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Lemma 11.22. \( {\eta }_{\Delta } = \sum {\left( -1\right) }^{\deg {\omega }_{i}}{\pi }^{ * }{\omega }_{i} \land {\rho }^{ * }{\tau }_{i} \) | Proof. We compute \( {\int }_{\Delta }{\pi }^{ * }{\tau }_{k} \land {\rho }^{ * }{\omega }_{l} \) in two ways. On the one hand, we can pull this integral back to \( M \) via the diagonal map \( \iota : M \rightarrow \Delta \subset M \times M \) :\n\n\[ \n{\int }_{\Delta }{\pi }^{ * }{\tau }_{k} \land {\rho }^{ * }{\ome... | Yes |
Lemma 11.23. The normal bundle \( {N}_{\Delta } \) of the diagonal \( \Delta \) in \( M \times M \) is isomorphic to the tangent bundle \( {T}_{\Delta } \) . | Proof. Since the diagonal map \( \iota : M \rightarrow M \times M \) sends \( M \) diffeomorphically onto \( \Delta ,{\iota }^{ * }{T}_{\Delta } = {T}_{M} \) . It follows from the commutative diagram\n\n\[ \left( {v, v}\right) \mapsto \left( {v, v}\right) \]\n\n\[ {\left. 0 \rightarrow {T}_{\Delta } \rightarrow {T}_{M ... | Yes |
Theorem 12.2 (Thom Isomorphism). For \( \pi : E \rightarrow M \) any vector bundle of rank \( n \) over \( M \) and \( \mathfrak{U} \) a good cover of \( M \) , \[ {H}_{cv}^{ * }\left( E\right) \simeq {H}^{* - n}\left( {\mathfrak{U},{\mathcal{H}}_{cv}^{n}}\right) \] where \( {\mathcal{H}}_{cv}^{n} \) is the presheaf \(... | We now deduce the orientable version of the Thom isomorphism from this. So suppose \( \pi : E \rightarrow M \) is an orientable vector bundle of rank \( n \) over \( M \) . This means there exist forms \( {\sigma }_{\alpha } \) on the sphere bundles \( {\left. S\left( E\right) \right| }_{{U}_{\alpha }} \) which restric... | No |
Proposition 12.3. The cohomology class of\n\n\\[ \n\\Phi = d\\left( {\\rho \\left( r\\right) \\cdot \\psi }\\right) \\in {\\Omega }_{cv}^{n}\\left( E\\right) \n\\]\n\nis the Thom class of the oriented vector bundle E. | Proof. Note that\n\n(12.3.1)\n\n\\[ \n\\Phi = {d\\rho }\\left( r\\right) \\cdot \\psi - \\rho \\left( r\\right) {\\pi }^{ * }e.\n\\]\n\nFor the same reasons as in the discussion following (6.40), \\( \\Phi \\) is a closed global form on \\( E \\) with compact support in the vertical direction. Its restriction to the fi... | Yes |
Theorem 12.5 (Whitney Product Formula for the Euler Class). If \( E \) and \( F \) are two oriented vector bundles, then \( e\left( {E \oplus F}\right) = e\left( E\right) e\left( F\right) \) . | Proof. By Proposition 6.19, the Thom class of \( E \oplus F \) is\n\n\[ \Phi \left( {E \oplus F}\right) = {\pi }_{1}{}^{ * }\Phi \left( E\right) \land {\pi }_{2}^{ * }\Phi \left( F\right) \]\n\nwhere \( {\pi }_{1} \) and \( {\pi }_{2} \) are the projections of \( E \oplus F \) onto \( E \) and \( F \) respectively. Let... | Yes |
Proposition 12.7. Let \( \pi : E \rightarrow M \) be any vector bundle and \( Z \) the zero locus of a transversal section. Then \( \mathbf{Z} \) is a submanifold of \( M \) and its normal bundle in \( M \) is \( {\left. {N}_{Z/M} \simeq E\right| }_{Z} \) . | Proof. Write \( S = s\left( M\right) \) for the image of the section \( s \) (see Figure 12.2). Because \( S \) intersects \( {S}_{0} \) transversally, \( S \cap {S}_{0} \) is a submanifold of \( S \) by the transversality theorem (Guillemin and Pollack [1, p. 28]). Under the diffeomorphism \( s : M \rightarrow S, Z \)... | Yes |
Proposition 12.8. Let \( \pi : E \rightarrow M \) be an oriented vector bundle over an oriented manifold \( M \) . Then the Euler class \( e\left( E\right) \) is Poincaré dual to the zero locus of a transversal section. | Proof. We will identify \( M \) with the image \( {S}_{0} \) of the zero section. If \( S \) is the image in \( E \) of the transversal section \( s : M \rightarrow E \), then the zero locus of \( s \) is \( Z = S \cap {S}_{0} \) . \( Z \) is a closed oriented submanifold of \( M \) and by Proposition 12.7, its normal ... | Yes |
Proposition 12.12 (Generalized Mayer-Vietoris Sequence for Compact Supports). Suppose the open cover \( \mathfrak{U} = \left\{ {U}_{\alpha }\right\} \) of the manifold \( M \) satisfies the local finite condition:\n\n(*) \n\n\[ \n\text{each open set}{U}_{\alpha }\text{intersects only finitely many}{U}_{\beta }\text{’s.... | Proof. We first show \( {\delta }^{2} = 0 \) . Let \( \omega \) be in \( \oplus {\Omega }_{c}^{ * }\left( {U}_{{\alpha }_{0}\ldots {\alpha }_{p}}\right) \) . Then \n\n\[ \n{\left( {\delta }^{2}\omega \right) }_{{\alpha }_{0}\ldots {\alpha }_{p - 2}} = \mathop{\sum }\limits_{\alpha }{\left( \delta \omega \right) }_{\alp... | Yes |
Theorem 13.2. Let \( \mathfrak{U} \) be a good cover on a connected topological space \( X \) and \( N\left( \mathfrak{U}\right) \) its nerve. If \( {\pi }_{1}\left( {N\left( \mathfrak{U}\right) }\right) = 0 \), then every locally constant presheaf on \( \mathfrak{U} \) is constant. | Proof. Suppose \( {\pi }_{1}\left( {N\left( \mathfrak{U}\right) }\right) = 0 \), i.e., every loop bounds some 2-chain. For each open set \( {U}_{\alpha } \), choose a path from \( {U}_{0} \) to \( {U}_{\alpha } \), say \( {U}_{0}{U}_{{\alpha }_{1}}\ldots {U}_{{\alpha }_{r}}{U}_{\alpha } \), and define \( {\psi }_{\alph... | Yes |
Theorem 13.4. Suppose the topological space \( X \) has a good cover \( \mathfrak{U} \) . Then the fundamental group of \( X \) is isomorphic to the fundamental group \( {\pi }_{1}\left( {N\left( \mathfrak{U}\right) }\right) \) of the nerve of the good cover. | Proof. Write \( {N}_{2}\left( \mathfrak{U}\right) \) for the 2-skeleton of the nerve \( N\left( \mathfrak{U}\right) \) . Let \( {U}_{i},{U}_{ij} \), and \( {U}_{ijk} \) be the barycenters of the vertices, edges, and faces of \( {N}_{2}\left( \mathfrak{U}\right) \) and let \( {N}_{2}^{\prime }\left( \mathfrak{U}\right) ... | Yes |
If \( K = \bigoplus {K}^{p, q} \) is a double complex with horizontal operator \( \delta \) and vertical operator \( d \), we can form a single complex out of it in the usual way, by letting \( K = \bigoplus {C}^{k} \), where \( {C}^{k} = {\bigoplus }_{p + q = k}{K}^{p, q} \), and defining the differential operator \( ... | \[ {K}_{p} = {\bigoplus }_{i \geq p}{\bigoplus }_{q \geq 0}{K}^{i, q} \] | Yes |
Theorem 14.6. Let \( K = {\bigoplus }_{n \in Z}{K}^{n} \) be a graded filtered complex with filtration \( \left\{ {K}_{p}\right\} \) and let \( {H}_{D}^{ * }\left( K\right) \) be the cohomology of \( K \) with filtration given by (14.5). Suppose for each dimension \( n \) the filtration \( \left\{ {K}_{p}^{n}\right\} \... | Proof. By treating the convergence question one dimension at a time, this proof reduces to the ungraded situation. To be absolutely sure, we will write out the details. As before,\n\n\[ {A}_{r} = {\bigoplus }_{p \in \mathbb{Z}}{i}^{r - 1}H\left( {K}_{p}\right) \]\n\nif \( r \geq p + 1 \), then \( {i}^{r}H\left( {K}_{p}... | Yes |
Proposition 14.17.1. In a short exact sequence of Abelian groups\n\n\[ 0 \rightarrow A\overset{f}{ \rightarrow }B\overset{g}{ \rightarrow }C \rightarrow 0, \]\n\nif \( C \) is free, then there exists a homomorphism \( s : C \rightarrow B \) such that \( g \circ s \) is the identity on \( C \) . | Proof. Define \( s \) appropriately on the generators of \( C \) and extend linearly. \( ▱ \) | No |
Corollary 14.17.2. Under the hypothesis of the proposition,\n\n(a) the map \( \\left( {f, s}\\right) : A \\oplus C \\rightarrow B \) is an isomorphism;\n\n(b) for any Abelian group \( G \) the induced sequence\n\n\[ \n0 \\rightarrow \\operatorname{Hom}\\left( {C, G}\\right) \\rightarrow \\operatorname{Hom}\\left( {B, G... | The proof is left to the reader. | No |
We now give a spectral sequence proof of the Künneth formula (5.9). Let \( M \) and \( F \) be two manifolds and \( \mathfrak{U} \) a good cover of \( M \) . Suppose \( F \) has finite-dimensional cohomology. By Leray's theorem (14.18), the spectral sequence of the trivial bundle\n\n\[ \begin{matrix} F \rightarrow M \t... | The proof of the Leray-Hirsch theorem is analogous. | No |
Example 14.21 (Orientability of a simply connected manifold). Let \( M \) be a simply connected manifold of dimension \( n \) and \( S\left( {T}_{M}\right) \) its unit tangent bundle. The spectral sequence of the fiber bundle \[ \begin{matrix} {S}^{n - 1} \rightarrow S\left( {T}_{M}\right) \\ \downarrow \\ M \end{matri... | This shows that there is an element in \( {C}^{0}\left( {{\pi }^{-1}\mathfrak{U},{\mathcal{H}}^{n - 1}}\right) \) which can be extended one step down toward being a \( D \) -cocycle. Therefore \( S\left( {T}_{M}\right) \) and also \( M \) are orientable. This gives an alternative proof of the orientability of a simply ... | Yes |
Proposition 15.1. Let \( K : {S}_{ * }\left( {\mathbb{R}}^{n}\right) \rightarrow {S}_{* + 1}\left( {\mathbb{R}}^{n}\right) \) be the cone construction. Then\n\n\[ \partial K - K\partial = {\left( -1\right) }^{q + 1} \]\n\non \( {S}_{q}\left( {\mathbb{R}}^{n}\right) \) for \( q \geq 1 \) . | Proof. The geometrical idea is clear from Figure 15.2. The proof itself is a routine matter of unravelling the definitions. We leave it to the reader. | No |
Proposition 15.2 (The Mayer-Vietoris Sequence for Singular Chains). The following sequence is exact\n\n\[ 0 \leftarrow {S}_{q}^{\mathbf{u}}\left( X\right) \overset{\varepsilon }{ \leftarrow }\mathop{\bigoplus }\limits_{{\alpha }_{0}}{S}_{q}\left( {U}_{{\alpha }_{0}}\right) \overset{\delta }{ \leftarrow }\mathop{\bigopl... | Although this sequence bears a formal resemblance to the generalized Mayer-Vietoris sequence for compact supports (Proposition 12.12), because we do not have partitions of unity at our disposal now, the second half of the proof of (12.12) does not apply. | No |
Lemma 15.3. Let\n\n\[ \n0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 \n\]\n\nbe a short exact sequence of differential complexes. If two out of the three complexes have zero homology, so does the third. | Proof. Consider the long exact sequence in homology\n\n\[ \n\cdots \rightarrow {H}_{q}\left( A\right) \rightarrow {H}_{q}\left( B\right) \rightarrow {H}_{q}\left( C\right) \rightarrow {H}_{q - 1}\left( A\right) \rightarrow \cdots . \n\] | Yes |
Corollary 15.6 (The Homology Mayer-Vietoris Sequence for Two Open Sets). Let \( X = U \cup V \) be the union of two open sets. Then there is a long exact sequence in homology\n\n\[ \cdots \rightarrow {H}_{q}\left( {U \cap V}\right) \overset{f}{ \rightarrow }{H}_{q}\left( U\right) \oplus {H}_{q}\left( V\right) \overset{... | Here \( f \) is the map induced by the signed inclusion \( a \mapsto \left( {-a, a}\right) \) and \( g \) is the sum \( \left( {a, b}\right) \mapsto a + b \) . | Yes |
Corollary 15.14.1. For any space \( X \) for which \( {H}_{q}\left( X\right) \) and \( {H}_{q - 1}\left( X\right) \) are finitely generated \( \mathbb{Z} \) -modules, | \[ {H}^{q}\left( X\right) \simeq {F}_{q} \oplus {T}_{q - 1} \] where \( {F}_{q} \) is the free part of \( {H}_{q}\left( X\right) \) and \( {T}_{q - 1} \) is the torsion part of \( {H}_{q - 1}\left( X\right) \). | Yes |
Example 15.15 (The cohomology of the unit tangent bundle of a sphere). The unit tangent bundle \( S\left( {T}_{{S}^{2}}\right) \) to the 2 -sphere in \( {\mathbb{R}}^{3} \) is a fiber bundle with fiber \( {S}^{1} \): | By (15.11) the \( {E}_{2} \) term of the spectral sequence is\n\n\[ {E}_{2}^{p, q} = {H}^{p}\left( {S}^{2}\right) \otimes {H}^{q}\left( {S}^{1}\right) \]\n\nFor dimensional reasons \( {d}_{3} = {d}_{4} = \cdots = 0 \), so \( {E}_{3} = {E}_{\infty } \). By Remark 14.20 the differential \( {d}_{2} \) in the diagram defin... | Yes |
Proposition 15.18. For any cover \( \mathfrak{U} \) of \( X \) the double complex \( {C}_{ * }\left( {\mathfrak{U},{S}_{ * }}\right) \) computes the singular homology of \( X \) : | \[ {H}_{D}\left\{ {{C}_{ * }\left( {\mathfrak{U},{S}_{ * }}\right) }\right\} = {H}_{ * }\left( X\right) \] | No |
Proposition 16.5. Let \( \pi : E \rightarrow X \) be a fibering. If \( X \) is simply connected and \( E \) is path connected, then the fibers are path connected. | Proof. Trivially the \( {E}_{2}^{0,0} \) term of the fibering survives to \( {E}_{\infty } \) . Hence\n\n\[ \n{E}_{2}^{0,0} = {E}_{\infty }^{0,0} = {H}^{0}\left( E\right) = \mathbb{Z} \n\] \n\nsince \( E \) is path connected. On the other hand, \n\n\[ \n{E}_{2}^{0,0} = {H}^{0}\left( {X,{H}^{0}\left( F\right) }\right) =... | No |
Proposition 17.1. (a) \( {\pi }_{q}\left( {X \times Y}\right) = {\pi }_{q}\left( X\right) \times {\pi }_{q}\left( Y\right) \) . | Proof. (a) is clear since every map from \( {I}^{q} \) into \( X \times Y \) is of the form \( \left( {{f}_{1},{f}_{2}}\right) \) where \( {f}_{1} \) is a map into \( X \) and \( {f}_{2} \) is a map into \( Y \) . Furthermore, since \( \left( {{f}_{1},{f}_{2}}\right) \left( {{g}_{1},{g}_{2}}\right) = \left( {{f}_{1}{g}... | Yes |
Proposition 17.2. \( {\pi }_{q - 1}\left( {\Omega X}\right) = {\pi }_{q}\left( X\right), q \geq 2 \) . | Sketch of Proof. Elements of \( {\pi }_{2}\left( X\right) \) are given by maps of the square \( {I}^{2} \) into \( X \) which send the boundary \( {I}^{2} \) to the base point \( * \) . Such a map may be viewed as a pencil of loops in \( X \), i.e., a map from the unit interval into \( {\Omega X} \) . Therefore, \( {\p... | No |
Proposition 17.3. The identity component \( H \) of a Lie group \( G \) is a normal subgroup of \( G \) . Therefore, \( {\pi }_{0}\left( G\right) = G/H \) is a group. | Proof. Let \( a, b \) be in \( H \) . Since the continuous image of a connected set is connected, \( {bH} \) is a connected set having a nonempty intersection with \( H \) . Hence \( {bH} \subset H \) . It follows that \( {abH} \subset {aH} \subset H \), so \( {ab} \) is in \( H \) . Similarly \( {a}^{-1}H \) is a conn... | Yes |
Proposition 17.8 Every continuous map \( f : M \rightarrow N \) between two manifolds is continuously homotopic to a differentiable map. | Proof. We first note that if \( f : M \rightarrow \mathbb{R} \) is a continuous function and \( \varepsilon \) a positive number, then there is a differentiable real-valued function \( h \) on \( M \) with \( \left| {f - h}\right| < \varepsilon \) . This is more or less clear from the fact that via its graph, \( f \) m... | Yes |
Proposition 17.9. \( {\pi }_{q}\left( {S}^{n}\right) = 0 \), for \( q < n \) . | Proof. Let \( f \) be a continuous map from \( {I}^{q} \) to \( {S}^{n} \), representing an element of \( {\pi }_{q}\left( {S}^{n}\right) \) . By the lemma above, we may assume \( f \) differentiable. Hence Sard’s theorem applies. Because \( q \) is strictly less than \( n \), the images of \( f \) are all critical val... | Yes |
Lemma 17.10.2 Two maps from \( {S}^{n} \) to \( {S}^{n} \) of the same degree can be deformed into each other. | To prove these lemmas we will deform any map \( f : {S}^{n} \rightarrow {S}^{n} \) into a normal form as follows. By the inverse function theorem \( f \) is a local diffeomorphism around a regular point. By Sard's theorem regular values exist. Let \( U \) be an open set around a regular value so that \( {f}^{-1}\left( ... | No |
Proposition 17.11. Attaching an n-cell to a space \( X \) does not alter the homotopy in dimensions strictly less than \( n - 1 \), but may kill elements in \( {\pi }_{n - 1}\left( X\right) \) ;\n\nmore precisely, the inclusion \( X \hookrightarrow X \cup {e}^{n} \) induces isomorphisms\n\n\[ \n{\pi }_{q}\left( X\right... | Proof. Assume \( q \leq n - 1 \) and let \( f : {S}^{q} \rightarrow X \cup {e}^{n} \) be a continuous basepoint preserving map. We would like first of all to show that \( f \) is homotopic to some map whose image does not contain all of \( {e}^{n} \) . If \( f \) is differentiable and \( X{ \cup }_{f}{e}^{n} \) is a ma... | Yes |
Proposition 17.12. Attaching an n-cell to a space \( X \) via a map \( f \) does not alter the homology except possibly in dimensions \( n - 1 \) and \( n \) . Writing \( {X}_{f} \) for \( X{ \cup }_{f}{e}^{n} \), there is an exact sequence\n\n\[ 0 \rightarrow {H}_{n}\left( X\right) \rightarrow {H}_{n}\left( {X}_{f}\ri... | Proof. Let \( U \) be \( {X}_{f} - \{ p\} \) where \( p \) is the origin of \( {e}^{n} \), and let \( V \) be \( \left\{ {x \in {e}^{n} \mid }\right. \) \( \parallel x\parallel < \frac{1}{2}\} \) . Then \( U \) is homotopic to \( X, V \) is contractible, and \( \{ U, V\} \) is an open cover of \( {X}_{f} \) . By the Ma... | Yes |
Proposition 17.13. Every \( {CW} \) complex is homotopy equivalent to a space with a good cover. | Hence the entire machinery of the spectral sequence that we have developed applies to \( {CW} \) complexes. This proposition follows from the nontrivial fact that every \( {CW} \) complex has the homotopy type of a simplicial complex (Gray [1, Cor. 16.44, p. 149 and Cor. 21.15, p. 206] or Lundell and Weingram [1, Cor. ... | No |
Theorem 17.15. Let \( f \) be a differentiable function on the manifold \( M \), and \( {M}_{a} \) the set \( {f}^{-1}\left( \left\lbrack {-\infty, a}\right\rbrack \right) \) . If \( {f}^{-1}\left( \left\lbrack {a, b}\right\rbrack \right) \) is compact and contains no critical points, then \( {M}_{a} \) has the same ho... | Outline of Proof. Choose a Riemannian structure \( \langle \) , \( \rangle {onM} \) . Then away from the critical points of \( f \), the gradient \( \nabla f \) of a differentiable function \( f \) is defined: it is the unique vector field on \( M \) such that for all vector fields \( Y \) on \( M \) ,\n\n\[ \left\lang... | No |
Theorem 17.16. Suppose \( {f}^{-1}\left( \left\lbrack {a, b}\right\rbrack \right) \) is compact and contains precisely one critical point in its interior, which is nondegenerate and of index \( k \) . Then \( {M}_{b} \) has the homotopy type of \( {M}_{a} \cup {e}^{k} \) . | Outline of a proof of THEOREM 17.16. Let \( c = f\left( p\right) \) be the critical value and \( \varepsilon \) a small positive number. By Theorem 17.15, \( {M}_{b} \) has the homotopy type of \( {M}_{c + \varepsilon } \), and \( {M}_{a} \) that of \( {M}_{c - \varepsilon } \), so it suffices to show that \( {M}_{c + ... | Yes |
Lemma 17.17. Let \( U \) be an open subset of \( {\mathbb{R}}^{n} \) and \( f \) any smooth real-valued function on \( U \) . Then for almost all \( a = \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) in \( {\mathbb{R}}^{n} \), the function \( {f}_{a}\left( x\right) = \) \( f\left( x\right) + {a}_{1}{x}_{1} + \cdots + {a}_{... | Proof. Recall that we denote the Jacobian matrix of a function \( h \) by \( D\left( h\right) \) . Define \( g\left( x\right) = \left( {\partial f/\partial {x}_{1},\ldots ,\partial f/\partial {x}_{n}}\right) \) . Note that the Hessian of \( f \) is precisely the Jacobian of \( g \), and \( x \) is a nondegenerate criti... | Yes |
Proposition 17.18. Let \( M \) be a manifold of dimension \( n \) in \( {\mathbb{R}}^{r} \) . For almost all \( a = \left( {{a}_{1},\ldots ,{a}_{r}}\right) \) in \( {\mathbb{R}}^{r} \), the function \( f\left( x\right) = {a}_{1}{x}_{1} + \cdots + {a}_{r}{x}_{r} \) is a Morse function on \( M \) . | Proof. Let \( {x}_{1},\ldots ,{x}_{r} \) be the coordinate functions on \( {\mathbb{R}}^{r} \) . Every point \( x \) in \( M \) has a neighborhood \( U \) in \( M \) on which some \( n \) of \( {x}_{1},\ldots ,{x}_{r} \) form a coordinate system. (Proof: Since \( {T}_{x}M \rightarrow {T}_{x}{\mathbb{R}}^{r} \) is injec... | Yes |
Every compact manifold \( M \) has the homotopy type of a finite CW complex. | Proof. By Whitney's embedding theorem (see de Rham [1, p. 12]), we may assume that \( M \) is a submanifold of some Euclidean space. Let \( f \) be a Morse function on \( M \) (the existence of \( f \) is guaranteed by Proposition 17.18). By the Morse lemma, the critical points of \( f \) are isolated. Since \( M \) is... | Yes |
Theorem 17.20. Let \( X \) be a path-connected space. Then \( {H}_{1}\left( X\right) \) is the Abelianization of \( {\pi }_{1}\left( X\right) \), i.e., if \( \left\lbrack {{\pi }_{1}\left( X\right) ,{\pi }_{1}\left( X\right) }\right\rbrack \) is the commutator subgroup of \( {\pi }_{1}\left( X\right) \), then \( {H}_{1... | We will assume this theorem as known. Its proof may be found in, for instance, Greenberg [1, p. 48]. | No |
Theorem 17.21 (Hurewicz Isomorphism Theorem). Let \( X \) be a simply connected path-connected CW complex. Then the first nontrivial homotopy and homology occur in the same dimension and are equal, i.e., given a positive integer \( n \geq 2 \), if \( {\pi }_{q}\left( X\right) = 0 \) for \( 1 \leq q < n \), then \( {H}_... | Proof. To start the induction, consider the case \( n = 2 \) . The \( {E}^{2} \) term of the homology spectral sequence of the path fibration is\n\n\n\nThus\n\n\[ \n{H}_{2}\left( X\right) = {H}_{1}\left( {\Omega X}\r... | Yes |
Proposition 17.22. (a) The definition of the Hopf invariant is independent of the choice of \( \omega \) . | Proof. (a) Let \( {\omega }^{\prime } \) be another \( \left( {n - 1}\right) \) -form on \( {S}^{{2n} - 1} \) such that \( {f}^{ * }\alpha = d{\omega }^{\prime } \) . Then \( 0 = d\left( {\omega - {\omega }^{\prime }}\right) \) . Hence\n\n\[ \n{\int }_{{S}^{{2n} - 1}}\omega \land {d\omega } - {\int }_{{S}^{{2n} - 1}}{\... | Yes |
We now show that \( \mathbb{R}{P}^{\infty } \) has no higher homotopy, i.e., \( {\pi }_{q}\left( {\mathbb{R}{P}^{\infty }}\right) = 0 \) for \( q > 1 \) . | Take \( {\pi }_{15}\left( {\mathbb{R}{P}^{\infty }}\right) \) for example. Suppose \( f : {S}^{15} \rightarrow \mathbb{R}{P}^{\infty } \) represents an element of \( {\pi }_{15}\left( {\mathbb{R}{P}^{\infty }}\right) \) . Since the image \( f\left( {S}^{15}\right) \) is compact, it must lie in a finite union of the \( ... | Yes |
Example 18.3. (The infinite complex projective space). Applying the telescoping construction to the sequences\n\n\\[ \n\\cdots \\subset {S}^{{2n} + 1} \\subset {S}^{{2n} + 3} \\subset \\cdots \n\\]\n\n\\[ \n{S}^{1} \\downarrow \\; \\downarrow \n\\]\n\n\\[ \n\\cdots \\subset \\mathbb{C}{P}^{n} \\subset \\mathbb{C}{P}^{n... | Since \\( {S}^{\\infty } \\) has no homotopy in any dimension, it follows from the homotopy sequence of the fibering that\n\n\\[ \n{\\pi }_{k}\\left( {\\mathbb{C}{P}^{\\infty }}\\right) = \\left\\{ \\begin{array}{ll} \\mathbb{Z} & \\text{ when }k = 2 \\\\ 0 & \\text{ otherwise. } \\end{array}\\right. \n\\]\n\nTherefore... | Yes |
Let \( {S}^{{2n} + 1} \) be the unit sphere in \( {\mathbb{C}}^{n + 1} \). Since \( {S}^{1} \) acts freely on \( {S}^{{2n} + 1} \), so does any subgroup of \( {S}^{1} \). For example, \( {\mathbb{Z}}_{5} \) acts on \( {S}^{{2n} + 1} \) by \[ {e}^{{2\pi i}/5} : \left( {{z}_{0},\ldots ,{z}_{n}}\right) \mapsto \left( {{e}... | Next we shall compute the cohomology of a lens space, say \( L\left( {n,5}\right) \). Since the lens space \( L\left( {n,5}\right) \) is not simply connected, the defining fibration \( {\mathbf{Z}}_{5} \rightarrow {S}^{{2n} + 1} \rightarrow L\left( {n,5}\right) \) is of little use in the computation of the cohomology. ... | Yes |
Proposition 18.13. Let \( \pi : E \rightarrow M \) be a fibration with fiber \( F \) in the differentiable category. An element \( \omega \) in \( {H}^{q}\left( F\right) \) is transgressive if and only if it is the restriction of a global form \( \psi \) on \( E \) such that \( {d\psi } = {\pi }^{ * }\tau \) for some f... | Proof of Proposition 18.13. Let \( \mathfrak{U} \) be a good cover of \( M \) . If \( \omega \) is transgressive, then by (14.12) it can be extended to a cochain \( \alpha = {\alpha }_{0} + \cdots + {\alpha }_{q} \) in the double complex \( {C}^{ * }\left( {{\pi }^{-1}\mathfrak{U},{\Omega }^{ * }}\right) \) such that \... | Yes |
Proposition 18.14. In mod 2 cohomology, if \( \alpha \) is a transgressive, so is \( {\alpha }^{2} \) . | Proof. Let \( \psi \) be the singular cochain on \( E \) given by Prop. 18.13. Since \( \psi \) restricts to \( \alpha \) on a fiber, \( {\psi }^{2} \) restricts to \( {\alpha }^{2} \) . With \( {\mathbb{Z}}_{2} \) coefficients,\n\n\[ d\left( {\psi }^{2}\right) = \left( {d\psi }\right) \psi \pm {\psi d\psi } = {2\psi d... | Yes |
Proposition 18.19 (Postnikov Approximation). Every connected CW complex can be approximated by a twisted product of Eilenberg-MacLane spaces; more precisely, for each \( n \), there is a sequence of fibrations \( {Y}_{q} \rightarrow {Y}_{q - 1} \) with the \( K\left( {{\pi }_{q}, q}\right) \) ’s as fibers and commuting... | Proof of Proposition 18.19. To construct \( {Y}_{n} \) we kill off all homotopy of \( X \) in dimensions \( \geq n + 1 \) by attaching cells of dimensions \( \geq n + 2 \) . Then\n\n\[ \n{\pi }_{k}\left( {Y}_{n}\right) = \left\{ \begin{array}{ll} 0, & k \geq n + 1 \\ {\pi }_{k}, & k = 1,2,\ldots, n. \end{array}\right.\... | Yes |
Theorem 19.5. Let \( M \) be a simply connected manifold and \( \mathcal{M} \) its minimal model. Then the dimension of the vector space \( {\pi }_{q}\left( M\right) \otimes \mathbb{Q} \) is the number of generators of the minimal model \( \mathcal{M} \) in dimension \( q \) . | To make this theorem plausible, we will say a few words about the computation of the rational cohomology of \( M \) . The idea is to compute it from the Postnikov towers of \( M \), whose fibers are the Eilenberg-MacLane spaces \( K\left( {{\pi }_{q}, q}\right) \) . Now there are two things to remember about the ration... | No |
Example 20.3 (Tautological bundles on a projective space). Let \( V \) be a complex vector space of dimension \( n \) and \( P\left( V\right) \) its projectivization:\n\n\[ P\left( V\right) = \{ 1\text{-dimensional subspaces of } V \} .\n\nOn \( P\left( V\right) \) there are several God-given vector bundles: the produc... | Consider the composition\n\n\[ \sigma : S \hookrightarrow P\left( V\right) \times V \rightarrow V \]\n\nof the inclusion followed by the projection. The inverse image of any point \( v \) is\n\n\[ \sigma^{-1}\left( v \right) = \{ \left( \ell, v \right) \mid v \in \ell \} \]\n\nIf \( v \neq 0, \sigma^{-1}\left( v \right... | Yes |
Recall that if \( V \) is a vector space with basis \( \left\{ {{v}_{1},\ldots ,{v}_{n}}\right\} \), then the exterior power \( {\Lambda }^{p}V \) is the vector space with basis \( {\left\{ {v}_{{i}_{1}} \land \cdots \land {v}_{{i}_{p}}\right\} }_{1 \leq {i}_{1} < \cdots < {i}_{p} \leq n} \) . So if \( E \) is the dire... | Hence \[ c\left( {{\Lambda }^{p}E}\right) = \prod \left( {1 + {c}_{1}\left( {{L}_{{i}_{1}} \otimes \cdots \otimes {L}_{{i}_{p}}}\right) }\right) \;\text{by the Whitney product formula} \] \[ = \prod \left( {1 + {x}_{{i}_{1}} + \cdots + {x}_{{i}_{p}}}\right) \;\text{by (20.1), with}{x}_{i} = {c}_{1}\left( {L}_{i}\right)... | Yes |
Let \( L \) be a complex line bundle. By (20.2), \[ {c}_{1}\left( {L}^{ * }\right) = - {c}_{1}\left( L\right) \] | Next consider a direct sum of line bundles \[ E = {L}_{1} \oplus \cdots \oplus {L}_{n} \] By the Whitney product formula \[ c\left( E\right) = c\left( {L}_{1}\right) \cdots c\left( {L}_{n}\right) = \left( {1 + {c}_{1}\left( {L}_{1}\right) }\right) \cdots \left( {1 + {c}_{1}\left( {L}_{n}\right) }\right) . \] On the oth... | Yes |
Example 21.13 (The Chern classes of the complex projective space). By analogy with the definition of a differentiable manifold, we say that a second countable, Hausdorff space \( M \) is a complex manifold of dimension \( n \) if every point has a neighborhood \( {U}_{\alpha } \) homeomorphic to some open ball in \( {\... | (1) Tensoring the tautological sequence with \( {S}^{ * } \), we get\n\n\[ \n0 \rightarrow \mathbb{C} \rightarrow {S}^{ * } \otimes {\mathbb{C}}^{n + 1} \rightarrow {S}^{ * } \otimes Q \rightarrow 0.\n\]\n\nBy the Whitney product formula\n\n\[ \nc\left( T\right) = c\left( {{S}^{ * } \otimes Q}\right) = c\left( {{S}^{ *... | Yes |
Proposition 21.15. The associated flag bundle \( {Fl}\left( E\right) \) of a vector bundle is the split manifold \( F\left( E\right) \) constructed earlier. | Proof. We first show this for \( E = V \) a vector space of dimension 3, viewed as a rank 3 vector bundle over a point.\n\n\n\nIn what follows all lines and planes go through the origin. A point in \( P\left( V\right... | Yes |
Proposition 21.16. \( {H}^{ * }\left( {P\left( E\right) }\right) = {H}^{ * }\left( M\right) \left\lbrack {c\left( S\right), c\left( Q\right) }\right\rbrack /\left( {c\left( S\right) c\left( Q\right) = {\pi }^{ * }c\left( E\right) }\right) \). | Proof. The idea is to eliminate the generators \( {c}_{1}\left( Q\right) ,\ldots ,{c}_{n - 1}\left( Q\right) \) by using the relation \( c\left( S\right) c\left( Q\right) = {\pi }^{ * }c\left( E\right) \). Let \( x = {c}_{1}\left( {S}^{ * }\right) ,{y}_{i} = {c}_{i}\left( Q\right) \), and \( {c}_{i} = {\pi }^{ * }{c}_{... | Yes |
Let \( L : \mathbb{C} \rightarrow \mathbb{C} \) be given by multiplication by the complex number \( \lambda = \alpha + {i\beta } \). | Since\n\n\[ \left( {\alpha + {i\beta }}\right) \left( {{x}_{1} + i{x}_{2}}\right) = \left( {\alpha {x}_{1} - \beta {x}_{2}}\right) + i\left( {\beta {x}_{1} + \alpha {x}_{2}}\right) ,\]\n\n as a linear map from \( {\mathbb{R}}^{2} \) to \( {\mathbb{R}}^{2},{L}_{\mathbb{R}} \) is given by\n\n\[ \left( \begin{array}{l} {x... | Yes |
Lemma 22.4. Let \( A \) be an \( n \) by \( n \) complex matrix. There is a \( {2n} \) by \( {2n} \) matrix \( B \), independent of \( A \), such that \( {A}_{\mathbb{R}} \otimes \mathbb{C} \) is similar to \( \left( \begin{array}{ll} A & 0 \\ 0 & A \end{array}\right) \) via \( B \) . | Proof. In the 1 by 1 case, this is a matter of diagonalizing\n\n\[ \n{\left( \alpha + i\beta \right) }_{\mathbb{R}} \otimes \mathbb{C} = \left( \begin{array}{rr} \alpha & - \beta \\ \beta & \alpha \end{array}\right) \n\]\n\nCorresponding to the eigenvalues \( \alpha + {i\beta } \) and \( \alpha - {i\beta } \) are the e... | Yes |
The Pontrjagin class of a complex manifold \( M \) is defined to be that of the underlying real manifold \( {M}_{\mathbb{R}} \) . Let \( T \) be the holomorphic tangent bundle to \( M \) . Then the tangent bundle to \( {M}_{\mathbb{R}} \) is the realization of \( T \) and | \[ p\left( M\right) = p\left( {T}_{\mathbb{R}}\right) = c\left( {{T}_{\mathbb{R}} \otimes \mathbb{C}}\right) = c\left( {T \oplus \bar{T}}\right) = c\left( T\right) c\left( \bar{T}\right) . \] So if the total Chern class of the complex manifold \( M \) is \( c\left( M\right) = \prod \left( {1 + {x}_{i}}\right) \) , then... | Yes |
Proposition 23.1. The cohomology of the complex Grassmannian \( {G}_{k}\left( V\right) \) has Poincaré polynomial \[ {P}_{t}\left( {{G}_{k}\left( V\right) }\right) = \frac{\left( {1 - {t}^{2}}\right) \cdots \left( {1 - {t}^{2n}}\right) }{\left( {1 - {t}^{2}}\right) \cdots \left( {1 - {t}^{2k}}\right) \left( {1 - {t}^{2... | Proof. The flag manifold \( F\left( V\right) \) may be obtained from the Grassmannian \( {G}_{k}\left( V\right) \) by a series of flag constructions as follows. Let \( \widehat{Q} \) be the pullback of \( Q \) to the flag bundle \( F\left( S\right) \). A point of \( F\left( S\right) \) is a pair \( \left( {\Lambda ,{L}... | Yes |
Proposition 23.2. Let \( V \) be a complex vector space of dimension \( n \) .\n\n(a) As a ring\n\n\[ \n{H}^{ * }\left( {{G}_{k}\left( V\right) }\right) = \frac{\mathbb{R}\left\lbrack {{c}_{1}\left( S\right) ,\ldots ,{c}_{n - k}\left( S\right) ,{c}_{1}\left( Q\right) ,\ldots ,{c}_{k}\left( Q\right) }\right\rbrack }{\le... | Proof. In the proof of Proposition 23.1, we saw that the flag manifold \( F\left( V\right) \) is obtained from the Grassmannian by two flag constructions\n\n\n\nBy (21.18) the cohomology ring of the flag manifold is\... | Yes |
Proposition 23.3. Let \( A = {\bigoplus }_{i = 0}^{\infty }{A}_{i} \) be a graded algebra over a field \( k \), and \( x \) a homogeneous element of degree \( n \) in \( A \) . If \( x \) is not a zero-divisor, then\n\n\[ \n{P}_{t}\left( {A/{xA}}\right) = {P}_{t}\left( A\right) \left( {1 - {t}^{n}}\right) \n\] | Proof. Because \( x \) is not a zero-divisor, multiplication by \( x \) is an injection. Hence for each integer \( i \) there is an exact sequence of vector spaces\n\n\[ \n0 \rightarrow {A}_{i}\overset{x}{ \rightarrow }{A}_{i + n} \rightarrow {\left( A/xA\right) }_{i + n} \rightarrow 0. \n\]\n\nBy the additivity of the... | Yes |
Proposition 23.4. Let \( A \) be a graded algebra over a field \( k \) and \( {a}_{1},\ldots ,{a}_{r}a \) regular sequence of homogeneous elements of degrees \( {n}_{1},\ldots ,{n}_{r} \) . Then\n\n\[ \n{P}_{t}\left( {A/\left( {{a}_{i},\ldots ,{a}_{r}}\right) }\right) = {P}_{t}\left( A\right) \left( {1 - {t}^{{n}_{1}}}... | Proof. This is an immediate consequence of Proposition 23.3 and induction on \( r \) . | No |
Lemma 23.5. Let \( A \) be a graded algebra over a field \( k \) . If \( {a}_{1},\ldots ,{a}_{r} \) is a regular sequence of homogeneous elements of positive degrees in \( A \), so is any permutation of \( {a}_{1},\ldots ,{a}_{r} \) . | Proof. Since any permutation is a product of transpositions of adjacent elements, it suffices to show that \( {a}_{1},\ldots ,{a}_{i - 1},{a}_{i + 1},{a}_{i},\ldots ,{a}_{r} \) is a regular sequence. For this it is enough to show that in the ring \( A/\left( {{a}_{1},\ldots ,{a}_{i - 1}}\right) \), the images of \( {a}... | Yes |
Lemma 23.6. If \( {a}_{1},\ldots ,{a}_{r}, b \) and \( {a}_{1},\ldots ,{a}_{r}, c \) are regular sequences in a ring \( A \), then so is \( {a}_{1},\ldots ,{a}_{r},{bc} \) . | Proof. It suffices to check that \( {bc} \) is not a zero-divisor in \( A/\left( {{a}_{1},\ldots ,{a}_{r}}\right) \) . This is clear since by hypothesis neither \( b \) nor \( c \) is a zero-divisor in \( A/\left( {{a}_{1},\ldots ,{a}_{r}}\right) \) . | Yes |
Proposition 23.7. The homogeneous terms of\n\n\\[ \n\\left( {1 + {x}_{1} + \\cdots + {x}_{j}}\\right) \\left( {1 + {y}_{1} + \\cdots + {y}_{k}}\\right) - 1 \n\\] \n\nform a regular sequence in \\( A = \\mathbb{R}\\left\\lbrack {{x}_{1},\\ldots ,{x}_{j},{y}_{1},\\ldots ,{y}_{k}}\\right\\rbrack \\) . | Proof. The proof proceeds by induction on \\( j \\) and \\( k \\) . Suppose \\( j = 1 \\) and \\( k = 1 \\) . Then \\( \\mathbb{R}\\left\\lbrack {{x}_{1},{y}_{1}}\\right\\rbrack /\\left( {{x}_{1} + {y}_{1}}\\right) = \\mathbb{R}\\left\\lbrack {x}_{1}\\right\\rbrack \\) and the image of \\( {x}_{1}{y}_{1} \\) in \\( \\m... | Yes |
Lemma 23.8. Let \( E \) be a rank \( k \) complex vector bundle over a differentiable manifold \( M \) of finite type. There exist on \( M \) finitely many smooth sections of \( E \) which span the fiber at every point. | Proof. Let \( {\left\{ {U}_{i}\right\} }_{i \in I} \) be a finite good cover for \( M \) . Since \( {U}_{i} \) is contractible, \( {\left. E\right| }_{{U}_{i}} \) is trivial and so we can find \( k \) sections \( {s}_{i,1},\ldots ,{s}_{i, k} \) over \( {U}_{i} \) which form a basis of the fiber above any point in \( {U... | Yes |
Proposition 23.9. Let \( E \) be a rank \( k \) complex vector bundle over a differentiable manifold \( M \) of finite type. Suppose there are \( n \) global sections of \( E \) which span the fiber at every point. Then there is a map \( f \) from \( M \) to some Grassmannian \( {G}_{k}\left( {\mathbb{C}}^{n}\right) \)... | Proof. Let \( {s}_{1},\ldots ,{s}_{n} \) be \( n \) spanning sections of \( E \) and let \( V \) be the complex vector space with basis \( {s}_{1},\ldots ,{s}_{n} \) . Since \( {s}_{1},\ldots ,{s}_{n} \) are spanning sections, for each point \( p \) in \( M \) the evaluation map\n\n\[ \n{\mathrm{{ev}}}_{p} : V \rightar... | Yes |
Lemma 23.9.1. Given a manifold \( M \) of dimension \( m \), if \( n \geq k + \frac{m}{2} \) and \( f \) and \( g : M \rightarrow {G}_{k}\left( {\mathbb{C}}^{n}\right) \) are two maps such that \( {f}^{-1}Q \simeq {g}^{-1}Q \), then \( f \) and \( g \) are homotopic. | A proof of this lemma based on obstruction theory may be found in Steen-rod [1, §19] and Husemoller [1, §7.6]. | No |
Theorem 23.10. Let \( M \) be a manifold having a finite good cover and let \( k \) be a positive integer. For \( n \) sufficiently large, the classifying map of a vector bundle induces a one-to-one correspondence\n\n\[ \n{\operatorname{Vect}}_{k}\left( {M;\mathbb{C}}\right) \simeq \left\lbrack {M,{G}_{k}\left( {\mathb... | Proof. By the homotopy property of vector bundles (Theorem 6.8), there is a map\n\n\[ \n\alpha : \left\lbrack {M,{G}_{k}\left( {\mathbb{C}}^{n}\right) }\right\rbrack \rightarrow {\operatorname{Vect}}_{k}\left( {M;\mathbb{C}}\right) \n\]\n\ngiven by the pullback of the universal quotient bundle over \( {G}_{k}\left( {\m... | Yes |
Proposition 23.11. Every natural transformation from the isomorphism classes of complex vector bundles over a manifold of finite type to the de Rham cohomology can be given as a polynomial in the Chern classes. | Proof. Let \( T \) be a natural transformation from the functor \( {\operatorname{Vect}}_{k}\left( {;\mathbb{C}}\right) \) to the functor \( {H}^{ * }\left( \right) \) in the category of manifolds of finite type. By Proposition 23.9 and the naturality of \( T \), if \( E \) is any rank \( k \) complex vector bundle ove... | Yes |
Proposition 1.2. \( \mathbb{Z}\left\lbrack \zeta \right\rbrack \) is the ring of algebraic integers in the field \( \mathbb{Q}\left( \zeta \right) \) . Therefore \( \mathbb{Z}\left\lbrack \zeta \right\rbrack \) is a Dedekind domain (so we have unique factorization into prime ideals, etc.). | Proof. Let \( \mathcal{O} \) denote the algebraic integers of \( \mathbb{Q}\left( \zeta \right) \) . Clearly \( \mathbb{Z}\left\lbrack \zeta \right\rbrack \subseteq \mathcal{O} \) . We must show the reverse inclusion. | No |
Lemma 1.3. Suppose \( r \) and \( s \) are integers with \( \left( {p,{rs}}\right) = 1 \) . Then \( \left( {{\zeta }^{r} - 1}\right) / \) \( \left( {{\zeta }^{s} - 1}\right) \) is a unit of \( \mathbb{Z}\left\lbrack \zeta \right\rbrack \) . | Proof. Writing \( r \equiv {st}\left( {\;\operatorname{mod}\;p}\right) \) for some \( t \), we have\n\n\[ \frac{{\zeta }^{r} - 1}{{\zeta }^{s} - 1} = \frac{{\zeta }^{st} - 1}{{\zeta }^{s} - 1} = 1 + {\zeta }^{s} + \cdots + {\zeta }^{s\left( {t - 1}\right) } \in \mathbb{Z}\left\lbrack \zeta \right\rbrack . \]\n\nSimilar... | Yes |
Lemma 1.4. The ideal \( \left( {1 - \zeta }\right) \) is a prime ideal of \( \mathcal{O} \) and \( {\left( 1 - \zeta \right) }^{p - 1} = \left( p\right) \) . Therefore \( p \) is totally ramified in \( \mathbb{Q}\left( \zeta \right) \) . | Proof. Since \( {X}^{p - 1} + {X}^{p - 2} + \cdots + X + 1 = \mathop{\prod }\limits_{{i = 1}}^{{p - 1}}\left( {X - {\zeta }^{i}}\right) \), we let \( X = 1 \) to obtain \( p = \prod \left( {1 - {\zeta }^{i}}\right) \) . From Lemma 1.3, we have the equality of ideals \( \left( {1 - \zeta }\right) = \left( {1 - {\bar{\ze... | Yes |
Proposition 1.5. Let \( \varepsilon \) be a unit of \( \mathbb{Z}\left\lbrack {\zeta }_{p}\right\rbrack \) . Then there exist \( {\varepsilon }_{1} \in \mathbb{Q}\left( {\zeta + {\zeta }^{-1}}\right) \) and \( r \in \mathbb{Z} \) such that \( \varepsilon = {\zeta }^{r}{\varepsilon }_{1} \) . | Proof of Proposition 1.5. Let \( \alpha = \varepsilon /\bar{\varepsilon } \) . Then \( \alpha \) is an algebraic integer since \( \bar{\varepsilon } \) is a unit. Also, all conjugates of \( \alpha \) have absolute value 1 (this follows easily from the fact that complex conjugation commutes with the other elements of th... | No |
Lemma 1.6. If \( \alpha \) is an algebraic integer all of whose conjugates have absolute value 1, then \( \alpha \) is a root of unity. | Proof. The coefficients of the irreducible polynomials for all powers of \( \alpha \) are rational integers which can be given bounds depending only on the degree of \( \alpha \) over \( \mathbb{Q} \) . It follows that there are only finitely many irreducible polynomials which can have a power of \( \alpha \) as a root... | No |
Lemma 1.7. The ideals \( \left( {x + {\zeta }^{i}y}\right), i = 0,1,\ldots, p - 1 \), are pairwise relatively prime. | Proof. Suppose \( \mathcal{P} \) is a prime ideal with \( \mathcal{P}\left| \left( {x + {\zeta }^{i}y}\right) \right| \) and \( \mathcal{P} \mid \left( {x + {\zeta }^{j}y}\right) \), where \( i \neq j \) . Then \( \mathcal{P} \mid \left( {{\zeta }^{i}y - {\zeta }^{j}y}\right) = \left( \text{unit}\right) \left( {1 - \ze... | Yes |
Lemma 1.8. Let \( \alpha \in \mathbb{Z}\left\lbrack \zeta \right\rbrack \) . Then \( {\alpha }^{p} \) is congruent mod \( p \) to a rational integer (note this congruence is mod \( p \), so it is much stronger than a congruence \( {\;\operatorname{mod}\;1} - \zeta ) \) . | Proof. Let \( \alpha = {b}_{0} + {b}_{1}\zeta + \cdots + {b}_{p - 2}{\zeta }^{p - 2} \) . Then \( {\alpha }^{p} \equiv {b}_{0}^{p} + {\left( {b}_{1}\zeta \right) }^{p} + \cdots + \) \( {\left( {b}_{p - 2}{\zeta }^{p - 2}\right) }^{p} = {b}_{0}^{p} + {b}_{1}^{p} + \cdots + {b}_{p - 2}^{p}\left( {\;\operatorname{mod}\;p}... | Yes |
Lemma 1.9. Suppose \( \alpha = {a}_{0} + {a}_{1}\zeta + \cdots + {a}_{p - 1}{\zeta }^{p - 1} \) with \( {a}_{i} \in \mathbb{Z} \) and at least one \( {a}_{i} = 0 \) . If \( n \in \mathbb{Z} \) and \( n \) divides \( \alpha \) then \( n \) divides each \( {a}_{j} \) . Similarly, suppose all \( {a}_{i} \in {\mathbb{Z}}_{... | Proof. Since \( 1 + \zeta + \cdots + {\zeta }^{p - 1} = 0 \), we may use any subset of \( \left\{ {1,\zeta ,\ldots ,{\zeta }^{p - 1}}\right\} \) with \( p - 1 \) elements as a basis of the \( \mathbb{Z} \) -module \( \mathbb{Z}\left\lbrack \zeta \right\rbrack \) . Since at least one \( {a}_{i} = 0 \), the other \( {a}_... | No |
Lemma 2.2. Let \( k \) be a number field with \( {r}_{2} \) pairs of complex embeddings. Then \( d\left( k\right) = \) discriminant of \( k \) has sign \( {\left( -1\right) }^{{r}_{2}} \) . | Proof. Let \( \left\{ {{\alpha }_{1},\ldots ,{\alpha }_{m}}\right\} \) be a \( \mathbb{Z} \) -module basis for the ring of integers of \( k \) . Then\n\n\[ d\left( k\right) = {\left( \det {\left( {\alpha }_{i}^{\sigma }\right) }_{\sigma, i}\right) }^{2}, \]\n\nwhere \( \sigma \) runs through all embeddings of \( k \) i... | Yes |
Proposition 2.3. p ramifies in \( \mathbb{Q}\left( {\zeta }_{m}\right) \Leftrightarrow p \) divides \( m \) . | Proof. If \( p \) divides \( m \) then \( \mathbb{Q}\left( {\zeta }_{p}\right) \subseteq \mathbb{Q}\left( {\zeta }_{m}\right) \). Since \( p \) ramifies in \( \mathbb{Q}\left( {\zeta }_{p}\right) \), it ramifies in \( \mathbb{Q}\left( {\zeta }_{m}\right) \). Conversely, suppose \( p \) does not divide \( m = \prod {p}_... | Yes |
Proposition 2.4. If \( \left( {m, n}\right) = 1 \) then \( \mathbb{Q}\left( {\zeta }_{n}\right) \cap \mathbb{Q}\left( {\zeta }_{m}\right) = \mathbb{Q} \) . | Proof. Let \( K = \mathbb{Q}\left( {\zeta }_{m}\right) \cap \mathbb{Q}\left( {\zeta }_{n}\right) \) . If \( K \neq \mathbb{Q} \) then there is some prime, call it \( p \) , which ramifies in \( K \) (this follows from the fact that \( \left| {d\left( K\right) }\right| > 1 \) . See Lemma 14.3). By the previous propositi... | Yes |
Theorem 2.5. \( \deg \left( {\mathbb{Q}\left( {\zeta }_{n}\right) /\mathbb{Q}}\right) = \phi \left( n\right) \) and \( \operatorname{Gal}\left( {\mathbb{Q}\left( {\zeta }_{n}\right) /\mathbb{Q}}\right) \simeq {\left( \mathbb{Z}/n\mathbb{Z}\right) }^{ \times } \), with \( a{\;\operatorname{mod}\;n} \) corresponding to t... | Proof. Since \( \mathbb{Q}\left( {\zeta }_{m}\right) \) is normal over \( \mathbb{Q} \), Proposition 2.4 implies that if \( \left( {m, n}\right) = 1 \) then \( \deg \left( {\mathbb{Q}\left( {\zeta }_{mn}\right) /\mathbb{Q}}\right) = \deg \left( {\mathbb{Q}\left( {\zeta }_{m}\right) /\mathbb{Q}}\right) \cdot \deg \left(... | No |
Theorem 2.6. \( \mathbb{Z}\left\lbrack {\zeta }_{n}\right\rbrack \) is the ring of algebraic integers of \( \mathbb{Q}\left( {\zeta }_{n}\right) \) . | Proof. We need the following result (for a proof see Lang [1], p. 68):\n\nSuppose \( K \) and \( E \) are two number fields which are linearly disjoint \( \left( { \Leftrightarrow \deg \left( {{KE}/\mathbb{Q}}\right) = \deg \left( {K/\mathbb{Q}}\right) \cdot \deg \left( {E/\mathbb{Q}}\right) }\right) \) and whose discr... | No |
Proposition 2.8. Suppose \( n \) has at least two distinct prime factors. Then \( 1 - {\zeta }_{n} \) is a unit of \( \mathbb{Z}\left\lbrack {\zeta }_{n}\right\rbrack \) and \( \mathop{\prod }\limits_{\substack{{0 < j < n} \\ {\left( {j, n}\right) = 1} }}\left( {1 - {\zeta }_{n}^{j}}\right) = 1 \) . | Proof. Since \( {X}^{n - 1} + {X}^{n - 2} + \cdots + X + 1 = \mathop{\prod }\limits_{{j = 1}}^{{n - 1}}\left( {X - {\zeta }_{n}^{j}}\right) \), we may let \( X = 1 \) to obtain \( n = \mathop{\prod }\limits_{{j = 1}}^{{n - 1}}\left( {1 - {\zeta }_{n}^{j}}\right) \) . If \( {p}^{a} \) is the exact power of \( p \) divid... | Yes |
Lemma 2.9. Suppose \( p \nmid n \) and \( a \in \mathbb{Z} \) . Then \( p \mid {\Phi }_{n}\left( a\right) \Leftrightarrow \) the multiplicative order of \( a{\;\operatorname{mod}\;p} \) is \( n \) (i.e., \( {a}^{n} \equiv 1\left( {\;\operatorname{mod}\;p}\right) \) and \( n \) is minimal). | Proof. Suppose \( p \mid {\Phi }_{n}\left( a\right) \) . Since \( {X}^{n} - 1 = \mathop{\prod }\limits_{{d \mid n}}{\Phi }_{d}\left( X\right) \), we have \( {a}^{n} \equiv 1\left( {\;\operatorname{mod}\;p}\right) \) . Let \( k \) be the order of \( a\left( {\;\operatorname{mod}\;p}\right) \) . Then \( k \mid n \) . Sup... | Yes |
Corollary 2.11. For any \( n \geq 1 \) there are infinitely many primes \( p \equiv 1\left( {\;\operatorname{mod}\;n}\right) \) . | Proof. Suppose there are only finitely many, say \( {p}_{1},\ldots ,{p}_{r} \) . Let \( M = n{p}_{1}\cdots {p}_{r} \) and let \( N \in \mathbb{Z} \) . Then \( {\Phi }_{n}\left( {NM}\right) \equiv {\Phi }_{n}\left( 0\right) \equiv \pm 1\left( {\;\operatorname{mod}\;M}\right) \), therefore \( {\;\operatorname{mod}\;{p}_{... | Yes |
Lemma 2.12. Suppose \( p \nmid n \) and let \( \mathcal{P} \) be a prime of \( \mathbb{Q}\left( {\zeta }_{n}\right) \) lying above \( p \) . Then the \( n \) th roots of unity are distinct \( {\;\operatorname{mod}\;\mathcal{P}} \) . | Proof. The result follows immediately from the equation\n\n\[ n = \mathop{\prod }\limits_{{j = 1}}^{{n - 1}}\left( {1 - {\zeta }_{n}^{j}}\right) \] | No |
Proposition 2.15. (a) If \( n = {p}^{m} \) then \( \mathbb{Q}\left( {\zeta }_{n}\right) /\mathbb{Q}{\left( {\zeta }_{n}\right) }^{ + } \) is ramified at the prime above \( p \) and at the archimedean primes, and unramified at the other primes. | Proof. In both cases, the archimedean primes ramify since \( \mathbb{Q}\left( {{\zeta }_{n} + {\zeta }_{n}^{-1}}\right) \) is totally real and \( \mathbb{Q}\left( {\zeta }_{n}\right) \) is totally complex. Part (a) is true since \( p \) is totally ramified in \( \mathbb{Q}\left( {\zeta }_{{p}^{m}}\right) \) and is the ... | Yes |
Proposition 2.16. \( \mathbb{Z}\left\lbrack {{\zeta }_{n} + {\zeta }_{n}^{-1}}\right\rbrack \) is the ring of integers of \( \mathbb{Q}\left( {{\zeta }_{n} + {\zeta }_{n}^{-1}}\right) \) . | Proof. Suppose \( \alpha = {a}_{0} + {a}_{1}\left( {{\zeta }_{n} + {\zeta }_{n}^{-1}}\right) + \cdots + {a}_{N}{\left( {\zeta }_{n} + {\zeta }_{n}^{-1}\right) }^{N} \) is an algebraic integer, with \( N \leq \frac{1}{2}\phi \left( n\right) - 1 \) and with \( {a}_{i} \in \mathbb{Q} \) . By removing those terms with \( {... | Yes |
Lemma 3.1. If \( G \) is a finite abelian group, then \( G \simeq \widehat{G} \) (noncanonically). | Proof. \( G \) may be written a direct sum of groups of the form \( \mathbb{Z}/m\mathbb{Z} \) . Therefore \( \widehat{G} \) is the product of groups of the form \( {\left( \mathbb{Z}/m\mathbb{Z}\right) }^{ \land } \) . But if \( \chi \in {\left( \mathbb{Z}/m\mathbb{Z}\right) }^{ \land } \), then \( \chi \left( 1\right)... | No |
Corollary 3.2. \( \widehat{\widehat{G}} \simeq G \) (canonically). | Proof. Let \( g \in G \) . Then \( g : \widehat{G} \rightarrow {\mathbb{C}}^{ \times } \) by \( g : \chi \rightarrow \chi \left( g\right) \) . Suppose \( \chi \left( g\right) = 1 \) for all \( \chi \in \widehat{G} \) . Let \( H \) be the subgroup of \( G \) generated by \( g \) . Then \( \widehat{G} \) acts as a set of... | Yes |
Proposition 3.4. \( {\left( {H}^{ \bot }\right) }^{ \bot } = H \) (we equate \( \widehat{\widehat{G}} = G \) ). | Proof. As in the preceding proof, a straightforward calculation shows both groups have the same order. If \( h \in H \) then \( h : \chi \rightarrow \chi \left( h\right) \) maps \( {H}^{ \bot } \rightarrow 1 \) . Therefore \( H \subseteq {H}^{ \bot \bot } \) . Therefore they are equal. | No |
Theorem 3.5. Let \( X \) be a group of Dirichlet characters and \( K \) the associated field. Let \( p \) be a prime number with ramification index \( e \) in \( K \). Then \( e = \# \left( {X}_{p}\right) \). | Proof. Let \( n \) be the least common multiple of the conductors of the characters of \( X \), so \( K \subseteq \mathbb{Q}\left( {\zeta }_{n}\right) \). Let \( n = {p}^{a} \cdot m \) with \( p \nmid m \). Form the field \( L = \) \( K\left( {\zeta }_{m}\right) = K \cdot \mathbb{Q}\left( {\zeta }_{m}\right) \). (See d... | Yes |
Corollary 3.6. Let \( \chi \) be a Dirichlet character and \( K \) the associated field. Then \( p \) ramifies in \( K \Leftrightarrow \chi \left( p\right) = 0 \) (equivalently \( p \mid f \) ). | Proof. \( p \) ramifies in \( L/\mathbb{Q} \Leftrightarrow {X}_{p} \neq 1 \Leftrightarrow \exists \chi \in X \) with \( {\chi }_{p} \neq 1 \Leftrightarrow \exists \chi \in X \) with \( p \mid {f}_{\chi } \Leftrightarrow \exists \chi \in X \) with \( \chi \left( p\right) = 0 \) . | Yes |
Proposition 3.8. Let \( G \) be any finite abelian group. Then there exist fields \( L \) and K such that\n\n(a) \( \operatorname{Gal}\left( {L/K}\right) \simeq G \), and\n\n(b) \( L/K \) is unramified at all primes (including the archimedean primes).\n\nWe may also make \( L/\mathbb{Q} \) abelian and \( K/\mathbb{Q} \... | Proof. By the structure theorem for finite abelian groups, we may write\n\n\[ G \simeq \mathbb{Z}/{n}_{1}\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/{n}_{r}\mathbb{Z} \]\n\nfor some integers \( {n}_{1},\ldots ,{n}_{r} \) . Let \( {p}_{1},\ldots ,{p}_{r} \) be distinct primes satisfying \( {p}_{i} \equiv 1 \) \( \left( {... | Yes |
Corollary 3.9. Given any finite abelian group \( G \), there exists a cyclic extension \( K \) of \( \mathbb{Q} \) such that the ideal class group of \( K \) contains a subgroup isomorphic to \( G \) . | Proof. We shall use one of the main results of class field theory:\n\nLet \( K \) be a number field and let \( H \) be the maximal unramified (at all primes, finite and infinite) abelian extension of \( K \) . Then \( \operatorname{Gal}\left( {H/K}\right) \) is isomorphic to the ideal class group of \( K \) (the field ... | Yes |
Lemma 3.10. If \( A \) is a finite abelian group and \( B \) is a subgroup, then \( A \) contains a subgroup isomorphic to \( A/B \) . | Proof. This result could be proved via the structure theory of finite abelian groups. Another proof is the following:\n\n\[ A/B \simeq {\left( A/B\right) }^{ \land } \simeq {B}^{ \bot } \subseteq \widehat{A} \simeq A. \] | No |
Theorem 3.11 (Conductor-Discriminant Formula). Let \( K \) be the number field associated to the group \( X \) of Dirichlet characters. Then the discriminant of \( K \) is given by\n\n\[ d\left( K\right) = {\left( -1\right) }^{{r}_{2}}\mathop{\prod }\limits_{{\chi \in X}}{f}_{\chi } \] | This theorem can be very useful for computing discriminants of abelian number fields. For example, consider the real subfield of \( \mathbb{Q}\left( {\zeta }_{p}\right) \) . The group of characters consists of the trivial character of conductor 1 and \( \left( {p - 3}\right) /2 \) other characters, all of conductor \( ... | Yes |
Proposition 4.1. Let \( F \) be any multiple of \( f \) . Then\n\n\[ \n{B}_{n,\chi } = {F}^{n - 1}\mathop{\sum }\limits_{{a = 1}}^{F}\chi \left( a\right) {B}_{n}\left( \frac{a}{F}\right) .\n\] | Proof.\n\n\[ \n\mathop{\sum }\limits_{{n = 0}}^{\infty }{F}^{n - 1}\mathop{\sum }\limits_{{a = 1}}^{F}\chi \left( a\right) {B}_{n}\left( \frac{a}{F}\right) \frac{{t}^{n}}{n!} = \mathop{\sum }\limits_{{a = 1}}^{F}\chi \left( a\right) \frac{t{e}^{\left( {a/F}\right) {Ft}}}{{e}^{Ft} - 1}.\n\]\n\nLet \( g = F/f \) and \( a... | Yes |
Theorem 4.3. Let \( X \) be a group of Dirichlet characters, \( K \) the associated field, and \( {\zeta }_{K}\left( s\right) \) the Dedekind zeta function of \( K \). Then \[ {\zeta }_{K}\left( s\right) = \mathop{\prod }\limits_{{\chi \in X}}L\left( {s,\chi }\right) \] | Proof. It suffices to consider the Euler factors corresponding to each prime p. Suppose \[ \left( p\right) = {\left( {\mathcal{P}}_{1}\ldots {\mathcal{P}}_{g}\right) }^{e} \] is the prime factorization of \( p \) in \( K \), and each \( \mathcal{P} \) has residue class degree \( f \), \( N\mathcal{P} = {p}^{f} \). Then... | Yes |
Corollary 4.4. \( L\left( {1,\chi }\right) \neq 0 \) . | Proof. Let \( K \) be the field belonging to \( \chi \) . It is well known that the zeta function of \( K \) has a (simple) pole at \( s = 1 \) . Let \( b \) be the order of \( \chi \) . Then\n\n\[ \n{\zeta }_{K}\left( s\right) = \mathop{\prod }\limits_{{a = 0}}^{{b - 1}}L\left( {s,{\chi }^{a}}\right) = \zeta \left( s\... | Yes |
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