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Lemma 76.11. Given any short exact sequence\n\n\\[ \n0 \rightarrow \left( {{X}_{i},{f}_{i}}\right) \overset{{u}_{i}}{ \rightarrow }\left( {{Y}_{i},{g}_{i}}\right) \overset{{v}_{i}}{ \rightarrow }\left( {{Z}_{i},{h}_{i}}\right) \rightarrow 0 \n\\]\n\nof inverse systems of abelian groups there exists a natural exact sequ... | Proof. We consider the following commutative diagram\n\n\n\nThe horizontal sequences are evidently exact. We can view the rows as chain complexes. The desired natural exact sequence now follows from the snake lemma... | Yes |
Proposition 76.12. Let \( X \) be a topological space and let \( {X}_{1} \subset {X}_{2} \subset {X}_{3} \subset \ldots \) be a sequence of subsets such that \( X = \mathop{\bigcup }\limits_{{i \in \mathbb{N}}}{X}_{i} \) and such that one the following three conditions holds:\n\n(1) every compact subset of \( X \) is c... | Proof. It follows easily from the Finiteness Theorem 36.14 and Lemma 25.8 we only have to prove the proposition under the hypothesis (1). Therefore suppose that we are given a topological space \( X \) and subspaces \( {X}_{1} \subset {X}_{2} \subset {X}_{3} \subset \ldots \) with \( X = \mathop{\bigcup }\limits_{{i \i... | Yes |
Proposition 76.13. Let \( X \) be a CW-complex and let \( n \in \mathbb{N} \). For every \( k > n \) and every abelian group \( G \) the inclusion \( i : {X}^{k} \rightarrow X \) induces an isomorphism\n\n\[ \n{i}^{ * } : {\mathrm{H}}^{n}\left( {X;G}\right) \rightarrow {\mathrm{H}}^{n}\left( {{X}^{k};G}\right) .\n\] | Proof. By Proposition 76.12 there exists a natural short exact sequence\n\n\[ \n0 \rightarrow \mathop{\lim }\limits_{\substack{ \leftarrow \\ {j \in \mathbb{N}} }}{}^{1}{\mathrm{H}}^{n - 1}\left( {{X}_{j};G}\right) \overset{\tau }{ \rightarrow }{\mathrm{H}}^{n}\left( {X;G}\right) \rightarrow \mathop{\lim }\limits_{\sub... | Yes |
Lemma 77.1. Let \( X \) be a topological space and let \( G \) be an abelian group.\n\n(1) A cochain \( \varphi \in {\mathrm{C}}^{i}\left( {X;G}\right) \) has compact support if and only if there exists a compact\n\nsubset \( K \) such that \( \varphi \) lies in the image of the map \( {\mathrm{C}}^{i}\left( {X, X \sma... | Proof. Let \( X \) be a topological space and let \( G \) be an abelian group. The first statement of the lemma follows immediately from the above remark. | No |
Lemma 77.2. Let \( X \) be a topological space and let \( G \) be an abelian group. Given any \( \varphi \in {\mathrm{H}}_{\mathrm{c}}^{i}\left( {X;G}\right) \) there exists a compact subset \( K \) such that \( \varphi \) lies in the image of the map \( {\mathrm{H}}^{i}\left( {X, X \smallsetminus K;G}\right) \rightarr... | Proof. Let \( \left\lbrack \varphi \right\rbrack \in {\mathrm{H}}_{\mathrm{c}}^{i}\left( {X;G}\right) \) . By Lemma 77.1 (1) there exists a compact subset \( K \) such that \( \varphi \) lies in the image of the map \( {\mathrm{C}}^{i}\left( {X, X\overline{\smallsetminus K;G}}\right) \rightarrow {\mathrm{C}}_{\mathrm{c... | Yes |
Lemma 77.3. Let \( G \) be an abelian group.\n\n(1) Let \( f : X \rightarrow Y \) be a map between topological spaces. If \( f \) is a proper map, then the cochain map \( {f}^{ * } : {\mathrm{C}}^{ * }\left( {Y;G}\right) \rightarrow {\mathrm{C}}^{ * }\left( {X;G}\right) \) restricts to a cochain map\n\n\[ \n{f}^{ * } :... | Proof. Let \( f : X \rightarrow Y \) be a proper map and let \( G \) be an abelian group. Let \( \varphi : {\mathrm{C}}_{n}\left( Y\right) \rightarrow G \) be a cocycle with compact support. We pick a compact subset \( K \) such that \( \varphi \) vanishes on \( Y \smallsetminus K \) . Then \( {f}^{ * }\varphi \) vanis... | Yes |
Proposition 77.4. Let \( X \) be a topological space and let \( G \) be an abelian group. Then the above map  defines a natural isomorphism. | Proof. We need to show that the map \( \Phi \) in the proposition is a bijection.\n\nWe first show that \( \Phi \) is surjective. So let \( \varphi \in {\mathrm{H}}_{\mathrm{c}}^{i}\left( {X;G}\right) \) . By Lemma 77.2 there exists a compact subset \( L \) such that \( \varphi \) lies in the image of \( {\mathrm{H}}^{... | No |
Lemma 77.5. Given a topological space \( X \) that is Hausdorff the above construction defines for each \( n \in {\mathbb{N}}_{0} \) a covariant functor category where the objects are open subsets of \( X \) and where the morphisms are inclusions \( \rightarrow \) abelian groups. | Proof (*). Let \( X \) be a topological space that is Hausdorff and let \( U \subset V \subset W \) be open subsets. Let \( n \in {\mathbb{N}}_{0} \). We have to show that \( {i}_{{UW} * } = {i}_{{VW} * } \circ {i}_{{UV} * } : {\mathrm{H}}_{\mathrm{c}}^{n}\left( U\right) \rightarrow {\mathrm{H}}_{\mathrm{c}}^{n}\left( ... | Yes |
Proposition 77.6. Let \( X \) be a topological space that is Hausdorff. Let \( {X}_{i}, i \in \mathbb{N} \) be a sequence of subsets such that the following holds:\n\n(1) the sequence is nested, i.e. for each \( i \in \mathbb{N} \) we have \( {X}_{i} \subset {X}_{i + 1} \), \n\n(2) each \( {X}_{i} \) is open in \( X \)... | Proof (*). Using Proposition 77.4 we see that we have to show that the map\n\n\[ \Phi : \mathop{\lim }\limits_{\underset{i \in \mathbb{N}}{ \rightarrow }}\mathop{\lim }\limits_{\underset{K \in \mathcal{K}\left( {X}_{i}\right) }{ \rightarrow }}{\mathrm{H}}^{n}\left( {{X}_{i},{X}_{i} \smallsetminus K}\right) \overset{ \c... | No |
Lemma 77.7. Let \( X \) be a topological space. The bounded cochains \( {\mathrm{C}}_{\mathrm{b}}^{ * }\left( X\right) \) form a subcomplex of the cochain complex \( \left( {{\mathrm{C}}^{ * }\left( X\right) ,\delta }\right) \) . | Proof. Suppose that \( \rho \in {\mathrm{C}}^{n}\left( {X;\mathbb{R}}\right) \) is a cochain that is \( C \) -bounded. For any \( \varphi : {\Delta }^{n + 1} \rightarrow X \) we obtain that\n\n\[ \left| {\left( {\delta \rho }\right) \left( \varphi \right) }\right| = \left| {\rho \left( {\partial \varphi }\right) }\righ... | Yes |
Proposition 77.8. (1) Let \( f : X \rightarrow Y \) be a map between topological spaces. The map \( {f}^{ * } : {\mathrm{C}}^{ * }\left( {Y;\mathbb{R}}\right) \rightarrow \) \( {\mathrm{C}}^{ * }\left( {X;\mathbb{R}}\right) \) on singular cochains restricts to a cochain map \( {f}^{ * } : {\mathrm{C}}_{\mathrm{b}}^{ * ... | (1) It is obvious that the pullback of a bounded cohomology class is again bounded. The first statement is an immediate consequence of this observation. | No |
Corollary 77.10. Let \( X \) be a connected CW-complex such that \( {\pi }_{1}\left( X\right) \) is amenable, e.g. such that \( {\pi }_{1}\left( X\right) \) is solvable. Then \( {\mathrm{H}}_{\mathrm{b}}^{n}\left( {X;\mathbb{R}}\right) = 0 \) for any \( n \geq 1 \) . | Proof. We consider the map \( f : X \rightarrow \left\{ {x}_{0}\right\} \) that is given by sending every point in \( X \) to the point \( {x}_{0} \) . Then for any \( n \geq 1 \) we have\n\n\[ \begin{array}{l} {\mathrm{H}}_{\mathrm{b}}^{n}\left( {X;\mathbb{R}}\right) \underset{ \cong }{\overset{{f}^{ * }}{ \rightlefth... | Yes |
Proposition 77.12. Let \( X \) be the wedge of two circles. Then \( {\mathrm{H}}_{\mathrm{b}}^{2}\left( {X;\mathbb{R}}\right) \) and \( {\mathrm{H}}_{\mathrm{b}}^{3}\left( {X;\mathbb{R}}\right) \) are vector spaces of uncountable dimension. | Proof. The statement regarding \( {\mathrm{H}}_{\mathrm{b}}^{2}\left( {X;\mathbb{R}}\right) \) is proved in EF97. An alternative fairly straightforward proof, using quasi-morphisms as introduced on page 600, is given in Roll09, Corollary 2.3]. Furthermore the statement regarding \( {\mathrm{H}}_{\mathrm{b}}^{3}\left( {... | No |
(1) Let \( f : M \rightarrow N \) be a diffeomorphism between compact oriented connected \( k \) -dimensional smooth manifolds and let \( \omega \) be a smooth differential \( k \) -form on \( N \) . Then\n\n\[ \mathop{\int }\limits_{M}{f}^{ * }\omega \; = \;\epsilon \cdot \mathop{\int }\limits_{N}\omega \;\text{where}... | Proof. The first statement is [Lee02, Proposition 16.6]. | No |
Proposition 78.4. Let \( M \) be an \( n \) -dimensional smooth manifold. Then the following hold:\n\n(1) If \( f : M \rightarrow N \) is a smooth map to another smooth manifold, then\n\n\[ \n{f}^{ * } : {\mathrm{H}}_{\mathrm{{dR}}}^{k}\left( N\right) \rightarrow {\mathrm{H}}_{\mathrm{{dR}}}^{k}\left( M\right)\n\]\n\n\... | (1) The first statement is an immediate consequence of Proposition 78.2 (6) together with Lemma 73.5 (1). | Yes |
Proposition 78.5. Let \( M \) be a smooth manifold without boundary and let \( N \) be any smooth manifold. Let \( f \) and \( g \) be two smooth maps between \( M \) and \( N \) . If there exists a smooth homotopy between \( f \) and \( g \), then for any \( k \in {\mathbb{N}}_{0} \) we have\n\n\[ \n{f}^{ * } = {g}^{ ... | The key to the proof of the proposition is the following lemma.\n\nLemma 78.6. Let \( M \) | No |
Lemma 78.6. Let \( M \) be a smooth manifold without boundary. For \( r = 0,1 \) we denote by \( {i}_{r} : M \rightarrow M \times \left\lbrack {0,1}\right\rbrack \) the inclusion maps given by \( {i}_{r}\left( z\right) = \left( {z, r}\right) \) . Then for any \( k \in {\mathbb{N}}_{0} \) we have\n\n\[ \n{i}_{0}^{ * } =... | Proof of Lemma 7.8.6. Let \( M \) be an \( m \) -dimensional smooth manifold without boundary. By Lemma 73.5 it suffices to show that there exists a cochain homotopy\n\n\[ \n{P}_{ * } : {\mathrm{C}}_{\mathrm{{dR}}}^{ * }\left( {M \times \left\lbrack {0,1}\right\rbrack }\right) \rightarrow {\mathrm{C}}_{\mathrm{{dR}}}^{... | Yes |
Lemma 78.7. (Poincaré Lemma) If \( U \subset {\mathbb{R}}^{n} \) is an open star-shaped set, then\n\n\[{\mathrm{H}}_{\mathrm{{dR}}}^{k}\left( U\right) = \left\{ \begin{array}{ll} \mathbb{R}, & \text{ if }k = 0, \\ 0, & \text{ otherwise. } \end{array}\right.\] | Proof. By definition of a star-shaped subset of \( {\mathbb{R}}^{n} \) there exists an \( x \in U \) such that for every \( y \in U \) the segment \( \{ {xt} + y\left( {1 - t}\right) \mid t \in \left\lbrack {0,1}\right\rbrack \} \) lies in \( U \) . Let \( i : \{ x\} \rightarrow U \) be the inclusion map and let \( p :... | Yes |
Corollary 78.8. If \( M \) is an \( n \) -dimensional smooth manifold that is diffeomorphic to an open star-shaped subset of \( {\mathbb{R}}^{n} \), e.g. that is diffeomorphic to \( {\mathbb{R}}^{n} \) or the open \( n \) -ball \( {B}^{n} \), then | \[ {H}_{\mathrm{{dR}}}^{k}\left( M\right) = \left\{ \begin{array}{ll} \mathbb{R}, & \text{ if }k = 0, \\ 0, & \text{ otherwise. } \end{array}\right. \] | Yes |
Lemma 78.11. Let \( W \) be a smooth manifold, let \( U \subset W \) be an open subset and let \( \varphi \) be a smooth form on \( U \) . We define\n\n\[ \epsilon \left( {\varphi, W}\right) \mathrel{\text{:=}} \text{the form on}W\text{that is given by}\varphi \text{on}U\text{and that is zero on}W \smallsetminus U\text... | Proof. To show that a form \( \mu \) on a smooth manifold \( W \) is smooth it suffices to show that there exists an open cover \( {\left\{ {W}_{i}\right\} }_{i \in I} \) such that the restriction of \( \mu \) to each \( {W}_{i} \) is smooth. In our case we consider the open cover of \( U \) given by \( U \) and by \( ... | Yes |
Lemma 78.12. For any \( n \geq 1 \) we have\n\n\[{\mathrm{H}}_{\mathrm{{dR}}}^{k}\left( {S}^{n}\right) \cong \left\{ \begin{array}{ll} \mathbb{R}, & \text{ if }k = 0, n, \\ 0, & \text{ otherwise. } \end{array}\right.\] | Proof. By induction we will prove the following statement: for any \( n \geq 1 \) we have\n\n\[{\mathrm{H}}_{\mathrm{{dR}}}^{k}\left( {S}^{n}\right) \cong \left\{ {\begin{array}{ll} \mathbb{R}, & \text{ if }k = 0, \\ 0, & \text{ otherwise } \end{array} \oplus \left\{ \begin{array}{ll} \mathbb{R}, & \text{ if }k = n, \\... | Yes |
Proposition 78.13. Let \( M \) be a smooth manifold and let \( {\left\{ {U}_{i}\right\} }_{i \in \mathbb{N}} \) be a sequence of open subsets of \( M \) that satisfy the following three conditions:\n\n(1) for each \( i \in \mathbb{N} \) the closure \( {\bar{U}}_{i} \) of \( {U}_{i} \) is compact,\n\n(2) for each \( i \... | Proof. For each \( k \in {\mathbb{N}}_{0} \) we consider the inverse system\n\n\[ {\mathrm{C}}_{\mathrm{{dR}}}^{k}\left( {U}_{1}\right) \leftarrow {\mathrm{C}}_{\mathrm{{dR}}}^{k}\left( {U}_{2}\right) \leftarrow {\mathrm{C}}_{\mathrm{{dR}}}^{k}\left( {U}_{3}\right) \leftarrow \ldots \]\n\nwhere all the maps are induced... | Yes |
(1) Singular cohomology with real coefficients is a smooth cohomology theory. | (1) It follows from Lemma 73.13, Theorem 74.15, together with Proposition 74.12 and Proposition 76.12 (1), that singular cohomology with real coefficients is a smooth cohomology theory. | Yes |
Theorem 79.2. Let \( {\mathcal{H}}^{ * } \) and \( {\mathcal{K}}^{ * } \) be smooth cohomology theories and let \( t : {\mathcal{H}}^{ * } \rightarrow {\mathcal{K}}^{ * } \) be a natural transformation. Suppose \( t \) has the following three properties:\n\n(i) the natural transformation \( t \) is compatible with the ... | Proof. Let \( {\mathcal{H}}^{ * } \) and \( {\mathcal{K}}^{ * } \) be smooth cohomology theories, which means they both satisfy\n\n\( \left( \alpha \right) \) smooth homotopy invariance,\n\n\( \left( \beta \right) \) the Mayer-Vietoris sequence,\n\n\( \left( \gamma \right) \) the limit property.\n\nFurthermore let \( t... | Yes |
Proposition 79.3. Smooth singular cohomology with real coefficients is a smooth cohomology theory. | SKETCH OF PROOF. We need to verify that smooth singular cohomology with real coefficients satisfies the properties of a smooth cohomology theory.\n\n(0) It is clear that smooth singular cohomology with real coefficients is a contravariant functor from the category of smooth manifolds to the category of \( {\mathbb{N}}_... | No |
Proposition 79.4. There exists a natural isomorphism from smooth singular cohomology with real coefficients to ordinary singular cohomology with real coefficients. In particular for any smooth manifold \( M \) and any \( n \in {\mathbb{N}}_{0} \) we have a natural isomorphism\n\n\[ \n{\mathrm{H}}^{n}\left( {M;\mathbb{R... | Proof. For any smooth manifold \( M \) we have the natural inclusion\n\n\[ \n{\mathrm{C}}_{ * }^{\text{smooth }}\left( M\right) \rightarrow {\mathrm{C}}_{ * }\left( M\right)\n\]\n\nwhich evidently respects the boundary maps. These maps thus give rise to natural maps\n\n\[ \n{\mathrm{C}}^{ * }\left( {M;\mathbb{R}}\right... | Yes |
Lemma 79.5. Let \( M \) be a smooth manifold, let \( \omega \) be a smooth differential \( \left( {n - 1}\right) \) -form on \( M \) and let \( \sigma : {\Delta }^{n} \rightarrow M \) be a smooth singular n-simplex. Then\n\n\[ \n{\int }_{\sigma }{d\omega } = {\int }_{\partial \sigma }\omega \n\] | Sketch of the proof. The idea of the proof is of course to use Stokes’ Theorem 78.3. The fact that \( {\Delta }^{n} \), or equivalently \( {\Delta }_{n} \) are not smooth manifolds in an obvious way is a slight nuisance. Here is one way how we can circumvent this problem. In Lee02, Chapter 16] or alternatively [Wall16,... | No |
Theorem 79.6. (De Rham Theorem) There exists a natural isomorphism \( u = {u}_{M} \) from de Rham cohomology to singular cohomology with real coefficients. In particular for any smooth manifold \( M \) and any \( k \in {\mathbb{N}}_{0} \) we have a natural isomorphism\n\n\[ \n{u}_{M} : {\mathrm{H}}_{\mathrm{{dR}}}^{k}\... | Proof. We denote by \( s \) the natural isomorphism from real-valued singular cohomology to real-valued smooth singular cohomology that we had constructed in the proof of Proposition [79.4]\n\nNow we construct a natural isomorphism \( t \) from de Rham cohomology to real-valued smooth singular cohomology. For any smoot... | Yes |
Corollary 79.8. If \( M \) is a closed, oriented connected, non-empty \( n \) -dimensional smooth manifold, then the map\n\n\[ \n{\mathrm{H}}_{\mathrm{{dR}}}^{n}\left( M\right) \rightarrow \mathbb{R} \]\n\n\[ \n\left\lbrack \omega \right\rbrack \mapsto {\int }_{M}\omega \]\n\nis an isomorphism. | Proof. By Proposition 78.4 (7) the map is well-defined and it is an epimorphism. On the other hand, by Proposition 75.15 and Theorem 79.6 we know that \( {\mathrm{H}}_{\mathrm{{dR}}}^{n}\left( M\right) \cong \mathbb{R} \), which implies that the map is an isomorphism. | Yes |
Lemma 79.13. Let \( M \) be a smooth manifold. For any \( k, l \in {\mathbb{N}}_{0} \) the map\n\n\[ \land : {\mathrm{H}}_{\mathrm{{dR}}}^{k}\left( M\right) \times {\mathrm{H}}_{\mathrm{{dR}}}^{l}\left( M\right) \; \rightarrow \;{\mathrm{H}}_{\mathrm{{dR}}}^{k + l}\left( M\right) \]\n\n\[ \left( {\left\lbrack \omega \r... | Proof. By [Tu11, Proposition 4.7] we know that for any smooth differential \( k \) -form \( \omega \) and any smooth differential \( l \) -form \( \sigma \) we have\n\n\( \left( *\right) \)\n\n\[ d\left( {\omega \land \sigma }\right) = {d\omega } \land \sigma + {\left( -1\right) }^{k} \cdot \omega \land {d\sigma }. \]\... | Yes |
Proposition 79.14. The maps\n\n\[ \nM \mapsto \left( {{\mathrm{H}}_{\mathrm{{dR}}}^{ * }\left( M\right) , \land }\right) \]\n\n\[ \n\left( {f : M \rightarrow N}\right) \mapsto \left( {{f}^{ * } : {\mathrm{H}}_{\mathrm{{dR}}}^{ * }\left( N\right) \rightarrow {\mathrm{H}}_{\mathrm{{dR}}}^{ * }\left( M\right) }\right) \]\... | Proof.\n\n(1) First note that it follows almost immediately from Lemma 78.1 that for a given smooth manifold \( M \) the wedge product turns \( {\mathrm{H}}_{\mathrm{{dR}}}^{ * }\left( M\right) \) into a graded superalgebra. It follows immediately from the definitions that the multiplicatively neutral element in de Rha... | Yes |
Theorem 80.1. (Eilenberg-Zilber)\n\n(1) Given any topological spaces \( X \) and \( Y \) there exist natural chain maps\n\n\[ \Upsilon : {\mathrm{C}}_{ * }\left( X\right) \otimes {\mathrm{C}}_{ * }\left( Y\right) \rightarrow {\mathrm{C}}_{ * }\left( {X \times Y}\right) \]\n\nand\n\n\[ \Theta : {\mathrm{C}}_{ * }\left( ... | 80.2. Proof of the Eilenberg-Zilber Theorem 80.1. The proof of the Eilenberg-Zilber Theorem 80.1 breaks up into three parts: we first construct \( \Upsilon \), then we construct \( \Theta \) and finally we show that these maps are unique in the above sense and that they have the desired properties.\n\nWe start out with... | No |
Theorem 80.2. Given any two topological spaces \( X \) and \( Y \) and any \( p, q \in {\mathbb{N}}_{0} \) there exists a homomorphism \[ {\Upsilon }_{p, q} : {\mathrm{C}}_{p}\left( X\right) \otimes {\mathrm{C}}_{q}\left( Y\right) \rightarrow {\mathrm{C}}_{p + q}\left( {X \times Y}\right) \] such that the following con... | Proof. We prove the existence of the maps \( {\Upsilon }_{p, q} \) by induction on \( p \cdot q \) . For clarity we henceforth suppress the subscript from \( {\Upsilon }_{p, q} \) . Clearly for \( p = 0 \) or \( q = 0 \) the maps from (1) have the properties (2) and (3). So now suppose that we are given \( p, q \in {\m... | No |
Theorem 80.3. Given any two topological spaces \( X \) and \( Y \) and any \( k \in {\mathbb{N}}_{0} \) there exists a homomorphism\n\n\[ \n{\Theta }_{k} : {\mathrm{C}}_{k}\left( {X \times Y}\right) \rightarrow {\left( {\mathrm{C}}_{ * }\left( X\right) \otimes {\mathrm{C}}_{ * }\left( Y\right) \right) }_{k} = {\bigoplu... | In the proof of Theorem 80.3 we will need the following lemma.\n\nLemma 80.4. Let \( X \) | No |
Lemma 80.4. Let \( X \) and \( Y \) be two acyclic topological spaces. For every \( k \geq 1 \) we have\n\n\[ \n{\mathrm{H}}_{k}\left( {{\mathrm{C}}_{ * }\left( X\right) \otimes {\mathrm{C}}_{ * }\left( Y\right) }\right) = 0.\n\]\n\nFurthermore we have an isomorphism\n\n\[ \n\epsilon : {\mathrm{H}}_{0}\left( {{\mathrm{... | Proof. The lemma is an immediate consequence of our hypotheses, Proposition 41.5 and the Künneth Theorem 58.7 for Chain Complexes. (Note that this Künneth Theorem is purely algebraic.) | No |
Proposition 80.6. Let \( X \) and \( Y \) be topological spaces and let \( A \subset X \) and \( B \subset Y \) be subsets.\n\n(1) The Eilenberg-Zilber map \( \Upsilon \) induces a natural chain homotopy equivalence\n\n\[ \n{\mathrm{C}}_{ * }\left( {X, A}\right) \otimes {\mathrm{C}}_{ * }\left( {Y, B}\right) \rightarro... | Proof. Let \( X \) and \( Y \) be topological spaces and let \( A \subset X \) and \( B \subset Y \) be subsets.\n\n(1) We consider the following diagram\n\n\[ \n\begin{array}{l} 0 \rightarrow {\mathrm{C}}_{ * }\left( A\right) \otimes {\mathrm{C}}_{ * }\left( Y\right) \rightarrow {\mathrm{C}}_{ * }\left( X\right) \otim... | Yes |
Theorem 80.8. (The Künneth Theorem) Let \( \left( {X, A}\right) \) and \( \left( {Y, B}\right) \) be pairs of topological spaces. We assume that one of the following two statements holds:\n\n(1) \( A \) and \( B \) are open subsets (note that \( A \) and \( B \) can possibly be empty), or\n\n(2) \( \left( {X, A}\right)... | Proof. We consider the following maps\n\n\[ \begin{matrix} 0 \rightarrow \mathop{\bigoplus }\limits_{{p + q = n}}{\mathrm{H}}_{p}\left( {X, A}\right) \otimes {\mathrm{H}}_{q}\left( {Y, B}\right) \rightarrow {\mathrm{H}}_{n}\left( {{\mathrm{C}}_{ * }\left( {X, A}\right) \otimes {\mathrm{C}}_{ * }\left( {Y, B}\right) }\r... | No |
(1) Let \( M \) be an \( m \) -dimensional topological manifold and let \( N \) be an \( n \) -dimensional topological manifold. The topological space \( M \times N \) is an \( \left( {m + n}\right) \) -dimensional topological manifold where the boundary is given by\n\n\[ \partial \left( {M \times N}\right) = \left( {\... | Proof. Let \( M \) be an \( m \) -dimensional topological manifold and let \( N \) be an \( n \) -dimensional topological manifold. We consider the topological space \( M \times N \) . By Proposition 3.12 we remarked that the product of two Hausdorff spaces is again Hausdorff. Thus we see that \( M \times N \) is Hausd... | Yes |
Proposition 80.10. Let \( M \) be an \( m \) -dimensional topological manifold and let \( N \) be an \( n \) -dimensional topological manifold. Suppose \( M \) is equipped with an orientation \( {\left\{ {\mu }_{x}\right\} }_{x \in M \smallsetminus \partial M} \) and that \( N \) is equipped with an orientation \( {\le... | Proof. Let \( M \) be an \( m \) -dimensional topological manifold and let \( N \) be an \( n \) -dimensional topological manifold.\n\n(1) The first statement of the proposition is a straightforward consequence of the definitions and the naturality of the cross product, see Lemma 80.7. For completeness' sake we sketch ... | Yes |
Proposition 80.12. The maps \( \nabla \) that we had just defined have all the properties that we had mentioned in Theorem 80.2, i.e. the maps \( \nabla : {\mathrm{C}}_{ * }\left( X\right) \otimes {\mathrm{C}}_{ * }\left( Y\right) \rightarrow {\mathrm{C}}_{ * }\left( {X \times Y}\right) \) are natural chain maps that o... | Proof. It follows immediately from the definition that the maps \( {\nabla }_{mn} \) are natural and that on the 0-level they agree with the canonical maps. It is elementary (albeit rather painful) to verify that these maps are chain maps. We refer to [Dol56, p. 184] and [EM53] for details and proofs. | No |
Theorem 80.13. (Acyclic Model Theorem) Let Top be the category of topological spaces and let \( \mathcal{C} \) be the category of generalized \( {}^{1156} \) chain complexes. Let \( G : \) Top \( \rightarrow \mathcal{C} \) be a covariant functor. Furthermore let \( S \) be a natural transformation from the functor \( X... | SKETCH OF PROOF. The proof of this theorem is very similar to the \ | No |
Proposition 80.14. Let \( {C}_{ * } \) and \( {D}_{ * } \) be two chain complexes. If \( {C}_{ * } \) and \( {D}_{ * } \) are almost acyclic and if \( {C}_{ * } \) is free, i.e. if each chain group \( {C}_{n} \) is a free abelian group, then \( {C}_{ * } \otimes {D}_{ * } \) is also almost acyclic. | Proof. Let \( n \in {\mathbb{N}}_{0} \) . Since \( {C}_{ * } \) is free we can apply the Künneth Theorem for Chain Complexes 58.7 which gives us the following short exact sequence:\n\n\[ 0 \rightarrow {\bigoplus }_{p + q = n}{\mathrm{H}}_{p}\left( {C}_{ * }\right) \otimes {\mathrm{H}}_{q}\left( {D}_{ * }\right) \overse... | Yes |
Lemma 80.15. Let \( \\left( {\\left( {{\\mathrm{C}}_{ * },{\\partial }_{ * }}\\right) ,\\epsilon }\\right) \) be an augmented chain complex. If the chain complex \( \\left( {{\\mathrm{C}}_{ * },{\\partial }_{ * }}\\right) \) is almost acyclic, then the augmented chain complex \( \\left( {\\left( {{\\mathrm{C}}_{ * },{\... | Proof \( \\left( *\\right) \) . We denote by \( {\\widetilde{C}}_{ * } \) the generalized chain complex\n\n\\[ \n\\ldots \\overset{{\\partial }_{3}}{ \\rightarrow }{\\mathrm{C}}_{2}\\overset{{\\partial }_{2}}{ \\rightarrow }{\\mathrm{C}}_{1}\\overset{{\\partial }_{1}}{ \\rightarrow }{\\mathrm{C}}_{0}\\overset{\\epsilon... | Yes |
Theorem 80.16. (Acyclic Model Theorem) Let \( \mathcal{K} \) be a category with a collection of objects \( \mathcal{M} \) . Furthermore let \( F, G : \mathcal{K} \rightarrow \mathcal{A} \) be two covariant functors from \( \mathcal{K} \) to the category of augmented chain complexes. If \( F \) is free relative to \( \m... | Proof. The proof is, ultimately very much the same as the proof of Theorem 80.5. We decline to provide a proof and we follow the lead given by Munkres [Mun84, p. 185], namely like Munkres we assign the proof as an exercise to the reader. Alternatively the reader can find the proof in [Spa95, p. 169], [Vic94, Theorem 5.... | No |
Proposition 80.17. Given any two diagonal approximations \( \Phi \) and \( \Psi \) there exists a natural chain homotopy equivalence between \( \Phi \) and \( \Psi \) . | Proof. We let \( \mathcal{K} \) be the category of non-empty topological spaces. As on page 1976 we consider the collection\n\n\[ \mathcal{M} \mathrel{\text{:=}} \left\{ {{\Delta }^{k} \mid k \in {\mathbb{N}}_{0}}\right\} = \text{the set of all standard simplices.} \]\n\nWe consider the covariant functors\n\n\[ \begin{... | Yes |
Lemma 81.1. Let \( {\mathcal{C}}_{ * } \) and \( {\mathcal{D}}_{ * } \) be chain complexes. For any commutative ring \( R \) the map\n\n\[ \n{\mathrm{H}}^{p}\left( {{\mathcal{C}}_{ * };R}\right) \times {\mathrm{H}}^{q}\left( {\mathcal{D};R}\right) \rightarrow {\mathrm{H}}^{p + q}\left( {{\mathcal{C}}_{ * } \otimes {\ma... | Proof. We leave the verification of the lemma to the reader. | No |
Lemma 81.3. Let \( X \) be a topological space and let \( R \) be a commutative ring. Then the map \[ \cup : {\mathrm{H}}^{p}\left( {X;R}\right) \times {\mathrm{H}}^{q}\left( {X;R}\right) \rightarrow {\mathrm{H}}^{p + q}\left( {X;R}\right) \] \[ \left( {\left\lbrack \varphi \right\rbrack ,\left\lbrack \psi \right\rbrac... | Proof. Let \( \varphi \in {\mathrm{C}}^{p}\left( {X;R}\right) \) and \( \psi \in {\mathrm{C}}^{q}\left( {X;R}\right) \) be cocycles. (1) It follows immediately from Lemma 81.2 that \( \varphi \cup \psi \) is again a cocycle. (2) If \( {\varphi }^{\prime } = \varphi + {\delta \tau } \) is another representative of \( \l... | Yes |
Proposition 81.6.\n\n(1) There exist diagonal approximations.\n\n(2) Given any two diagonal approximations \( \Phi \) and \( \Psi \) there exists a natural chain homotopy equivalence between \( \Phi \) and \( \Psi \) . | Proof.\n\n(1) We had just given two examples of diagonal approximations.\n\n(2) The uniqueness statement was proved in Proposition 81.6. | Yes |
Proposition 81.7. Let \( X \) be a topological space and let \( R \) be a commutative ring.\n\n(1) The cup product on \( {\mathrm{H}}^{ * }\left( {X;R}\right) \) is \( R \) -bilinear and associative. | Proof. Let \( X \) be a topological space.\n\n(1) It is clear from the second definition of the cup product that it is \( R \) -bilinear. It is also almost obvious, using the second definition, that the cup product is associative, but for completeness’ sake we carry out the argument. Thus let \( \alpha \in {\mathrm{C}}... | Yes |
Proposition 81.8. Let \( X \) be a topological space and let \( R \) be a commutative ring. For any \( a \in {\mathrm{H}}^{k}\left( {X;R}\right) \) and \( b \in {\mathrm{H}}^{l}\left( {X;R}\right) \) we have\n\n\[ a \cup b = {\left( -1\right) }^{kl} \cdot b \cup a. \] | Proof (*). We give a proof of the proposition following [Hat02, p. 216-217]. In Hatcher's book more motivations for the steps in the proof are given.\n\nSo let \( \varphi \in {\mathrm{C}}^{k}\left( {X;R}\right) \) and \( \psi \in {\mathrm{C}}^{l}\left( {X;R}\right) \) be two cocycles. Given a singular \( n \) -simplex ... | Yes |
Lemma 81.9. Let \( X \) be a topological space. For any \( \varphi \in {\mathrm{H}}^{1}\left( {X;\mathbb{Z}}\right) \) we have \( \varphi \cup \varphi = 0 \) . | Proof. Let \( X \) be a topological space and let \( \varphi \in {\mathrm{C}}^{1}\left( {X;\mathbb{Z}}\right) \) be a cocycle. We need to show that \( \varphi \cup \varphi \in {\mathrm{C}}^{2}\left( {X;\mathbb{Z}}\right) \) is a coboundary. We consider the 1-dimensional cochain\n\n\[ \mu : {\mathrm{C}}_{1}\left( X\righ... | Yes |
Let \( R \) be a commutative ring.\n\n(1) Let \( f : X \rightarrow Y \) be a map between topological spaces. For any \( c \in {\mathrm{H}}^{p}\left( {Y;R}\right) \) and \( d \in {\mathrm{H}}^{q}\left( {Y;R}\right) \) we have\n\n\[ \n{f}^{ * }\left( c\right) \cup {f}^{ * }\left( d\right) = {f}^{ * }\left( {c \cup d}\rig... | The lemma follows basically immediately from the definitions. But for completeness’ sake we write down the proof. So let \( c = \left\lbrack \varphi \right\rbrack \in {\mathrm{H}}^{p}\left( {Y;R}\right) \) and let \( b = \left\lbrack \psi \right\rbrack \in {\mathrm{H}}^{q}\left( {Y;R}\right) \) . Furthermore let \( \si... | Yes |
Lemma 81.11. For any wedge of finitely many spheres all cup products in degrees \( \geq 1 \) are zero. | Proof. Let \( {S}^{{r}_{1}} \vee \cdots \vee {S}^{{r}_{k}} \) be the wedge of finitely many spheres and let \( R \) be a commutative ring. For each \( i \in \{ 1,\ldots, k\} \) we denote by \( {p}_{i} : {S}^{{r}_{1}} \vee \cdots \vee {S}^{{r}_{k}} \rightarrow {S}^{{r}_{i}} \) the obvious projection map. By Proposition ... | Yes |
Corollary 81.12. Let \( X \) and \( Y \) be topological spaces. If \( X \) and \( Y \) are homotopy equivalent, then for any commutative ring \( R \) there exists an isomorphism\n\n\[ \left( {{\mathrm{H}}^{ * }\left( {Y;R}\right) , \cup }\right) \overset{ \cong }{ \rightarrow }\left( {{\mathrm{H}}^{ * }\left( {X;R}\rig... | Proof. Let \( f : X \rightarrow Y \) be a homotopy equivalence. In Lemma 73.13 we saw that the induced maps \( {f}^{ * } : {\mathrm{H}}^{n}\left( {Y;R}\right) \rightarrow {\mathrm{H}}^{n}\left( {X;R}\right) \) on cohomology groups are isomorphisms. But it follows from Lemma 81.10 that these isomorphisms preserve the ri... | Yes |
Lemma 81.14. Let \( T \) be the 2-dimensional torus. We use the above notation. The cup product \( {\mathrm{H}}^{1}\left( {T;\mathbb{Z}}\right) \times {\mathrm{H}}^{1}\left( {T;\mathbb{Z}}\right) \rightarrow {\mathrm{H}}^{2}\left( {T;\mathbb{Z}}\right) \) with respect to the basis \( \alpha ,\beta \) of \( {\mathrm{H}}... | \[ \left( \begin{array}{ll} \alpha \cup \alpha & \overline{\alpha \cup \beta } \\ \beta \cup \alpha & \beta \cup \beta \end{array}\right) = \left( \begin{array}{rr} 0 & 1 \\ - 1 & 0 \end{array}\right) \] Thus we have shown that the cup product on cohomology is in general non-zero and that it is in general not commutati... | Yes |
Lemma 81.16. Let \( \sum \) be the surface of genus 2. We use the above notation. The cup product \( {\mathrm{H}}^{1}\left( {\sum ;\mathbb{Z}}\right) \times {\mathrm{H}}^{1}\left( {\sum ;\mathbb{Z}}\right) \rightarrow {\mathrm{H}}^{2}\left( {\sum ;\mathbb{Z}}\right) \) with respect to the basis \( {\alpha }_{1},{\beta ... | \[ \left( \begin{matrix} {\alpha }_{1} \cup {\alpha }_{1} & {\alpha }_{1} \cup {\beta }_{1} & {\alpha }_{1} \cup {\alpha }_{2} & {\alpha }_{1} \cup {\beta }_{2} \\ {\beta }_{1} \cup {\alpha }_{1} & {\beta }_{1} \cup {\beta }_{1} & {\beta }_{1} \cup {\alpha }_{2} & {\beta }_{1} \cup {\beta }_{2} \\ {\alpha }_{2} \cup {\... | Yes |
(1) The torus \( {\sum }_{1} = {S}^{1} \times {S}^{1} \) is flexible. | (1) We first show that the torus \( {S}^{1} \times {S}^{1} \) is flexible. So let \( k \in \mathbb{Z} \) . We consider the map \( f : {S}^{1} \rightarrow {S}^{1} \) that is given by \( z \mapsto {z}^{k} \) . We claim that \( f \times \mathrm{{id}} : {S}^{1} \times {S}^{1} \rightarrow {S}^{1} \times {S}^{1} \) has degre... | Yes |
Lemma 81.18. Under the identifications \( {\mathrm{H}}^{i}\left( {{\mathbb{{RP}}}^{2};{\mathbb{F}}_{2}}\right) = {\mathbb{F}}_{2} \) the cup product\n\n\[ \cup : {\mathrm{H}}^{1}\left( {{\mathbb{{RP}}}^{2};{\mathbb{F}}_{2}}\right) \times {\mathrm{H}}^{1}\left( {{\mathbb{{RP}}}^{2};{\mathbb{F}}_{2}}\right) \rightarrow {... | Proof. By the discussion above it suffices to show that \( \eta \left( {\left\lbrack \varphi \right\rbrack \cup \left\lbrack \varphi \right\rbrack }\right) \neq 0 \) . Put differently, by definition of \( \eta \) we need to show that\n\n\[ \left( {\varphi \cup \varphi }\right) \left( {\sigma }_{1}\right) + \left( {\var... | No |
Lemma 82.1. Let \( \\left( {X, A, B}\\right) \) be an excisive triad of topological spaces. Given any commutative ring \( R \) the natura\n\n\[ \n\\overline{{\\mathrm{C}}^{ * }}\\left( {X, A \\cup B;R}\\right) \\; \\rightarrow \\;{\\mathrm{C}}^{ * }\\left( {X,\\{ A, B\\} ;R}\\right) \n\]\n\ninduces for every \( n \\in ... | Proof. We have the following commutative diagram of short exact sequences of chain complexes\n\n\[ \n\\begin{array}{l} 0 \\rightarrow {\\mathrm{C}}_{ * }^{\\{ A, B\\} }\\left( {A \\cup B}\\right) \\rightarrow {\\mathrm{C}}_{ * }\\left( X\\right) \\rightarrow \\frac{{\\mathrm{C}}_{ * }\\left( X\\right) }{{\\mathrm{C}}_{... | Yes |
Lemma 82.2. Let \( \\left( {X, A, B}\\right) \) be a triad of topological spaces and let \( R \) be a commutative ring. Then the map\n\n\[ \n{\\mathrm{C}}^{p}\\left( {X, A;R}\\right) \\times {\\mathrm{C}}^{q}\\left( {X, B;R}\\right) \\rightarrow {\\mathrm{C}}^{p + q}\\left( {X,\\{ A, B\\} ;R}\\right)\n\]\n\n\[ \n\\left... | Proof. The lemma follows easily from the following claim.\n\nClaim. Given \( \\varphi \\in {\\mathrm{C}}^{p}\\left( {X, A;R}\\right) \) and \( \\psi \\in {\\mathrm{C}}^{q}\\left( {X, B;R}\\right) \) the homomorphism\n\n\[ \n{\\mathrm{C}}_{p + q}\\left( X\\right) \\rightarrow R\n\]\n\n\[ \n\\left( {\\sigma : {\\Delta }^... | Yes |
Proposition 82.3. Let \( \\left( {X, A, B}\\right) \) be an excisive triad of topological spaces and let \( R \) be a commutative ring. For every \( \\varphi \\in {\\mathrm{H}}^{k}\\left( {X, A;R}\\right) \) and \( \\psi \\in {\\mathrm{H}}^{l}\\left( {X, B;R}\\right) \) we have\n\n\[ \n\\varphi \\cup \\psi = {\\left( -... | Proof. In Exercise 82.1 we will see that a modification of the proof of Proposition 81.8 provides us with a proof of Proposition 82.3. | No |
Corollary 82.5. Let \( \left( {X, A, B}\right) \) be an excisive triad and let \( R \) be a commutative ring. Then the following diagram commutes:\n\n\[ \n{\mathrm{H}}^{p}\left( {X, A;R}\right) \times {\mathrm{H}}^{q}\left( {X, B;R}\right) \overset{ \cup }{ \rightarrow }{\mathrm{H}}^{p + q}\left( {X, A \cup B;R}\right)... | Proof. The first part of the corollary follows immediately from Proposition 82.4 (1) applied to the obvious map \( f : \left( {X,\varnothing ,\varnothing }\right) \rightarrow \left( {X, A, B}\right) \) of excisive triads.\n\nWe turn to the proof of the second part. We denote by \( \sigma : {\mathrm{H}}^{p}\left( {X, A;... | Yes |
Lemma 82.6. Let \( \left( {X, A}\right) \) be a pair of topological spaces and let \( R \) be a commutative ring. We denote by \( i : A \rightarrow X \) the inclusion map and we denote by \( \delta \) the connecting homomorphisms in the long exact sequence in cohomology corresponding to the pair \( \left( {X, A}\right)... | Proof \( \left( *\right) \) . For no particular reason we prove the second statement. The first statement can either be proved the same way or it can be deduced from the second statement using Proposition 82.3. We pick cocycles \( \varphi \in {\mathrm{C}}^{p}\left( {X;R}\right) \) and \( \psi \in {\mathrm{C}}^{q}\left(... | Yes |
Lemma 82.7. Let \( X \) be a path-connected topological space and let \( R \) be a commutative ring.\n\n(1) We have\n\n\[ \n{\mathrm{H}}^{k}\left( {\sum \left( X\right) ;R}\right) \cong \left\{ \begin{array}{ll} R, & \text{ if }k = 0, \\ 0, & \text{ if }k = 1, \\ {\mathrm{H}}^{k - 1}\left( {X;R}\right) , & \text{ if }k... | Proof. We denote by \( p : X \times \left\lbrack {-1,1}\right\rbrack \rightarrow \sum \left( X\right) \) the obvious projection map. We denote by \( N \mathrel{\text{:=}} \left\lbrack {X\times \{ 1\} }\right\rbrack \) the \ | No |
Lemma 82.8. We continue with the above notation. Then for any \( \alpha ,\beta \in {\mathrm{H}}^{ * }\left( {X;R}\right) \) and \( \varphi ,\psi \in {\mathrm{H}}^{ * }\left( {Y;R}\right) \) in degrees \( \geq 1 \) the following statements hold in \( {\mathrm{H}}^{ * }\left( {X \vee Y;R}\right) \) :\n\n\[ \begin{matrix}... | Proof (*). The statement regarding the \ | No |
Lemma 83.1. Let \( X \) be a topological space and let \( R \) be a commutative ring. Furthermore let \( \varphi \in {\mathrm{C}}^{k}\left( {X;R}\right) \) and let \( \sigma \in {\mathrm{C}}_{l}\left( {X;R}\right) \) . Then\n\n\[ \partial \left( {\varphi \cap \sigma }\right) = {\left( -1\right) }^{k} \cdot \left( {-{\d... | Proof. For \( k > l \) the statement is trivial. So assume that \( k \leq l \) . Clearly it suffices to prove the equality for the case that \( \sigma : {\Delta }^{l} \rightarrow X \) is a singular \( l \) -simplex. This case follows from an elementary calculation. Indeed, we have\n\n(a)\n\n\[ \varphi \cap \partial \si... | Yes |
Lemma 83.2. Let \( X \) be a topological space and let \( R \) be a commutative ring. Then for any \( k, l \in {\mathbb{N}}_{0} \) the map\n\n\[ \n{\mathrm{H}}^{k}\left( {X;R}\right) \times {\mathrm{H}}_{l}\left( {X;R}\right) \rightarrow {\mathrm{H}}_{l - k}\left( {X;R}\right)\n\]\n\n\[ \n\left( {\left\lbrack \varphi \... | Proof. For \( k > l \) the statement is trivial. So assume that \( k \leq l \) . The statement now follows easily from Lemma 83.1. Indeed, let \( \varphi \in {\mathrm{C}}^{k}\left( {X;R}\right) \) be a cocycle and let \( \sigma \in {\mathrm{C}}_{l}\left( {X;R}\right) \) be a cycle.\n\n(1) It follows from Lemma 83.1 tha... | Yes |
Lemma 83.4. Let \( X \) be a topological space and let \( R \) be a commutative ring.\n\n(1) The composition\n\n\[ \n{\mathrm{H}}^{k}\left( {X;R}\right) \times {\mathrm{H}}_{k}\left( {X;R}\right) \overset{ \cap }{ \rightarrow }{\mathrm{H}}_{0}\left( {X;R}\right) \xrightarrow[\text{ from page }]{\text{ augmentation }\ep... | Proof. Both statements follow immediately from recalling all the relevant definitions. | No |
Lemma 83.5. Let \( \\left( {X, A, B}\\right) \) be a triad of topological spaces, let \( k \\leq l \) and let \( R \) be a commutative ring. Then the map\n\n\[ \n\\cap : {\\mathrm{C}}^{k}\\left( {X, A;R}\\right) \\times {\\mathrm{C}}_{l}\\left( {X,\\{ A, B\\} ;R}\\right) \\rightarrow {\\mathrm{C}}_{l - k}\\left( {X, B;... | Proof. We first show that the cap product on (co-) chains is well-defined. By definition\n\nof\n\n\[ \n{\\mathrm{C}}_{n}\\left( {X,\\{ A, B\\} ;R}\\right) \\mathrel{\\text{:=}} \\frac{{\\mathrm{C}}_{n}\\left( X\\right) \\otimes R}{{\\mathrm{C}}_{n}^{\\{ A, B\\} }\\left( {A \\cup B}\\right) \\otimes R} = \\frac{{\\mathr... | Yes |
Lemma 83.6. If \( \left( {X, A, B}\right) \) is an excisive triad of topological spaces, then given any commutative ring \( R \) the obvious map\n\n\[ \n{\mathrm{C}}_{ * }\left( {X,\{ A, B\} ;R}\right) \; \rightarrow \;{\mathrm{C}}_{ * }\left( {X, A \cup B;R}\right) \n\]\n\ninduces for every \( n \in {\mathbb{N}}_{0} \... | Proof. In the proof of Lemma 82.1 we already showed that the map\n\n\[ \n{\mathrm{C}}_{ * }\left( {X,\{ A, B\} ;\mathbb{Z}}\right) \rightarrow {\mathrm{C}}_{ * }\left( {X, A \cup B;\mathbb{Z}}\right) \n\]\n\non chain complexes is a chain homotopy equivalence. By tensoring all maps and chain homotopies with \( R \) we s... | Yes |
Lemma 83.7. Let \( X \) be a topological space and let \( R \) be a commutative ring.\n\n(1) Let \( \varphi \in {\mathrm{C}}^{k}\left( {X;R}\right) ,\psi \in {\mathrm{C}}^{l}\left( {X;R}\right) \) and let \( \sigma \in {\mathrm{C}}_{n}\left( {X;R}\right) \) . Then\n\n\[ \varphi \cap \left( {\psi \cap \sigma }\right) = ... | Proof. The case \( k + l > n \) is trivial. So suppose that \( k + l \leq n \).\n\n(1) So let \( \varphi \in {\mathrm{C}}^{k}\left( {X;R}\right) ,\psi \in {\mathrm{C}}^{l}\left( {X;R}\right) \) . It suffices to consider the case that \( \sigma : {\Delta }^{n} \rightarrow X \) is a singular \( n \) -simplex. We have\n\n... | Yes |
Lemma 83.8. Let \( R \) be a commutative ring.\n\n(1) If \( f : \left( {X, A, B}\right) \rightarrow \left( {\widetilde{X},\widetilde{A},\widetilde{B}}\right) \) is a map between two excisive triads, then for any \( k, l \in {\mathbb{N}}_{0} \) the following diagram commutes \( {}^{1201} \)\n\n\[ \n{\mathrm{H}}^{k}\left... | Proof. The first statement follows immediately from the definitions. We leave the details to the skeptical reader. The second statement is an immediate consequence of the first statement. | No |
Lemma 83.10. Let \( R \) be a commutative ring.\n\n(1) Let \( \left( {X, B, A}\right) \) be a triple of topological spaces. For any \( n, k \in {\mathbb{N}}_{0} \) the following diagram commutes:\n\n\[ \begin{matrix} {\mathrm{H}}^{k}\left( {B, A;R}\right) \times {\mathrm{H}}_{n - 1}\left( {B, A;R}\right) \xrightarrow[]... | Proof \( \left( *\right) \) .\n\n(1) We pick \( \psi \in {\mathrm{H}}^{k}\left( {B, A;R}\right) \) and \( \tau \in {\mathrm{H}}_{n}\left( {B, A;R}\right) \) . We make the following observations:\n\n(a) We can find \( \sigma \in {\mathrm{C}}_{n}\left( {B;R}\right) \) such that the image of \( \sigma \) in \( {\mathrm{C}... | No |
Lemma 83.11. Let \( M \) be a topological manifold. We denote by \( r : \mathrm{D}M \rightarrow M \) the folding map from page 1164 and given \( l \in {\mathbb{N}}_{0} \) we denote by \( D : {\mathrm{H}}_{l}\left( {M,\partial M;\mathbb{Z}}\right) \rightarrow {\mathrm{H}}_{l}\left( {\mathrm{D}M;\mathbb{Z}}\right) \) the... | Proof. We will show that the desired equality actually holds already on the (co-) chain level. So let \( \phi \in {\mathrm{C}}^{k}\left( {M;\mathbb{Z}}\right) \) and let \( \sigma : {\Delta }^{l} \rightarrow M \) be a singular \( l \) -simplex. If \( k > l \), then both sides of the lemma are trivially zero. Hence we c... | Yes |
Proposition 84.2. Let \( X \) be a compact oriented connected \( m \) -dimensional topological manifold and let \( Y \) be a compact oriented connected \( n \) -dimensional topological manifold. | We denote by \( p : X \times Y \rightarrow X \) and \( q : X \times Y \rightarrow Y \) the obvious projection maps. Then the following two statements hold:\n\n(1)\n\n\[ \underset{ \in {\mathrm{H}}^{m}\left( {X \times Y,\partial X \times Y}\right) }{\underbrace{{p}^{ * }\left( {\left\lbrack X\right\rbrack }^{ * }\right)... | Yes |
The cohomology of \( {S}^{m} \times {S}^{n} \) is given as follows:\n\n\[ \n{H}^{ * }\left( {{S}^{m} \times {S}^{n};\mathbb{Z}}\right) = \mathbb{Z} \cdot 1 \oplus \mathbb{Z} \cdot {p}^{ * }\left( {\left\lbrack {S}^{m}\right\rbrack }^{ * }\right) \oplus \mathbb{Z} \cdot {q}^{ * }\left( {\left\lbrack {S}^{n}\right\rbrack... | (1) Using Proposition 43.4, the Künneth Theorem 58.8 for topological spaces and the Universal Coefficient Theorem 75.13 for Cohomology Groups one can easily determine the isomorphism type of \( {\mathrm{H}}^{ * }\left( {{S}^{m} \times {S}^{n};\mathbb{Z}}\right) \). Furthermore, using Lemma 74.7 and the homology classes... | Yes |
Lemma 84.4. The topological spaces \( {S}^{2} \vee {S}^{2} \vee {S}^{4} \) and \( {S}^{2} \times {S}^{2} \) are not even homotopy equivalent. | Proof. We had just seen in Lemma 84.3 that there exist cohomology classes \( \alpha \) and \( \beta \) in \( {\mathrm{H}}^{2}\left( {{S}^{2} \times {S}^{2};\mathbb{Z}}\right) \) with \( \alpha \cup \beta \neq 0 \) . On the other hand in Lemma 81.11 we saw that the cup product\n\n\[ \cup : {\mathrm{H}}^{2}\left( {{S}^{2... | Yes |
Lemma 84.5. There is no map \( f : {S}^{4} \rightarrow {S}^{2} \times {S}^{2} \) of non-zero degree. | Proof. Let \( f : {S}^{4} \rightarrow {S}^{2} \times {S}^{2} \) be a map. We have to show that \( \deg \left( f\right) = 0 \) . By Lemma 75.17 it suffices to show that \( {f}^{ * }\left( {\left\lbrack {S}^{2} \times {S}^{2}\right\rbrack }^{ * }\right) = 0 \) . In Lemma 84.3 we showed that there exist cohomology classes... | Yes |
Lemma 84.6. Let \( {f}_{i} : {\mathrm{C}}_{ * }^{i} \rightarrow {D}_{ * }^{i}, i = 1,\ldots, k \) be chain maps between chain complexes. Then the map\n\n\[ \n{f}_{1} \otimes \cdots \otimes {f}_{k} : {\mathrm{C}}_{ * }^{1} \otimes \cdots \otimes {\mathrm{C}}_{ * }^{k} \rightarrow {D}_{ * }^{1} \otimes \cdots \otimes {D}... | Proof. Since in the proof of the Product Theorem 84.1 there are lots of steps later on which can make us nervous, let us convince ourselves that the statement is indeed correct. We have\n\n\[ \n\partial \left( {\left( {{f}_{1} \otimes \ldots \otimes {f}_{k}}\right) \left( {{c}_{1} \otimes \ldots \otimes {c}_{k}}\right)... | Yes |
Lemma 84.7. Let \( {A}_{ * } \) and \( {B}_{ * } \) be chain complexes. The unique map\n\n\[ \n\tau : {A}_{ * } \otimes {B}_{ * } \rightarrow {B}_{ * } \otimes {A}_{ * }\n\]\n\nthat for \( a \in {A}_{k} \) and \( b \in {B}_{l} \) is given by\n\n\[ \na \otimes b \mapsto {\left( -1\right) }^{kl} \cdot b \otimes a\n\]\n\n... | Proof. The difficult bit about the lemma is coming up with the statement. The verification is now almost trivial. So let \( a \in {A}_{k} \) and \( b \in {B}_{l} \) . Then we have\n\n\[ \n\tau \left( {\partial \left( {a \otimes b}\right) }\right) = \tau \left( {\partial a \otimes b + {\left( -1\right) }^{k} \cdot a \ot... | Yes |
Lemma 84.8. Let \( {C}_{ * } \) be a chain complex and let \( k \in {\mathbb{N}}_{0} \) . The following statements hold:\n\n(1) Let \( \varphi : {\mathrm{C}}_{k} \rightarrow \mathbb{Z} \) be a cocycle. The map \( {\Pi }_{k}\left( \varphi \right) : {C}_{ * } \rightarrow \mathbb{Z}\left\lbrack k\right\rbrack \) is a chai... | (1) Let \( \varphi : {\mathrm{C}}_{k} \rightarrow \mathbb{Z} \) be a cocycle. We consider the following diagram:\n\n\n\nThe hypothesis that \( \varphi \) is a cocycle means that \( {\delta \varphi } = \varphi \circ... | Yes |
Lemma 84.9. Let \( {C}_{ * } \) be a chain complex and let \( k \in \mathbb{Z} \) . The maps\n\n\[ \n{\eta }_{m} : {\left( \mathbb{Z}\left\lbrack k\right\rbrack \otimes {\mathrm{C}}_{ * }\right) }_{m} = \mathbb{Z} \otimes {\mathrm{C}}_{m - k}\; \rightarrow \;{\left( {\mathrm{C}}_{ * }\left\lbrack k\right\rbrack \right)... | Proof. It follows from Lemma 57.3 (3) that each \( {\eta }_{m} \) is an isomorphism. But we still need to verify that the maps \( {\eta }_{m} \) form a chain map \( \eta : \mathbb{Z}\left\lbrack k\right\rbrack \otimes {C}_{ * } \rightarrow {\mathrm{C}}_{ * }\left\lbrack k\right\rbrack \) . Thus let \( k \in {\mathbb{N}... | Yes |
Lemma 84.10. Let \( W \) be a topological space and let \( \alpha \in {\mathrm{C}}^{k}\left( {W;\mathbb{Z}}\right) \) and \( \beta \in {\mathrm{C}}^{l}\left( {W;\mathbb{Z}}\right) \) be cocycles. Then we have the following equality of chain maps:\n\n\[ \n{\left( -1\right) }^{kl} \cdot {\Pi }_{k + l}\left( \underbrace{\... | Proof. We introduce the following notation:\n\n(1) We denote by \( \mu : \mathbb{Z} \otimes \mathbb{Z} \rightarrow \mathbb{Z} \) the obvious isomorphism of abelian groups given by \( \mu \left( {a \otimes b}\right) = a \cdot b. \)\n\n(2) We denote by \( {\Pi }_{k, l} : {\left( {\mathrm{C}}_{ * }\left( W\right) \otimes ... | Yes |
Proposition 84.12. Let \( W \) be a topological space and let \( \varphi \in {\mathrm{C}}^{m}\left( {W;\mathbb{Z}}\right) \) be a cocycle. Then we have the following equality of chain maps\n\n\[ \varphi \widetilde{ \circ } - = \eta \circ \left( {{\Pi }_{m}\left( \varphi \right) \otimes \mathrm{{id}}}\right) \circ {\Del... | Proof. Let \( W \) be a topological space, let \( \varphi \in {\mathrm{C}}^{m}\left( W\right) \) be a cocycle and let \( \sigma : {\Delta }^{n} \rightarrow W \) be a singular \( n \) -simplex. We have the following equalities in \( {\mathrm{C}}_{ * }\left( W\right) {\left\lbrack m\right\rbrack }_{n} = {\mathrm{C}}_{n -... | Yes |
Lemma 84.14. Let \( {C}_{ * } \) and \( {D}_{ * } \) be chain complexes and let \( k, l \in {\mathbb{N}}_{0} \) . The map\n\n\[ \n{\Psi }_{k, l} : {\mathrm{C}}_{ * }\left\lbrack k\right\rbrack \otimes {D}_{ * }\left\lbrack l\right\rbrack \rightarrow \left( {{C}_{ * } \otimes {D}_{ * }}\right) \left\lbrack {k + l}\right... | Proof. First note that it is clear that \( {\Psi }_{k, l} \) preserves degrees and that on each degree it is an isomorphism. It remains to show that the map \( {\Psi }_{k, l} \) is a chain map. Let \( {c}_{m} \in {\mathrm{C}}_{m} \) and \( {d}_{n} \in {D}_{n} \) . We calculate that\n\nsince \( {c}_{m} \) has degree \( ... | Yes |
Lemma 84.18. Let \( {\mathcal{C}}_{ * } \) and \( {\mathcal{C}}_{ * }^{\prime } \) be two chain complexes. We assume that all chain groups of \( \mathcal{C} \) are finitely generated free abelian groups. Then the map\n\n\[ \Omega : \operatorname{Hom}\left( {{\mathcal{C}}_{ * },\mathbb{Z}}\right) \otimes \operatorname{H... | Proof (*). We start out the proof of the lemma with the following elementary well-known observation:\n\n\( \left( *\right) \) For finitely many abelian groups \( {A}_{1},\ldots ,{A}_{k} \) we have a canonical isomorphism\n\n\[ \operatorname{Hom}\left( {{A}_{1} \oplus \cdots \oplus {A}_{k},\mathbb{Z}}\right) = \operator... | Yes |
(1) The tensor product of two superalgebras is again a superalgebra.\n\n(2) The tensor product of superalgebras is associative, i.e. for any three superalgebras \( A, B \) and \( C \) the map\n\n\[ \left( {A \otimes B}\right) \otimes C \rightarrow A \otimes \left( {B \otimes C}\right) \]\n\nthat is induced by\n\n\[ \le... | Proof (*). Let \( A, B \) and \( C \) be superalgebras and let \( {a}_{k} \in {A}_{k},{a}_{l} \in {A}_{l},{b}_{m} \in {B}_{m} \) and \( {b}_{n} \in {B}_{n} \).\n\n(1) We have the following equalities:\n\n\[ \left( {{a}_{k} \otimes {b}_{l}}\right) \cdot \left( {{a}_{m} \otimes {b}_{n}}\right) = {\left( -1\right) }^{lm} ... | Yes |
Proposition 84.21. Let \( X \) and \( Y \) be two topological spaces. We denote by \( p : X \times Y \rightarrow X \) and \( q : X \times Y \rightarrow Y \) the obvious projection maps.\n\n(1) The map\n\n\[ \Phi : {\mathrm{H}}^{ * }\left( {X;\mathbb{Z}}\right) \otimes {\mathrm{H}}^{ * }\left( {Y;\mathbb{Z}}\right) \rig... | Proof. Let \( X \) and \( Y \) be two topological spaces.\n\n(1) Let \( {\varphi }_{k} \in {\mathrm{H}}^{k}\left( {X;\mathbb{Z}}\right) ,{\psi }_{l} \in {\mathrm{H}}^{l}\left( {Y;\mathbb{Z}}\right) ,{\varphi }_{m} \in {\mathrm{H}}^{m}\left( {X;\mathbb{Z}}\right) \) and \( {\psi }_{n} \in {\mathrm{H}}^{n}\left( {Y;\math... | Yes |
Proposition 84.22. Let \( n \in {\mathbb{N}}_{0} \) . We denote by \( T = {\left( {S}^{1}\right) }^{n} \) the \( n \) -dimensional torus. For \( i = 1,\ldots, n \) we denote by \( {p}_{i} : T \rightarrow {S}^{1} \) the projection onto the \( i \) -th factor. Then there exists a unique isomorphism \[ \psi : \Lambda \lef... | Proof. We consider the following isomorphism of superalgebras \[ \Lambda \left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \underset{ \uparrow }{\overset{ \cong }{ \rightleftharpoons }}\mathbb{Z}\left\lbrack {x}_{1}\right\rbrack /\left( {x}_{1}^{2}\right) \otimes \ldots \otimes \mathbb{Z}\left\lbrack {x}_{k}\right\rb... | Yes |
For the n-dimensional torus \( T = {\left( {S}^{1}\right) }^{n} \) the Lusternik-Schnirelmann category \( \operatorname{cat}\left( T\right) \) equals \( n \) . | Given any \( n \in {\mathbb{N}}_{0} \) we have\n\nsee page 2014 by Proposition 82.11\n\n\( n\overset{ \downarrow }{ \geq } \) Lusternik-Schnirelmann category \( \operatorname{cat}\left( {\left( {S}^{1}\right) }^{n}\right) \overset{ \downarrow }{ \geq } \) cup length \( \operatorname{cl}\left( {\left( {S}^{1}\right) }^{... | Yes |
Theorem 84.25. Let \( \\mathcal{C} \) and \( {\\mathcal{C}}^{\\prime } \) be chain complexes. If \( \\mathcal{C} \) is a free chain complex and if all homology groups of \( \\mathcal{C} \) are finitely generated, then for every \( n \\in {\\mathbb{N}}_{0} \) there exists a natural isomorphism\n\n\\[ \n{\\bigoplus }_{k ... | The proof of Theorem 84.25 is evidently very similar to the proof of the Künneth Theorem 84.17 for cochain complexes. We refer to [Mun84, Theorem 60.6] for a proof. | No |
Lemma 85.2. Let \( n \in {\mathbb{N}}_{0} \) and let \( X \) be a subset of \( {\mathbb{R}}^{n} \) . If \( X \) is a neighborhood retract in \( {\mathbb{R}}^{n} \), then the topological space \( X \) is weakly locally contractible. | Proof (*). Let \( W \) be a neighborhood of \( X \) that admits a retraction \( r : W \rightarrow X \) . Let \( Q \in X \) and let \( U \) be a neighborhood of \( Q \) in \( X \) . Since \( W \) is a neighborhood of \( X \) there exists an \( s > 0 \) such that \( {B}_{s}^{n}\left( Q\right) \subset {r}^{-1}\left( U\rig... | Yes |
Theorem 85.3. (Borsuk’s Theorem) Let \( n \in {\mathbb{N}}_{0} \) and let \( X \) be a closed subset of \( {\mathbb{R}}^{n} \) . If the topological space \( X \) is weakly locally contractible, then \( X \) is a neighborhood retract in \( {\mathbb{R}}^{n} \) . | 85.2. Proof of the Borsuk's Theorem 85.3. Before we provide the proof of Borsuk's Theorem 85.3 it is convenient to introduce one definition and to prove one lemma.\n\nDefinition. Let \( k \in {\mathbb{N}}_{0} \) and let \( n \in \mathbb{N} \) . A \( \frac{1}{{2}^{k}} \) -cube is a subset \( C \) of \( {\mathbb{R}}^{n} ... | No |
Lemma 85.4. Let \( x \in {\mathbb{R}}^{n} \) and let \( r > 0 \) . If \( k \in {\mathbb{N}}_{0} \) is chosen such that \( \frac{1}{{2}^{k}} < \frac{1}{3\sqrt{n}} \cdot r \), then there exist finitely many \( \frac{1}{{2}^{k}} \) -cubes \( {W}_{1},\ldots ,{W}_{m} \) such that\n\n\[ x \in \text{interior of}\left( {{W}_{1... | Proof (*). Let \( x = \left( {{x}_{1},\ldots ,{x}_{n}}\right) \in {\mathbb{R}}^{n} \) and let \( r > 0 \) . Furthermore let \( k \in {\mathbb{N}}_{0} \) such that \( \frac{1}{{2}^{k}} < \frac{1}{3\sqrt{n}} \cdot r \) . We write \( \epsilon = \frac{1}{{2}^{k}} \) . For \( i = 1,\ldots, n \) we denote by \( {y}_{i} \) th... | Yes |
Theorem 85.5. (Borsuk’s Theorem) Let \( X \) be a topological space which is weakly locally contractible and which admits an embedding \( f : X \rightarrow {\mathbb{R}}^{n} \) .\n\n(1) If \( f\left( X\right) \) is closed, then \( X \) is an ENR.\n\n(2) If \( X \) is regionally compact (e.g. if \( X \) is compact), then... | Proof.\n\n(1) This statement follows immediately from Borsuk’s Theorem 85.5\n\n(2) Now assume that \( X \) is regionally compact. If \( X \) is compact, then we know of course by Lemma 2.17 (2) that \( f\left( X\right) \) is compact and we can apply (1). Slightly more interestingly, if \( X \) is regionally compact, th... | No |
(1) If \( K \subset {\mathbb{R}}^{n} \) is a compact neighborhood retract, then \( K \) is the retract of a finite \( n \) - dimensional simplicial complex. | (1) Let \( K \) be a compact subset of \( {\mathbb{R}}^{n} \) which is a retract of some open neighborhood \( V \) . The idea is to define a simplicial complex \( X \) to be the union of finitely many sufficiently small cubes. To do so we set \( r \mathrel{\text{:=}} d\left( {K,{\mathbb{R}}^{n} \smallsetminus V}\right)... | Yes |
Lemma 85.9. Let \( X \) be a CW-complex. If \( X \) is countable, regionally compact and finite-dimensional, then \( X \) admits an embedding into some \( {\mathbb{R}}^{n} \) . | Proof of Lemma 85.9. The theorem in its full generality is proved in FrPi90b, Theorem A. In the discussion below we will only deal with the much simpler case that \( X \) is actually a finite CW-complex. We prove this case by induction on the dimension of the CW-complex \( X \) . The case \( \dim \left( X\right) = - 1 ... | Yes |
Lemma 85.10. If \( X \) is a compact ENR, then for every field \( \mathbb{F} \) the homology of \( X \) is \( \mathbb{F} \) -finite. | Proof. Let \( X \) be a compact ENR and let \( \mathbb{F} \) be a field. It follows from Propositions 85.6 that \( X \) is a retract of a finite simplicial complex \( Y \) . Recall that this means that there exists a map \( i : X \rightarrow Y \) and a map \( r : Y \rightarrow X \) with \( r \circ i = {\operatorname{id... | Yes |
Theorem 85.11. (Lefschetz Fixed Point Theorem III) Let \( X \) be a compact ENR (e.g. \( X \) could be a compact topological manifold or a finite CW-complex) and let \( \varphi : X \rightarrow X \) be a map. If the \( \mathbb{F} \) -Lefschetz number \( \Lambda \left( {\varphi ,\mathbb{F}}\right) \) is non-zero for some... | Proof. Let \( X \) be a compact ENR. By Proposition 85.6 we know that \( X \) is a retract of a finite simplicial complex \( Y \) . Recall that this means that there exists a map \( i : X \rightarrow Y \) and a map \( r : Y \rightarrow X \) with \( r \circ i = {\operatorname{id}}_{X} \) . By the very harmless Exercise ... | Yes |
Proposition 85.13. Let \( M \) be a compact 0-connected topological manifold. The following statement holds:\n\n(1) There exists an \( l \in {\mathbb{N}}_{0} \) such that for every \( k > l \) we have \( {\mathrm{H}}_{k}\left( M\right) = 0 \) and such that for every abelian group \( G \) we have \( {\mathrm{H}}_{k}\lef... | Proof of Proposition 85.13. \( {}^{1227} \) Let \( M \) be a compact 0-connected topological manifold, let \( \mathbb{F} \) be a field, let \( G \) be an abelian group and let \( n \in \mathbb{N} \) . We pick a base point \( {x}_{0} \in M \) . It follows from Propositions 85.6 and 85.8 (1) that \( M \) is a retract of ... | Yes |
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