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Corollary 7.14 (Dimension invariance) Let \( U \subseteq {\mathbf{R}}^{n} \) and \( V \subseteq {\mathbf{R}}^{m} \) be non-empty open sets. If \( U \) and \( V \) are homeomorphic then \( n = m \) . | Proof. Assume that \( m < n \) . From Corollary 7.13 applied to \( V \), considered as a subset of \( {\mathbf{R}}^{n} \) via the inclusion \( {\mathbf{R}}^{m} \subseteq {\mathbf{R}}^{n} \), it follows that \( V \) is open in \( {\mathbf{R}}^{n} \) . This contradicts that \( V \) is contained in a proper subspace. | Yes |
A knot in \( {\mathbf{R}}^{3} \) is a subset \( \sum \subseteq {\mathbf{R}}^{3} \) that is homeomorphic to \( {S}^{1} \). The corresponding knot-complement is the open set \( U = {\mathbf{R}}^{3} - \sum \). We show:\n\n\[ \n{H}^{p}\left( U\right) \cong \left\{ \begin{array}{ll} \mathbf{R} & \text{ if }0 \leq p \leq 2 \... | According to Theorem 7.8, it is sufficient to show this for the \ | No |
Proposition 7.16 Let \( \sum \subseteq {\mathbf{R}}^{n}\left( {n \geq 2}\right) \) be homeomorphic to \( {S}^{n - 1} \) and let \( {U}_{1} \) and \( {U}_{2} \) be the interior and exterior domains of \( \sum \) . Then\n\n\[ \n{H}^{p}\left( {U}_{1}\right) \cong \left\{ {\begin{array}{ll} \mathbf{R} & \text{ if }p = 0 \\... | Proof. The case \( p = 0 \) follows from Theorem 7.10. Set \( W = {\mathbf{R}}^{n} - {D}^{n} \) . For \( p > 0 \) there are isomorphisms\n\n\[ \n{H}^{p}\left( {U}_{1}\right) \oplus {H}^{p}\left( {U}_{2}\right) \cong {H}^{p}\left( {{\mathbf{R}}^{n} - \sum }\right) \cong {H}^{p}\left( {{\mathbf{R}}^{n} - {S}^{n - 1}}\rig... | Yes |
The \( n \)-dimensional sphere \( {S}^{n} = \left\{ {x \in {\mathbf{R}}^{n + 1} \mid \parallel x\parallel = 1}\right\} \) is an \( n \)-dimensional smooth manifold. | We define an atlas with \( 2\left( {n + 1}\right) \) charts \( \left( {{U}_{\pm i},{h}_{\pm i}}\right) \) where\n\n\[ \n{U}_{+i} = \left\{ {x \in {S}^{n} \mid {x}_{1} > 0}\right\} ,\;{U}_{-i} = \left\{ {x \in {S}^{n} \mid {x}_{i} < 0}\right\} \]\n\nand \( {h}_{\pm 1} : {U}_{\pm 1} \rightarrow {\dot{D}}^{n} \) is the ma... | No |
The \( n \) -sphere \( {S}^{n} \) is a smooth submanifold of \( {\mathbf{R}}^{n + 1} \). | In fact the charts \( \left( {{U}_{\pm i},{h}_{\pm i}}\right) \) from Example 8.5 can easily be extended to diffeomorphisms satisfying (1). | No |
Lemma 8.12 Let \( {M}^{n} \) be an \( n \) -dimensional smooth manifold. For \( p \in M \) there exist smooth maps\n\n\[ \n{\phi }_{p} : M \rightarrow \mathbf{R},\;{f}_{p} \cdot M \rightarrow {\mathbf{R}}^{n} \n\]\n\nsuch that \( {\phi }_{p}\left( p\right) > 0 \), and \( {f}_{p} \) maps the open set \( M - {\phi }_{p}^... | Proof. Choose a chart \( h : V \rightarrow {V}^{\prime } \) with \( p \in V \) . By Lemma A. 7 we can find a function \( \psi \in {C}^{\infty }\left( {{\mathbf{R}}^{n},\mathbf{R}}\right) \) with compact support \( {\operatorname{supp}}_{{\mathbf{R}}^{n}}\left( \psi \right) \subseteq {V}^{\prime } \), such that \( \psi ... | Yes |
Proposition 8.15 If \( g : N \rightarrow M \) is a continuous map between smooth manifolds \( N \) and \( M \), then \( g \) is smooth if and only if the homomorphism\n\n\[ \n{g}^{ * } : {C}^{0}\left( {M,\mathbf{R}}\right) \rightarrow {C}^{0}\left( {N,\mathbf{R}}\right)\n\]\n\ngiven by \( {g}^{ * }\left( \psi \right) =... | Proof. \ | No |
Lemma 9.3 Let \( f : {M}^{m} \rightarrow {N}^{n} \) be a smooth map and \( p \in M \) . (i) There is a linear map \( {D}_{p}f : {T}_{p}M \rightarrow {T}_{f\left( p\right) }N \) given in terms of representing curves by \[ {D}_{p}f\left( \left\lbrack \alpha \right\rbrack \right) = \left\lbrack {f \circ \alpha }\right\rbr... | Proof. Remark 9.1.(ii) applied to \( F = g \circ f \circ {h}^{-1} \), defined on the open set \( h\left( {U \cap {f}^{-1}\left( V\right) }\right) \), shows that the bottom map in the diagram is linear and given on representing curves as stated there. Since \( {\Phi }_{h} \) and \( {\Phi }_{g} \) are linear isomorphisms... | Yes |
Lemma 9.6 Let \( {g}_{i} : {W}_{i} \rightarrow N \) be a family of local parametrizations with \( N = \) \( \bigcup {g}_{i}\left( {W}_{i}\right) \) . If \( {g}_{i}^{ * }\left( \omega \right) \) is smooth for all \( i \), then \( \omega \) is smooth. | Proof. Let \( g : W \rightarrow N \) be any local parametrization and \( z \in W \) . We show that \( {g}^{ * }\left( \omega \right) \) is smooth close to \( z \) . Choose an index \( i \) with \( g\left( z\right) \in {g}_{i}\left( {W}_{i}\right) \) . Close to \( z \)\n\nwe can write \( g = {g}_{i} \circ {g}_{i}^{-1} \... | Yes |
Lemma 9.10 Let \( \mathcal{V} = {\left( {V}_{i}\right) }_{i \in I} \) be an open cover of the smooth submanifold \( {M}^{n} \) . Suppose that all \( {V}_{\mathrm{i}} \) have orientations and that the restrictions of the orientations from \( {V}_{i} \) and \( {V}_{j} \) to \( {V}_{i} \cap {V}_{j} \) coincide for all \( ... | The proof is a typical application of a smooth partition of unity in the following form: | No |
Theorem 9.11 Let \( \mathcal{V} = {\left( {V}_{i}\right) }_{i \in I} \) be an open cover of the smooth manifold \( {M}^{n} \subseteq \)\n\n\( {\mathbf{R}}^{l} \) . Then there exist smooth functions \( {\phi }_{\imath } : M \rightarrow \left\lbrack {0,1}\right\rbrack \left( {\imath \in I}\right) \) that satisfy\n\n(i) \... | Proof. Since \( M \) has the topology induced by \( {\mathbf{R}}^{l} \), we can choose an open set \( {U}_{i} \subseteq {\mathbf{R}}^{l} \) with \( {U}_{i} \cap M = {V}_{i} \) for each \( \imath \in I \) . By applying Theorem A. 1 to \( U = \mathop{\bigcup }\limits_{{i \in I}}{U}_{i} \) we get smooth functions \( {\psi... | Yes |
Proposition 9.14 If \( \left\{ {{h}_{i} : {U}_{i} \rightarrow {U}_{i}^{\prime } \mid i \in I}\right\} \) is a positive atlas on \( {M}^{n} \), then \( {M}^{n} \) has a uniquely determined orientation, so all \( {h}_{1} \) are oriented charts. | Proof. For \( i \in I \) we orient \( {U}_{i} \) so that \( {h}_{i} \) is an orientation-preserving diffeomorphism. By Example 9.13, the two orientations on \( {U}_{i} \cap {U}_{j} \) defined by the restriction from \( {U}_{i} \) and \( {U}_{j} \) coincide. The assertion follows from Lemma 9.10. | Yes |
Proposition 9.16 If \( {M}^{n} \) is an oriented Riemannian manifold then \( {M}^{n} \) has a uniquely determined orientation form \( {\operatorname{vol}}_{M} \) with\n\n\[ \n{\operatorname{vol}}_{M}\left( {{b}_{1},\ldots ,{b}_{n}}\right) = 1 \n\]\n\nfor every positively oriented orthonormal basis of a tangent space \(... | Proof. Let the orientation be given by the orientation form \( \omega \in {\Omega }^{n}\left( {M}^{n}\right) \) . Consider two positively oriented orthonormal bases \( {b}_{1},\ldots ,{b}_{n} \) and \( {b}_{1}^{\prime },\ldots ,{b}_{n}^{\prime } \) in the same tangent space \( {T}_{p}M \) . There exists an orthogonal \... | Yes |
Define an \( \left( {n - 1}\right) \) -form \( {\omega }_{0} \in {\Omega }^{n - 1}\left( {\mathbf{R}}^{n}\right) \) by\n\n\[ \n{\omega }_{0x}\left( {{w}_{1},\ldots ,{w}_{n - 1}}\right) = \det \left( {x,{w}_{1},\ldots ,{w}_{n - 1}}\right) \in {\operatorname{Alt}}^{n - 1}\left( {\mathbf{R}}^{n}\right) ,\n\]\n\nfor \( x \... | Since \( {\omega }_{0x}\left( {{e}_{1},\ldots ,{\bar{e}}_{i},\ldots ,{e}_{n}}\right) = {\left( -1\right) }^{i - 1}{x}_{1} \), we have\n\n\[ \n{\omega }_{0} = \mathop{\sum }\limits_{{i = 1}}^{n}{\left( -1\right) }^{i - 1}{x}_{i}d{x}_{1} \land \ldots \land \widehat{d{x}_{i}} \land \ldots \land d{x}_{n}\n\] | Yes |
For the antipodal map\n\n\\[ A : {S}^{n - 1} \\rightarrow {S}^{n - 1};\\;{Ax} = - x \\]\n\nwe have\n\n\\[ {A}^{ * }\\left( {\\operatorname{vol}}_{{S}^{n - 1}}\\right) = {\\left( -1\\right) }^{n}{\\operatorname{vol}}_{{S}^{n - 1}} \\]\n\nand \\( A \\) is orientation-preserving if and only if \\( n \\) is even. In this c... | For \\( x \\in {S}^{n - 1} \\), \n\n\\[ {T}_{x}{S}^{n - 1}\\overset{{D}_{x}A}{ \\rightarrow }{T}_{Ax}{S}^{n - 1} \\]\n\nis a linear isometry. Hence there exists a Riemannian structure on \\( {\\mathbf{{RP}}}^{n - 1} \\) characterized by the requirement that the isomorphism\n\n\\[ {T}_{x}{S}^{n - 1}\\overset{{D}_{x}\\pi... | Yes |
Lemma 9.21 For every \( {p}_{0} \in {M}^{n} \subseteq {\mathbf{R}}^{n + k} \) there exists an open neighborhood \( W \) of \( {p}_{0} \) on \( M \) and smooth normal vector fields \( {Y}_{j}\left( {1 \leq j \leq k}\right) \) on \( W \) such that \( {Y}_{1}\left( p\right) ,\ldots ,{Y}_{k}\left( p\right) \) form an ortho... | Proof. On a coordinate patch around \( {p}_{0} \in M \), there exist smooth tangent vector fields \( {X}_{1},\ldots ,{X}_{n} \), which at every point \( p \) yield a basis of \( {T}_{p}M \), cf. Remark 9.4. Choose a basis \( {V}_{1},\ldots ,{V}_{k} \) of \( {T}_{{p}_{0}}{M}^{ \bot } \) . Since the \( \left( {n + k}\rig... | Yes |
Proposition 9.22 Let \( {M}^{n} \subseteq {\mathbf{R}}^{n + 1} \) be a smooth submanifold of codimension 1 .\n\n(i) There is a 1-1 correspondence between smooth normal vector fields \( Y \) on \( M \) and \( n \) -forms in \( {\Omega }^{n}\left( M\right) \) . It associates to \( Y \) the \( n \) -form \( \omega = {\ome... | Proof. If \( p \in M \) then \( Y\left( p\right) = 0 \) if and only if \( {\omega }_{p} = 0 \) . Since \( {\omega }_{Y} \) depends linearly on \( Y \), the map \( Y \rightarrow {\omega }_{Y} \) must be injective. If \( Y \) is a Gauss map, then \( {\omega }_{Y} \) is an orientation form and it can be seen that \( {\ome... | Yes |
Proposition 9.25 For any compact differentiable manifold \( {M}^{n} \) all cohomology spaces \( {H}^{d}\left( M\right) \) are finite-dimensional. | Proof. We may assume that \( {M}^{n} \) is a smooth submanifold of \( {\mathbf{R}}^{n + k} \) by Theorem 8.11, and that \( \left( {V, i, r}\right) \) is a tubular neighborhood. Since \( M \) is compact we can find finitely many open balls \( {U}_{1},\ldots ,{U}_{r} \) in \( {\mathbf{R}}^{n + k} \) such that their union... | Yes |
Proposition 9.26 Let \( {M}_{1} \) and \( {M}_{2} \) be smooth submanifolds of Euclidean spaces.\n\n(i) If \( {f}_{0},{f}_{1} : {M}_{1} \rightarrow {M}_{2} \) are two homotopic smooth maps, then\n\n\[ \n{H}^{d}\left( {f}_{0}\right) = {H}^{d}\left( {f}_{1}\right) : {H}^{d}\left( {M}_{2}\right) \rightarrow {H}^{d}\left( ... | Proof. Choose tubular neighborhoods \( \left( {{V}_{\nu },{i}_{\nu },{r}_{\nu }}\right) \) of \( {M}_{\nu },\nu = 1,2 \) . Lemma 6.3 implies that \( {i}_{2} \circ {f}_{0} \circ {r}_{1} \simeq {i}_{2} \circ {f}_{1} \circ {r}_{1} \) . Hence \( {H}^{d}\left( {{i}_{2} \circ {f}_{0} \circ {r}_{1}}\right) = {H}^{d}\left( {{i... | Yes |
Corollary 9.28 If \( {M}^{n} \subseteq {\mathbf{R}}^{n + k} \) is a smooth submanifold and \( \left( {V, i, r}\right) \) an open tubular neighborhood, then \( {H}^{d}\left( i\right) : {H}^{d}\left( V\right) \rightarrow {H}^{d}\left( M\right) \) is an isomorphism with \( {H}^{d}\left( r\right) \) as its inverse. | Proof. We have \( r \circ i = {\operatorname{id}}_{M} \) and \( i \circ r \simeq {\operatorname{id}}_{V} \), as \( V \) contains the line segment between \( x \) and \( r\left( x\right) \) for all \( x \in V \) . By Proposition 9.26.(i) we can conclude that \( {H}^{d}\left( r\right) \) and \( {H}^{d}\left( \iota \right... | Yes |
Example 9.29 For \( n \geq 1 \) we have\n\n\[ \n{H}^{d}\left( {S}^{n}\right) \cong \left\{ \begin{array}{ll} \mathbf{R} & \text{ if }d = 0, n \\ 0 & \text{ otherwise. } \end{array}\right.\n\] | Let \( i : {S}^{n} \rightarrow {\mathbf{R}}^{n + 1} - \{ 0\} \) be the inclusion and define \( r : {\mathbf{R}}^{n + 1} - \{ 0\} \rightarrow {S}^{n} \) by \( r\left( x\right) = \frac{z}{\parallel z\parallel } \) Then \( r \circ i = {\mathrm{{id}}}_{{S}^{ * }},\;i \circ r \simeq {\mathrm{{id}}}_{{\mathbf{R}}^{* + 1}-\{ ... | No |
Lemma 10.1 Let \( \phi : V \rightarrow W \) be a diffeomorphism between open subsets \( V \) and \( W \) of \( {\mathbf{R}}^{n} \), and assume that the Jacobi determinant \( \det \left( {{D}_{x}\phi }\right) \) is of constant sign \( \delta = \pm 1 \) for \( x \in V \) . For \( \omega \in {\Omega }_{c}^{n}\left( W\righ... | Proof. If \( \omega \) is written in the form\n\n\[ \n\omega = f\left( x\right) d{x}_{1} \land \ldots \land d{x}_{n} \n\]\n\nwith \( f \in {C}_{c}^{\infty }\left( {W,\mathbf{R}}\right) \), it follows from Example 3.13.(ii) that\n\n\[ \n{\phi }^{ * }\left( \omega \right) = f\left( {\phi \left( x\right) }\right) \det \le... | Yes |
Proposition 10.2 For an arbitrary oriented \( n \) -dimensional smooth manifold \( {M}^{n} \) there exists a unique linear map\n\n\[ \n{\int }_{M} : {\Omega }_{c}^{n}\left( {M}^{n}\right) \rightarrow \mathbf{R} \n\]\n\nwith the following property: If \( \omega \in {\Omega }_{c}^{n}\left( {M}^{n}\right) \) has support c... | Proof. First consider \( \omega \in {\Omega }_{c}^{n}\left( {M}^{n}\right) \) with \ | No |
Lemma 10.6 Let \( N \subseteq {M}^{n} \) be a domain with smooth boundary. Then \( \partial N \) is an \( \left( {n - 1}\right) \) -dimensional smooth submanifold of \( {M}^{n} \). Suppose \( {M}^{n}\left( {n \geq 2}\right) \) is oriented. There is an induced orientation of \( \partial N \) with the following property:... | Proof. Every smooth chart \( \left( {U, h}\right) \) in \( M \) that satisfies \( h\left( {U \cap N}\right) = h\left( U\right) \cap {\mathbf{R}}_{ - }^{n} \) can be restricted to a chart \( \left( {U \cap \partial N,{h}_{ \mid }}\right) \) on \( \partial N \) :\n\n\[ {h}_{ \mid } : U \cap \partial N \rightarrow h\left(... | Yes |
The volume of \( {S}^{n - 1} \) can be calculated by applying Stokes’ theorem to \( {D}^{n} \) with the standard orientation of \( {\mathbf{R}}^{n} \) and the \( \left( {n - 1}\right) \) -form on \( {\mathbf{R}}^{n} \) given by\n\n\[ \n{\omega }_{0} = \mathop{\sum }\limits_{{i = 1}}^{n}{\left( -1\right) }^{i - 1}{x}_{i... | Since \( {\omega }_{0 \mid {S}^{n - 1}} = {\operatorname{vol}}_{{S}^{n - 1}} \) and \( d{\omega }_{0} = {nd}{x}_{1} \land \ldots \land d{x}_{n} \) we have that\n\n\[ \n\operatorname{Vol}\left( {S}^{n - 1}\right) = {\int }_{{S}^{n - 1}}{\omega }_{0} = {\int }_{{D}^{n}}d{\omega }_{0} = n\operatorname{Vol}\left( {D}^{n}\r... | Yes |
Corollary 10.14 For a connected compact smooth manifold \( {M}^{n} \), integration over \( M \) induces an isomorphism\n\n\[ {\int }_{M} : {H}^{n}\left( {M}^{n}\right) \overset{ \cong }{ \rightarrow }\mathbf{R} \] | In (8) it is obvious that the integral is non-zero and hence surjective. It follows from Corollary 10.9 that the image of \( d \) is contained in the kernel of the integral. We show the converse inclusion. | No |
Lemma 10.16 Let \( {\left( {U}_{\alpha }\right) }_{\alpha \in A} \) be an open cover of the connected manifold \( M \), and let \( p, q \in M \) . There exist indices \( {\alpha }_{1},\ldots ,{\alpha }_{k} \) such that\n\n(i) \( p \in {U}_{{\alpha }_{1}} \) and \( q \in {U}_{{\alpha }_{k}} \)\n\n(ii) \( {U}_{{\alpha }_... | Proof. For a fixed \( p \) we define \( V \) to be the set of \( q \in M \), for which there exists a finite sequence of indices \( {\alpha }_{1},\ldots ,{\alpha }_{k} \) from \( A \), such that (i) and (ii) are satisfied. It is obvious that \( V \) is both open and closed in \( M \) and that \( V \) contains \( p \) .... | Yes |
Lemma 10.17 Let \( U \subseteq M \) be an open set diffeomorphic to \( {\mathbf{R}}^{n} \) and let \( W \subseteq U \) be non-empty and open. For every \( \omega \in {\Omega }_{c}^{n}\left( M\right) \) with \( \operatorname{supp}\omega \subseteq U \), there exists a \( \kappa \in {\Omega }_{c}^{n - 1}\left( M\right) \)... | Proof. It suffices to prove the lemma when \( M = U \), and by diffeomorphism invariance it is enough to consider the case where \( M = U = {\mathbf{R}}^{n} \) .\n\nChoose \( {\omega }_{1} \in {\Omega }_{c}^{n}\left( {\mathbf{R}}^{n}\right) \) with \( \operatorname{supp}{\omega }_{1} \subseteq W \) and \( {\int }_{{\ma... | Yes |
Lemma 10.18 Assume that \( {M}^{n} \) is connected and let \( W \subseteq M \) be non-empty and open. For every \( \omega \in {\Omega }_{c}^{n}\left( M\right) \) there exists \( {a\kappa } \in {\Omega }_{c}^{n - 1}\left( M\right) \) with \( \operatorname{supp}\left( {\omega - {d\kappa }}\right) \subseteq \) w. | Proof. Suppose that \( \operatorname{supp}\omega \subseteq {U}_{1} \) for some open set \( {U}_{1} \subseteq M \) diffeomorphic to \( {\mathbf{R}}^{n} \) . We apply Lemma 10.16 to find open sets \( {U}_{2},\ldots ,{U}_{k} \), diffeomorphic to \( {\mathbf{R}}^{n} \) , such that \( {U}_{i - 1} \cap {U}_{i} \neq \varnothi... | Yes |
Proposition 11.1 Let \( f : {N}^{n} \rightarrow {M}^{n} \) be a smooth map between compact \( n \) - dimensional oriented manifolds with \( M \) connected. There exists a unique \( \deg \left( f\right) \in \) \( \mathbf{R} \) such that (2) holds for all \( \omega \in {\Omega }^{n}\left( M\right) \) . We call \( \deg \l... | Proof. We write \( N \) as a disjoint union of its connected components \( {N}_{1},\ldots ,{N}_{k} \) and denote the restriction of \( f \) to \( {N}_{j} \) by \( {f}_{j} \) . We have already defined \( \deg \left( {f}_{j}\right) \) ; we set\n\n(3)\n\n\[ \deg \left( f\right) = \mathop{\sum }\limits_{{j = 1}}^{k}\deg \l... | Yes |
Corollary 11.2 \( \deg \left( f\right) \) depends only on the homotopy class of \( f : N \rightarrow M \) . | Proof. By (3) we can restrict ourselves to the case where \( N \) is connected. The assertion then follows from diagram (1), since \( {H}^{n}\left( f\right) \) depends only on the homotopy class of \( f \) . | Yes |
Corollary 11.3 Suppose \( {N}^{n}\overset{f}{ \rightarrow }{M}^{n}\overset{g}{ \rightarrow }{P}^{n} \) are smooth maps between \( n \) - dimensional compact oriented manifolds and that \( M \) and \( P \) are connected. Then \[ \deg \left( {gf}\right) = \deg \left( f\right) \deg \left( g\right) \] | Proof. For \( \omega \in {\Omega }^{n}\left( P\right) \) , \[ \deg \left( {gf}\right) {\int }_{P}\omega = {\int }_{N}{\left( gf\right) }^{ * }\left( \omega \right) = {\int }_{N}{f}^{ * }\left( {{g}^{ * }\left( \omega \right) }\right) \] \[ = \deg \left( f\right) {\int }_{M}{g}^{ * }\left( \omega \right) = \deg \left( f... | Yes |
Lemma 11.8 Let \( p \in {M}^{n} \) be a regular value for the smooth map \( f : {N}^{n} \rightarrow \) \( {M}^{n} \), with \( {N}^{n} \) compact. Then \( {f}^{-1}\left( p\right) \) consists of finitely many points \( {q}_{1},\ldots ,{q}_{k} \) . Moreover, there exist disjoint open neighborhoods \( {V}_{i} \) of \( {q}_... | Proof. For each \( q \in {f}^{-1}\left( p\right) ,{D}_{q}f : {T}_{q}N \rightarrow {T}_{q}M \) is an isomorphism. From the inverse function theorem we know that \( f \) is a local diffeomorphism around \( q \) . In particular \( q \) is an isolated point in \( {f}^{-1}\left( p\right) \) . Compactness of \( N \) implies ... | Yes |
Theorem 11.9 In the situation above, and for every regular value \( p \) ,\n\n\[ \deg \left( f\right) = \mathop{\sum }\limits_{{q \in {f}^{-1}\left( p\right) }}\operatorname{Ind}\left( {f, q}\right) \]\n\nIn particular \( \deg \left( f\right) \) is an integer. | Proof. Let \( {q}_{i},{V}_{i} \), and \( U \) be as in Lemma 11.8. We may assume that \( U \) and hence \( {V}_{\mathrm{i}} \) connected. The diffeomorphism \( {f}_{\mid {V}_{\mathrm{i}}} : {V}_{\mathrm{i}} \rightarrow U \) is positively or negatively oriented, depending on whether \( \operatorname{Ind}\left( {f;{q}_{i... | Yes |
Proposition 11.11 Let \( F : {P}^{n + 1} \rightarrow {M}^{n} \) be a smooth map between oriented smooth manifolds, with \( {M}^{n} \) compact and connected. Let \( X \subseteq P \) be a compact domain with smooth boundary \( {N}^{n} = \partial X \), and suppose \( N \) is the disjoint union of submanifolds \( {N}_{1}^{... | Proof. Let \( f = {F}_{\mid N} \) so that\n\n\[ \deg \left( f\right) = \mathop{\sum }\limits_{{i = 1}}^{k}\deg \left( {f}_{i}\right) \]\n\nOn the other hand, if \( \omega \in {\Omega }^{n}\left( M\right) \) has \( {\int }_{M}\omega = 1 \), then\n\n\[ \deg \left( f\right) = {\int }_{N}{f}^{ * }\left( \omega \right) = {\... | Yes |
Theorem 11.14 With the notation above we have:\n\n(i) (Gauss)\n\n\\[ \n\\operatorname{lk}\\left( {J, K}\\right) = \\frac{1}{4\\pi }{\\int }_{0}^{a}{\\int }_{0}^{b}\\frac{\\det \\left( {\\alpha \\left( u\\right) - \\beta \\left( v\\right) ,{\\alpha }^{\\prime }\\left( u\\right) ,{\\beta }^{\\prime }\\left( v\\right) }\\... | Proof. We apply formula (2) to the map \\( \\Psi = {\\Psi }_{J, K} \\) and the volume form \\( \\omega = {\\operatorname{vol}}_{{S}^{2}} \\) (with integral \\( {4\\pi } \\) ) to get\n\n(12)\n\n\\[ \n\\operatorname{lk}\\left( {J, K}\\right) = \\deg \\left( \\Psi \\right) = \\frac{1}{4\\pi }{\\int }_{J \\times K}{\\Psi }... | Yes |
Lemma 11.17 Suppose \( F \in {C}^{\infty }\left( {{\mathbf{R}}^{n},{\mathbf{R}}^{n}}\right) \) has the origin as its only zero. Then\n\n\[ F : {\mathbf{R}}^{n} - \{ 0\} \rightarrow {\mathbf{R}}^{n} - \{ 0\} \]\n\ninduces multiplication by \( \iota \left( {F;0}\right) \) on \( {H}^{n - 1}\left( {{\mathbf{R}}^{n}-\{ 0\} ... | Proof. Let \( i : {S}^{n - 1} \rightarrow {\mathbf{R}}^{n} - \{ 0\} \) be the inclusion map and \( r : {\mathbf{R}}^{n - 1} - \{ 0\} \rightarrow {S}^{n - 1} \) the retraction \( r\left( x\right) = x/\parallel x\parallel \) . We have \( \iota \left( {F;0}\right) = \deg {F}_{1} \), where \( {F}_{1} = r \circ F \circ i \)... | Yes |
Lemma 11.20 If \( {p}_{0} \) is a non-degenerate singularity, then\n\n\[ \iota \left( {X,{p}_{0}}\right) = \operatorname{sign}\left( {\det {D}_{0}F}\right) \in \{ \pm 1\} . \] | Proof. By shrinking \( U \) we may assume that \( h \) maps \( U \) diffeomorphically onto an open set \( {U}_{0} \subseteq {\mathbf{R}}^{n} \), which is star-shaped around 0, and that \( F \) is a diffeomorphism from \( {U}_{0} \) to an open set. As in the proof of Lemma 11.18 we can define a homotopy\n\n\[ G : {U}_{0... | Yes |
Theorem 11.22 Let \( F \in {C}^{\infty }\left( {U,{\mathbf{R}}^{n}}\right) \) be a vector field on an open set \( U \subseteq {\mathbf{R}}^{n} \) , with only isolated zeros. Let \( R \subseteq U \) be a compact domain with smooth boundary \( \partial R \), and assume that \( F\left( p\right) \neq 0 \) for \( p \in \par... | Proof. Let \( {p}_{1},\ldots ,{p}_{k} \) be the zeros in \( R \) for \( F \), and choose disjoint closed balls \( {D}_{j} \subseteq R - \partial R \), with centers \( {p}_{j} \) . Define\n\n\[ {f}_{\jmath } : \partial {D}_{j} \rightarrow {S}^{n - 1};\;{f}_{j}\left( x\right) = F\left( x\right) /\parallel F\left( x\right... | Yes |
Corollary 11.24 In the situation of Theorem 11.22, suppose for every \( p \in \partial R \) that the vector \( F\left( p\right) \) points outward. Let \( g : \partial R \rightarrow {S}^{n - 1} \) be the Gauss map which to \( p \in \partial R \) associates the outward pointing unit normal vector to \( \partial R \) . Th... | Proof. By Corollary 11.2 it sufficies to show that \( f \) and \( g \) are homotopic. Since \( f\left( p\right) \) and \( g\left( p\right) \) belong to the same open half-space of \( {\mathbf{R}}^{n} \), the desired homotopy can be defined by\n\n\[ \frac{\left( {1 - t}\right) f\left( p\right) + {tg}\left( p\right) }{\p... | Yes |
Lemma 11.25 Suppose \( F \in {C}^{\infty }\left( {{\mathbf{R}}^{n},{\mathbf{R}}^{n}}\right) \) has the origin as its only zero. Then there exists an \( \widetilde{F} \in {C}^{\infty }\left( {{\mathbf{R}}^{n},{\mathbf{R}}^{n}}\right) \), with only non-degenerate zeros, that coincides with \( F \) outside a compact set. | Proof. We choose a function \( \phi \in {C}^{\infty }\left( {{\mathbf{R}}^{n},\left\lbrack {0,1}\right\rbrack }\right) \) with\n\n\[ \phi \left( x\right) = \left\{ \begin{array}{ll} 1 & \text{ if }\parallel x\parallel \leq 1 \\ 0 & \text{ if }\parallel x\parallel \geq 2 \end{array}\right. \]\n\nWe want to define \( \wi... | Yes |
Corollary 11.26 Let \( X \) be a smooth vector field on the compact manifold \( {M}^{n} \) with isolated singularities. Then there exists a smooth vector field \( \widehat{X} \) on \( M \) having only non-degenerate zeros and with | Proof. We choose disjoint coordinate patches which are diffeomorphic to \( {\mathbf{R}}^{n} \) around the finitely many zeros of \( X \), and apply Lemma 11.25 on the interior of each of them to obtain \( \widetilde{X} \) . The formula then follows from (14). | No |
Theorem 11.27 Let \( {M}^{n} \subseteq {\mathbf{R}}^{n + k} \) be a compact smooth submanifold and let \( {N}_{e} \) be a tubular neighborhood of radius \( \epsilon > 0 \) around \( M \) . Denote by \( g : \partial {N}_{\epsilon } \rightarrow {S}^{n + k - 1} \) the outward pointing Gauss map. If \( X \) is an arbitrary... | Proof. By Corollary 11.26 one may assume that \( X \) only has non-degenerate zeros. From the construction of the tubular neighborhood we have a smooth projection \( \pi : N \rightarrow M \) from an open tubular neighborhood \( N \) with \( {N}_{\epsilon } \subseteq N \subseteq {\mathbf{R}}^{n + k} \), and can define a... | Yes |
Theorem 12.1 (Poincaré-Hopf) Let \( X \) be a smooth vector field on a compact manifold \( M \) . If \( X \) has only isolated zeros then\n\n\[ \operatorname{Index}\left( X\right) = \chi \left( M\right) \] | By the final result of Chapter 11 it is sufficient to show the formula for just one such vector field \( X \) on \( M \) . We shall do so by making use of a Morse function on \( {M}^{n} \) . | No |
Proposition 12.2 Suppose that \( p \in M \) is a critical point for \( f \in {C}^{\infty }\left( {M,\mathbf{R}}\right) \) .\n\n(i) There exists a quadratic form \( {d}_{p}^{2}f \) on \( {T}_{p}M \) characterized by the equation\n\n\[ \n{d}_{p}^{2}f\left( {{\alpha }^{\prime }\left( 0\right) }\right) = {\left( f \circ \a... | Proof. Set \( h \circ \alpha \left( t\right) = \gamma \left( t\right) = \left( {{\gamma }_{1}\left( t\right) ,\ldots ,{\gamma }_{n}\left( t\right) }\right) \) and \( \phi = f \circ {h}^{-1} \) . A direct calculation yields\n\n\[ \n{\left( f \circ \alpha \right) }^{\prime }\left( t\right) = {\left( \phi \circ \gamma \ri... | Yes |
Example 12.5 Let \( f : {\mathbf{R}}^{n} \rightarrow \mathbf{R} \) be the function\n\n\[ f\left( x\right) = c - {x}_{1}^{2} - {x}_{2}^{2} - \ldots - {x}_{\lambda }^{2} + {x}_{\lambda + 1}^{2} + \ldots {x}_{n}^{2}, \]\n\nwhere \( c \in \mathbf{R},\lambda \in \mathbf{Z} \) and \( 0 \leq \lambda \leq n \) . | Since\n\n\[ {\operatorname{grad}}_{\mathbf{x}}\left( f\right) = 2\left( {-{x}_{1},\ldots , - {x}_{\lambda },{x}_{\lambda + 1},\ldots ,{x}_{n}}\right) \]\n\n0 is the only critical point of \( f \) . We find that\n\n\[ \left( {\frac{{\partial }^{2}f}{\partial {x}_{i}\partial {x}_{j}}\left( 0\right) }\right) = \operatorna... | Yes |
Theorem 12.6 Let \( p \in {M}^{n} \) be a non-degenerate critical point for \( f \in {C}^{\infty }\left( {M,\mathbf{R}}\right) \). There exists a \( {C}^{\infty } \)-chart \( h : U \rightarrow h\left( U\right) \subseteq {\mathbf{R}}^{n} \) with \( p \in U \) and \( h\left( p\right) = 0 \) such that\n\n\[ f \circ {h}^{-... | Proof. After replacing \( f \) with \( f - f\left( p\right) \) we may assume that \( f\left( p\right) = 0 \). Since the problem is local and diffeomorphism invariant, we may also assume that \( f \in {C}^{\infty }\left( {W,\mathbf{R}}\right) \), where \( W \) is an open convex neighborhood of 0 in \( {\mathbf{R}}^{n} \... | No |
Lemma 12.8 Every Morse function on \( M \) admits a gradient-like vector field. | Proof. We can find a \( {C}^{\infty } \) -atlas \( {\left( {U}_{\alpha },{h}_{\alpha }\right) }_{\alpha \in A} \) for \( M \) which satisfies the following two conditions:\n\n(i) Every critical point of \( f \) belongs to just one of the coordinate patches \( {U}_{\alpha } \).\n\n(ii) For any \( \alpha \in A \) either ... | Yes |
Lemma 12.9 Let \( f \) be a Morse function on \( M \) and \( X \) a smooth tangent vector field such that \( {d}_{p}f\left( {X\left( p\right) }\right) > 0 \) for every \( p \in M \) that is not a critical point for \( f \) . Let \( {p}_{0} \in M \) be a critical point for \( f \) of index \( \lambda \) . If \( X\left( ... | Proof. We choose a gradient-like vector field \( \widetilde{X} \) . By Definition 12.7.(ii) and Example 12.5,\n\n\[ \iota \left( {\widetilde{X};{p}_{0}}\right) = {\left( -1\right) }^{\lambda } \]\n\nLet \( U \) be an open neighborhood of \( {p}_{0} \) that is diffeomorphic to \( {\mathbf{R}}^{n} \) and chosen so small ... | Yes |
Theorem 12.11 Let \( {M}^{n} \) be a compact differentiable manifold and \( X \) a smooth tangent vector field on \( {M}^{n} \) with isolated singularities. Let \( f \in {C}^{\infty }\left( {M,\mathbf{R}}\right) \) be a Morse function and \( {c}_{\lambda } \) the number of critical points of index \( \lambda \) for \( ... | Proof. It is a consequence of Theorem 11.27 that any two tangent vector fields with isolated singularities have the same index. Thus we may assume that \( X \) is gradient-like for \( f \) . The zeros for \( X \) are exactly the critical points of \( f \), and the claimed formula follows from Lemma 12.9. | Yes |
Proposition 12.14 In the situation of Lemma 12.13 suppose that \( M\left( {a - \epsilon }\right) \) has finite-dimensional cohomology. Then the same will be true for \( M\left( {a + \epsilon }\right) \), and\n\n\[ \chi \left( {M\left( {a + \epsilon }\right) }\right) = \chi \left( {M\left( {a - \epsilon }\right) }\right... | Proof. For \( U = {U}_{1} \cup \ldots \cup {U}_{r} \), Lemma 12.13.(ii) and Corollary 6.10 imply that\n\n\[ {H}^{p}\left( U\right) \simeq \left\{ \begin{array}{ll} 0 & \text{ if }p \neq 0 \\ {\mathbf{R}}^{r} & \text{ if }p = 0 \end{array}\right. \]\n\nThis gives \( \chi \left( U\right) = r \) . Condition (iii) of Lemma... | Yes |
Lemma 12.15 Let \( U \) and \( V \) be open subsets of a smooth manifold. If \( U, V \) and \( U \cap V \) have finite dimensional de Rham cohomology, the same is true for \( U \cup V \) , and \[ \chi \left( {U \cup V}\right) = \chi \left( U\right) + \chi \left( V\right) - \chi \left( {U \cap V}\right) . \] | Proof. We use the long exact Mayer-Vietoris sequence \[ \cdots \rightarrow {H}^{p - 1}\left( {U \cap V}\right) \rightarrow {H}^{p}\left( {U \cup V}\right) \rightarrow {H}^{p}\left( U\right) \oplus {H}^{p}\left( V\right) \rightarrow {H}^{p}\left( {U \cap V}\right) \rightarrow \cdots \] First we conclude that \( \dim {H}... | Yes |
Theorem 12.16 If \( f \) is a Morse function on the compact manifold \( {M}^{n} \), then\n\n\[ \chi \left( {M}^{n}\right) = \mathop{\sum }\limits_{{\lambda = 0}}^{n}{\left( -1\right) }^{\lambda }{c}_{\lambda } \]\n\nwhere \( {c}_{\lambda } \) denotes the number of critical points for \( f \) of index \( \lambda \) . | Proof. Let \( {a}_{1} < {a}_{2} < \ldots < {a}_{k - 1} < {a}_{k} \) be the critical values. Choose real numbers \( {b}_{0} < {a}_{1},{b}_{j} \in \left( {{a}_{j},{a}_{j + 1}}\right) \) for \( 1 \leq j \leq k - 1 \) and \( {b}_{k} > {a}_{k} \) . Lemma 12.12 shows that the dimensions of \( {H}^{d}\left( {M\left( {b,}\righ... | Yes |
Corollary 12.17 If \( {M}^{n} \) is compact and of odd dimension \( n \) then \( \chi \left( {M}^{n}\right) = 0 \) . | Proof. Let \( f \) be a Morse function on \( M \) . Then \( - f \) is also a Morse function, and \( - f \) has the same critical points as \( f \) . If a critical point \( p \) has index \( \lambda \) with respect to \( f \), then \( p \) has index \( n - \lambda \) with respect to \( - f \) . Theorem 12.16 applied to ... | Yes |
Consider the torus \( T \) in \( {\mathbf{R}}^{3} \). The height function \( f : T \rightarrow \mathbf{R} \) is a Morse function with the four indicated critical points. The Gauss curvature of \( T \) is positive at \( p \) and \( s \), but negative at \( q \) and \( r \) (cf. Example 12.18). Hence \( f \) is a Morse f... | Theorem 12.16 gives \( \chi \left( T\right) = 0 \). Since we know that \( \dim {H}^{0}\left( T\right) = \dim {H}^{2}\left( T\right) = 1 \), we can calculate \( \dim {H}^{1}\left( T\right) = 2 \). | Yes |
Example 12.20 (Morse function on \( {\mathbf{{RP}}}^{n} \) ) Real functions on \( {\mathbf{{RP}}}^{n} \) are equivalent to even functions \( f : {S}^{n} \rightarrow \mathbf{R} \), i.e. \( f\left( x\right) = f\left( {-x}\right) \) for all \( x \in {S}^{n} \) . Let us try\n\n\[ \n f\left( \mathbf{x}\right) = \mathop{\sum... | The differential of \( f \) at \( \mathbf{x} \) is given by \n\n\[ \n {d}_{x}f\left( {{v}_{0},\ldots ,{v}_{n}}\right) = 2\mathop{\sum }\limits_{{i = 0}}^{n}{a}_{i}{x}_{i}{v}_{i} \n\] \n\nwhere \( v = \left( {{v}_{0},{v}_{1},\ldots ,{v}_{n}}\right) \in {T}_{x}{S}^{n} \) so that \n\n\[ \n \mathop{\sum }\limits_{{i = 0}}^... | Yes |
Lemma 13.2\n\\[ \n{H}_{c}^{q}\left( {\\mathbf{R}}^{n}\\right) = \\left\\{ \\begin{array}{ll} \\mathbf{R} & \\text{ if }q - n \\\\ 0 & \\text{ otherwise. } \\end{array}\\right.\n\\]\n | Proof. The above remarks give the result for \\( q = 0 \\) and \\( q = n \\), so we may assume that \\( 0 < q < n \\) . We identify \\( {\\mathbf{R}}^{n} \\) with \\( {S}^{n} - \\left\\{ {p}_{0}\\right\\} \\), e.g. by stereographic projection, and can thus instead prove that\n\n\\[ \n{H}_{c}^{q}\left( {{S}^{n} - \\left... | Yes |
Theorem 13.3 (Mayer-Victoris) With the above notation there is an exact sequence\n\n\[ \cdots \rightarrow {H}_{c}^{q}\left( {{U}_{1} \cap {U}_{2}}\right) \overset{{J}_{ * }}{ \rightarrow }{H}_{c}^{q}\left( {U}_{1}\right) \oplus {H}_{c}^{q}\left( {U}_{2}\right) \overset{{I}_{ * }}{ \rightarrow }{H}_{c}^{q}\left( U\right... | Proof. This follows from Theorem 4.9 applied to (4). | No |
Theorem 13.5 (Poincaré duality) For an oriented smooth \( n \) -dimensional manifold, \( {D}_{M}^{p} \) is an isomorphism for all \( p \) . | The proof is based upon a series of lemmas. | No |
Lemma 13.6 Suppose \( V \subseteq U \subseteq {M}^{n} \) are open subsets. Then the diagram\n\n\n\ncommutes. | Proof. Let \( \omega \in {\Omega }^{p}\left( U\right) ,\tau \in {\Omega }_{c}^{n - p}\left( V\right) \) be closed forms representing cohomology classes \( \left\lbrack \omega \right\rbrack \) and \( \left\lbrack \tau \right\rbrack \) . Then\n\n\[ \n{D}_{V}^{p} \circ {H}^{p}\left( \iota \right) \left( \left\lbrack \omeg... | Yes |
Proposition 13.11 There is a long exact sequence\n\n\[ \cdots \rightarrow {H}^{q - 1}\left( M\right) \overset{\delta }{ \rightarrow }{H}_{c}^{q}\left( U\right) \overset{{i}_{ * }}{ \rightarrow }{H}^{q}\left( N\right) \overset{{j}^{ * }}{ \rightarrow }{H}^{q}\left( M\right) \rightarrow \cdots \] | Proof of Proposition 13.11. By Lemma 13.12.(i) we have a short exact sequence of chain complexes\n\n\[ 0 \rightarrow {\Omega }^{ \bullet }\left( {N, M}\right) \rightarrow | No |
Example 14.1 (The Riemann sphere and Hopf fibration) Since \( \mathbf{C} \times \mathbf{R} \) can be identified with \( {\mathbf{R}}^{3} \), the unit sphere \( {S}^{2} \) can be written as\n\n\[ \n{S}^{2} = \left\{ {\left( {z, t}\right) \in \mathbf{C} \times \mathbf{R}\left| {\;{\left| z\right| }^{2} + {t}^{2} = 1}\rig... | \[ \n{\psi }_{ + }\left( {z, t}\right) = \frac{z}{1 - t},\;{\psi }_{ - }\left( {z, t}\right) = \frac{z}{1 + t}. \n\]\n\nThe \( {\psi }_{ \pm } \) are diffeomorphisms with inverses\n\n\[ \n{\psi }_{ \pm }^{-1}\left( w\right) = \left( {\frac{2w}{{\left| w\right| }^{2} + 1}, \pm \frac{{\left| w\right| }^{2} - 1}{{\left| w... | Yes |
Theorem 14.2 The cohomology of \( {\mathbf{{CP}}}^{n} \) is\n\n\[ \n{H}^{2j}\left( {\mathbf{{CP}}}^{n}\right) = \mathbf{R}\;\text{ for }\;0 \leq \jmath \leq n \]\n\n\[ \n{H}^{k}\left( {\mathbb{{CP}}}^{n}\right) = 0\;\text{ otherwise. } \]\n | Proof. The embedding \( {C}^{n} \subset {C}^{n + 1} \) induces an embedding of \( C{P}^{n - 1} \) into \( C{P}^{n} \) , and we can use Proposition 13.11 on the pair \( \left( {{\mathbf{{CP}}}^{n},{\mathbf{{CP}}}^{n - 1}}\right) \) . We can assume the result for \( {\mathbf{{CP}}}^{n - 1} \) and that \( n \geq 2 \), sin... | Yes |
Theorem 14.3 The cohomology algebra \( {H}^{ \bullet }\left( {\mathbf{{CP}}}^{n}\right) \) is a truncated polynomial algebra | \[ {H}^{ * }\left( {\mathbf{{CP}}}^{n}\right) = \mathbf{R}\left\lbrack c\right\rbrack /\left( {c}^{n + 1}\right) \] where \( \mathrm{c} \) is a non-zero class in degree 2, and \( \left( {c}^{n + 1}\right) \) the ideal generated by \( {c}^{n + 1} \) . Proof. We use induction over \( n \), so suppose the theorem proved f... | Yes |
Lemma 14.5 If \( V \) is a finite-dimensional \( \mathrm{C} \) -vector space and \( F : V \rightarrow V \) is a C-linear map, then\n\n\[ \det \left( {rF}\right) = {\left| \det F\right| }^{2} \] | Proof. We use induction on \( m = {\dim }_{\mathbb{C}}V \) . If \( m = 1 \) then \( F \) is multiplication by some \( z \in \mathbf{C} \) . The matrix for \( {rF} \), with respect to a basis of the form \( b,{ib} \) for \( {rV} \), is\n\n\[ \left( \begin{matrix} x & - y \\ y & x \end{matrix}\right) \]\n\nwhere \( x = \... | Yes |
Corollary 14.6 If \( V \) is an \( m \) -dimensional C-vector space then \( {rV} \) has a natural orientation with the property that any basis \( {b}_{1},\ldots ,{b}_{m} \) over \( \mathbb{C} \) gives rise to a positive basis \( \left\{ {{b}_{1},\mathrm{i}{b}_{1},{b}_{2}, i{b}_{2},\ldots ,{b}_{m}, i{b}_{m}}\right\} \) ... | Proof. Let \( {b}_{1}^{\prime },\ldots ,{b}_{m}^{\prime } \) be another basis of \( V \) . We can apply Lemma 14.5 to the C-linear map \( F \) determined by \( F\left( {b}_{j}\right) = {b}_{j}^{\prime }\left( {1 \leq j \leq m}\right) \) . Since \( \det \left( {rF}\right) > 0 \), the assertion follows. | No |
Proposition 14.7 Let \( V \) be an \( m \) -dimensional C-vector space with hermitian inner product \( \langle \) , \( \rangle \) . Then\n\n(i) \( g\left( {{v}_{1},{v}_{2}}\right) = \operatorname{Re}\left( {{v}_{1},{v}_{2}}\right) \) defines an inner product on \( {rV} \), and\n\n\[ \omega \left( {{v}_{1},{v}_{2}}\righ... | Proof. We leave (i) to the reader. An orthonormal basis \( {b}_{1},\ldots ,{b}_{m} \) of \( V \) with respect to \( \langle \) , \( \rangle {givesrisetothepositivelyorientedorthonormalbasisof}{rV} \) with respect to \( g \) ,\n\n\[ {b}_{1}, i{b}_{1},{b}_{2}, i{b}_{2},\ldots ,{b}_{m}, i{b}_{m}. \]\n\nLet \( {\epsilon }_... | No |
Theorem 14.8 The \( \omega = {\left\{ {\omega }_{p}\right\} }_{p \in {\mathbb{{CP}}}^{n}} \) define a closed 2 -form on \( {\mathbb{{CP}}}^{n} \) and \( g = \) \( {\left\{ {g}_{p}\right\} }_{p \in {\mathbb{{CP}}}^{n}} \) is a Riemannian metric on \( {\mathbb{{CP}}}^{n} \) (the Fubini-Study metric). Moreover,\n\n\[ \n{\... | Proof. Let \( p \in {\mathbf{{CP}}}^{n} \) and \( v \in {S}^{{2n} + 1} \) with \( \pi \left( v\right) = p \) . Choose \( s : U \rightarrow {S}^{{2n} + 1} \) with \( \pi \circ s = {\operatorname{id}}_{U} \) and \( s\left( p\right) = v \) as in Lemma 14.4. We will show that\n\n(10)\n\n\[ \n{\omega }_{\mid U} = {s}^{ \bul... | Yes |
Corollary 14.9 Let \( \omega \) be the closed 2-form on \( {\mathrm{{CP}}}^{n} \) constructed in Theorem 14.8. The \( j \) -th exterior power \( {\omega }^{j} \) represents a basis element of \( {H}^{2j}\left( {\mathbf{{CP}}}^{n}\right) \) when \( 1 \leq \jmath \leq n \) . | Proof. The class in \( {H}^{2n}\left( {\mathbb{{CP}}}^{n}\right) \cong \mathbf{R} \) determined by volcp- is non-trivial. Since \( \left\lbrack \omega \right\rbrack \in {H}^{2}\left( {\mathbb{{CP}}}^{n}\right) \) we have\n\n\[{\left\lbrack \omega \right\rbrack }^{n} = n!\left\lbrack {\operatorname{vol}}_{{\mathbf{{CP}}... | Yes |
Example 14.10 (The Hopf fibration again) Let \( {z}_{\nu } = {x}_{\nu } + i{y}_{\nu },\nu = 0,1 \) . The Hopf fibration \( \eta \) from (3) is the restriction to \( {S}^{3} \subseteq {\mathbf{R}}^{4} \) of the map \( h : {\mathbf{R}}^{4} \rightarrow {\mathbf{R}}^{3} \) given by\n\n\[ h\left( {{x}_{0},{y}_{0},{x}_{1},{y... | with Jacobian matrix\n\n\[ 2\left( \begin{matrix} {x}_{1} & {y}_{1} & {x}_{0} & {y}_{0} \\ {y}_{1} & - {x}_{1} & - {y}_{0} & {x}_{0} \\ {x}_{0} & {y}_{0} & - {x}_{1} & - {y}_{1} \end{matrix}\right) \]\n\nIf \( v \in {S}^{3} \) has real coordinates \( \left( {{x}_{0},{y}_{0},{x}_{1},{y}_{1}}\right) \), then \( {iv} \) w... | Yes |
Example 15.5 (The tangent bundle) Let \( {M}^{n} \subset {\mathbf{R}}^{n + k} \) be a smooth manifold. Consider\n\n\[ \n{TM} = \left\{ {\left( {p, v}\right) \in M \times {\mathbf{R}}^{n + k} \mid v \in {T}_{p}M}\right\} ,\;\pi \left( {p, v}\right) = p.\n\]\n\nThe fiber over \( p \in M \) is the tangent space \( {T}_{p}... | Let \( b \in M \) . Choose a parametrization \( \left( {U, g}\right) \) around \( b, g : W \rightarrow U,\;W \subseteq {\mathbf{R}}^{n} \) and let\n\n\[ \nh : U \times {\mathbf{R}}^{n} \rightarrow {\pi }^{-1}\left( U\right) ;\;h\left( {x, v}\right) = D{g}_{{g}^{-1}\left( z\right) }\left( v\right) .\n\]\n\nThis gives th... | No |
Lemma 15.10 A (smooth) continuous map \( \widehat{f} : E\left( \xi \right) \rightarrow E\left( \eta \right) \) of (smooth) vector bundles over \( B \), which map the fiber \( {F}_{b}\left( \xi \right) \) isomorphically onto the fiber \( {F}_{b}\left( \eta \right) \), is a (smooth) isomorphism. | Proof. Since \( \widehat{f} \) is a bijection, it is sufficient to show that \( {\widehat{f}}^{-1} \) is a (smooth) homomorphism of vector bundles (over \( {\mathrm{{id}}}_{B} \) ). We need to check that \( {\widehat{f}}^{-1} \) is continuous (smooth). Since \( \widehat{f} \) is a fiberwise isomorphism, it is enough to... | Yes |
Proposition 15.13 Every vector bundle over a compact B has an inner product. | Proof. Choose local trivializations\n\n\[ \n{h}_{1} : {U}_{i} \times {\mathbf{R}}^{n} \rightarrow {\pi }_{\xi }^{-1}\left( {U}_{i}\right) \]\n\nwhere \( {U}_{1},\ldots ,{U}_{r} \) cover \( B \), and choose a partition of unity \( {\left\{ {\alpha }_{i}\right\} }_{i = 1}^{r} \) with \( \operatorname{supp}\left( {\alpha ... | Yes |
Lemma 15.16 Let \( \xi \) and \( \eta \) be (smooth) vector bundles with inner product over the compact space \( B \), and let \( \widehat{f} : \xi \rightarrow \eta \) be an isomorphism. Then there exists an \( \epsilon > 0 \) such that every homomorphism \( \widehat{g} : \xi \rightarrow \eta \) that satisfies \( \begi... | Proof. If \( \xi \) and \( \eta \) are trivial, then after choice of frames, \( \widehat{f} \) and \( \widehat{g} \) are represented by maps \( \operatorname{ad}\left( \widehat{f}\right) : B \rightarrow G{L}_{n}\left( \mathbf{R}\right) \) and \( \operatorname{ad}\left( \widehat{g}\right) : B \rightarrow {M}_{n}\left( \... | Yes |
Lemma 15.17 If two smooth vector bundles \( \xi \) and \( \eta \) over the compact manifold B are isomorphic as continuous bundles, then they are smoothly isomorphic. | Proof. We choose a cover \( {U}^{1},\ldots ,{U}^{r} \) of \( B \) and smooth local orthonormal frames \( {\mathbf{s}}^{i} = \left( {{s}_{1}^{i},\ldots ,{s}_{n}^{i}}\right) \) and \( {\mathbf{t}}^{i} = \left( {{t}_{1}^{i},\ldots ,{t}_{n}^{i}}\right) \) for \( \xi \) and \( \eta \), over \( {U}^{i} \) . A continuous isom... | Yes |
Theorem 15.18 Every vector bundle \( \xi \) over a compact base space \( B \) has a complement \( \eta \), i.e. \( \xi \oplus \eta \cong {\varepsilon }^{N} \) (for a suitably large \( N \) ). | Proof. Choose an open cover \( {U}^{1},\ldots ,{U}^{r} \) of \( B \) admitting trivializations \( {h}_{1} \) of \( {\xi }_{\mid {U}_{i}} \), and let \( \left\{ {\alpha }_{1}\right\} \) be a partition of unity with \( \operatorname{supp}\left( {\alpha }_{1}\right) \subset {U}^{i} \) . Denote by \( {f}^{i} \) the composi... | Yes |
The normal bundle to the unit sphere \( {S}^{2} \subseteq {\mathbf{R}}^{3} \) is trivial, since the outward directed unit normal vector defines a global frame. We also know that \( {\tau }_{{S}^{2}} \oplus {\nu }_{{S}^{2}} = {\varepsilon }^{3} \), such that\n\n\[ \left\lbrack {\tau }_{{S}^{2}}\right\rbrack + \left\lbra... | Indeed, if \( \left\lbrack {\tau }_{{S}^{2}}\right\rbrack \) were equal to \( \left\lbrack {\varepsilon }^{2}\right\rbrack \), then there would exist a section \( s \in \Gamma \left( {\tau }_{{S}^{2}}\right) \) with \( s\left( x\right) \neq 0 \) for all \( x \in {S}^{2} \) . However, Theorem 7.3 implies that \( {\tau }... | Yes |
Corollary 15.22 Every vector bundle over a contractible base space is trivial. | Proof. With our assumption \( {\operatorname{id}}_{B} \simeq f \), where \( f \) is the constant map with value \( f\left( B\right) = \{ b\} \) . Hence \( {f}^{ * }\left( \xi \right) \cong \xi \) . But \( {f}^{ * }\left( \xi \right) \) is trivial by construction when \( f \) is constant. | Yes |
Lemma 16.2 Let \( V, W \) and \( U \) be \( R \) -modules, and let \( f : V \times W \rightarrow U \) be any \( R \) -bilinear map. Then there exists a unique \( R \) -linear map \( \bar{f} : V{ \otimes }_{R}W \rightarrow U \), with \( f = \bar{f} \circ \pi \) | Proof. Since the set \( V \times W \) is a basis for the \( R \) -module \( R\left\lbrack {V \times W}\right\rbrack, f \) extends to an \( R \) -linear map \( f : R\left\lbrack {V \times W}\right\rbrack \rightarrow U \) . The bilinearity of \( f \) implies that \( \widehat{f}\left( {R\left( {V, W}\right) }\right) = 0 \... | Yes |
Lemma 16.3 Let \( V \) and \( {V}^{\prime } \) be free \( R \) -modules with bases \( B \) and \( {B}^{\prime } \) . Then \( V{ \otimes }_{R}{V}^{\prime } \) is a free \( R \) -module with basis \( \left\{ {b{ \otimes }_{R}{b}^{\prime } \mid b \in B,{b}^{\prime } \in {B}^{\prime }}\right\} \) . | Proof. The bilinearity of \( \pi : V \times {V}^{\prime } \rightarrow V{ \otimes }_{R}{V}^{\prime } \) shows that the stated set generates \( V{ \otimes }_{R}{V}^{\prime } \) ; so suppose that\n\n(3)\n\n\[ \sum {r}_{ij}{b}_{i}{ \otimes }_{R}{b}_{j}^{\prime } = 0 \]\n\n\nis a (finite) relation. Let \( {\varphi }_{0} : V... | Yes |
Lemma 16.9 For (smooth) vector bundles \( \xi \) and \( \eta \) there are isomorphisms \( \xi \cong {\xi }^{* * } \) and \( {\xi }^{ * } \otimes \eta \cong \operatorname{Hom}\left( {\xi ,\eta }\right) \) . | Proof. For finite-dimensional vector spaces there are natural isomorphisms\n\n\[ V \rightarrow {V}^{* * },\;{V}^{ * } \otimes W \rightarrow \operatorname{Hom}\left( {V, W}\right) \]\n\ndefined without any reference to basis. This gives maps of vector bundles (over the identity)\n\n\[ \xi \rightarrow {\xi }^{* * },\;{\x... | Yes |
For every smooth vector bundle \( \xi \) over a compact smooth manifold \( M \) , \( {\Omega }^{0}\left( \xi \right) \) is a direct \( {\Omega }^{0}\left( M\right) \) -summand in a finitely generated free \( {\Omega }^{0}\left( M\right) \) -module. | By Theorem 15.18 there is a complement \( \eta \) to \( \xi ,\xi \oplus \eta \cong {\varepsilon }^{k + l} \) . Then\n\n\[ \n{\Omega }^{0}\left( \xi \right) \oplus {\Omega }^{0}\left( \eta \right) \cong {\Omega }^{0}\left( {\xi \oplus \eta }\right) \cong {\Omega }^{0}\left( {\varepsilon }^{k + l}\right) \n\]\n\nand \( {... | Yes |
Lemma 16.12 Let \( {P}_{1} \) and \( {P}_{2} \) be finitely generated projective \( R \) -modules. Then there are isomorphisms\n\n\[ \n{P}_{1} \cong {P}_{1}^{\bullet \bullet },\;{\operatorname{Hom}}_{R}\left( {{P}_{1},{P}_{2}}\right) \cong {P}_{1}^{ \bullet }{ \otimes }_{R}{P}_{2} \]\n\nwhere \( {P}_{1}^{ * } = {\opera... | Proof. One first proves the assertions for finitely generated free modules, where the argument is completely similar to the case of vector spaces. The general case follows easily upon choosing complements \( {P}_{1} \oplus {Q}_{1} = {R}^{{n}_{1}},{P}_{2} \oplus {Q}_{2} = {R}^{{n}_{2}} \) . Details are left as an exerci... | No |
Let \( {M}^{n} \subset {\mathbf{R}}^{n + k} \) be a smooth manifold. One can define a connection on its tangent bundle as follows: a section \( s \in {\Omega }^{0}\left( \tau \right) \) can be considered as a smooth function \( s : M \rightarrow {\mathbf{R}}^{n + k} \) with \( s\left( p\right) \in {T}_{p}M \), and we s... | It is easy to see that (3) is satisfied. | No |
Example 17.3 Suppose \( \xi \oplus \eta \cong {\varepsilon }^{n + k} \), and let \( i : \xi \rightarrow {\varepsilon }^{n + k} \) and \( j : {\varepsilon }^{n + k} \rightarrow \xi \) be the inclusion and the projection on the first factor, respectively. We give the trivial bundle the connection \( {\nabla }_{0} \) from... | \[ {\Omega }^{0}\left( \xi \right) \overset{{i}_{ * }}{ \rightarrow }{\Omega }^{0}\left( {\varepsilon }^{n + k}\right) \;\text{ and }\;{\Omega }^{0}\left( {\varepsilon }^{n + k}\right) \overset{{j}_{ * }}{ \rightarrow }{\Omega }^{0}\left( \xi \right) \] and the composition \[ {\Omega }^{0}\left( \xi \right) \overset{{i... | Yes |
Lemma 17.6 There is a unique R-linear operator \( {d}^{\nabla } : {\Omega }^{j}\left( \xi \right) \rightarrow {\Omega }^{j + 1}\left( \xi \right) \) that satisfies\n\n(i) \( {d}^{\nabla } = \nabla \) when \( j = 0 \)\n\n(ii) \( {d}^{\nabla }\left( {\omega \land t}\right) = {d\omega } \land t + {\left( -1\right) }^{i}\o... | Proof. Let \( \tau \in {\Omega }^{j}\left( M\right) \) and \( s \in {\Omega }^{0}\left( \xi \right) \) and set \( {d}^{\nabla }\left( {\tau \otimes s}\right) = {dt} \land s + {\left( -1\right) }^{j}\tau \land \nabla s \) . One checks that \( {d}^{\nabla } \) is \( {\Omega }^{0}\left( M\right) \) -balanced in the sense ... | Yes |
Lemma 17.8 The composition \( {d}^{\nabla } \circ {d}^{\nabla } : {\Omega }^{i}\left( \xi \right) \rightarrow {\Omega }^{i + 2}\left( \xi \right) \) maps \( t \) to \( t \land F\nabla \) . | Proof. Let \( \omega \otimes s \in {\Omega }^{i}\left( M\right) { \otimes }_{{\Omega }^{0}\left( M\right) }{\Omega }^{0}\left( \xi \right) \) . By Lemma 17.6,\n\n\[ \n{d}^{\nabla } \circ {d}^{\nabla }\left( {\omega \otimes s}\right) = {d}^{\nabla }\left( {{d\omega } \otimes s + {\left( -1\right) }^{i}\omega \land \nabl... | Yes |
Lemma 17.11 Under the identification \( \alpha : {\xi }^{ * } \otimes \eta \overset{ \cong }{ \rightarrow }\operatorname{Hom}\left( {\xi ,\eta }\right) ,{\nabla }_{{\xi }^{ * } \otimes \eta } = \) \( \nabla \operatorname{Hom}\left( {\xi ,\eta }\right) \) | Proof. There is a commutative diagram of vector bundles over \( M \)\n\n\[\n\xi \otimes {\xi }^{ * } \otimes \eta \overset{\mathrm{{id}} \otimes q}{ \rightarrow }\xi \otimes \mathrm{{Hom}}\left( {\xi ,\eta }\right)\n\]\n\n\[ \n{\varepsilon }_{M}^{1} \otimes \eta \overset{\text{ mult }}{ \rightarrow }\eta\n\]\n\nand a c... | Yes |
Theorem 17.13 (Bianchi’s identity) We have \( {d}^{\nabla }{F}^{\nabla } = 0 \), where \( {d}^{\nabla } \) is associated to the connection \( \nabla = {\nabla }_{\operatorname{Hom}\left( {\xi ,\xi }\right) } \) . | Proof. Use the local forms (10) and (17) to get\n\n\[ \n{F}^{\nabla } = {dA} - A \land A \n\]\n\n\[ \n{d}^{\nabla }{F}^{\nabla } = - d\left( {A \land A}\right) + {F}^{\nabla } \land A - A \land {F}^{\nabla } \n\]\n\n\[ \n= - d\left( {A \land A}\right) + {dA} \land A - A \land {dA} = 0. \n\] | Yes |
Theorem 17.14 For \( \phi \in {\Omega }^{i}\left( {\operatorname{Hom}\left( {\xi ,\xi }\right) }\right) \), \[ d\operatorname{Tr}\left( \phi \right) = \operatorname{Tr}\left( {{d}^{\nabla }\phi }\right) \] where \( {d}^{\nabla } \) is associated to \( \nabla = {\nabla }_{\operatorname{Hom}\left( {\xi ,\xi }\right) } \)... | Proof. Let \( s \in {\Omega }^{0}\left( \xi \right) ,{s}^{ * } \in {\Omega }^{0}\left( {\xi }^{ * }\right) ,\omega \in {\Omega }^{i}\left( M\right) \) and suppose \[ \phi = \omega \otimes s \otimes {s}^{ \bullet } \in {\Omega }^{i}\left( M\right) { \otimes }_{{\Omega }^{0}\left( M\right) }{\Omega }^{0}\left( \xi \right... | Yes |
Lemma 18.2 The cohomology class \( \left\lbrack {P\left( {F\nabla }\right) }\right\rbrack \) in \( {H}^{ \bullet }\left( {M,\mathbb{C}}\right) \) is independent of the choice of connection. | Proof of Lemma 18.2. Let \( {\nabla }_{0},{\nabla }_{1} \) be two connections on \( \xi \), and \( \pi : M \times \mathbf{R} \rightarrow M \) the projection onto the first factor. Let \( {\widehat{\nabla }}_{\nu } = {\pi }^{ \circ }\left( {\nabla }_{\nu }\right) \) be the induced connections on \( {\pi }^{ \bullet }\le... | Yes |
Theorem 18.4 The integration homomorphism maps \( {c}_{1}\left( {H}_{1}\right) \) to -1 . | Proof. Apply the two positively oriented stereographic charts \( {\psi }_{ - } \) and \( {\bar{\psi }}_{ + } \) on \( {S}^{2} = {\mathbf{{CP}}}^{1} \) from Example 14.1. In Example 17.9 we calculated the pre-image of the curvature form \( {F}^{\nabla } \) under \( g = {\left( {\psi }_{ - }\right) }^{-1} \) to be\n\n\[ ... | Yes |
Theorem 18.5 Let \( f.N \rightarrow M \) be a smooth map and \( \xi \) a complex vector bundle on \( M \) . For every invariant polynomial we have \( {f}^{ \bullet }\left\lbrack {P\left( \xi \right) }\right\rbrack = \left\lbrack {P\left( {{f}^{ \bullet }\left( \xi \right) }\right) }\right\rbrack \) . | Proof. We give \( {f}^{ \bullet }\left( \xi \right) \) the connection \( {f}^{ \bullet }\left( \nabla \right) \) of Lemma 17.10. By formula (17.13), \( {f}^{ \bullet }\left( {F}^{\nabla }\right) = {F}^{{f}^{ \bullet }\left( \nabla \right) } \) . Hence \( {f}^{ \bullet }\left( {P\left( {F}^{\nabla }\right) }\right) = P\... | Yes |
Theorem 18.6 For a sum of complex vector bundles,\n\n(i) \( {\operatorname{ch}}_{k}\left( {{\xi }_{0} \oplus {\xi }_{1}}\right) = {\operatorname{ch}}_{k}\left( {\xi }_{0}\right) + {\operatorname{ch}}_{k}\left( {\xi }_{1}\right) \)\n\n(ii) \( {c}_{k}\left( {{\xi }_{0} \oplus {\xi }_{1}}\right) = \mathop{\sum }\limits_{{... | Proof. Choose complex connections \( {\nabla }_{\nu } \) on \( {\xi }_{\nu } \) . We identify \( {\Omega }^{i}\left( {{\xi }_{0} \oplus {\xi }_{1}}\right) \) with \( {\Omega }^{i}\left( {\xi }_{0}\right) \oplus {\Omega }^{i}\left( {\xi }_{1}\right) \) ; then\n\n\[{\nabla }_{\mathbf{0}} \oplus {\nabla }_{\mathbf{1}} : {... | Yes |
Theorem 18.7 For a tensor product of complex vector bundles,\n\n\[ \n{\mathrm{{ch}}}_{k}\left( {{\xi }_{0} \otimes {\xi }_{1}}\right) = \mathop{\sum }\limits_{{\nu = 0}}^{k}{\mathrm{{ch}}}_{\nu }\left( {\xi }_{0}\right) {\mathrm{{ch}}}_{k - \nu }\left( {\xi }_{1}\right) \n\]\n\nwhere \( {\operatorname{ch}}_{0}\left( {\... | Proof. The tensor product of linear maps, applied fiberwise, defines a map of vector bundles\n\n\[ \n\operatorname{Hom}\left( {{\xi }_{0},{\xi }_{0}}\right) \otimes \operatorname{Hom}\left( {{\xi }_{1},{\xi }_{1}}\right) \rightarrow \operatorname{Hom}\left( {{\xi }_{0} \otimes {\xi }_{1},{\xi }_{0} \otimes {\xi }_{1}}\... | Yes |
Theorem 18.9 There exists precisely one set of cohomology classes \( {c}_{k}\left( \xi \right) \in \) \( {H}^{2\mathbf{k}}\left( {M;\mathbf{C}}\right) \), depending only on the isomorphism class of \( \xi \), and such that\n\n(i) \( I\left( {{c}_{1}\left( {H}_{1}\right) }\right) = - 1,{c}_{k}\left( {H}_{n}\right) = 0 \... | The uniqueness part of Theorem 18.9 rests on the so-called splitting principle, whose proof is deferred to Chapter 20. | No |
Given a line \( L \subset {\mathbf{C}}^{n + 1} \), consider the map\n\n\[ \n{g}_{L} : \operatorname{Hom}\left( {L,{L}^{ \bot }}\right) \rightarrow {\mathbf{{CP}}}^{n} \n\]\n\nwhich maps an element \( \phi \in \operatorname{Hom}\left( {L,{L}^{ \bot }}\right) \) into the graph of \( \phi \) . Its image is the open set \(... | Let \( {H}^{ \bot } \) be the \( n \) -plane bundle over \( {\mathbf{{CP}}}^{n} \) with total space\n\n\[ \nE\left( {H}^{ \bot }\right) = \left\{ {\left( {L, u}\right) \in {\mathbf{{CP}}}^{n} \times {\mathbf{C}}^{n + 1} \mid u \in {L}^{ \bot }}\right\} \n\]\n\nThen \( H \oplus {H}^{ \bot } \) is the trivial \( \left( {... | Yes |
One of the main applications of characteristic classes is to the question of whether a given closed manifold is (diffeomorphic to) the boundary of a compact manifold. We refer the reader to [Milnor-Stasheff] for the general theory and just present an example. We show that \( {\mathbf{{CP}}}^{2n} \) is not the boundary ... | Indeed, suppose this was the case. By Stokes's theorem,\n\n\[ \n{\int }_{\partial R}\omega = {\int }_{R}{d\omega } = 0 \n\] \n\nfor any closed form \( \omega \in {\Omega }^{4n}\left( R\right) \) . But we can exhibit a closed \( {4n} \) -dimensional form on \( R \) which contradicts this as follows. The tangent bundle o... | Yes |
Lemma 19.2 For any choice of inner products and metric connections \( {g}_{\nu },{\nabla }_{\nu } \) \( \left( {\nu = 0,1}\right) \) on the smooth real vector bundle \( \xi \) over \( M \), there is an inner product \( \widetilde{g} \) on \( \widetilde{\xi } \) and a metric connection \( \widetilde{\nabla } \) compatib... | Proof. We can pull back by \( {\pi }^{ * } \) the metric \( {g}_{\nu } \) and the metric connections \( {\nabla }_{\nu } \) to \( \widetilde{\xi } \) . Let \( \left\{ {{\rho }_{0},{\rho }_{1}}\right\} \) be a partition of unity on \( M \times \mathbf{R} \) subordinate to the cover \( M \times \left( {-\infty ,3/4}\righ... | Yes |
Corollary 19.3 The cohomology class \( \left\lbrack {\operatorname{Pf}\left( {F}^{\nabla }\right) }\right\rbrack \in {H}^{2k}\left( M\right) \) is independent of the metric and the compatible metric connection. | Proof. Let \( \left( {{g}_{0},{ \bigtriangledown }_{0}}\right) \) and \( \left( {{g}_{1},{ \bigtriangledown }_{1}}\right) \) be two different choices and let \( \left( {\widetilde{g},\widetilde{ \bigtriangledown }}\right) \) be the metric and connection of the previous lemma. Then \( {\iota }_{\nu }^{ \bullet }\left( {... | Yes |
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