \times V \rightarrow \mathbb{R} \) such that\n\ni) \( \omega \left( {{iu},{iv}}\right) = \omega \left( {u, v}\right) \), \n\nii) \( \omega \le... | Proof. Clearly, the Riemann form \( \omega \) can be considered as a constant two-form. Recall that \( \mathop{\bigwedge }\limits^{2}{V}^{ * } \cong {H}^{2}\left( {X,\mathbb{R}}\right) \) . Conditions i) and ii) ensure that \( \omega \) is in fact a Kähler form and iii) is equivalent to \( \omega \in {H}^{2}\left( {X,\... | Yes |
Corollary 5.3.7 If \( X \) is projective, the natural homomorphism \( \operatorname{Div}\left( X\right) \rightarrow \) \( \operatorname{Pic}\left( X\right) \) (see Section 2.3) is surjective. | Proof. Let \( L \in \operatorname{Pic}\left( X\right) \) be an ample line bundle. In particular, \( L \) is positive and we may apply Serre vanishing 5.2.7. If \( M \in \operatorname{Pic}\left( X\right) \) is any line bundle then \( \chi \left( {X, M \otimes {L}^{k}}\right) = {h}^{0}\left( {X, M \otimes {L}^{k}}\right)... | Yes |
Lemma 6.1.4 The first-order deformation of the complex structure I on the manifold \( M \) (defining the complex manifold \( X = \left( {M, I}\right) \) ) induced by a one-parameter family \( {F}_{t} \) of diffeomorphism of \( M \) is determined by the map | \[ {\phi }_{1} = \bar{\partial }\left( {\left( {\left. \frac{d{F}_{t}}{dt}\right| }_{t = 0}\right) }^{1,0}\right) : {T}^{0,1}X \rightarrow {T}^{1,0}X. \] | Yes |
Proposition 6.1.5 Let \( X \) be a complex manifold. There is a natural bijection between all first-order deformations of \( X \) and elements of \( {H}^{1}\left( {X,{\mathcal{T}}_{X}}\right) \) . | Proof. Write \( X = \left( {M, I}\right) \) . Then the first order deformations of \( I \) correspond to \( \bar{\partial } \) -closed elements in \( {\mathcal{A}}^{0,1}\left( {\mathcal{T}}_{X}\right) \) and by the previous lemma those that define isomorphic ones differ by elements of the image of \( \bar{\partial } : ... | Yes |
Lemma 6.1.8 The operator \( \Delta \) anti-commutes with \( \bar{\partial } \), i.e. \( \Delta \circ \bar{\partial } = - \bar{\partial } \circ \Delta \) . | Proof. We use the fact that \( \Omega \in {\mathcal{A}}^{n,0}\left( X\right) \) is holomorphic, i.e. \( \bar{\partial } \) -closed. If we write in local coordinates \( \Omega = {fd}{z}_{1} \land \ldots \land d{z}_{n} \), then \( f \) is a holomorphic function. Then\n\n\[ \eta \left( {g \cdot d{\bar{z}}_{I} \otimes \fra... | Yes |
Proposition 6.1.11 Let \( X \) be a Calabi-Yau manifold and let \( v \in {H}^{1}\left( {X,{\mathcal{T}}_{X}}\right) \) . Then there exists a formal power series \( {\phi }_{1}t + {\phi }_{2}{t}^{2} + \ldots \) with \( {\phi }_{i} \in {\mathcal{A}}^{0,1}\left( {\mathcal{T}}_{X}\right) \) satisfying the Maurer-Cartan equ... | Proof. We begin with \( {\phi }_{1} \in {\mathcal{A}}^{0,1}\left( {\mathcal{T}}_{X}\right) \) representing \( v \) such that \( \eta \left( {\phi }_{1}\right) \in \) \( {\mathcal{A}}^{n - 1,1}\left( X\right) \) is harmonic ( \( \partial \) -closed would be enough). Then one finds \( {\phi }_{2} \) as above. Let us supp... | Yes |
Proposition 6.2.2 (Ehresmann) Let \( \pi : \mathcal{X} \rightarrow S \) be a proper family of differentiable manifolds. If \( S \) is connected, then all fibres are diffeomorphic. | Proof. By connecting two points of the base by an arc, we may assume that the base \( S \) is an interval \( \left( {-\varepsilon ,1 + \varepsilon }\right) \) . Then one has to show that the fibres \( {\mathcal{X}}_{0} \) and \( {\mathcal{X}}_{1} \) are diffeomorphic. Locally in \( \mathcal{X} \), the morphism \( \pi \... | Yes |
Theorem 6.2.13 (Kuranishi) Any compact complex manifold admits a versal deformation. | Note that for a versal deformation \( \mathcal{X} \rightarrow S \) the Kodaira-Spencer map \( {T}_{0}S \rightarrow \) \( {H}^{1}\left( {X,{\mathcal{T}}_{X}}\right) \) is bijective. | No |
Theorem 10.3 (Riemann [202]) Let \( S \) be a compact Riemann surface of genus \( p \) , then, for any divisor \( D \) on \( S \) , \[ \dim L\left( D\right) \geq \deg \left( D\right) - p + 1. \] | Let’s look at the situation we discussed earlier, and let \( D \) be the divisor defined by our system of \( m \) points \( {p}_{1},\cdots ,{p}_{m} \) on \( S \) , \[ D = \mathop{\sum }\limits_{{l = 1}}^{m}{p}_{l} \] where \( m > p \) . Then \( \deg \left( D\right) = m \), and Riemann’s inequality in Theorem 10.3 gives... | No |
Theorem 13.3 Let \( F \) be a mapping from the unit ball \( B \) in \( {C}^{m} \) into \( {C}^{m - \alpha } \) . Suppose that:\n\n- F has two continuous Fréchet derivatives, both bounded by \( M \) .\n\n- There exists a mapping \( L \) with domain \( B \) and range in the space \( \mathcal{B}\left( {{C}^{m},{C}^{m - \a... | The proof of this theorem (see [210]) uses a family of suitably-defined smoothing operators \( {T}_{n} \) indexed by an integer \( n \) and the successive approximations (13.21) and the initial estimate \( {f}_{0} = 0 \) . | No |
Theorem 14.2 (Hodge decomposition theorem [109]) Let \( M \) be a compact Riemannian manifold, then there is a continuous linear mapping\n\n\[ G : {\mathcal{E}}^{k}\left( M\right) \rightarrow {\mathcal{E}}^{k}\left( M\right) \]\n\nand an orthogonal decomposition of the form\n\n\[ {\mathcal{E}}^{k}\left( M\right) = {\ma... | The operator \( G \) in this theorem is referred to as a Green’s operator. We define the orthogonal projection (the harmonic projection)\n\n\[ H : {\mathcal{E}}^{k}\left( M\right) \rightarrow {\mathcal{H}}^{k}\left( M\right) \]\n\nto be the orthogonal projection onto the harmonic differential forms (first term in the s... | Yes |
Theorem 14.3 (Kodaira vanishing theorem [126]) Let \( X \) be an \( n \) -dimensional compact complex manifold, and let \( E \) be a holomorphic line bundle on \( X \), then:\n\n(a) If \( E \otimes {K}^{ * } \) is a positive line bundle, then\n\n\[ \n{H}^{q}\left( {X,\mathcal{O}\left( E\right) }\right) = 0, q > 0.\n\]\... | \[ {}^{13} \] See, for instance, Wells [239], Chap. III, Sect. 4. for a proof of this. It uses Čech cohomology to represent the sheaf cohomology group \( {H}^{2}\left( {X,\mathbf{Z}}\right) \) . | No |
Theorem 14.4 (Hirzebruch-Riemann-Roch [104]) Let \( E \) be a holomorphic vector bundle on a projective algebraic manifold \( X \), then there exists a homogeneous polynomial \( p\left( {{X}_{1},\ldots ,{X}_{r},{Y}_{1},\ldots ,{Y}_{n}}\right) \) with rational coefficients such that\n\n\[ \chi \left( {X, E}\right) = \in... | Hirzebruch has a very explicit formula for the polynomial \( p \) in this theorem in terms of Todd polynomials for the tangent bundle and Chern classes of the holomorphic vector bundle \( E \) (see Hirzebruch’s monograph [104]). | Yes |
Lemma 15.2 Let \( K \) be a compact subset of \( X \), then there is an integer \( N \) and a mapping\n\n\[ F : X \rightarrow {\mathbf{C}}^{N} \]\n\nwhich is one-to-one and regular on \( K \) . | We outline the proof of this lemma. Since \( X \) has global local coordinates, there is a finite covering of \( K \) by open sets \( {U}_{\alpha } \) with functions \( {f}_{1}^{\alpha },\ldots ,{f}_{n}^{\alpha } \in \mathcal{O}\left( X\right) \) such that the mappings \( {f}^{\alpha } = \left( {{f}_{1}^{\alpha },\ldot... | Yes |
Lemma 15.4 Let \( K \) be a compact subset of \( X \) such that \( K = {\widehat{K}}_{X} \), then, for any neighborhood \( U \) of \( K \) in \( X \), there exists an analytic polyhedron \( P \) such that\n\n\[ K \subset P \subset U\text{.} \] | To see this, assume that \( U \) is relatively compact \( {}^{8} \) in \( X \) . For each \( x \in \partial U \), since \( K = {\widehat{K}}_{X} \), we can find a holomorphic function \( f \in \mathcal{O}\left( X\right) \) such that \( \left| f\right| < 1 \) on \( K \) and \( \mid f\left( x\right) > 1 \) . By the compa... | Yes |
Lemma 15.5 Let \( K \) be compact in \( X \), and let \( P \) be an analytic polyhedron of order \( N + 1 \) in \( X \) with \( K \subset P \) . If \( N \geq {2n} \), then there exists an analytic polyhedron \( {P}^{\prime } \) of order \( N \) such that\n\n\[ K \subset {P}^{\prime } \subset P \subset X.\text{ } \] | We will outline the construction of \( {P}^{\prime } \) here. Suppose that \( P \) is defined by\n\n\[ P = \left\{ {x \in X : \left| {{f}_{j}\left( x\right) }\right| < 1,\text{ for }{f}_{j} \in \mathcal{O}\left( X\right), j = 1,\ldots, N + 1}\right\} .\n\]\n\nLet\n\n\[ {c}_{0} < {c}_{1} < {c}_{2} < {c}_{3} < 1 \]\n\n(1... | No |
Theorem 15.5 (Grauert [86]) Let \( D \) be a relatively compact strongly pseudoconvex domain in a complex manifold \( X \), then \( D \) is holomorphically convex. | Grauert's proof of this theorem consists of two fundamental steps. The first step we formulate as a lemma (although it has the strong merit of being a theorem in its own right).\n\nLemma 15.6 The cohomolo | No |
Lemma 15.7 For each point \( p \in \partial D \), there exists a holomorphic function \( f \in \) \( \mathcal{O}\left( D\right) \) such that\n\n\[ \mathop{\lim }\limits_{{x \rightarrow p}}\left| {f(x}\right| = \infty \] | It is clear from this lemma that \( D \) must be holomorphically convex, and that concludes the proof of the theorem, assuming these two lemmas. | No |
Lemma 15.8 Let \( V \) and \( U \) be two open subsets of \( X \) with \( V \subset \subset U \), then the natural restriction mapping\n\n\[ r : \mathcal{O}\left( U\right) \rightarrow \mathcal{O}\left( V\right) \]\n\nis a continuous linear mapping of Fréchet spaces, and moreover, it is a completely continuous mapping. | This is a generalization of theorems of Montel and Vitali from classical function theory to this more abstract setting. See, for instance, a proof of this in Gunning and Rossi's monograph [93] (Proposition 1, Chap. VIII, pp. 234-235). | No |
Theorem 15.7 (Leray [142]) If the simplices of the covering \( \mathfrak{U} \) have the property that\n\n\[ \n{H}^{q}\left( {\left| \sigma \right| ,\mathcal{F}}\right) = 0 \n\]\n\nthen\n\n\[ \n{H}^{q}\left( {X,\mathcal{F}}\right) \cong {H}^{q}\left( {\mathfrak{U},\mathcal{F}}\right), q \geq 0. \n\] | This theorem is proved \( {}^{11} \) and used in Henri Cartan’s lectures in 1954 [35] on the finite-dimensionality of cohomology groups on compact complex manifolds, as mentioned above, and we will see below how Grauert used it in the same way in the following paragraphs. The key to using this theorem is to find an ope... | No |
Theorem 15.9 (Schwartz [211]) Let \( {F}_{1} \) and \( {F}_{2} \) be Fréchet spaces, and let\n\n\[ u : {F}_{1} \rightarrow {F}_{2} \]\n\nbe a surjective continuous linear mapping and\n\n\[ v : {F}_{1} \rightarrow {F}_{2} \]\n\nbe a completely continuous mapping, then the vector space\n\n\[ {F}_{2}/\left( {u\left( {F}_{... | Schwartz devotes two pages to his proof, using the relatively new (at that time) formalism of locally convex topological vector spaces. A more detailed proof can be found in Gunning and Rossi ([93], pp. 294-295). See also the exposition by Serre in the Cartan seminar from 1954 [214]. | No |
Real projective \( n \) -space \( {\mathbb{{RP}}}^{n} \) is defined to be the space of all lines through the origin in \( {\mathbb{R}}^{n + 1} \) . Each such line is determined by a nonzero vector in \( {\mathbb{R}}^{n + 1} \) , unique up to scalar multiplication, and \( {\mathbb{{RP}}}^{n} \) is topologized as the quo... | Since \( \partial {D}^{n} \) with antipodal points identified is just \( {\mathbb{{RP}}}^{n - 1} \), we see that \( {\mathbb{{RP}}}^{n} \) is obtained from \( {\mathbb{{RP}}}^{n - 1} \) by attaching an \( n \) -cell, with the quotient projection \( {S}^{n - 1} \rightarrow {\mathbb{{RP}}}^{n - 1} \) as the attaching map... | Yes |
Complex projective \( \mathbf{n} \) -space \( {\mathbb{{CP}}}^{n} \) is the space of complex lines through the origin in \( {\mathbb{C}}^{n + 1} \), that is,1-dimensional vector subspaces of \( {\mathbb{C}}^{n + 1} \) . As in the case of \( {\mathbb{{RP}}}^{n} \), each line is determined by a nonzero vector in \( {\mat... | From this description of \( {\mathbb{{CP}}}^{n} \) as the quotient of \( {D}_{ + }^{2n} \) under the identifications \( v \sim {\lambda v} \) for \( v \in {S}^{{2n} - 1} \) it follows that \( {\mathbb{{CP}}}^{n} \) is obtained from \( {\mathbb{{CP}}}^{n - 1} \) by attaching a cell \( {e}^{2n} \) via the quotient map \(... | Yes |
Let \( X \) be the union of a torus with \( n \) meridional disks. To obtain a CW structure on \( X \), choose a longitudinal circle in the torus, intersecting each of the meridional disks in one point. These intersection points are then the 0 -cells, the 1-cells are the rest of the longitudinal circle and the boundary... | \n\nequivalent space \( Y \) consisting of \( {n2} \) -spheres, each tangent to its two neighbors, a ’necklace with \( n \) beads.’ The third space \( Z \) in the figure, a strand of \( n \) beads with a string joining... | Yes |
Let us rederive the result in Example 0.8 that a sphere with two points identified is homotopy equivalent to \( {S}^{1} \vee {S}^{2} \) . | The sphere with two points identified can be obtained by attaching \( {S}^{2} \) to \( {S}^{1} \) by a map that wraps a closed arc \( A \) in \( {S}^{2} \) around \( {S}^{1} \) , as shown in the figure. Since \( A \) is contractible, this attaching map is homotopic to a constant map, and attaching \( {S}^{2} \) to \( {... | Yes |
In similar fashion we can see that the necklace in Example 0.9 is homotopy equivalent to the wedge sum of a circle with \( n \) 2-spheres. | The necklace can be obtained from a circle by attaching \( n \) 2-spheres along arcs, so the necklace is homotopy equivalent to the space obtained by attaching \( n \) 2-spheres to a circle at points. Then we can slide these attaching points around the circle until they all coincide, producing the wedge sum. | Yes |
Here is an application of the earlier fact that collapsing a contractible subcomplex is a homotopy equivalence: If \( \left( {X, A}\right) \) is a CW pair, consisting of a cell complex \( X \) and a subcomplex \( A \), then \( X/A \simeq X \cup {CA} \), the mapping cone of the inclusion \( A \hookrightarrow X \) . | For we have \( X/A = \left( {X \cup {CA}}\right) /{CA} \simeq X \cup {CA} \) since \( {CA} \) is a contractible subcomplex of \( X \cup {CA} \) . | Yes |
If \( \left( {X, A}\right) \) is a CW pair and \( A \) is contractible in \( X \), that is, the inclusion \( A \hookrightarrow X \) is homotopic to a constant map, then \( X/A \simeq X \vee {SA} \) . | Namely, by the previous example we have \( X/A \simeq X \cup {CA} \), and then since \( A \) is contractible in \( X \), the mapping cone \( X \cup {CA} \) of the inclusion \( A \hookrightarrow X \) is homotopy equivalent to the mapping cone of a constant map, which is \( X \vee {SA} \) . | Yes |
A pair \( \left( {X, A}\right) \) has the homotopy extension property if \( A \) has a mapping cylinder neighborhood in \( X \) | To verify the homotopy extension property, notice first that \( I \times I \) retracts onto \( I \times \{ 0\} \cup \partial I \times I \), hence \( B \times I \times I \) retracts onto \( B \times I \times \{ 0\} \cup B \times \partial I \times I \), and this retraction induces a retraction of \( {M}_{f} \times I \) o... | Yes |
Proposition 0.16. If \( \left( {X, A}\right) \) is a CW pair, then \( X \times \{ 0\} \cup A \times I \) is a deformation retract of \( X \times I \), hence \( \left( {X, A}\right) \) has the homotopy extension property. | Proof: There is a retraction \( r : {D}^{n} \times I \rightarrow {D}^{n} \times \{ 0\} \cup \partial {D}^{n} \times I \), for example the radial projection from the point \( \left( {0,2}\right) \in {D}^{n} \times \mathbb{R} \). Then setting \( {r}_{t} = {tr} + \left( {1 - t}\right) \mathbb{1} \) gives a deformation ret... | Yes |
Proposition 0.18. If \( \left( {{X}_{1}, A}\right) \) is a CW pair and we have attaching maps \( f, g : A \rightarrow {X}_{0} \) that are homotopic, then \( {X}_{0}{ \sqcup }_{f}{X}_{1} \simeq {X}_{0}{ \sqcup }_{g}{X}_{1} \) rel \( {X}_{0} \) . | Proof: If \( F : A \times I \rightarrow {X}_{0} \) is a homotopy from \( f \) to \( g \), consider the space \( {X}_{0}{ \sqcup }_{F}\left( {{X}_{1} \times I}\right) \) . This contains both \( {X}_{0}{ \sqcup }_{f}{X}_{1} \) and \( {X}_{0}{ \sqcup }_{g}{X}_{1} \) as subspaces. A deformation retraction of \( {X}_{1} \ti... | Yes |
Corollary 0.21. A map \( f : X \rightarrow Y \) is a homotopy equivalence iff \( X \) is a deformation retract of the mapping cylinder \( {M}_{f} \) . Hence, two spaces \( X \) and \( Y \) are homotopy \( \parallel \) equivalent iff there is a third space containing both \( X \) and \( Y \) as deformation retracts. | Proof: In the diagram at the right the maps \( i \) and \( j \) are the inclu-\nsions and \( r \) is the canonical retraction, so \( f = {ri} \) and \( i \simeq {jf} \) . Since \( j \) and \( r \) are homotopy equivalences, it follows that \( f \) is a homotopy equivalence iff \( i \) is a homotopy equivalence, since t... | Yes |
Proposition 1.5. The map \( {\beta }_{h} : {\pi }_{1}\left( {X,{x}_{1}}\right) \rightarrow {\pi }_{1}\left( {X,{x}_{0}}\right) \) defined by \( {\beta }_{h}\left\lbrack f\right\rbrack = \left\lbrack {h \cdot f \cdot \bar{h}}\right\rbrack \) is an isomorphism. | Proof: If \( {f}_{t} \) is a homotopy of loops based at \( {x}_{1} \) then \( h \cdot {f}_{t} \cdot \bar{h} \) is a homotopy of loops based at \( {x}_{0} \), so \( {\beta }_{h} \) is well-defined. Further, \( {\beta }_{h} \) is a homomorphism since \( {\beta }_{h}\left\lbrack {f \cdot g}\right\rbrack = \left\lbrack {h ... | Yes |
Proposition 1.6. A space \( X \) is simply-connected iff there is a unique homotopy class | Proof: Path-connectedness is the existence of paths connecting every pair of points, so we need be concerned only with the uniqueness of connecting paths. Suppose \( {\pi }_{1}\left( X\right) = 0 \) . If \( f \) and \( g \) are two paths from \( {x}_{0} \) to \( {x}_{1} \), then \( f \simeq f \cdot \bar{g} \cdot g \sim... | Yes |
Theorem 1.9. Every continuous map \( h : {D}^{2} \rightarrow {D}^{2} \) has a fixed point, that is, a point \( x \) with \( h\left( x\right) = x \) . | Proof: Suppose on the contrary that \( h\left( x\right) \neq x \) for all \( x \in {D}^{2} \) . Then we can define a map \( r : {D}^{2} \rightarrow {S}^{1} \) by letting \( r\left( x\right) \) be the point of \( {S}^{1} \) where the ray in \( {\mathbb{R}}^{2} \) starting at \( h\left( x\right) \) and passing through \(... | Yes |
Proposition 1.12. \( {\pi }_{1}\left( {X \times Y}\right) \) is isomorphic to \( {\pi }_{1}\left( X\right) \times {\pi }_{1}\left( Y\right) \) if \( X \) and \( Y \) are path-connected. | Proof: A basic property of the product topology is that a map \( f : Z \rightarrow X \times Y \) is continuous iff the maps \( g : Z \rightarrow X \) and \( h : Z \rightarrow Y \) defined by \( f\left( z\right) = \left( {g\left( z\right), h\left( z\right) }\right) \) are both continuous. Hence a loop \( f \) in \( X \t... | Yes |
Lemma 1.19. If \( {\varphi }_{t} : X \rightarrow Y \) is a homotopy and \( h \) is the path \( {\varphi }_{t}\left( {x}_{0}\right) \) formed by the images of a basepoint \( {x}_{0} \in X \), then the three maps in the diagram at the right satisfy \( {\varphi }_{0 * } = {\beta }_{h}{\varphi }_{1 * } \) . | Proof: Let \( {h}_{t} \) be the restriction of \( h \) to the interval \( \left\lbrack {0, t}\right\rbrack \) , with a reparametrization so that the domain of \( {h}_{t} \) is still \( \left\lbrack {0,1}\right\rbrack \) . Explicitly, we can take \( {h}_{t}\left( s\right) = h\left( {ts}\right) \) . Then if \( f \) is a ... | Yes |
Example 1.22. Let \( X \) be the graph shown in the figure, consisting of the twelve edges of a cube. The seven heavily shaded edges form a maximal tree \( T \subset X \), a contractible subgraph containing all the vertices of \( X \). We claim that \( {\pi }_{1}\left( X\right) \) is the free product of five copies of ... | To deduce this from van Kampen’s theorem, choose for each edge \( {e}_{\alpha } \) of \( X - T \) an open neighborhood \( {A}_{\alpha } \) of \( T \cup {e}_{\alpha } \) in \( X \) that deformation retracts onto \( T \cup {e}_{\alpha } \). The intersection of two or more \( {A}_{\alpha } \)’s deformation retracts onto \... | Yes |
Proposition 1.26. The inclusion \( X \hookrightarrow Y \) induces a surjection \( {\pi }_{1}\left( {X,{x}_{0}}\right) \rightarrow {\pi }_{1}\left( {Y,{x}_{0}}\right) \) whose kernel is \( N \) . Thus \( {\pi }_{1}\left( Y\right) \approx {\pi }_{1}\left( X\right) /N \) . | Proof: Let us expand \( Y \) to a slightly larger space \( Z \) that deformation retracts onto \( Y \) and is more convenient for applying van Kampen’s theorem. The space \( Z \) is obtained from \( Y \) by attaching rectangular strips \( {S}_{\alpha } = I \times I \), with the lower edge \( I \times \{ 0\} \) attached... | Yes |
Corollary 1.27. The surface \( {M}_{g} \) is not homeomorphic, or even homotopy equivalent, to \( {M}_{h} \) if \( g \neq h \) . | Proof: The abelianization of \( {\pi }_{1}\left( {M}_{g}\right) \) is the direct sum of \( {2g} \) copies of \( \mathbb{Z} \) . So if \( {M}_{g} \simeq {M}_{h} \) then \( {\pi }_{1}\left( {M}_{g}\right) \approx {\pi }_{1}\left( {M}_{h}\right) \), hence the abelianizations of these groups are isomorphic, which implies \... | Yes |
If \( G = \left\langle {a \mid {a}^{n}}\right\rangle = {\mathbb{Z}}_{n} \) then \( {X}_{G} \) is \( {S}^{1} \) with a cell \( {e}^{2} \) attached by the map \( z \mapsto {z}^{n} \), thinking of \( {S}^{1} \) as the unit circle in \( \mathbb{C} \) . When \( n = 2 \) we get \( {X}_{G} = {\mathbb{{RP}}}^{2} \), but for \(... | For example, when \( n = 3 \) one can construct a neighborhood \( N \) of \( {S}^{1} \) in \( {X}_{G} \) by taking the product of the graph \( Y \) with the interval \( I \), and then identifying the two ends of this product via a one-third twist as shown in the figure. The boundary of \( N \) consists of a single circ... | Yes |
Proposition 1.30. Given a covering space \( p : \widetilde{X} \rightarrow X \), a homotopy \( {f}_{t} : Y \rightarrow X \), and a map \( {\widetilde{f}}_{0} : Y \rightarrow \widetilde{X} \) lifting \( {f}_{0} \), then there exists a unique homotopy \( {\widetilde{f}}_{t} : Y \rightarrow \widetilde{X} \) of \( {\widetil... | Proof: For the covering space \( p : \mathbb{R} \rightarrow {S}^{1} \) this is property (c) in the proof of Theorem 1.7 , and the proof there applies to any covering space. | No |
Proposition 1.31. The map \( {p}_{ * } : {\pi }_{1}\left( {\widetilde{X},{\widetilde{x}}_{0}}\right) \rightarrow {\pi }_{1}\left( {X,{x}_{0}}\right) \) induced by a covering space \( p : \left( {\widetilde{X},{\widetilde{x}}_{0}}\right) \rightarrow \left( {X,{x}_{0}}\right) \) is injective. The image subgroup \( {p}_{ ... | Proof: An element of the kernel of \( {p}_{ * } \) is represented by a loop \( {\widetilde{f}}_{0} : I \rightarrow \widetilde{X} \) with a homotopy \( {f}_{t} : I \rightarrow X \) of \( {f}_{0} = p{\widetilde{f}}_{0} \) to the trivial loop \( {f}_{1} \) . By the remarks preceding the proposition, there is a lifted homo... | Yes |
Proposition 1.32. The number of sheets of a covering space \( p : \left( {\widetilde{X},{\widetilde{x}}_{0}}\right) \rightarrow \left( {X,{x}_{0}}\right) \) \( \parallel \) with \( X \) and \( \widetilde{X} \) path-connected equals the index of \( {p}_{ * }\left( {{\pi }_{1}\left( {\widetilde{X},{\widetilde{x}}_{0}}\ri... | Proof: For a loop \( g \) in \( X \) based at \( {x}_{0} \), let \( \widetilde{g} \) be its lift to \( \widetilde{X} \) starting at \( {\widetilde{x}}_{0} \) . A product \( h \cdot g \) with \( \left\lbrack h\right\rbrack \in H = {p}_{ * }\left( {{\pi }_{1}\left( {\widetilde{X},{\widetilde{x}}_{0}}\right) }\right) \) h... | Yes |
Proposition 1.33. Suppose given a covering space \( p : \left( {\widetilde{X},{\widetilde{x}}_{0}}\right) \rightarrow \left( {X,{x}_{0}}\right) \) and a map \( f : \left( {Y,{y}_{0}}\right) \rightarrow \left( {X,{x}_{0}}\right) \) with \( Y \) path-connected and locally path-connected. Then a lift \( f : \left( {Y,{y}_... | Proof: The ’only if’ statement is obvious since \( {f}_{ * } = {p}_{ * }{\widetilde{f}}_{ * } \) . For the converse, let \( y \in Y \) and let \( y \) be a path in \( Y \) from \( {y}_{0} \) to \( y \) . The path \( {fy} \) in \( X \) starting at \( {x}_{0} \) has a unique lift \( \widetilde{fy} \) starting at \( {\wid... | Yes |
Proposition 1.34. Given a covering space \( p : \widetilde{X} \rightarrow X \) and a map \( f : Y \rightarrow X \) with two lifts \( {\widetilde{f}}_{1},{\widetilde{f}}_{2} : Y \rightarrow \widetilde{X} \) that agree at one point of \( Y \), then if \( Y \) is connected, these two lifts must agree on all of \( Y \) . | Proof: For a point \( y \in Y \), let \( U \) be an open neighborhood of \( f\left( y\right) \) in \( X \) for which \( {p}^{-1}\left( U\right) \) is a disjoint union of open sets \( {\widetilde{U}}_{\alpha } \) each mapped homeomorphically to \( U \) by \( p \), and let \( {\widetilde{U}}_{1} \) and \( {\widetilde{U}}... | Yes |
Theorem 1.38. Let \( X \) be path-connected, locally path-connected, and semilocally simply-connected. Then there is a bijection between the set of basepoint-preserving isomorphism classes of path-connected covering spaces \( p : \left( {\widetilde{X},{\widetilde{x}}_{0}}\right) \rightarrow \left( {X,{x}_{0}}\right) \)... | Proof: It remains only to prove the last statement. We show that for a covering space \( p : \left( {\widetilde{X},{\widetilde{x}}_{0}}\right) \rightarrow \left( {X,{x}_{0}}\right) \), changing the basepoint \( {\widetilde{x}}_{0} \) within \( {p}^{-1}\left( {x}_{0}\right) \) corresponds exactly to changing \( {p}_{ * ... | Yes |
Proposition 1.39. Let \( p : \left( {\widetilde{X},{\widetilde{x}}_{0}}\right) \rightarrow \left( {X,{x}_{0}}\right) \) be a path-connected covering space of the path-connected, locally path-connected space \( X \), and let \( H \) be the subgroup \( {p}_{ * }\left( {{\pi }_{1}\left( {\widetilde{X},{\widetilde{x}}_{0}}... | Proof: We observed earlier in the proof of the classification theorem that changing the basepoint \( {\widetilde{x}}_{0} \in {p}^{-1}\left( {x}_{0}\right) \) to \( {\widetilde{x}}_{1} \in {p}^{-1}\left( {x}_{0}\right) \) corresponds precisely to conjugating \( H \) by an element \( \left\lbrack y\right\rbrack \in {\pi ... | Yes |
Proposition 1.40. If an action of a group \( G \) on a space \( Y \) satisfies \( \left( *\right) \), then:\n\n(a) The quotient map \( p : Y \rightarrow Y/G, p\left( y\right) = {Gy} \), is a normal covering space.\n\n(b) \( G \) is the group of deck transformations of this covering space \( Y \rightarrow Y/G \) if \( Y... | Proof: Given an open set \( U \subset Y \) as in condition \( \left( *\right) \), the quotient map \( p \) simply identifies all the disjoint homeomorphic sets \( \{ g\left( U\right) \mid g \in G\} \) to a single open set \( p\left( U\right) \) in \( Y/G \) . By the definition of the quotient topology on \( Y/G, p \) r... | Yes |
Let \( Y \) be the closed orientable surface of genus 11, an ’11-hole torus’ as shown in the figure. This has a 5-fold rotational symmetry, generated by a rotation of angle \( {2\pi }/5 \) . Thus we have the cyclic group \( {\mathbb{Z}}_{5} \) acting on \( Y \), and the condition \( \left( *\right) \) is obviously sati... | In particular, we see that \( {\pi }_{1}\left( {M}_{3}\right) \) contains the ’larger’ group \( {\pi }_{1}\left( {M}_{11}\right) \) as a normal subgroup of index 5, with quotient \( {\mathbb{Z}}_{5} \) . This example obviously generalizes by replacing the two holes in each ’arm’ of \( {M}_{11} \) by \( m \) holes and t... | No |
In Example 1.35 we constructed a contractible 2-complex \( {\widetilde{X}}_{m, n} = \) \( {T}_{m, n} \times \mathbb{R} \) as the universal cover of a finite 2-complex \( {X}_{m, n} \) that was the union of the mapping cylinders of the two maps \( {S}^{1} \rightarrow {S}^{1}, z \mapsto {z}^{m} \) and \( z \mapsto {z}^{n... | We described a decomposition of \( {\widetilde{X}}_{m, n} \) into rectangles, with \( {X}_{m, n} \) the quotient of one rectangle. These rectangles in fact define a cell structure on \( {\widetilde{X}}_{m, n} \) lifting a cell structure on \( {X}_{m, n} \) with two vertices, three edges, and one 2-cell. The group \( {G... | Yes |
For \( G = {\mathbb{Z}}_{2} = \left\langle {x \mid {x}^{2}}\right\rangle ,{X}_{G} \) is \( {\mathbb{{RP}}}^{2} \) and \( {\widetilde{X}}_{G} = {S}^{2} \). | More generally, for \( {\mathbb{Z}}_{n} = \left\langle {x \mid {x}^{n}}\right\rangle ,{X}_{G} \) is \( {S}^{1} \) with a disk attached by the map \( z \mapsto {z}^{n} \) and \( {\widetilde{X}}_{G} \) consists of \( n \) disks \( {D}_{1},\cdots ,{D}_{n} \) with their boundary circles identified. A generator of \( {\math... | Yes |
Example 1.48. If \( G = {\mathbb{Z}}_{2} * {\mathbb{Z}}_{2} = \left\langle {a, b \mid {a}^{2},{b}^{2}}\right\rangle \) then the Cayley graph is a union of an infinite sequence of circles each tangent to its two neighbors. | We obtain \( {\widetilde{X}}_{G} \) from this graph by making each circle the equator of a 2-sphere, yielding an infinite sequence of tangent 2-spheres. Elements of the index-two normal subgroup \( \mathbb{Z} \subset {\mathbb{Z}}_{2} * {\mathbb{Z}}_{2} \) generated by \( {ab} \) act on \( {\widetilde{X}}_{G} \) as tran... | Yes |
Lemma 2.1. The composition \( {\Delta }_{n}\left( X\right) \overset{{\partial }_{n}}{ \rightarrow }{\Delta }_{n - 1}\left( X\right) \overset{{\partial }_{n - 1}}{ \rightarrow }{\Delta }_{n - 2}\left( X\right) \) is zero. | Proof: We have \( {\partial }_{n}\left( \sigma \right) = \mathop{\sum }\limits_{i}{\left( -1\right) }^{i}\sigma \mid \left\lbrack {{v}_{0},\cdots ,{\widehat{v}}_{i},\cdots ,{v}_{n}}\right\rbrack \), and hence\n\n\[{\partial }_{n - 1}{\partial }_{n}\left( \sigma \right) = \mathop{\sum }\limits_{{j < i}}{\left( -1\right)... | Yes |
Example 2.2. \( X = {S}^{1} \), with one vertex \( v \) and one edge \( e \) . Then \( {\Delta }_{0}\left( {S}^{1}\right) \) | and \( {\Delta }_{1}\left( {S}^{1}\right) \) are both \( \mathbb{Z} \) and the boundary map \( {\partial }_{1} \) is zero since \( \partial e = v - v \) . The groups \( {\Delta }_{n}\left( {S}^{1}\right) \) are 0 for \( n \geq 2 \) since there are no simplices in these dimensions. Hence \[ {H}_{n}^{\Delta }\left( {S}^{... | Yes |
Example 2.3. \( X = T \), the torus with the \( \Delta \) -complex structure pictured earlier, having one vertex, three edges \( a, b \), and \( c \), and two 2-simplices \( U \) and \( L \) . As in the previous example, \( {\partial }_{1} = 0 \) so \( {H}_{0}^{\Delta }\left( T\right) \approx \mathbb{Z} \) . Since \( {... | Thus \[ {H}_{n}^{\Delta }\left( T\right) \approx \left\{ \begin{array}{ll} \mathbb{Z} \oplus \mathbb{Z} & \text{ for }n = 1 \\ \mathbb{Z} & \text{ for }n = 0,2 \\ 0 & \text{ for }n \geq 3 \end{array}\right. \] | Yes |
Example 2.4. \( X = {\mathbb{{RP}}}^{2} \), as pictured earlier, with two vertices \( v \) and \( w \), three edges \( a, b \), and \( c \), and two 2-simplices \( U \) and \( L \) . Then \( \operatorname{Im}{\partial }_{1} \) is generated by \( w - v \), so \( {H}_{0}^{\Delta }\left( X\right) \approx \mathbb{Z} \) wit... | Further, \( \operatorname{Ker}{\partial }_{1} \approx \mathbb{Z} \oplus \mathbb{Z} \) with basis \( a - b \) and \( c \), and \( \operatorname{Im}{\partial }_{2} \) is an index-two subgroup of \( \operatorname{Ker}{\partial }_{1} \) since we can choose \( c \) and \( a - b + c \) as a basis for \( \operatorname{Ker}{\p... | Yes |
Proposition 2.6. Corresponding to the decomposition of a space \( X \) into its path- \( \parallel \) components \( {X}_{\alpha } \) there is an isomorphism of \( {H}_{n}\left( X\right) \) with the direct sum \( { \oplus }_{\alpha }{H}_{n}\left( {X}_{\alpha }\right) \) . | Proof: Since a singular simplex always has path-connected image, \( {C}_{n}\left( X\right) \) splits as the direct sum of its subgroups \( {C}_{n}\left( {X}_{\alpha }\right) \) . The boundary maps \( {\partial }_{n} \) preserve this direct sum decomposition, taking \( {C}_{n}\left( {X}_{\alpha }\right) \) to \( {C}_{n ... | Yes |
If \( X \) is nonempty and path-connected, then \( {H}_{0}\left( X\right) \approx \mathbb{Z} \) . Hence for \( \parallel \) any space \( X,{H}_{0}\left( X\right) \) is a direct sum of \( \mathbb{Z} \) ’s, one for each path-component of \( X \) . | Proof: By definition, \( {H}_{0}\left( X\right) = {C}_{0}\left( X\right) /\operatorname{Im}{\partial }_{1} \) since \( {\partial }_{0} = 0 \) . Define a homomorphism \( \varepsilon : {C}_{0}\left( X\right) \rightarrow \mathbb{Z} \) by \( \varepsilon \left( {\mathop{\sum }\limits_{i}{n}_{i}{\sigma }_{i}}\right) = \matho... | Yes |
Proposition 2.8. If \( X \) is a point, then \( {H}_{n}\left( X\right) = 0 \) for \( n > 0 \) and \( {H}_{0}\left( X\right) \approx \mathbb{Z} \) . | Proof: In this case there is a unique singular \( n \) -simplex \( {\sigma }_{n} \) for each \( n \), and \( \partial \left( {\sigma }_{n}\right) = \) \( \mathop{\sum }\limits_{i}{\left( -1\right) }^{i}{\sigma }_{n - 1} \), a sum of \( n + 1 \) terms, which is therefore 0 for \( n \) odd and \( {\sigma }_{n - 1} \) for... | Yes |
Theorem 2.13. If \( X \) is a space and \( A \) is a nonempty closed subspace that is a deformation retract of some neighborhood in \( X \), then there is an exact sequence\n\n\[ \cdots \rightarrow {\widetilde{H}}_{n}\left( A\right) \overset{{i}_{ * }}{ \rightarrow }{\widetilde{H}}_{n}\left( X\right) \overset{{j}_{ * }... | The map \( \partial \) will be constructed in the course of the proof. The idea is that an element \( x \in {\widetilde{H}}_{n}\left( {X/A}\right) \) can be represented by a chain \( \alpha \) in \( X \) with \( \partial \alpha \) a cycle in \( A \) whose homology class is \( \partial x \in {\widetilde{H}}_{n - 1}\left... | Yes |
In the long exact sequence of reduced homology groups for the pair \( \left( {{D}^{n},\partial {D}^{n}}\right) \), the maps \( {H}_{i}\left( {{D}^{n},\partial {D}^{n}}\right) \overset{\partial }{ \rightarrow }{\widetilde{H}}_{i - 1}\left( {S}^{n - 1}\right) \) are isomorphisms for all \( i > 0 \) since the remaining te... | Thus we obtain the calculation\n\n\[ \n{H}_{i}\left( {{D}^{n},\partial {D}^{n}}\right) \approx \left\{ \begin{array}{ll} \mathbb{Z} & \text{ for }i = n \\ 0 & \text{ otherwise } \end{array}\right.\n\] | Yes |
Proposition 2.22. For good pairs \( \left( {X, A}\right) \), the quotient map \( q : \left( {X, A}\right) \rightarrow \left( {X/A, A/A}\right) \) induces isomorphisms \( {q}_{ * } : {H}_{n}\left( {X, A}\right) \rightarrow {H}_{n}\left( {X/A, A/A}\right) \approx {\widetilde{H}}_{n}\left( {X/A}\right) \) for all \( n \) ... | Proof: Let \( V \) be a neighborhood of \( A \) in \( X \) that deformation retracts onto \( A \) . We have a commutative diagram\n\n\[ \n{H}_{n}\left( {X/A, A/A}\right) \rightarrow {H}_{n}\left( {X/A, V/A}\right) \leftarrow {H}_{n}\left( {X/A - A/A, V/A - A/A}\right) \]\n\nThe upper left horizontal map is an isomorphi... | Yes |
Corollary 2.24. If the CW complex \( X \) is the union of subcomplexes \( A \) and \( B \), then the inclusion \( \left( {B, A \cap B}\right) \hookrightarrow \left( {X, A}\right) \) induces isomorphisms \( {H}_{n}\left( {B, A \cap B}\right) \rightarrow {H}_{n}\left( {X, A}\right) \) for all \( n \) . | Proof: Since CW pairs are good, Proposition 2.22 allows us to pass to the quotient spaces \( B/\left( {A \cap B}\right) \) and \( X/A \) which are homeomorphic, assuming we are not in the trivial case \( A \cap B = \varnothing \) . | No |
For a wedge sum \( { \vee }_{\alpha }{X}_{\alpha } \), the inclusions \( {i}_{\alpha } : {X}_{\alpha } \hookrightarrow { \vee }_{\alpha }{X}_{\alpha } \) induce an isomorphism \( { \oplus }_{\alpha }{i}_{\alpha * } : { \oplus }_{\alpha }{\widetilde{H}}_{n}\left( {X}_{\alpha }\right) \rightarrow {\widetilde{H}}_{n}\left... | Proof: Since reduced homology is the same as homology relative to a basepoint, this follows from the proposition by taking \( \left( {X, A}\right) = \left( {\mathop{\coprod }\limits_{\alpha }{X}_{\alpha },\mathop{\coprod }\limits_{\alpha }\left\{ {x}_{\alpha }\right\} }\right) \) . | Yes |
Theorem 2.26. If nonempty open sets \( U \subset {\mathbb{R}}^{m} \) and \( V \subset {\mathbb{R}}^{n} \) are homeomorphic, I then \( m = n \) . | Proof: For \( x \in U \) we have \( {H}_{k}\left( {U, U-\{ x\} }\right) \approx {H}_{k}\left( {{\mathbb{R}}^{m},{\mathbb{R}}^{m}-\{ x\} }\right) \) by excision. From the long exact sequence for the pair \( \left( {{\mathbb{R}}^{m},{\mathbb{R}}^{m}-\{ x\} }\right) \) we get \( {H}_{k}\left( {{\mathbb{R}}^{m},{\mathbb{R}... | Yes |
Proposition 2.29. \( {\mathbb{Z}}_{2} \) is the only nontrivial group that can act freely on \( {S}^{n} \) if \( n \) is even. | Proof: Since the degree of a homeomorphism must be \( \pm 1 \), an action of a group \( G \) on \( {S}^{n} \) determines a degree function \( d : G \rightarrow \{ \pm 1\} \) . This is a homomorphism since \( \deg {fg} = \deg f\deg g \) . If the action is free, then \( d \) sends every nontrivial element of \( G \) to \... | Yes |
We can use this result to construct a map \( {S}^{n} \rightarrow {S}^{n} \) of any given degree, for each \( n \geq 1 \) . | Let \( q : {S}^{n} \rightarrow { \vee }_{k}{S}^{n} \) be the quotient map obtained by collapsing the complement of \( k \) disjoint open balls \( {B}_{i} \) in \( {S}^{n} \) to a point, and let \( p : {\bigvee }_{k}{S}^{n} \rightarrow {S}^{n} \) identify all the summands to a single sphere. Consider the composition \( ... | Yes |
In the case of \( {S}^{1} \), the map \( f\left( z\right) = {z}^{k} \), where we view \( {S}^{1} \) as the unit circle in \( \mathbb{C} \), has degree \( k \). | This is evident in the case \( k = 0 \) since \( f \) is then constant. The case \( k < 0 \) reduces to the case \( k > 0 \) by composing with \( z \mapsto {z}^{-1} \), which is a reflection, of degree -1 . To compute the degree when \( k > 0 \), observe first that for any \( y \in {S}^{1},{f}^{-1}\left( y\right) \) co... | Yes |
Proposition 2.33. \( \deg {Sf} = \deg f \), where \( {Sf} : {S}^{n + 1} \rightarrow {S}^{n + 1} \) is the suspension of the \( \parallel \operatorname{map}f : {S}^{n} \rightarrow {S}^{n} \) . | Proof: Let \( C{S}^{n} \) denote the cone \( \left( {{S}^{n} \times I}\right) /\left( {{S}^{n} \times 1}\right) \) with base \( {S}^{n} = {S}^{n} \times 0 \subset C{S}^{n} \) , so \( C{S}^{n}/{S}^{n} \) is the suspension of \( {S}^{n} \) . The map \( f \) induces \( {Cf} : \left( {C{S}^{n},{S}^{n}}\right) \rightarrow \... | Yes |
Lemma 2.34. If \( X \) is a \( {CW} \) complex, then:\n\n(a) \( {H}_{k}\left( {{X}^{n},{X}^{n - 1}}\right) \) is zero for \( k \neq n \) and is free abelian for \( k = n \), with a basis in one-to-one correspondence with the \( n \) -cells of \( X \) .\n\n(b) \( {H}_{k}\left( {X}^{n}\right) = 0 \) for \( k > n \) . In ... | Proof: Statement (a) follows immediately from the observation that \( \left( {{X}^{n},{X}^{n - 1}}\right) \) is a good pair and \( {X}^{n}/{X}^{n - 1} \) is a wedge sum of \( n \) -spheres, one for each \( n \) -cell of \( X \) . Here we are using Proposition 2.22 and Corollary 2.25.\n\nTo prove (b), consider the long ... | Yes |
Theorem 2.35. \( {H}_{n}^{CW}\left( X\right) \approx {H}_{n}\left( X\right) \) . | Proof: From the diagram above, \( {H}_{n}\left( X\right) \) can be identified with \( {H}_{n}\left( {X}^{n}\right) /\operatorname{Im}{\partial }_{n + 1} \) . Since \( {j}_{n} \) is injective, it maps \( \operatorname{Im}{\partial }_{n + 1} \) isomorphically onto \( \operatorname{Im}\left( {{j}_{n}{\partial }_{n + 1}}\r... | Yes |
Let \( {M}_{g} \) be the closed orientable surface of genus \( g \) with its usual CW structure consisting of one \( 0 - \mathrm{{cell}},{2g1} - \mathrm{{cells}} \), and one 2-cell attached by the product of commutators \( \left\lbrack {{a}_{1},{b}_{1}}\right\rbrack \cdots \left\lbrack {{a}_{g},{b}_{g}}\right\rbrack \)... | As observed above, \( {d}_{1} \) must be 0 since there is only one 0 -cell. Also, \( {d}_{2} \) is 0 because each \( {a}_{i} \) or \( {b}_{i} \) appears with its inverse in \( \left\lbrack {{a}_{1},{b}_{1}}\right\rbrack \cdots \left\lbrack {{a}_{g},{b}_{g}}\right\rbrack \), so the maps \( {\Delta }_{\alpha \beta } \) a... | Yes |
The closed nonorientable surface \( {N}_{g} \) of genus \( g \) has a cell structure with one \( 0 - \mathrm{{cell}},{g1} - \mathrm{{cells}} \), and one 2-cell attached by the word \( {a}_{1}^{2}{a}_{2}^{2}\cdots {a}_{g}^{2} \). | Again \( {d}_{1} = 0 \), and \( {d}_{2} : \mathbb{Z} \rightarrow {\mathbb{Z}}^{g} \) is specified by the equation \( {d}_{2}\left( 1\right) = \left( {2,\cdots ,2}\right) \) since each \( {a}_{i} \) appears in the attaching word of the 2-cell with total exponent 2 , which means that each \( {\Delta }_{\alpha \beta } \) ... | Yes |
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