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Corollary 36.4. Let \( n > 1 \) . Let \( C \) be a subset of \( {\mathbf{R}}^{n} \) homeomorphic to \( {S}^{n - 1} \) . Then \( {\mathbf{R}}^{n} - C \) has precisely two components, of which \( C \) is the common boundary. | Proof. Step 1. We show first that if \( U \) is a connected open set in \( {S}^{n} \) , where \( n > 1 \), no point of \( U \) separates \( U \) .\n\nLet \( p \in U \) and suppose \( U - p \) is not connected. We derive a contradiction. Choose an open \( \epsilon \) -ball \( {B}_{\epsilon } \) centered at \( p \) and l... | Yes |
Theorem 36.5 (Invariance of domain). Let \( U \) be open in \( {\mathbf{R}}^{n} \) ; let \( f : U \rightarrow {\mathbf{R}}^{n} \) be continuous and injective. Then \( f\left( U\right) \) is open in \( {\mathbf{R}}^{n} \) and \( f \) is an imbedding. | Proof. Without loss of generality, we can replace \( {\mathbf{R}}^{n} \) by \( {S}^{n} \) . Step 1. Given a point \( y \) of \( f\left( U\right) \), we show that \( f\left( U\right) \) contains a neighborhood of \( y \) . This proves that \( f\left( U\right) \) is open in \( {S}^{n} \) . Let \( x \) be the point of \( ... | Yes |
Theorem 37.1. Let \( p : X \rightarrow Y \) be a quotient map. If \( p \) is a closed map, and if \( X \) is normal, then \( Y \) is normal. | Proof. If \( x \) is a point of \( X \), then \( x \) is closed in \( X \), so the one-point set \( p\left( x\right) \) is closed in \( Y \) (because \( p \) is a closed map). Thus \( Y \) is a \( {T}_{\mathrm{i}} \) -space.\n\nLet \( A \) and \( B \) be disjoint closed sets in \( Y \) . Then \( {p}^{-1}\left( A\right)... | Yes |
Theorem 37.2. If \( X \) and \( Y \) are normal, then the adjunction space \( X{ \cup }_{f}Y \) is normal. | Proof. As usual, \( A \) is closed in \( X \) and \( f : A \rightarrow Y \) is continuous. Let \( B \) and \( C \) be disjoint closed sets in \( X{ \cup }_{f}Y \) . Let\n\n\[ \n{B}_{X} = {p}^{-1}\left( B\right) \cap X;\;{C}_{X} = {p}^{-1}\left( C\right) \cap X;\n\]\n\n\[ \n{B}_{Y} = {p}^{-1}\left( B\right) \cap Y;\;{C}... | Yes |
Lemma 37.3. Let \( X \) be a set which is the union of the topological spaces \( \left\{ {X}_{\alpha }\right\} \) . (a) If there is a topological space \( {X}_{T} \) having \( X \) as its underlying set, and each \( {X}_{\alpha } \) is a subspace of \( {X}_{T} \), then \( X \) has a topology, of which the \( {X}_{\alph... | Proof. (a) Let us define a topological space \( {X}_{C} \) whose underlying set is \( X \) by declaring a set \( A \) to be closed in \( {X}_{C} \) if and only if its intersection with each \( {X}_{\alpha } \) is a closed set of \( {X}_{\alpha } \) . The collection of such sets contains arbitrary intersections and fini... | Yes |
Theorem 37.4. Let \( X \) be a space that is the countable union of certain closed subspaces \( {X}_{n} \) . Suppose the topology of \( X \) is coherent with the spaces \( {X}_{n} \) . Then if each \( {X}_{i} \) is normal, so is \( X \) . | Proof. If \( p \) is a point of \( X \), then \( \{ p\} \cap {X}_{i} \) is closed in \( {X}_{i} \) for each \( i \), so \( \{ p\} \) is closed in \( X \) . Thus \( X \) is a \( {T}_{1} \) -space.\n\nLet \( A \) and \( B \) be disjoint closed sets in \( X \) . Define \( {Y}_{0} = A \cup B \), and for \( n > 0 \), define... | Yes |
Lemma 38.1. Let \( X \) be a \( {CW} \) complex with open cells \( {e}_{\alpha } \) . A function \( f : X \rightarrow Y \) is continuous if and only if \( f \mid {\bar{e}}_{\alpha } \) is continuous for each \( \alpha \) . A function \( F : X \times I \rightarrow Y \) is continuous if and only if \( F \mid \left( {{\wi... | ▱ | No |
Consider the torus as a quotient space of a rectangle, as usual. See Figure 38.1. We can express \( T \) as a CW complex having a single open 2-cell (the image under \( \pi \) of the interior of the rectangle), two open 1-cells (the images of the open edges), and one 0-cell (the image of the vertices). | Conditions (1)-(3) hold at once. | No |
Let \( K \) and \( L \) be simplicial complexes; suppose \( K \) is locally finite. The space \( X = \left| K\right| \times \left| L\right| \) can be expressed as a CW complex by taking the sets (Int \( \sigma \) ) \( \times \) (Int \( \tau \) ) as its cells, for \( \sigma \in K \) and \( \tau \in L \) . | In this case the characteristic maps\n\n\[ \n{f}_{\alpha } : {B}^{m} \rightarrow \sigma \times \tau \n\]\n\ncan be taken to be homeomorphisms. Furthermore, in this case \( \operatorname{Bd}\left( {\sigma \times \tau }\right) \) equals a union of open cells of lower dimension. Condition (3) is a consequence of Exercise ... | No |
Definition. Let \( X \) be a CW complex. Let \( Y \) be a subspace of \( X \) that equals a union of open cells of \( X \) . Suppose that for each open cell \( {e}_{\alpha } \) of \( X \) contained in \( Y \), its closure is also contained in \( Y \) . Then we shall show that \( Y \) is a closed set in \( X \), and tha... | Clearly, \( Y \) is Hausdorff. If \( {e}_{\alpha } \) is an open \( m \) -cell of \( X \) contained in \( Y \), then its characteristic map \( {f}_{\alpha } : {B}^{m} \rightarrow X \) carries \( {B}^{m} \) onto \( {\bar{e}}_{\alpha } \), which is contained in \( Y \) by hypothesis. The open cells of \( X \) that inters... | Yes |
We show that the space \( X \) cannot be triangulated; hence in particular, it is not triangulable as a CW complex. | Suppose \( h : \left| K\right| \rightarrow X \) is a triangulation. First we write \( X \) as the disjoint union\n\n\[X = \left( {A - C}\right) \cup C \cup {e}_{3}.\]\n\nNow if \( x \in {e}_{3} \), then \( {H}_{3}\left( {X, X - x}\right) \) is infinite cyclic, because \( x \) has a neighborhood homeomorphic to an open ... | Yes |
Theorem 38.2. (a) Suppose \( X \) is a \( {CW} \) complex of dimension \( p \) . Then \( X \) is homeomorphic to an adjunction space formed from \( {X}^{p - 1} \) and a topological sum \( \sum {B}_{\alpha } \) of closed p-balls, by means of a continuous map \( g : \sum \operatorname{Bd}{B}_{\alpha } \rightarrow {X}^{p ... | Proof. (a) For each cell \( {e}_{\alpha } \) of \( X \) of dimension \( p \), one is given the characteristic map \( {f}_{\alpha } : {B}^{p} \rightarrow {\bar{e}}_{\alpha } \) . Let \( {B}_{\alpha } = {B}^{p} \times \{ \alpha \} \), and let \( \sum {B}_{\alpha } \) be the topological sum of these disjoint \( p \) -ball... | Yes |
Theorem 38.3. (a) Let \( X \) be a \( {CW} \) complex. Then \( {X}^{p} \) is a closed subspace of \( {X}^{p + 1} \) for each \( p \), and \( X \) is the coherent union of the spaces \( {X}^{\mathrm{e}} \subset \) \( {X}^{1} \subset \cdots \) . It follows that \( X \) is normal. | Proof. (a) Suppose \( C \cap {X}^{p} \) is closed in \( {X}^{p} \) for each \( p \) . Then \( C \cap {\bar{e}}_{\alpha } \) is ciosed in \( {\bar{e}}_{\alpha } \) for each cell \( {e}_{\alpha } \) of dimension at most \( p \) . Since \( p \) is arbitrary, we con-ciude that \( C \) is closed in \( X \) . Thus \( X \) ha... | No |
Consider the case where \( X \) is the space of a simplicial complex \( K \), and the open cells of \( X \) are the open simplices of \( K \) . Let \( {H}_{p} \) denote ordinary simplicial homology. We compute \( {H}_{p}\left( {{X}^{p},{X}^{p - 1}}\right) \) . | The simplicial chain group \( {C}_{i}\left( {{K}^{\left( p\right) },{K}^{\left( p - 1\right) }}\right) \) vanishes if \( t \neq p \), and it equals the chain group \( {C}_{p}\left( {K}^{\left( p\right) }\right) = {C}_{p}\left( K\right) \) when \( t = p \) . Therefore, \[ {H}_{p}\left( {{X}^{p},{X}^{p - 1}}\right) = {H}... | Yes |
Lemma 39.1. Given an open p-cell \( {e}_{\alpha } \) of \( X \), any characteristic map for \( {\varepsilon }_{\alpha } \) ,\n\n\[ \n{f}_{\alpha } : \left( {{B}^{p},{S}^{p - 1}}\right) \rightarrow \left( {{\bar{e}}_{\alpha },{\dot{e}}_{\alpha }}\right) ,\n\]\n\ninduces an isomorphism in relative homology. | Proof. If \( p = 0 \), the result is trivial. Let \( p > 0 \) . The point \( \mathbf{0} \) is the center of \( {B}^{p} \) ; let \( {\widehat{e}}_{\alpha } \) denote \( {f}_{\alpha }\left( \mathbf{0}\right) \) . Note that because \( {f}_{\alpha } \) is a quotient map, so is its restriction\n\n\[ \n{f}_{\alpha }^{\prime ... | Yes |
Lemma 39.2. Let the map\n\n\[ f : {X}^{p - 1} \cup \sum {B}_{\alpha } \rightarrow {X}^{p} \]\n\nexpress \( {X}^{p} \) as the adjunction space obtained from \( {X}^{p - 1} \) and a topological sum of p-balls \( \sum {B}_{\alpha } \) via a map \( g : \sum {S}_{\alpha } \rightarrow {X}^{p - 1} \), where \( {S}_{\alpha } =... | Proof. The proof is similar to that of the preceding lemma. The restriction \( {f}^{\prime } \) of \( f \) to the space\n\n\[ {X}^{p - 1} \cup \sum \left( {{B}_{\alpha } - {\mathbf{0}}_{\alpha }}\right) \]\n\nwhere \( {\mathbf{0}}_{\alpha } \) is the center of \( {B}_{\alpha } \), is a quotient map. Furthermore, there ... | Yes |
Theorem 39.3. The group \( {H}_{i}\left( {{X}^{p},{X}^{p - 1}}\right) \) vanishes for \( i \neq p \), and is free abelian for \( i = p \). If \( \gamma \) generates \( {H}_{p}\left( {{B}^{p},{S}^{p - 1}}\right) \), then the elements \( {\left( {f}_{\alpha }\right) }_{ * }\left( \gamma \right) \) form a basis for \( {H}... | Proof. The preceding lemma tells us that\n\n\[ \n{H}_{i}\left( {{X}^{p},{X}^{p - 1}}\right) \simeq {H}_{i}\left( {\sum {B}_{\alpha },\sum {S}_{\alpha }}\right) , \n\]\n\nwhere \( \sum {B}_{\alpha } \) is a topological sum of \( p \) -balls and \( {S}_{\alpha } = \operatorname{Bd}{B}_{\alpha } \). Because the sets \( {B... | Yes |
Theorem 39.5. Let \( X \) be filtered by the subspaces \( {X}_{0} \subset X, \subset \cdots \) ; suppose that \( X \) is the space of a simplicial complex \( K \), and each subspace \( {X}_{p} \) is the space of a subcomplex of \( K \) of dimension at most \( p \) . Let \( {H}_{i} \) denote simplicial homology. Suppose... | Indeed. \( {H}_{p}\left( {{X}_{p},{X}_{p - 1}}\right) \) is the subgroup of \( {C}_{p}\left( K\right) \) consisting of all \( p \) -chains of \( K \) carried by \( {X}_{\rho } \) whose boundaries are carried by \( {X}_{\rho } - 1 \) .\n\nProof. Any compact set in \( X \) lies in a finite subcomplex of \( K \), so it li... | Yes |
Theorem 40.1. The space \( {P}^{n} \) is a CW complex having one cell in each dimension \( 0 \leq j \leq n \) ; its \( j \) -skeleton is \( {P}^{j} \) . | Proof. The space \( {P}^{0} \) is obtained from the 2-point space \( {S}^{0} \) by identifying these two points. Thus \( {P}^{0} \) consists of a single point.\n\nWe proceed by induction. Suppose we restrict the map \( p : {S}^{n} \rightarrow {P}^{n} \) to the closed upper hemisphere \( {E}_{ + }^{n} \) of \( {S}^{n} \... | Yes |
Theorem 40.2. The space \( {\mathrm{{CP}}}^{n} \) is a \( {CW} \) complex of dimension \( {2n} \) . It has one open cell in each even dimension \( {2j} \) for \( 0 \leq {2j} \leq {2n} \), and \( {\mathrm{{CP}}}^{j} \) is its \( {2j} \) - skeleton. | Proof. The space \( \mathbf{C}{P}^{0} \) is a single point. In general, we show that \( \mathbf{C}{P}^{n} - \) \( C{P}^{n - 1} \) is an open \( {2n} \) -cell, which we denote by \( {e}_{2n} \) . Consider the subset of \( {S}^{{2n} + 1} \) consisting of all points \( z = \left( {{z}_{1},\ldots ,{z}_{n + 1},0,\ldots }\ri... | Yes |
The group \( {H}_{i}\left( {\mathbf{C}{P}^{a}}\right) \) is infinite cyclic if \( i \) is even and \( 0 \leq i \leq {2n} \) ; it vanishes otherwise. The group \( {H}_{i}\left( {\mathbf{C}{P}^{\infty }}\right) \) is infinite cyclic if \( i \) is even and \( i \geq 0 \) ; it vanishes otherwise. | The cellular chain group \( {D}_{i}\left( {\mathbf{C}{P}^{n}}\right) \) is infinite cyclic if \( i \) is even and \( 0 \leq i \leq {2n} \) ; otherwise, it vanishes. Therefore, every chain of this chain complex is a cycle, and no chain bounds. A similar computation applies to \( \mathbf{C}{P}^{\infty }. \cdot ▱ | No |
Lemma 40.4. Let \( p : {S}^{n} \rightarrow {P}^{n} \) be the quotient map \( \left( {n \geq 1}\right) \) . Let \( j : {P}^{n} \rightarrow \) \( \left( {{P}^{n},{P}^{n - 1}}\right) \) be inclusion. The composite homomorphism \[ {H}_{n}\left( {S}^{n}\right) \overset{{p}_{ * }}{ \rightarrow }{H}_{n}\left( {P}^{n}\right) \... | Chain-level proof. We assume that \( {S}^{n} \) is triangulated so that the antipodal map \( a : {S}^{n} \rightarrow {S}^{n} \) is simplicial, and that \( {P}^{n} \) is triangulated so that \( p : {S}^{n} \rightarrow {P}^{n} \) is simplicial. (See Lemma 40.7 following.) We use simplicial homology, Let \( {c}_{n} \) be ... | Yes |
Theorem 40.5. The homomorphism\n\n\\[ \n{\\partial }_{ * } : {H}_{n + 1}\\left( {{P}^{n + 1},{P}^{n}}\\right) \\rightarrow {H}_{n}\\left( {{P}^{n},{P}^{n - 1}}\\right) \n\\]\n\nis zero if \\( n \\) is even, and carries a generator to twice a generator if \\( n \\) is odd. | Proof. The map \\( {p}^{\\prime } : \\left( {{E}_{ + }^{n + 1},{S}^{n}}\\right) \\rightarrow \\left( {{P}^{n + 1},{P}^{n}}\\right) \\) is a characteristic map for the open \\( n + 1 \\) cell of \\( {P}^{n + 1} \\) ; therefore, it induces a homology isomorphism. Consider the commutative diagram\n\n\\[ \n{H}_{n + 1}\\lef... | Yes |
Theorem 40.6. The homology of projective space is as follows:\n\n\[ \n{\widetilde{H}}_{i}\left( {P}^{{2n} + 1}\right) \simeq \left\{ \begin{array}{ll} \mathbf{Z}/2 & \text{ if }i\text{ is odd and }0 < i < {2n} + 1, \\ \mathbf{Z} & \text{ if }i = {2n} + 1, \\ 0 & \text{ otherwise. } \end{array}\right. \]\n\n\[ \n{\widet... | Proof. The cellular chain group \( {D}_{i}\left( {P}^{\infty }\right) \) is infinite cyclic for \( i \geq 0 \), and the augmented chain complex has the form\n\n\[ \n: \rightarrow {D}_{2i}\left( {P}^{\infty }\right) \overset{2}{ \rightarrow }{D}_{{2i} - 1}\left( {P}^{\infty }\right) \overset{0}{ \rightarrow }\cdots \ove... | Yes |
Lemma 40.7. The spaces \( {S}^{n} \) and \( {P}^{n} \) may be triangulated so that the antipodal map \( a : {S}^{n} \rightarrow {S}^{n} \) and the projection map \( p : {S}^{n} \rightarrow {P}^{n} \) are simplicial. | Proof. Step 1. We show first there is a complex \( L \) in \( {\mathbf{R}}^{n + 1} \) such that each reflection map\n\n\[ \n{\rho }_{i}\left( {{x}_{1},\ldots ,{x}_{i},\ldots ,{x}_{n + 1}}\right) = \left( {{x}_{1},\ldots , - {x}_{i},\ldots ,{x}_{n + 1}}\right) \n\] \ninduces a linear isomorphism of \( L \) with itself; ... | Yes |
Theorem 40.8. The space \( L\left( {n, k}\right) \) is a CW complex with one cell in each dimension \( 0,1,2,3 \) . | Proof. We first show that the quotient map \( p \) is closed, so that \( L\left( {n, k}\right) \) is Hausdorff (in fact, normal). Let \( A \) be closed in \( {B}^{3} \) . The saturation \( {p}^{-1}p\left( A\right) \) of \( A \) is the union of the set \( A \), the following subsets of \( {S}^{2} \) :\n\n\[ f\left( {{E}... | Yes |
Theorem 41.1. Let \( f \) be a homomorphism; let \( \widetilde{f} \) be the dual homomorphism.\n\n(c): If \( f \) is surjective, then \( \widetilde{f} \) is injective. That is, exactness of\n\n\[ B\overset{f}{ \rightarrow }C \rightarrow 0 \]\n\nimplies exactness of\n\n\[ \operatorname{Hom}\left( {B, G}\right) \overset{... | Proof. (a) and (b) are immediate. To prove (c), suppose \( f \) is surjective. Let \( \psi \in \operatorname{Hom}\left( {C, G}\right) \) and suppose \( \widetilde{f}\left( \psi \right) = 0 = \psi \circ f \) . Then \( \psi \left( {f\left( b\right) }\right) = 0 \) for every \( b \in B \) . As \( b \) ranges over \( B \),... | Yes |
Theorem 41.2. If the sequence\n\n\\( {x}^{\prime } = 1 \\)\n\\[ \nA\\overset{f}{ \\rightarrow }B\\overset{g.}{ \\rightarrow }C \\rightarrow 0 \n\\]\nM\n\nis exact, then the dual sequence\n\n\\[ \n\\operatorname{Hom}\\left( {A, G}\\right) \\overset{\\widetilde{f}}{ \\leftarrow }\\operatorname{Hom}\\left( {B, G}\\right) ... | Proof. Injectivity of \\( \\widetilde{g} \\) follows from the preceding theorem. We check exactness at \\( \\operatorname{Hom}\\left( {B, G}\\right) \\) . Because \\( h = g \\circ f \\) is the zero homomorphism, so is \\( \\widetilde{h} = \\widetilde{f} \\circ \\widetilde{g} \\) . On the other hand, supposing \\( \\wid... | Yes |
Theorem 41.3. (a) One has the following isomorphisms:\n\n\[ \n\operatorname{Hom}\left( {{ \oplus }_{\alpha \in J}{A}_{\alpha }, G}\right) \simeq {\Pi }_{\alpha \in J}\operatorname{Hom}\left( {{A}_{\alpha }, G}\right) ,\n\]\n\n\[ \n\operatorname{Hom}\left( {A,{\Pi }_{\alpha \in J}{G}_{\alpha }}\right) \cong {\Pi }_{\alp... | Proof: Property (a) follows immediately from standard facts of algebra concerning homomorphisms of products. | No |
Consider the complex \( K \) pictured in Figure 42.1. Let us compute the coboundaries of a few cochains. Let \( \left\{ {v}_{i}\right\} \) denote the set of vertices; let \( \left\{ {e}_{i}\right\} \) denote the edges, oriented as indicated; let \( \left\{ {\sigma }_{i}\right\} \) denote the 2-simplices, oriented as in... | \[ \delta {e}_{5}^{ * } = {\sigma }_{1}^{ * } - {\sigma }_{2}^{ * } \] | Yes |
Consider the complex \( K \) pictured in Figure 42.3. We compute its cohomology groups. The general 0-cochain is a sum of the form \( {c}^{0} = \sum {n}_{i}{v}_{i}^{ * } \) . Since \( \left\langle {\delta {c}^{0},{e}_{i}}\right\rangle = \left\langle {{c}^{0},\partial {e}_{i}}\right\rangle \), we see that \( \delta {c}^... | Now let \( {c}^{1} \) be a 1-cochain; it is a cocycle, trivially. We show that \( {c}^{1} \) is cohomol-ogous to some multiple of \( {e}_{1}^{ * } \) . It suffices to show that \( {e}_{i}^{ * } \) is cohomologous to \( {e}_{1}^{ * } \) for each \( i \), and this can be done directly. For instance; \( {e}_{3}^{ * } \) i... | Yes |
Let \( S \) denote the Klein bottle, represented by the labelled rectangle of Figure 42.4. We show that \( {H}^{2}\left( S\right) \) is nontrivial, whereas we know that \( {H}_{2}\left( S\right) = 0 \) . | Orient the 2-simplices of \( L \) counterclockwise. Use the induced orientation of the 2-simplices of \( S \), and let \( \gamma \) denote their sum. Now \( \gamma \) is not a cycle, because \( \partial \gamma = 2{z}_{1} \) , where \( {z}_{1} = \left\lbrack {a, d}\right\rbrack + \left\lbrack {d, e}\right\rbrack + \left... | Yes |
Theorem 42.1. Let \( K \) be a complex. Then \( {H}^{ \circ }\left( {K;G}\right) \) equals the group of all 0 -cochains \( {c}^{0} \) such that \( \left\langle {{c}^{0}, v}\right\rangle = \left\langle {{c}^{0}, w}\right\rangle \) whenever \( v \) and \( w \) belong to the same component of \( \left| K\right| \) . | Proof. Note that \( {H}^{0}\left( {K;G}\right) \) equals the group of 0 -cocycles, because there are no coboundaries in dimension 0 . If \( v \) and \( w \) belong to the same component of \( \left\lbrack K\right\rbrack \), then there is a 1-chain \( {c}_{1} \) of \( K \) such that \( \partial {c}_{1} = v - w \) . Then... | Yes |
Theorem 42.2. If \( \left| K\right| \) is connected, then \( {\widetilde{H}}^{0}\left( {K;G}\right) = 0 \) . More generally, for any complex \( K \) ,\n\n\[ \n{H}^{0}\left( {K;G}\right) \cong {\widetilde{H}}^{0}\left( {K;G}\right) \oplus G.\n\] | Proof. If \( \left| K\right| \) is connected, then \( {\widetilde{H}}_{0}\left( K\right) \) vanishes, so \( {C}_{1}\left( K\right) \rightarrow {C}_{0}\left( K\right) \rightarrow \) \( \mathbf{Z} \rightarrow 0 \) is exact. It follows that\n\n\[ \n{C}^{1}\left( {K;G}\right) \leftarrow {C}^{0}\left( {K;G}\right) \leftarro... | No |
Theorem 43.1. Let \( K \) be a complex; let \( {K}_{0} \) be a subcomplex. There exists an exact sequence \[ \cdots \leftarrow {H}^{p}\left( {{K}_{0};G}\right) \leftarrow {H}^{p}\left( {K;G}\right) \leftarrow {H}^{p}\left( {K,{K}_{0};G}\right) \overset{{\delta }^{ * }}{ \leftarrow }{H}^{p - 1}\left( {{K}_{0};G}\right) ... | Proof. This theorem follows from applying the zig-zag lemma to the diagram  Since \( i \) and \( j \) commute with \( \theta \), the dual maps \( \widetilde{i} \) and \( \widetilde{j} \) commute with \( \delta \) . T... | Yes |
Consider the case of a square \( K \) modulo its boundary \( {K}_{0} \), as pictured in Figure 43.1. We treat the group of relative cochains as those cochains of \( K \) that are carried by \( K - {K}_{0} \). Both \( {\sigma }_{1}^{ * } \) and \( {\sigma }_{2}^{ * } \) are such cochains, and each is a cocycle (triviall... | \[ \delta {e}_{5}^{ * } = {\sigma }_{1}^{ * } - {\sigma }_{2}^{ * } \] Thus the group \( {H}^{2}\left( {K,{K}_{0}}\right) \) is infinite cyclic, and is generated by the cohomology class \( \left\{ {\sigma }_{1}^{ * }\right\} = \left\{ {\sigma }_{2}^{ * }\right\} \). | Yes |
Consider the Möbius band \( M \) modulo its edge \( E \), as pictured in Figure 43.2. We calculate the cohomology of \( \left( {M, E}\right) \) and of \( M \) . | Each of the cochains \( {\sigma }_{i}^{ * } \) is a cocycle (trivially), so they form a basis for the group \( {Z}^{2}\left( {M, E}\right) \) of relative 2-cocycles. Similarly, \( {e}_{1}^{ * },\ldots ,{e}_{6}^{ * } \) form a basis for the group \( {C}^{1}\left( {M, E}\right) \) of relative 1-cochains. (The other 1-sim... | Yes |
Theorem 44.1. Let \( \mathcal{C} \) and \( {\mathcal{C}}^{\prime } \) be chain complexes; let \( \phi : \mathcal{C} \rightarrow {\mathcal{C}}^{\prime } \) be \( a \) chain equivalence. Then \( {\phi }_{ * } \) and \( {\phi }^{ * } \) are isomorphisms of homology and cohomology, respectively. If \( \mathcal{O} \) and \(... | Proof. Since \( \phi \) is a chain equivalence, there is a chain map \( \psi : {\mathcal{C}}^{\prime } \rightarrow \mathcal{C} \) such that \( \phi \circ \psi \) and \( \psi \circ \phi \) are chain homotopic to identity maps. Then \( \widetilde{\psi } \circ \widetilde{\phi } \) and \( \widetilde{\phi } \circ \widetilde... | Yes |
Theorem 44.2. Let \( \left( {K,{K}_{0}}\right) \) be a simplicial pair. Then \( \eta \) induces a cohomology isomorphism\n\n\[ \n{H}^{p}\left( {\mathcal{C}\left( {K,{K}_{0}}\right) ;G}\right) \overset{{\eta }^{ * }}{ \leftarrow }{H}^{p}\left( {\mathcal{S}\left( {\left| K\right| ,\left| {K}_{0}\right| }\right) ;G}\right... | Proof. The chain map \( \eta \) carries the oriented simplex \( \left\lbrack {{v}_{0},\ldots ,{v}_{p}}\right\rbrack \) of \( K \) to the linear singular simplex \( l\left( {{v}_{0},\ldots ,{v}_{p}}\right) \) of \( K \), provided \( {v}_{0} < \ldots < {v}_{p} \) in the chosen ordering. Because the chain complexes involv... | Yes |
Theorem 45.1. Let \( \mathcal{C} \) and \( {\mathcal{C}}^{\prime } \) be free chain complexes. If \( \gamma : {H}_{\rho }\left( \mathcal{C}\right) \rightarrow \) \( {H}_{p}\left( {\mathcal{C}}^{\prime }\right) \) is a homomorphism defined for all \( p \), then there is a chain map \( \phi : \mathcal{C} \rightarrow {\ma... | Indeed, if \( \beta : {Z}_{\rho } \rightarrow {Z}_{\rho }^{\prime } \) is any homomorphism of cycle groups inducing \( \gamma \) , then \( \beta \) extends to a chain map \( \phi \) .\n\nProof. Let \( {Z}_{p} \) denote the \( p \) -cycles, and \( {B}_{p} \), the \( p \) -boundaries, in the chain complex \( \mathcal{C} ... | Yes |
Corollary 45.2. Suppose \( \{ \mathcal{O},\epsilon \} \) and \( \left\{ {{\mathcal{C}}^{\prime },{\epsilon }^{\prime }}\right\} \) are free augmented chain complexes. If \( \gamma : {\widetilde{H}}_{p}\left( \mathcal{O}\right) \rightarrow {\widetilde{H}}_{p}\left( {\mathcal{O}}^{\prime }\right) \) is a homomorphism def... | Proof. Consider the augmented chain complexes obtained from \( \mathcal{C} \) and \( {\mathcal{C}}^{\prime } \) ; they have \( \mathbf{Z} \) as their \( \left( {-1}\right) \) -dimensional groups and \( \epsilon ,{\epsilon }^{\prime } \), respectively, as the boundary operators from dimension 0 to dimension -1 . We defi... | Yes |
Lemma 45.3. Let\n\n\[ 0 \rightarrow \mathcal{C}\overset{\phi }{ \rightarrow }\mathcal{D} \rightarrow \mathcal{E} \rightarrow 0 \]\n\nbe an exact sequence of free chain complexes. If \( \phi \) induces homology isomorphisms in all dimensions, it induces cohomology isomorphisms as well. | Proof. The existence of the long exact sequence in homology and the fact that \( {\phi }_{ * } \) is an isomorphism imply that \( {H}_{p}\left( \mathcal{E}\right) = 0 \) for all \( p \) . To prove that \( {\phi }^{ * } \) is an isomorphism, it suffices to show that \( {H}^{p}\left( {\mathcal{E};G}\right) = 0 \) for all... | Yes |
Lemma 45.4. Let \( \mathcal{C} \) and \( \mathcal{D} \) be free chain complexes; let \( \phi : \mathcal{C} \rightarrow \mathcal{D} \) be a chain map. There is a free chain complex \( {\mathcal{D}}^{\prime } \) and injective chain maps \( i : \mathcal{C} \rightarrow {\mathcal{D}}^{\prime } \) and \( j : \mathcal{D} \rig... | Proof. The definition of \( {\mathcal{D}}^{\prime } \) is one we shall simply \ | No |
Theorem 45.5. Let \( \mathcal{O} \) and \( \mathcal{D} \) be free chain complexes; let \( \phi : \mathcal{O} \rightarrow \mathcal{D} \) be a chain map. If \( \phi \) induces homology isomorphisms in all dimensions, then \( \phi \) induces cohomology isomorphisms in all dimensions. | Proof. Given \( \phi \), let \( i : \mathcal{O} \rightarrow {\mathcal{D}}^{\prime } \) and \( j : \mathcal{D} \rightarrow {\mathcal{D}}^{\prime } \) be as in the preceding lemma. One has exact sequences of free chain complexes\n\n\[ 0 \rightarrow \mathcal{C}\overset{i}{ \rightarrow }{\mathcal{D}}^{\prime } \rightarrow ... | Yes |
Lemma 45.7. Let \( \mathcal{C} \) be a free chain complex. Then there is a natural exact sequence\n\n\[ 0 \leftarrow \operatorname{Hom}\left( {{H}_{\rho }\left( \mathcal{O}\right), G}\right) \overset{\kappa }{ \leftarrow }{H}^{\rho }\left( {\mathcal{O};G}\right) \leftarrow \ker \kappa \leftarrow 0.\n\]\nIt splits, but ... | Proof. We shall construct a homomorphism\n\n\[ {\lambda }^{ * } : \operatorname{Hom}\left( {{H}_{p}\left( \mathcal{C}\right), G}\right) \rightarrow {H}^{p}\left( {\mathcal{C};G}\right)\n\]\n\nsuch that \( \kappa \circ {\lambda }^{ * } \) is the identity. It follows that \( \kappa \) is surjective and that the sequence ... | Yes |
Theorem 45.8. Let \( \mathcal{O} \) be a free chain complex. If \( {H}_{p}\left( \mathcal{O}\right) \) is free for all \( p \) , then \( k \) .is an isomorphism for all \( p \) . | Proof. Let \( \lambda : \mathcal{C} \rightarrow \mathcal{E} \) be as in the preceding lemma. Since the homology of \( \mathcal{O} \) is: free, \( \mathcal{E} \) is a free chain complex and Theorem 45.5 applies. Since the chain map \( \lambda : \mathcal{O} \rightarrow \mathcal{E} \) induces homology isomorphisms \( {\la... | Yes |
Lemma 46.1. Let \( \mathcal{E} \) and \( \mathcal{F} \) be non-negative chain complexes. Suppose \( {E}_{p} \) is free for \( p > 0 \) and \( {H}_{p}\left( \mathcal{F}\right) = 0 \) for \( p > 0 \) . Then any two chain maps \( f, g : \mathcal{E} \rightarrow \mathcal{F} \) that agree in dimension 0 are chain homotopic. | Proof. Define \( D : {E}_{0} \rightarrow {F}_{1} \) to be zero. Then the equation \( \partial D + D\partial = \) \( g - f \) holds in dimension 0, because \( g = f \) in dimension 0 . Suppose \( D \) is defined in dimension \( p - 1 \), where \( p > 0 \) . Choose a basis for \( {E}_{p} \) . If \( e \) is a basis elemen... | Yes |
Theorem 46.2. Let \( \mathcal{C} \) and \( \mathcal{D} \) be free chain complexes that vanish below a certain dimension; let \( \phi : \mathcal{O} \rightarrow \mathcal{D} \) be a chain map. If \( \phi \) induces homology isomorphisms in all dimensions, then \( \phi \) is a chain equivalence. | Proof. We return to the chain complex \( {\mathcal{D}}^{\prime } \) defined in the proof of Lemma 45.4 The inclusion mapping\n\n\[ i : {C}_{p} \rightarrow {D}_{p}^{\prime } = {C}_{p} \oplus {D}_{p} \oplus {C}_{p - 1} \]\n\nis chain homotopic to \( j \circ \phi \), where \( j : {D}_{p} \rightarrow {D}_{p}^{\prime } \) i... | Yes |
Theorem 47.1. Let \( X \) be a \( {CW} \) complex; let \( \mathcal{D}\left( X\right) \) be its cellular chain complex. Then\n\n\[ \n{H}^{p}\left( {\mathcal{D}\left( X\right) ;G}\right) \simeq {H}^{p}\left( {X;G}\right) \n\]\n\nfor all p*and \( G \) . If \( X \) is a triangulable \( {CW} \) complex, triangulated by a co... | Proof. Both \( \mathcal{D}\left( X\right) \) and \( \mathcal{S}\left( X\right) \) are free chain complexes. Since their homology groups are isomorphic, so are their cohomology groups, by Corollary 45.6. In the case where \( X \) is triangulable, the inclusion map \( i : \mathcal{D}\left( X\right) \rightarrow \mathcal{O... | Yes |
Corollary 47.2. Let \( n > 0 \) . Then\n\n\[ \n{H}^{i}\left( {{S}^{n};G}\right) \simeq G\text{ for }i = 0\text{ and }i = n,\n\]\n\n\[ \n{H}^{i}\left( {{B}^{n},{S}^{n - 1};G}\right) \simeq G\text{for}i = n.\n\]\n\nThese cohomology groups vanish for other values of \( i \) . | Proof. The first statement follows from the fact that the cellular chain complex of \( {S}^{n} \) is infinite cyclic in dimensions 0 and \( n \) and vanishes otherwise, and all the boundary operators vanish. The second then follows from the long exact sequence in reduced cohomology, using the fact that the reduced coho... | Yes |
Let \( X \) denote either the torus \( T \) or the Klein bottle \( S \), expressed as a CW complex having one open cell in dimension 2, two in dimension 1, and one in dimension 0 . We computed the cellular chain complex of \( X \), in Example 2 of \( §{39} \), to be of the form\n\n\[ \cdots \rightarrow 0 \rightarrow \m... | Let \( \gamma \) generate \( {D}_{2}\left( X\right) \) ; let \( {w}_{1} \) and \( {z}_{1} \) be a basis for \( {D}_{1}\left( X\right) \) . We know \( {\partial }_{2} \) and \( {\partial }_{1} \) vanish in the case of the torus. Passing to the dual sequence, we compute\n\n\[ {H}^{2}\left( {T;G}\right) \simeq G,\;{H}^{1}... | Yes |
Generators for the cohomology of the torus. We represent \( T \) as a quotient, space of the rectangle, as in the preceding example. Let \( {w}_{1} \) and \( {z}_{1} \) be as in that example. The cochains \( {w}^{1} \) and \( {z}^{1} \) pictured in Figure 47.2 are cocycles of \( T \), by direct computation. | Furthermore, when evaluated on the cycles \( {w}_{1} \) and \( {z}_{1} \) that generate \( {D}_{1}\left( X\right) \), we have \[ \left\langle {{w}^{1},{w}_{1}}\right\rangle = 1\;\text{ and }\;\left\langle {{w}^{1},{z}_{1}}\right\rangle = 0, \] \[ \left\langle {{z}^{1},{w}_{1}}\right\rangle = 0\;\text{ and }\;\left\lang... | Yes |
Generators for the cohomology of the Klein bottle, with integer coefficients. We follow the pattern of the preceding example. Switch the labels \( d \) and \( \dot{e} \) on the right side of the rectangles in Figure 47.2 so they represent the Klein bottle. Then \( {w}^{1} \) still represents a cocycle; it generates \( ... | (More generally, the cochain \( \sum {n}_{i}{\sigma }_{i}^{ * } \) represents the non-zero element of \( {H}^{2}\left( S\right) \) if and only if \( \sum {n}_{i} \) is odd.) | No |
Generators for the cohomology of the Klein bottle, with \( \mathbf{Z}/2 \) coefficients. The cohomology groups are given by \[ {H}^{2}\left( {S;\mathbf{Z}/2}\right) \cong \mathbf{Z}/2,\;{H}^{\prime }\left( {S;\mathbf{Z}/2}\right) \cong \mathbf{Z}/2 \oplus \mathbf{Z}/2,\;{H}^{ \circ }\left( {S;\mathbf{Z}/2}\right) \cong... | The pattern of the preceding argument applies to show that the cochains \( {w}^{1} \) and \( {z}^{1} \) of Figure 47.2 generate the 1-dimensional cohomology. (You can erase the arrows if you like, since \( 1 = - 1 \) in the group \( \mathbf{Z}/2 \) . Thus there is no problem in making \( {z}^{1} \) a cocycle.) The coch... | No |
The cohomology of \( {P}^{2} \) with \( \mathbf{Z}/2 \) coefficients. One has\n\n\[ \n{H}^{i}\left( {{P}^{2};\mathbf{Z}/2}\right) \simeq \mathbf{Z}/2\;\text{ for }\;i = 0,1,2.\n\] | If \( \sigma \) is a 2-simplex, the cochain \( {\sigma }^{ * } \) generates the 2-dimensional group. And the co-\ncycle pictured in Figure 47.3 generates \( {H}^{1}\left( {{P}^{2};\mathbf{Z}/2}\right) \), for its value is 1 on the cycle\n\n\[ \n\left\lbrack {a, b}\right\rbrack + \left\lbrack {b, c}\right\rbrack + \left... | Yes |
Now suppose \( A \) is an additive group and \( R \) is a commutative ring with unity. We say \( A \) has the structure of module over \( R \) if there is a binary operation \( {}^{ \circ }R \times A \rightarrow A \) (written as scalar multiplication) such that for \( \alpha ,\beta \in R \) and \( a, b \in A \), we hav... | (1) \( \alpha \left( {a + b}\right) = {\alpha a} + {\alpha b} \). (2) \( \left( {\alpha + \beta }\right) a = {\alpha a} + {\beta a} \). (3) \( \alpha \left( {\beta a}\right) = \left( {\alpha \cdot \beta }\right) a \). (4) \( {1a} = a \). | Yes |
Given \( R \), it can always be considered as an \( R \) -module over itself. More generally, the cartesian product \( {R}^{n} \) becomes an \( R \) -module if we define | \[ \alpha \left( {{\beta }_{1},\ldots ,{\beta }_{n}}\right) = \left( {\alpha {\beta }_{1},\ldots ,\alpha {\beta }_{n}}\right) . \] | Yes |
Theorem 48.1. Cup product of cochains is bilinear and associative. The cochain \( {z}^{0} \) whose value is 1 on each singular 0 -simplex acts as a unity element. Furthermore, the following coboundary formula holds:\n\n(*) \n\n\[ \delta \left( {{c}^{p} \cup {c}^{q}}\right) = \left( {\delta {c}^{p}}\right) \cup {c}^{q} ... | Proof. Bilinearity is immediate, since two cochains are added by adding their values, and multiplication in \( R \) is distributive. Associativity is also immediate; the value of \( \left( {{c}^{p} \cup {c}^{q}}\right) \cup {c}^{r} \) on \( T : {\Delta }_{p + q + r} \rightarrow X \) equals the product of \n\n\[ \left\l... | Yes |
Theorem 48.2. The cochain cup product induces an operation\n\n\[ \n{H}^{p}\left( {X;R}\right) \times {H}^{q}\left( {X;R}\right) \overset{ \cup }{ \rightarrow }{H}^{p + q}\left( {X;R}\right) \n\]\n\nthat is bilinear and associative. The cohomology class \( \left\{ {z}^{0}\right\} \) acts as a unity element. | Proof. If \( {z}^{p} \) and \( {z}^{q} \) are cocycles, then their cup product is a cocycle as weli, since\n\n\[ \n\delta \left( {{z}^{p} \cup {z}^{q}}\right) = \delta {z}^{p} \cup {z}^{q} + {\left( -1\right) }^{p}{z}^{p} \cup \delta {z}^{q} = 0. \n\]\n\nThe cohomology class of this product depends only on the cohomolo... | Yes |
Theorem 49.1. Given an ordering of the vertices of \( K \), the corresponding simplicial cup product is bilinear and associative. The cochain \( {z}^{0} \) whose value is 1 on each vertex of \( K \) acts as a unity element. The coboundary formula \( \left( *\right) \) of Theorem 48.1 holds. If \( \eta : {C}_{p}\left( K... | Proof. The proofs are straightforward. Only the coboundary formula requires comment. One can prove it by the same computations we used in proving Theorem 48.1; only slight changes of notation are needed. Alternatively, one can use the fact that since \( \eta \) carries basis elements to basis elements, \( \eta \) is in... | No |
Theorem 49.2. The simplicial cup product induces an operation\n\n\[ \n{H}^{p}\left( {K;R}\right) \times {H}^{q}\left( {K;R}\right) \hookrightarrow {H}^{p + q}\left( {K;R}\right) \]\nthat is bilinear and associative. It is independent of the ordering of vertices of \( K \) . The cohomology class \( \left\{ {z}^{0}\right... | Proof. The existence of \( \cup \) follows from the coboundary formula as before. The chain map \( \eta \) induces an isomorphism \( {\eta }^{ * } \) of singular with simplicial cohomology that preserves cup products. Since \( {\eta }^{ * } \) is independent of the chosen ordering in \( K \), so is the cup product in s... | Yes |
Consider the torus \( T \) . Let \( {w}^{1} \) and \( {z}^{1} \) denote the cocycles pictured in Figure 49.1. We know that \( \alpha = \left\{ {w}^{t}\right\} \) and \( \beta = \left\{ {z}^{t}\right\} \) generate \( {H}^{t}\left( T\right) \) . If we orient each 2-simplex counterclockwise, then \( \Lambda = \left\{ {\si... | Order the vertices of \( T \) alphabetically. Using this ordering, we compute the \( \cdots \) value of \( {w}^{1} \cup {z}^{1} \) on each oriented 2 -simplex \( \sigma \) . Note that \( \left\langle {{w}^{1} \cup {z}^{1},\sigma }\right\rangle = 0 \) unless a face of \( \sigma \) is in the carrier of \( {w}^{1} \) and ... | Yes |
Consider now the Klein bottle \( S \) . Let us compute the cohomology ring with \( \mathbf{Z}/2 \) coefficients. We know that \( {H}^{1}\left( {S;\mathbf{Z}/2}\right) \) is generated by the cocycles \( {w}^{1} \) and \( {z}^{1} \) pictured in Figure 49.3. Furthermore, \( {H}^{2}\left( {S;\mathbf{Z}/2}\right) \) is gene... | Some of the computations we carried out in Example 1 apply without change, provided we reduce the coefficients modulo 2 . In particular,\n\n\[ \n{w}^{1} \cup {z}^{1} = {\left\lbrack g, h, i\right\rbrack }^{ * },\text{ and } \n\]\n\n\[ \n{w}^{1} \cup {y}^{1} = 0 \n\]\n\nwhere \( {y}^{1} \) is the cochain pictured in Fig... | Yes |
Consider the connected sum \( {P}^{2}\# {P}^{2} \) . We compute its cohomology ring with \( \mathbf{Z}/2 \) coefficients. | Let us express \( {P}^{2}\# {P}^{2} \) as a CW complex \( X \) having one cell in dimension 0 , one cell in dimension 2 , and two cells in dimension 1 . See Figure 49.4. Fundamental cycles for the 1-cells of \( X \) are \[ {w}^{1} = \left\lbrack {a, b}\right\rbrack + \left\lbrack {b, c}\right\rbrack + \left\lbrack {c, ... | Yes |
Example 4. Consider the space \( X \) pictured in Figure 49.5; it is the union of two topological circles and a topological 2-sphere with a point in common. It is called the wedge product \( {S}^{1} \vee {S}^{1} \vee {S}^{2} \) . The space \( X \) can be expressed as a CW complex with one cell in dimension 0 , one cell... | It follows that the homology groups and cohomology groups of \( X \) are isomorphic to those of \( T \) . However, their cohomology rings are not isomorphic. For it is easy to see that the cohomology ring of \( X \) is trivial. Consider the cocycles\n\n\[ \n{w}^{\\prime } = {\\left\\lbrack b, c\\right\\rbrack }^{ * }\\... | Yes |
Theorem 50.2. There is an isomorphism\n\n\[ \n\mathbf{Z} \otimes G \cong G \n\]\n\nthat maps \( n \otimes g \) to \( {ng} \) ; it is natural with respect to homomorphisms of \( G \) . | Proof. The function mapping \( \mathbf{Z} \times G \) to \( G \) that sends \( \left( {n, g}\right) \) to \( {ng} \) is bilinear, so it induces a homomorphism \( \phi : \mathbf{Z} \otimes G \rightarrow G \) sending \( n \otimes g \) to \( {ng} \) .\n\nLet \( \psi : G \rightarrow \mathbf{Z} \otimes G \) be defined by th... | Yes |
Lemma 50.3. Suppose the homomorphisms \( \phi : B \rightarrow C \) and \( {\phi }^{\prime } : {B}^{\prime } \rightarrow {C}^{\prime } \) are surjective. Then\n\n\[ \phi \otimes {\phi }^{\prime } : B \otimes {B}^{\prime } \rightarrow C \otimes {C}^{\prime } \]\n\n is surjective, and its kernel is the subgroup of \( B \o... | Proof. Let \( G \) denote the subgroup of \( B \otimes {B}^{\prime } \) generated by these elements \( b \otimes {b}^{\prime } \) . Clearly \( \phi \otimes {\phi }^{\prime } \) maps \( G \) to zero, so it induces a homomorphism\n\n\[ \Phi : \left( {B \otimes {B}^{\prime }}\right) /G \rightarrow C \otimes {C}^{\prime }.... | Yes |
Theorem 50.4. Suppose the sequence\n\n\[ A\overset{\phi }{ \rightarrow }B\overset{\psi }{ \rightarrow }C \rightarrow 0 \]\n\nis exact. Then the sequence\n\n\[ A \otimes G\xrightarrow[]{\phi \otimes {i}_{G}}B \otimes G\xrightarrow[]{\psi \otimes {i}_{G}}C \otimes G \rightarrow 0 \]\n\nis exact. If \( \phi \) is injectiv... | Proof. The preceding lemma implies that \( \psi \otimes {i}_{G} \) is surjective, and that its kernel is the subgroup \( D \) of \( B \otimes G \) generated by all elements of the form \( b \otimes g \) for \( b \in k \) ker \( \psi \) . The image of \( \phi \otimes {i}_{G} \) is the subgroup \( E \) generated by all e... | Yes |
Corollary 50.5. There is a natural isomorphism\n\n\[ \mathbf{Z}/m \otimes G \cong G/{mG}. \] | Proof. We take the exact sequence\n\n\[ 0 \rightarrow \mathbf{Z}\overset{m}{ \rightarrow }\mathbf{Z} \rightarrow \mathbf{Z}/m \rightarrow 0 \]\n\nand tensor it with \( G \), obtaining the exact sequence\n\n\[ \mathbf{Z} \otimes G\xrightarrow[]{m \otimes {i}_{G}}\mathbf{Z} \otimes G \rightarrow \mathbf{Z}/m \otimes G \r... | Yes |
Theorem 50.6. One has the following natural isomorphisms:\n\n(a) \( A \otimes B \cong B \otimes A \) . | (a) The map \( A \times B \rightarrow B \times A \) sending \( \left( {a, b}\right) \) to \( \left( {b, a}\right) \) induces an isomorphism of \( F\left( {A, B}\right) \) with \( F\left( {B, A}\right) \) that carries \( R\left( {A, B}\right) \) onto \( R\left( {B, A}\right) \) . | Yes |
Theorem 50.8. If \( A \) is free abelian with basis \( \left\{ {a}_{i}\right\} \) and \( B \) is free abelian with basis \( \left\{ {b}_{j}\right\} \), then \( A \otimes B \) is free abelian with basis \( \left\{ {{a}_{i} \otimes {b}_{j}}\right\} \) . | Proof. Let \( \left\langle {a}_{i}\right\rangle \) and \( \left\langle {b}_{j}\right\rangle \) denote the infinite cyclic subgroups of \( A \) and \( B \) generated by \( {a}_{i} \) and \( {b}_{j} \), respectively. Then\n\n\[ A = \oplus \left\langle {a}_{i}\right\rangle \;\text{ and }\;B = \oplus \left\langle {b}_{j}\r... | Yes |
Theorem 51.1. Let \( \mathcal{C} \) and \( \mathcal{D} \) be free chain complexes. If the chain map \( \phi : \mathcal{O} \rightarrow \mathcal{D} \) induces homology isomorphisms in all dimensions, so does the chain map\n\n\[ \phi \otimes {i}_{G} : \mathcal{C} \otimes G \rightarrow \mathcal{D} \otimes G. \] | Proof. Step 1. We first consider the case where we have a short exact sequence\n\n\[ 0 \rightarrow \mathcal{C} \rightarrow \mathcal{D} \rightarrow \mathcal{E} \rightarrow 0 \]\n\nof free chain complexes. We know that \( {H}_{p}\left( \mathcal{E}\right) = 0 \) for all \( p \), and we wish to prove that \( {H}_{p}\left( ... | Yes |
Theorem 52.1. There is a function that assigns, to each free resolution\n\n\[ \n0 \rightarrow R\xrightarrow[]{\phi }F\xrightarrow[]{\psi }A \rightarrow 0 \]\n\nof the abelian group \( A \), and to each abelian group \( B \), an exact sequence\n\n\[ \n0 \leftarrow \operatorname{Ext}\left( {A, B}\right) \overset{\pi }{ \... | We shall prove this theorem shortly. It will then be used to derive the other properties of the Ext functor, and to compute it. | No |
Lemma 52.2. Suppose one is given a homomorphism\n\n\n\nof free resolutions of \( A \) and \( {A}^{\prime } \), respectively, and a homomorphism \( \delta : {B}^{\prime } \rightarrow B \) . Then there is a unique homo... | Proof. Functoriality of Hom shows that the two right squares of the preceding diagram commute. Therefore, \( \operatorname{Hom}\left( {\alpha ,\delta }\right) \) induces a homomorphism \( \epsilon \) of cokernels.\n\nWe show \( \epsilon \) is independent of the choice of \( \alpha \) and \( \beta \) . Suppose \( \left\... | Yes |
Theorem 52.3. (a) There are natural isomorphisms\n\n\[ \n\operatorname{Ext}\left( {\oplus {A}_{\alpha }, B}\right) \simeq \Pi \operatorname{Ext}\left( {{A}_{\alpha }, B}\right) ,\n\]\n\n\[ \n\operatorname{Ext}\left( {A,\Pi {B}_{\alpha }}\right) \cong \Pi \operatorname{Ext}\left( {A,{B}_{\alpha }}\right) .\n\] | In the proof, we shall use the fact that direct sums and direct products of exact sequences are exact, and the fact that a direct sum (but not a direct product) of free abelian groups is free abelian.\n\n(a) Let \( 0 \rightarrow {R}_{\alpha } \rightarrow {F}_{\alpha } \rightarrow {A}_{\alpha } \rightarrow 0 \) be a fre... | Yes |
Corollary 53.3. Let \( \\mathcal{C} \) and \( \\mathcal{D} \) be free chain complexes; let \( \\phi : \\mathcal{C} \\rightarrow \\mathcal{D} \) be a chain map. If \( {\\phi }_{ * } : {H}_{i}\\left( \\mathcal{O}\\right) \\rightarrow {H}_{i}\\left( \\mathcal{D}\\right) \) is an isomorphism for \( i = p \) and \( i = p - ... | Proof. Apply naturality of the universal coefficient sequence and the Five-lemma. | No |
Lemma 53.4. Let \( \mathcal{O} \) be a chain complex. Let\n\n\[ \omega : \operatorname{Hom}\left( {{C}_{p}, F}\right) \rightarrow {\operatorname{Hom}}_{F}\left( {{C}_{p} \otimes F, F}\right) \]\n\nbe defined by the equation\n\n\[ \left\langle {\omega \left( f\right) ,{c}_{p} \otimes \alpha }\right\rangle = \left\langle... | Proof. Strictly speaking, we use the preceding formula to define \( \omega \left( f\right) \) :as a function on the cartesian product \( {C}_{p} \times F \), and note that it is bilinear. To check that \( \omega \left( f\right) \) is a linear transformation, we compute\n\n\[ \left\langle {\omega \left( f\right) ,\alpha... | Yes |
Theorem 53.5. Let \( \mathcal{O} \) be a free chain complex; let \( F \) be a field. Then there is a natural vector space isomorphism \[ {\operatorname{Hom}}_{F}\left( {{H}_{p}\left( {\mathcal{O};F}\right), F}\right) \leftarrow {H}^{p}\left( {\mathcal{O};F}\right) . \] | Proof. We imitate the proof of the universal coefficient theorem. First, we note that if \[ 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 \] is a short exact sequence of vector spaces over \( F \) and linear transformations, then for any vector space \( V \) over \( F \), the dual sequence \[ 0 \leftarrow {... | Yes |
Corollary 53.6. If \( \left( {X, A}\right) \) is a topological pair, there is a natural vector space isomorphism\n\n\[ \n{\operatorname{Hom}}_{F}\left( {{H}_{p}\left( {X, A;F}\right), F}\right) \leftarrow {H}^{p}\left( {X, A;F}\right) .\n\] | This theorem shows that if \( F \) is a field, then the vector space \( {H}^{p}\left( {X, A;F}\right) \) can be identified in a natural way with the dual vector space of the vector space \( {H}_{p}\left( {X, A;F}\right) \) . In the case where the dimension of \( {H}_{p}\left( {X, A;F}\right) \) is finite, this means th... | Yes |
Lemma 54.2. Given a homomorphism of free resolutions\n\n\n\nand a homomorphism \( \\delta : B \\rightarrow {B}^{\\prime } \\), there exists a unique homomorphism \( \\epsilon \) making the following diagram commute: ... | Proof. Functoriality of \( \\otimes \) shows the two right-hand squares of the preceding diagram commute. Therefore, \( \\alpha \\otimes \\delta \) induces a homomorphism \( \\epsilon \) of kernels.\n\nThe proof that \( {\\epsilon }^{\\prime } \) is independent of the choice of \( \\alpha \) and \( \\beta \) proceeds a... | Yes |
Lemma 54.3. There is a function assigning to each short exact sequence of abelian groups\n\n\[ 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 \]\nand each abelian group \( D \), an exact sequence\n\n\[ 0 \rightarrow D * A \rightarrow D * B \rightarrow D * C \rightarrow D \otimes A \rightarrow D \otimes B \ri... | Proof. This result is analogous to the theorem stated in Exercise 4 of §52. Let\n\n\[ 0 \rightarrow R\overset{\phi }{ \rightarrow }F \rightarrow D \rightarrow 0 \]\n\nbe a free resolution of \( D \) . Because \( R \) and \( F \) are free, we have horizontal exactness in the diagram\n\n\[ 0 \rightarrow R \otimes A \righ... | Yes |
Theorem 54.4. (a) There is a natural isomorphism\n\n\[ A * B \cong B * A. \] | (a) Apply the preceding lemma to the free resolution \( 0 \rightarrow R \rightarrow F \rightarrow \) \( A \rightarrow 0 \) of \( A \) . One obtains a six-term exact sequence. The first terms\n\n\[ 0 \rightarrow B * R \rightarrow B * F \]\n\nvanish because \( R \) and \( F \) are torsion-free. What remains is the exact ... | Yes |
Theorem 55.1 (The universal coefficient theorem for homology). Let \( \mathcal{C} \) be a free chain complex; let \( G \) be an abelian group. There is an exact sequence\n\n\[ 0 \rightarrow {H}_{p}\left( \mathcal{C}\right) \otimes G \rightarrow {H}_{p}\left( {\mathcal{C};G}\right) \rightarrow {H}_{p - 1}\left( \mathcal... | One can give a direct proof of this theorem that is very similar to the proof of the universal coefficient theorem for cohomology. Instead, we shall postpone the proof, and derive it from a more general theorem called the Künneth theorem, which we shall prove in \( §{58} \) . | No |
Corollary 55.3. Let \( \mathcal{C} \) and \( \mathcal{D} \) be free chain complexes; let \( \phi : \mathcal{C} \rightarrow \mathcal{D} \) be a chain map. If \( {\phi }_{ * } : {H}_{i}\left( \mathcal{O}\right) \rightarrow {H}_{i}\left( \mathcal{D}\right) \) is an isomorphism for \( i = p \) and \( i = p - 1 \) , then \n... | Proof. This result follows from naturality of the universal coefficient sequence and the Five-lemma. | No |
Theorem 56.2. The preceding theorem holds if the hypothesis that \( \mathcal{C} \) is finitely generated in each dimension is replaced by the hypothesis that @wan-ishes below a certain dimension, and the homology of \( \mathcal{O} \) is finitely generated in each dimension. | To prove this theorem, we need the following lemma:\n\nLemma 56.3. Let \( \mat | No |
Lemma 56.3. Let \( \mathcal{C} \) be a free chain complex such that \( {H}_{i}\left( \mathcal{C}\right) \) is finitely generated for each \( i \) . Then there is a free chain complex \( {\mathcal{C}}^{\prime } \) that is finitely generated in each dimension, whose homology is isomorphic to that of \( \mathcal{O} \) . I... | Proof. Let \( {\beta }_{p} \) be the betti number of \( {H}_{p}\left( \mathcal{O}\right) \), and let \( {t}_{1}^{\left( p\right) },\ldots ,{t}_{{k}_{p}}^{\left( p\right) } \) be its torsion coefficients. Let \( {U}_{p},{V}_{p} \), and \( {W}_{p} \) be free abelian groups, where \( {U}_{p} \) has rank \( {k}_{p - 1} \),... | Yes |
Lemma 58.1. Let \( \mathcal{C} \) and \( {\mathcal{C}}^{\prime } \) be chain complexes such that in each dimension the cycles form a direct summand in the chains. (This occurs, for instance, when \( Q \) and \( {Q}^{\prime } \) are free.) Then\n\n\[ \Theta : { \oplus }_{\rho + q = m}{H}_{\rho }\left( \mathcal{O}\right)... | Proof. We define a homomorphism \( \lambda \) in the opposite direction to \( \Theta \), such that \( \lambda \circ \theta \) equals the identity. This suffices.\n\nLet \( {Z}_{p} \) denote the group of \( p \) -cycles in \( \mathcal{O} \) ; let \( {Z}_{q}^{\prime } \) denote the \( q \) -cycles in \( {\mathcal{O}}^{\p... | Yes |
Corollary 58.3. Let \( \mathcal{O},{\mathcal{O}}^{\prime },\mathcal{D},{\mathcal{D}}^{\prime } \) be chain complexes with \( \mathcal{O} \) and \( \mathcal{D} \) -free; let \( \phi : \mathcal{C} \rightarrow \mathcal{D} \) and \( {\phi }^{\prime } : {\mathcal{C}}^{\prime } \rightarrow {\mathcal{D}}^{\prime } \) be chain... | Proof. We apply naturality of the Künneth sequence, and the Rive-lemma. | No |
Let \( K \) and \( L \) be simplicial complexes, with \( K \) or \( L \) locally finite. The results of the preceding section show that the chain complex \( \mathcal{O}\left( K\right) \otimes \mathcal{O}\left( L\right) \) can be used to compute the homology of \( \left| K\right| \times \left| L\right| \) . | Since \( \mathcal{O}\left( K\right) \) and \( \mathcal{O}\left( L\right) \) are free, tie Künneth theorem implies that\n\n\[ \n{H}_{m}\left( {\left| K\right| \times \left| L\right| }\right) \cong { \oplus }_{p + q = m}\left\lbrack {{H}_{p}\left( K\right) \otimes {H}_{q}\left( L\right) \oplus {H}_{p - 1}\left( K\right) ... | Yes |
In particular, for the product \( {S}^{r} \times {S}^{s} \), we have\n\n\[ \n{H}_{m}\left( {{S}^{r} \times {S}^{s}}\right) \cong { \oplus }_{p + q = m}{H}_{p}\left( {S}^{r}\right) \otimes {H}_{q}\left( {S}^{s}\right) .\n\] | Hence if \( r \neq s \) ,\n\n\[ \n{H}_{m}\left( {{S}^{r} \times {S}^{s}}\right) \simeq \left\{ \begin{array}{ll} \mathbf{Z} & \text{ if }m = 0, r, s, r + s, \\ 0 & \text{ otherwise. } \end{array}\right.\n\]\n\n\[ \n{H}_{m}\left( {{S}^{r} \times {S}^{r}}\right) \simeq \left\{ \begin{array}{ll} \mathbf{Z} & \text{ if }m ... | Yes |
Theorem 58.4. Suppose the chain complexes & and & are vector spaces over the field \( F \), and the boundary operators are vector space homomorphisms. Then \( {H}_{\rho }\left( \mathcal{E}\right) \) and \( {H}_{q}\left( {\mathcal{E}}^{\prime }\right) \) are vector spaces over \( F \), and there is a natural isomorphism... | Proof. The proof of the Künneth theorem proceeds unchanged through its first four steps, if \( \otimes \) is replaced throughout by \( { \otimes }_{F} \) . A change first appears in Step 5, where we take the sequence\n\n\[ \n0 \rightarrow {B}_{p} \rightarrow {Z}_{p} \rightarrow {H}_{p}\left( \mathcal{E}\right) \rightar... | Yes |
If \( K \) and \( L \) are simplicial complexes, with \( K \) or \( L \) locally finite, there is a vector space isomorphism\n\n\[ \oplus _{p + q = m}{H}_{p}\left( {K;F}\right) \otimes _{F}{H}_{q}\left( {L;F}\right) \rightarrow {H}_{m}\left( {\left| K\right| \times \left| L\right| ;F}\right) . \] | This fact follows from the preceding theorem, once we note that since for all \( i \) ,\n\n\[ {H}_{i}\left( {\mathcal{C}\left( K\right) \otimes \mathcal{C}\left( L\right) }\right) \simeq {H}_{i}\left( {\left| K\right| \times \left| L\right| }\right) , \]\n\nthe same holds with arbitrary coefficients \( F \), by Theorem... | No |
Lemma 59.1. If \( \{ \mathcal{O},\epsilon \} \) and \( \left\{ {{\mathcal{O}}^{\prime },{\epsilon }^{\prime }}\right\} \) are acyclic augmented chain complexes, and if \( \mathcal{C} \) is free, then \( \left\{ {\mathcal{C} \otimes {\mathcal{C}}^{\prime },\bar{\epsilon }}\right\} \) is acyclic. | Proof. We show that \( {H}_{m}\left( {\mathcal{C} \otimes {\mathcal{C}}^{\prime }}\right) \) is infinite cyclic if \( m = 0 \), and vanishes otherwise. This proves that \( \mathcal{O} \otimes {\mathcal{O}}^{\prime } \) is acyclic relative to any augmentation.\n\nWe can apply the Künneth theorem, since \( \mathcal{O} \)... | Yes |
Theorem 59.3 (The Künneth theorem for topological spaces). Given topological spaces \( X, Y \), there is an exact sequence\n\n\[ 0 \rightarrow { \oplus }_{p + q = m}{H}_{p}\left( X\right) \otimes {H}_{q}\left( Y\right) \rightarrow {H}_{m}\left( {X \times Y}\right) \rightarrow \]\n\n\[ { \oplus }_{p + q = m}{H}_{p - 1}\... | It is natural with respect to homomorphisms induced by continuous maps. It splits, but not naturally.\n\nThe monomorphism\n\n\[ {H}_{p}\left( X\right) \otimes {H}_{q}\left( Y\right) \rightarrow {H}_{m}\left( {X \times Y}\right) \]\nof this theorem is called the homology cross product. It equals the composite\n\n\[ {H}_... | No |
Theorem 59.5. Let \( {\pi }_{1} : X \times Y \rightarrow X \) and \( {\pi }_{2} : X \times Y \rightarrow Y \) be projections. Define\n\n\[ \nu : {S}_{m}\left( {X \times Y}\right) \rightarrow { \oplus }_{p + q = m}{S}_{p}\left( X\right) \otimes {S}_{q}\left( Y\right) \]\n\nby the equation\n\n\[ \nu \left( T\right) = \ma... | Proof. That \( \nu \) is natural and augmentation-preserving is easy to check. The fact that \( \nu \) is a chain map is a straightforward computation, about as messy as such computations usually are. In fact, when one computes directly, one finds that the expression for \( \bar{\partial }\nu \left( T\right) \) equals ... | Yes |
Lemma 60.1. The homomorphism \( \theta \) is a natural cochain map. | Proof. Let \( \phi \in \operatorname{Hom}\left( {{C}_{p}, R}\right) \) and \( \psi \in \operatorname{Hom}\left( {{C}_{q}^{\prime }, R}\right) \), where \( p + q = m \) . Let \( r + s = m + 1 \) . We compute\n\n\[ \left\langle {\theta \left( {\bar{\delta }\left( {\phi \otimes \psi }\right) }\right) ,{c}_{r} \otimes {c}_... | Yes |
Theorem 60.3 (The Künneth theorem for cohomology). Let \( \mathcal{C} \) and \( {\mathcal{C}}^{\prime } \) be chain complexes that vanish below a certain dimension. Suppose \( \mathcal{O} \) is free and finitely generated in each dimension. Then there is a natural exact sequence\n\n\[ 0 \hookrightarrow {\bigoplus }_{p ... | Proof. Let \( \mathcal{E} \) and \( {\mathcal{E}}^{\prime } \) be the chain complexes whose chain groups in dimen-, sion \( - p \) are defined by the equations\n\n\[ {E}_{-p} = \operatorname{Hom}\left( {{C}_{p},\mathbf{Z}}\right) \;\text{ and }\;{E}_{-p}^{\prime } = \operatorname{Hom}\left( {{C}_{p}^{\prime },\mathbf{Z... | Yes |
Corollary 60.4. In the preceding theorem, the hypothesis that @ be finitely generated in each dimension can be replaced by the hypothesis that \( {H}_{i}\left( \mathcal{C}\right) \) be finitely generated for each i. Similarly, the hypothesis that \( {\mathcal{C}}^{\prime } \) be finitely generated in each dimension can... | Proof. The proof is similar to that of Theorem 56.2. If \( \mathcal{O} \) is free and vanishes below a certain dimension, and \( {H}_{i}\left( \mathcal{O}\right) \) is finitely generated for each \( i \), we choose a free chain complex \( \mathcal{D} \) that vanishes below a certain dimension such that \( {H}_{i}\left(... | No |
Lemma 61.1. The cochain cross product is given by the formula\n\n\[ \n\\left\\langle {{c}^{p} \\times {c}^{q}, T}\\right\\rangle = \\left\\langle {{c}^{p},{\\pi }_{1} \\circ T \\circ l\\left( {{\\epsilon }_{0},\\ldots ,{\\epsilon }_{p}}\\right) }\\right\\rangle \\cdot \\left\\langle {{c}^{q},{\\pi }_{2} \\circ T \\circ... | This result holds for \( \\left( {\\mathbf{Z}, G}\\right) \) coefficients as well. In words, it says that the value of \( {c}^{p} \\times {c}^{q} \) on \( T \) equals the value of \( {c}^{p} \) on the front face of the first component of \( T \), times the value of \( {c}^{q} \) on the back face of the second component... | No |
Theorem 61.2. (a) If \( \lambda : X \times Y \rightarrow Y \times X \) is the map that reverses coordinates, then\n\n\[{\lambda }^{ * }\left( {{\beta }^{q} \times {\alpha }^{p}}\right) = {\left( -1\right) }^{pq}{\alpha }^{p} \times {\beta }^{q}.\] | Proof. (a) Consider the chain maps\n\n\[\\begin{matrix} {\\left( \\mathcal{S}\\left( X\\right) \\otimes \\mathcal{S}\\left( Y\\right) \\right) }_{m} \\leftarrow {S}_{m}\\left( {X \\times Y}\\right) \\\\ {\\left( \\omega \\right) }_{m} \\leftarrow {S}_{m}\\left( {Y \\times X}\\right) , \\end{matrix}\n\nwhere we define \... | No |
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