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Theorem 4.2.30. Let \( \\left( {X, A}\\right) \) be a CW-pair and let \( U \\subseteq A \) be such that \( \\left( {Y, B}\\right) = \) \( \\left( {X - U, A - U}\\right) \) is a CW-pair. Then the inclusion \( \\left( {Y, B}\\right) \\rightarrow \\left( {X, A}\\right) \) is excisive for cellular homology. | Proof. First observe that the hypothesis on \( U \) implies that \( U \) is a union of open cells of \( A \) . Let \( {F}_{n} \) be the free abelian group on the \( n \) -cells of \( X \) that are not contained in \( A \), which are exactly the \( n \) -cells of \( Y \) that are not contained in \( B \) . Then we have ... | Yes |
Theorem 4.2.33. Let \( X \) be a CW-complex with only finitely many cells in each dimension. Then for each \( n,{H}_{n}^{\text{cell }}\left( X\right) \) and \( {H}_{\text{cell }}^{n}\left( X\right) \) are finitely generated abelian groups. | Proof. \( {H}_{n}^{\text{cell }}\left( X\right) \) is a quotient of \( {Z}_{n}^{\text{cell }}\left( X\right) \), which is a subgroup of a finitely generated free abelian group, and hence itself is a finitely generated free abelian group, and similarly for \( {H}_{\text{cell }}^{n}\left( X\right) \) . | Yes |
Theorem 4.3.2. Let \( d = {\dim }_{\mathbb{R}}\mathbb{F} \) (so that \( d = 1 \) if \( \mathbb{F} = \mathbb{R} \) and \( d = 2 \) if \( \mathbb{F} = \mathbb{C} \) ). Then \( \mathbb{F}{P}^{n} \) has a CW-structure with one cell in dimension di for each \( i = 0,\ldots, n \) . | Proof. By induction on \( n \). For \( n = 0,\mathbb{F}{P}^{0} \) is just a point. Assume now the theorem is true for \( n - 1 \). We shall show that \( \mathbb{F}{P}^{n} - \mathbb{F}{P}^{n - 1} \) is a single cell of dimension \( {dn} \), which, by induction, completes the proof. Now \( \mathbb{F}{P}^{n} - \mathbb{F}{... | Yes |
Theorem 4.3.3. The homology of \( \mathbb{C}{P}^{n} \) is as follows:\n\n\[ \n{H}_{i}\left( {\mathbb{C}{P}^{n}}\right) = \left\{ \begin{array}{ll} 0 & i > {2n} \\ \mathbb{Z} & 0 \leq i \leq {2n}\text{ even } \\ 0 & 0 < i < {2n}\text{ odd. } \end{array}\right.\n\] | Proof. The cellular chain complex of \( \mathbb{C}{P}^{n} \) is\n\n\[ \n0 \rightarrow \mathbb{Z} \rightarrow 0 \rightarrow \mathbb{Z} \rightarrow \cdots \rightarrow \mathbb{Z} \rightarrow 0 \rightarrow \mathbb{Z} \rightarrow 0 \n\]\n\nwith \( \mathbb{Z} \) in every even dimension between 0 and \( {2n} \), and 0 otherwi... | Yes |
Lemma 5.1.3. For any \( n,\partial \left( {\partial {I}^{n}}\right) = 0 \) . | Proof. For \( n \leq 1 \) this is clear.\n\nFor \( n \geq 2,\partial \left( {\partial {I}^{n}}\right) \) is an element in the free abelian group generated by the \( \left( {n - 2}\right) \) - faces of \( {I}^{n} \), i.e., by the subsets, for each \( i \neq j \) and each \( {\varepsilon }_{i} = 0 \) or \( 1,{\varepsilon... | No |
Lemma 5.1.12. Let \( f : X \rightarrow Y \) be a map. Then finduces a chain map \( \left\{ {{f}_{n} : {C}_{n}\left( X\right) \rightarrow }\right. \) \( \left. {{C}_{n}\left( Y\right) }\right\} \) where \( {f}_{n} : {C}_{n}\left( X\right) \rightarrow {C}_{n}\left( Y\right) \) as follows. Let \( \Phi : {I}^{n} \rightarro... | Proof. This would be immediate if we were dealing with \( {Q}_{n}\left( X\right) \) and \( {Q}_{n}\left( Y\right) \) . But since \( {f\Phi } \) is degenerate wherever \( \Phi \) is, it is just about immediate for \( {C}_{n}\left( X\right) \) and \( {C}_{n}\left( Y\right) \) . Then the fact that we have maps on homology... | No |
Theorem 5.1.14. Singular homology satisfies Axioms 1 and 2. | Proof. Immediate from the definition of the induced map on singular cubes as composition. | No |
Theorem 5.1.15. Singular homology satisfies Axiom 3. | Proof. Immediate from the definition of the boundary map on singular cubes and from the definition of the induced map on singular cubes as composition. | No |
Theorem 5.1.16. Singular homology satisfies Axiom 4. | Proof. We have defined \( {C}_{n}\left( {X, A}\right) = {C}_{n}\left( X\right) /{C}_{n}\left( A\right) \) . Thus for every \( n \), we have a short exact sequence\n\n\[ 0 \rightarrow {C}_{n}\left( A\right) \rightarrow {C}_{n}\left( X\right) \rightarrow {C}_{n}\left( {X, A}\right) \rightarrow 0.\]\n\nIn other words, we ... | Yes |
Theorem 5.1.17. Singular homology satisfies Axiom 5. | Proof. For simplicity we consider the case of homotopic maps of spaces \( f : X \rightarrow Y \) and \( g : X \rightarrow Y \) (rather than maps of pairs). Then by definition, setting \( {f}_{0} = f \) and \( {f}_{1} = g \), there is a map \( F : X \times I \rightarrow Y \) with \( F\left( {x,0}\right) = {f}_{0}\left( ... | Yes |
Theorem 5.1.19. Let \( X \) be the space consisting of a single point. Then \( {H}_{0}\left( X\right) \cong \mathbb{Z} \) and \( {H}_{i}\left( X\right) = 0 \) for \( i \neq 0 \) . Thus singular homology satisfies the dimension axiom, Axiom 7, and has coefficient group \( \mathbb{Z} \) . | Proof. Let \( \Phi : {I}^{0} \rightarrow X \) be the unique map. Then \( {C}_{0}\left( X\right) \) is the free abelian group generated by \( \Phi \) . On the other hand, for any \( i > 0,\Phi : {I}^{i} \rightarrow X \) is a degenerate \( i \) -cube. Hence \( {C}_{i}\left( X\right) = \{ 0\} \) for \( i > 0 \) . Thus \( ... | Yes |
Theorem 5.1.24. For any singular chain \( c \) , \( \operatorname{supp}\left( c\right) \) is a compact subset of \( X \) . | Proof. For any \( \Phi : {I}^{n} \rightarrow X,\Phi \left( {I}^{n}\right) \) is a compact subset of \( X \) as it is the continuous image of a compact set. Then for any singular chain \( c \), supp \( \left( c\right) \) is a finite union of compact sets and hence is compact. | Yes |
Corollary 5.1.25. Let \( X \) be a union of components, \( X = \mathop{\bigcup }\limits_{{i \in I}}{X}_{i} \) . Then for any \( n \) , \( {H}_{n}\left( X\right) = {\bigoplus }_{i \in I}{H}_{n}\left( {X}_{i}\right) \) | Proof. This follows for any generalized homology theory from Lemma 3.2.1 if there are only finitely many components. But for singular homology theory, if \( c \in {C}_{n}\left( X\right) \) is any chain, then \( \operatorname{supp}\left( c\right) \) is compact, by Theorem 5.1.24, so is contained in \( \mathop{\bigcup }\... | Yes |
Lemma 5.1.26. (1) For any space \( X \), the group of singular \( n \) -chains \( {C}_{n}\left( X\right) \) is isomorphic to the free abelian group with basis the non-degenerate n-cubes. | Proof. This follows easily once we recall that \( {C}_{n}\left( X\right) = {Q}_{n}\left( X\right) /{D}_{n}\left( X\right) \) where \( {Q}_{n}\left( X\right) \) is the free abelian group on all \( n \) -cubes and \( {D}_{n}\left( X\right) \) is the free abelian group on the degenerate \( n \) -cubes | Yes |
Theorem 5.2.1. Let \( X \) be a space. Then \( {H}_{0}\left( X\right) \) is isomorphic to the free abelian group on the path components of \( X \) . | Proof. We assume \( X \) nonempty. We have already seen in Corollary 5.1.25 that if \( X = {X}_{1} \cup {X}_{2} \cup \cdots \) is a union of path components, then \( {H}_{i}\left( X\right) = {\bigoplus }_{k}{H}_{i}\left( {X}_{k}\right) \) . Thus it satisfies to prove the theorem in case \( X \) is path connected, so we... | Yes |
Lemma 5.2.2. Let \( f : I \rightarrow X \) and \( g : I \rightarrow X \) with \( f\left( 1\right) = g\left( 0\right) \) . Define \( h : I \rightarrow X \) by \( h\left( t\right) = f\left( {2t}\right) \) for \( 0 \leq t \leq \frac{1}{2} \), and \( h\left( t\right) = g\left( {{2t} - 1}\right) \) for \( \frac{1}{2} \leq t... | Proof. We exhibit a 2-cell \( C \) with \( \partial C = f + g - h.C : I \rightarrow I \rightarrow X \) is given by following \( f \) and then \( g \) along each of the heavy solid lines as indicated:\n\n\n\nThen \( \pa... | Yes |
Corollary 5.2.5. Let \( X \) be a path-connected space. The map \( \theta \) induces a bijection (of sets)\n\n\[ \left\{ \right. \text{free homotopy classes of maps:}\left. {{S}^{1} \rightarrow X}\right\} \rightarrow {H}_{1}\left( X\right) \text{.} \] | Proof. Immediate from Theorems 5.2.4 and 2.5.1. | No |
Theorem 5.2.7. Let \( d \) be any integer. Then for any integer \( n \geq 1 \), there exists a map \( f : {S}^{n} \rightarrow {S}^{n} \) of degree \( d \) . | Proof. Again the key step is the \( n = 1 \) case, and we provide an alternate proof of that (with the remainder of the proof being the same as in the previous proof).\n\nAgain we claim that \( f : {S}^{1} \rightarrow {S}^{1} \) by \( f\left( z\right) = {z}^{d} \) has degree \( d \) .\n\nTo prove that, we consider the ... | Yes |
Here is a pair of examples to show that the condition closure \( \left( U\right) \subseteq \) interior \( \left( A\right) \) cannot in general be relaxed to \( U \subseteq \operatorname{interior}\left( A\right) \) for the inclusion \( \left( {X - U, A - U}\right) \rightarrow \left( {X, A}\right) \) to be excisive. In e... | (a) Let \( X = {\mathbb{R}}^{2} \) and let \( A \) be the subset of \( {\mathbb{R}}^{2} \) that is on or below the graph of the function\n\n\[ f\left( x\right) = \left\{ \begin{array}{ll} \sin \left( \frac{1}{x}\right) & x > 0 \\ 1 & x \leq 0. \end{array}\right. \]\n\nNote that \( \partial A \) consists of the union of... | Yes |
Lemma 5.3.2. \( {C}_{n}\left( {X;G}\right) \) is isomorphic to \( {C}_{n}\left( X\right) \otimes G \) (and similarly for \( A \) ). Also, \( {C}_{n}\left( {X, A;G}\right) \) is isomorphic to \( {C}_{n}\left( {X, A}\right) \otimes G \) . | Proof. Clear from Definition 5.3.1 and the fact that for any two abelian groups \( A \) and \( B,\left( {A \oplus B}\right) \otimes G \approx \left( {A \otimes G}\right) \oplus \left( {B \otimes G}\right) \), and hence, in this situation, \( \left( {A \otimes G}\right) \approx \) \( \left( {\left( {A \oplus B}\right) \... | No |
Lemma 5.3.3. With the above identifications, \( {C}_{n}\left( {X;G}\right) \) is a chain complex with boundary map \( \partial \otimes 1 : {C}_{n}\left( {X;G}\right) \rightarrow {C}_{n - 1}\left( {X;G}\right) \), and similarly for \( {C}_{n}\left( {A;G}\right) \) and \( {C}_{n}\left( {X, A;G}\right) \) . | Proof. The only thing to check is that \( {\left( \partial \otimes 1\right) }^{2} = 0 \) . But \( {\left( \partial \otimes 1\right) }^{2} = {\partial }^{2} \otimes 1 = 0 \) . | Yes |
Lemma 5.3.4. There is a split short exact sequence\n\n\[ 0 \rightarrow {C}_{n}\left( {A;G}\right) \rightarrow {C}_{n}\left( {X;G}\right) \rightarrow {C}_{n}\left( {X, A;G}\right) \rightarrow 0, \]\n\nand hence \( {C}_{n}\left( {X;G}\right) \) is isomorphic to \( {C}_{n}\left( {A;G}\right) \oplus {C}_{n}\left( {X, A;G}\... | Proof. We have the short exact sequence\n\n\[ 0 \rightarrow {C}_{n}\left( A\right) \rightarrow {C}_{n}\left( X\right) \rightarrow {C}_{n}\left( {X, A}\right) \rightarrow 0. \]\n\nTensoring such a sequence with \( G \) does not in general produce an exact sequence. But if this sequence is split short exact, tensoring wi... | Yes |
Lemma 5.3.5. With the identification in Lemma 5.3.2, \( f : X \rightarrow Y \) induces \( {f}_{ * } \otimes 1 \) : \( {C}_{ * }\left( {X;G}\right) \rightarrow {C}_{ * }\left( {Y;G}\right) \), and similarly for \( f : \left( {X, A}\right) \rightarrow \left( {Y, B}\right) \) . | In concrete terms, if \( \left\{ {{\Phi }_{i} : {I}^{n} \rightarrow X}\right\} \) are singular \( n \) -cubes, \( \left( {{f}_{ * } \otimes 1}\right) \left( {\mathop{\sum }\limits_{i}{g}_{i}{\Phi }_{i}}\right) = \) \( \sum {g}_{i}\left( {f{\Phi }_{i}}\right) \) . | Yes |
Theorem 5.3.7. Singular homology with coefficients in \( G \) is an ordinary homology theory with coefficient group \( G \) . | Proof. First we check Axiom 7, the dimension axiom. If \( X \) consists of a single point, then \( {C}_{ * }\left( {X;G}\right) \) is isomorphic to\n\n\[ \cdots \rightarrow 0 \rightarrow 0 \rightarrow G \rightarrow 0 \rightarrow 0 \rightarrow \cdots \]\n\nwith homology as claimed.\n\nThe proof that this theory satisfie... | No |
Lemma 5.3.8. The map \( \tau : {C}_{n}\left( X\right) \rightarrow {C}_{n}\left( {X;G}\right) \) given by \( \tau \left( \Phi \right) = \Phi \otimes 1 \) where \( \Phi \) is a singular \( n \) -cube induces a map\n\n\[ \tau : {H}_{n}\left( X\right) \otimes G \rightarrow {H}_{n}\left( {X;G}\right) \] | Proof. Lemma 5.3.3 implies that \( \tau : {Z}_{n}\left( X\right) \rightarrow {Z}_{n}\left( {X;G}\right) \) and \( \tau : {B}_{n}\left( X\right) \rightarrow {B}_{n}\left( {X;G}\right) \) , where, as usual, \( {Z}_{n}\left( X\right) = \operatorname{Ker}\left( {\partial }_{n}\right) \) and \( {B}_{n}\left( X\right) = \ope... | Yes |
Theorem 5.3.9 (Universal coefficient theorem). (1) For any space \( X \) and abelian group \( G \), there is a split short exact sequence\n\n\[ 0 \rightarrow {H}_{n}\left( X\right) \otimes G\overset{\tau }{ \rightarrow }{H}_{n}\left( {X;G}\right) \rightarrow \operatorname{Tor}\left( {{H}_{n - 1}\left( X\right), G}\righ... | Proof. This is a purely algebraic fact about the homology of chain complexes, and we omit the proof. | No |
Example 5.3.12. By Lemma A.3.8, \( \operatorname{Tor}\left( {{\mathbb{Z}}_{2},{\mathbb{Z}}_{m}}\right) \approx {Z}_{2} \) for \( m \) even. Thus, from Theorem 4.3.4, for real projective spaces we have, for \( m \) even, | \[ {H}_{i}\left( {\mathbb{R}{P}^{n};{\mathbb{Z}}_{m}}\right) = \left\{ \begin{array}{ll} 0 & i > n \\ {\mathbb{Z}}_{m} & i = n\text{ odd } \\ {\mathbb{Z}}_{2} & i = n\text{ even } \\ {\mathbb{Z}}_{2} & 1 \leq i \leq n - 1 \\ {\mathbb{Z}}_{m} & i = 0. \end{array}\right. \] | Yes |
Corollary 5.3.13. Let \( f : X \rightarrow Y \) and suppose that \( {f}_{ * } : {H}_{n}\left( X\right) \rightarrow {H}_{n}\left( Y\right) \) is an isomorphism for all \( n \) . Then \( {f}_{ * } : {H}_{n}\left( {X;G}\right) \rightarrow {H}_{n}\left( {Y;G}\right) \) is an isomorphism for all \( n \) . | Proof. This follows directly from the universal coefficient theorem and the short five lemma. | Yes |
Lemma 5.4.2. The cross product induces a map\n\n\[ \n{C}_{j}\left( X\right) \otimes {C}_{k}\left( Y\right) \rightarrow {C}_{j + k}\left( {X \times Y}\right) \n\] | Proof. If either \( \Phi \) or \( \Psi \) is degenerate, so is \( \Phi \times \Psi \) . | No |
Lemma 5.4.3. In this situation, \n\n\\[\n\\partial \\left( {\\Phi \\times \\Psi }\\right) = \\left( {\\partial \\Phi }\\right) \\times \\Psi + {\\left( -1\\right) }^{j}\\Phi \\times \\left( {\\partial \\Psi }\\right) .\n\\]\n | Proof. Direct calculation, with careful attention to signs. | No |
Lemma 5.4.5. The cross product induces a map\n\n\[ \n{H}_{j}\left( X\right) \otimes {H}_{k}\left( Y\right) \rightarrow {H}_{j + k}\left( {X \times Y}\right) \n\] | Proof. First we show that we obtain a map\n\n\[ \n{Z}_{j}\left( X\right) \otimes {Z}_{k}\left( Y\right) \rightarrow {Z}_{j + k}\left( {X \times Y}\right) \n\]\n\nLet \( c \in {Z}_{j}\left( X\right) \) and \( d \in {Z}_{k}\left( Y\right) \) be singular cycles, so that \( \partial c = 0 \) and \( \partial d = 0 \) . Then... | Yes |
Theorem 5.4.6 (Künneth formula). (1) For any spaces \( X \) and \( Y \), there is a split short exact sequence\n\n\[ 0 \rightarrow {\left( {H}_{ * }\left( X\right) \otimes {H}_{ * }\left( Y\right) \right) }_{n} \rightarrow {H}_{n}\left( {X \times Y}\right) \rightarrow {\left( \mathrm{{Tor}}\left( {H}_{ * }\left( X\righ... | Proof. This is a purely algebraic result, whose proof we omit, but we again remark that it crucially uses the fact that \( {C}_{ * }\left( X\right) \) and \( {C}_{ * }\left( Y\right) \) are chain complexes of free abelian groups. | No |
Lemma 5.4.8. Let \( Y \) be a path connected space and let \( \pi : X \times Y \rightarrow X \) be projection on the first factor. For any element \( \alpha \) of \( {H}_{n}\left( X\right) \) , \[ {\pi }_{ * }\left( {\alpha \otimes {1}_{Y}}\right) = \alpha \] | Proof. Clear from the construction in Lemma 5.4.5. | No |
Lemma 5.5.3. The map \( {f}^{ * } : {C}^{ * }\left( Y\right) \rightarrow {C}^{ * }\left( X\right) \) induces a map \( {f}^{ * } : {H}^{ * }\left( Y\right) \rightarrow {H}^{ * }\left( X\right) \) . | Proof. It is routine to check that \( {f}^{ * }\left( {{Z}^{n}\left( Y\right) }\right) \subseteq {Z}^{n}\left( X\right) \) and \( {f}^{ * }\left( {{B}^{n}\left( Y\right) }\right) \subseteq {B}^{n}\left( X\right) \) . | No |
Theorem 5.5.4. Singular cohomology is an ordinary cohomology theory with \( \mathbb{Z} \) coefficients. | Proof. This proof entirely mimics the proof that singular homology is an ordinary homology theory with \( \mathbb{Z} \) coefficients. There is just one subtlety, Axiom 4, the exactness axiom. Exactness for homology followed from the short exactness of the sequence of singular chain complexes \[ 0 \rightarrow {C}_{ * }\... | Yes |
Theorem 5.5.7. Singular cohomology with coefficients in \( G \) is an ordinary cohomology theory with coefficient group \( G = {H}^{0}\left( {X;G}\right) \) . | Proof. Again this mirrors the proof for singular homology in Sect. 5.1. Again Axiom 4 uses the fact that, for every \( n \), the sequence \( 0 \rightarrow {C}^{n}\left( {X, A}\right) \rightarrow {C}^{n}\left( X\right) \rightarrow \) \( {C}^{n}\left( A\right) \rightarrow 0 \) is split exact. | No |
Theorem 5.5.8 (Universal coefficient theorem). (1) Let \( X \) be a space and let \( G \) be an abelian group. Suppose that \( X \) is of finite type or that \( G \) is of finitely generated. Then there is a split short exact sequence\n\n\[ 0 \rightarrow {H}^{n}\left( X\right) \otimes G \rightarrow {H}^{n}\left( {X;G}\... | Proof. Again we omit the purely algebraic argument, but we note that, while the cochain groups \( {C}^{ * }\left( X\right) \) are not in general free, they are torsion-free, and that fact, together with our additional hypotheses, suffices to be able to apply that argument. | No |
Lemma 5.5.11. The evaluation map e induces a map\n\n\[ e : {H}^{n}\left( X\right) \otimes {H}_{n}\left( X\right) \rightarrow \mathbb{Z} \]\n\nby \( e\left( {\left\lbrack \gamma \right\rbrack ,\left\lbrack c\right\rbrack }\right) = \gamma \left( c\right) \), where \( \gamma \) (resp. \( c \) ) is a representative of the... | Proof. We can restrict \( e \) to evaluate cocycles on cycles,\n\n\[ e : {Z}^{n}\left( X\right) \otimes {Z}_{n}\left( X\right) \rightarrow \mathbb{Z} \]\n\nby \( e\left( {\gamma, c}\right) = \gamma \left( c\right) \) . But then if \( c \) is a boundary, \( c = \partial d, e\left( {\gamma, c}\right) = e\left( {\gamma ,\... | Yes |
Theorem 5.5.12 (Universal coefficient theorem). (1) For any space \( X \) and abelian group \( G \), there is a split short exact sequence\n\n\[ 0 \rightarrow \operatorname{Ext}\left( {{H}_{n - 1}\left( X\right), G}\right) \rightarrow {H}^{n}\left( {X;G}\right) \overset{e}{ \rightarrow }\operatorname{Hom}\left( {{H}_{n... | Proof. Again this is a purely algebraic argument which we omit. | No |
Corollary 5.5.15. Let \( X \) be a space of finite type and suppose that \( {H}_{n}\left( X\right) \approx {F}_{n} \oplus {T}_{n} \) , where \( {F}_{n} \) is a free abelian group and \( {T}_{n} \) is a torsion group, for each \( n \) . Then\n\n\[ \n{H}^{n}\left( X\right) \approx {F}_{n} \oplus {T}_{n - 1} \n\]\n\nfor e... | Proof. This follows from the computation of Ext in Lemma A.3.12. | No |
The integral singular cohomology of \( \mathbb{R}{P}^{n} \) is as follows: | \[ {H}^{k}\left( {\mathbb{R}{P}^{n}}\right) = \left\{ \begin{array}{ll} 0 & k > n \\ \mathbb{Z} & k = n\text{ odd } \\ 0 & k = n\text{ even } \\ {\mathbb{Z}}_{2} & 1 \leq k \leq n - 1\text{ even } \\ 0 & 1 \leq k \leq n - 1\text{ odd } \\ \mathbb{Z} & k = 0, \end{array}\right. \] as we see from Corollary 5.5.15 and The... | Yes |
Theorem 5.5.19 (Universal coefficient theorem). (1) Let \( X \) be a space of finite type. For any abelian group \( G \) there is a split short exact sequence\n\n\[ 0 \rightarrow \operatorname{Ext}\left( {{H}^{n + 1}\left( X\right), G}\right) \rightarrow {H}_{n}\left( {X;G}\right) \overset{e}{ \rightarrow }\operatornam... | Proof. Again we omit this purely algebraic proof. | No |
Theorem 5.5.20. Let \( X \) be a space with finitely generated homology. Let \( \mathbb{F} \) be an arbitrary field. Then \( \chi \left( X\right) \) is given by\n\n\[ \chi \left( X\right) = \left\{ \begin{array}{l} \mathop{\sum }\limits_{{n = 0}}^{\infty }{\left( -1\right) }^{n}\operatorname{rank}{H}_{n}\left( {X;\math... | Proof. This follows directly from the universal coefficient theorems. (If \( \mathbb{F} \) is a field of characteristic zero, then all of these ranks are equal for every integer \( n \) . If \( \mathbb{F} \) does not have characteristic 0, that may not be the case, depending on the space \( X \), but nevertheless the a... | Yes |
Lemma 5.5.23. Let \( \pi : X \times Y \rightarrow X \) be projection on the first factor. For any element \( \alpha \) of \( {H}^{n}\left( X\right) \) , | \[ {\pi }^{ * }\left( \alpha \right) = \alpha \otimes {1}^{Y} \] | No |
Lemma 5.6.3. In this situation, \n\n\[ \n\delta \left( {f \times g}\right) = \left( {\delta f}\right) \times g + {\left( -1\right) }^{j}f \times \left( {\delta g}\right) . \n\] | Proof. Entirely analogous to the proof of Lemma 5.4.3. | No |
Lemma 5.6.4. The cross product induces a map\n\n\[ \times : {H}^{j}\left( X\right) \otimes {H}^{k}\left( Y\right) \rightarrow {H}^{j + k}\left( {X \times Y}\right) . \] | Proof. Entirely analogous to the proof of Lemma 5.4.5. | No |
Theorem 5.6.5. Let \( \alpha : {X}_{1} \rightarrow {X}_{2} \) and \( \beta : {Y}_{1} \rightarrow {Y}_{2} \) be maps. Then there are commutative diagrams\n\n\[ \n{H}_{j}\left( {X}_{1}\right) \otimes {H}_{k}\left( {Y}_{1}\right) \rightarrow {H}_{j + k}\left( {{X}_{1} \times {Y}_{1}}\right) \]\n\n\[ \n{\alpha }_{ * } \oti... | Proof. This follows directly from the covariance/contravariance of the maps on homology/cohomology and the naturality of the Eilenberg-Zilber maps. | Yes |
Let \( \alpha \in {H}^{j}\left( X\right) \) and \( \beta \in {H}^{k}\left( Y\right) \). Then\n\n\[ \alpha \times \beta = {\pi }_{1}{}^{ * }\left( \alpha \right) \cup {\pi }_{2}{}^{ * }\left( \beta \right) = \left( {\alpha \times {1}^{Y}}\right) \cup \left( {{1}^{X} \times \beta }\right) . \] | Proof. We prove the first of these. The last equality is just Lemma 5.5.23. To prove the first, let \( \bigtriangleup : X \times Y \rightarrow \left( {X \times Y}\right) \times \left( {X \times Y}\right) \) be the diagonal. Then, by definition,\n\n\[ {\pi }_{1}{}^{ * }\left( \alpha \right) \cup {\pi }_{2}{}^{ * }\left(... | Yes |
Theorem 5.6.14. (1) Let \( \alpha \in {H}^{j}\left( X\right) ,\beta \in {H}^{k}\left( X\right) ,\gamma \in {H}^{l}\left( Y\right) \), and \( \delta \in {H}^{m}\left( Y\right) \) . Then, if \( n = j + k + l + m \) , \[ \left( {\alpha \cup \beta }\right) \times \left( {\gamma \cap \delta }\right) = {\left( -1\right) }^{k... | This follows from the previous properties we have obtained with enough careful attention to detail (including signs). | No |
Theorem 5.6.17. Let \( X \) be a space and let \( C \) and \( D \) be subspaces of \( X \) . Assume that \( \{ X \times C, D \times X\} \) and \( \{ C, D\} \) are both excisive couples. Then there is a cup product\n\n\[ \cup : {H}^{j}\left( {X, C}\right) \otimes {H}^{k}\left( {X, D}\right) \rightarrow {H}^{j + k}\left(... | Proof. The condition that \( \{ X \times C, D \times X\} \) be excisive is necessary in order to apply the Eilenberg-Zilber theorem, and the condition that \( \{ C, D\} \) be excisive is necessary to obtain the analog of Lemma 5.6.12. Otherwise, the constructions are entirely analogous (though more complicated). | No |
Corollary 5.6.18. Let \( \left( {X, A}\right) \) be a pair. In part (1), assume that \( \{ X \times A, A \times X\} \) is an excisive couple.\n\n(1) There is a cup product\n\n\[ \cup : {H}^{j}\left( {X, A}\right) \otimes {H}^{k}\left( {X, A}\right) \rightarrow {H}^{j + k}\left( {X, A}\right) \]\n\nand a cap product\n\n... | Proof. (1) This is the special case \( C = D = A \) of Theorem 5.6.17. | No |
We take \( \mathbb{Z} \) coefficients. Let \( p, q \geq 1 \) . Then \( {H}^{p}\left( {S}^{p}\right) \cong \mathbb{Z} \) and we choose a generator \( \alpha \) . Also, \( {H}^{q}\left( {S}^{q}\right) \cong \mathbb{Z} \) and we choose a generator \( \beta \) . Now consider \( {H}^{ * }\left( {{S}^{p} \times {S}^{q}}\righ... | But by Lemma 5.6.12 this gives\n\n\[
\widetilde{\gamma } = \widetilde{\alpha } \cup \widetilde{\beta }
\] | Yes |
Example 5.7.2. Again we take \( \\mathbb{Z} \) coefficients. Let \( p, q \\geq 1 \) . Let \( Y = {S}^{p} \\vee {S}^{q} \\vee {S}^{p + q} \) , i.e., the union of \( {S}^{p},{S}^{q} \), and \( {S}^{p + q} \) with all three spaces identified at one point. | Let \( Z = {S}^{p} \\vee {S}^{q} \\subset Y \) and note that we have a retraction \( f : Y \\rightarrow Z \) given by collapsing \( {S}^{p + q} \) to the identification point. Then \( {f}^{ * } : {H}^{n}\\left( Z\\right) \\rightarrow {H}^{n}\\left( Y\\right) \) is an isomorphism for \( n = p, q \) . Let \( \\alpha \) b... | Yes |
Theorem 5.7.3. Let \( X = {S}^{p} \times {S}^{q} \) and \( Y = {S}^{p} \vee {S}^{q} \vee {S}^{p + q} \) . If \( p \) and \( q \) are not both 0, then \( X \) and \( Y \) are not homotopy equivalent. | Proof. If \( p = 0 \) or \( q = 0 \) this is trivial.\n\nSuppose \( p, q \geq 1 \) . Then by Examples 5.7.1 and 5.7.2 \( X \) and \( Y \) have nonisomorphic cohomology rings, so by Corollary 5.6.16 they are not homotopy equivalent. | Yes |
Corollary 5.7.5. Let \( n > m \geq 1 \) . If \( f : \mathbb{R}{P}^{n} \rightarrow \mathbb{R}{P}^{m} \) is any map, then \( {f}_{ * } : {H}_{1}\left( {\mathbb{R}{P}^{n}}\right) \rightarrow {H}_{1}\left( {\mathbb{R}{P}^{m}}\right) \) is the zero map. | Proof. If \( m = 1 \), then \( {H}_{1}\left( {\mathbb{R}{P}^{n}}\right) = {\mathbb{Z}}_{2} \) and \( {H}_{1}\left( {\mathbb{R}{P}^{m}}\right) = \mathbb{Z} \) and the only map from \( {\mathbb{Z}}_{2} \) to \( \mathbb{Z} \) is the zero map. Suppose \( m > 1 \) . Then \( {H}_{1}\left( {\mathbb{R}{P}^{n}}\right) = {\mathb... | Yes |
Theorem 6.1.8. If \( M \) is an \( n \) -manifold with nonempty boundary then \( \operatorname{int}\left( M\right) \) is an \( n \) -manifold and \( \partial M \) is an \( \left( {n - 1}\right) \) -manifold. | Proof. The first statement is clear. As for the second, if \( x \in \partial M \) and \( {\varphi }_{x} : {\mathbb{R}}_{ + }^{n} \rightarrow U \) is a homeomorphism with \( x \in {\varphi }_{x}\left( {\partial {\mathbb{R}}_{ + }^{n}}\right) \), then \( {\varphi }_{x} \mid \partial {\mathbb{R}}_{ + }^{n} \) is a homeomo... | Yes |
Lemma 6.2.1. Let \( M \) be an n-manifold and let \( x \in M \) be arbitrary. Then \( {H}_{n}\left( {M, M - x;G}\right) \) is isomorphic to \( G \) . | Proof. Let \( \left( {{U}_{\alpha },{\varphi }_{\alpha }}\right) \) be a coordinate patch with \( x \in {\varphi }_{\alpha } \) . Let \( p = {\varphi }_{\alpha }^{-1}\left( x\right), p \in {\mathbb{R}}^{n} \) . Then we have maps\n\n\[ \left( {{\mathbb{R}}^{n},{\mathbb{R}}^{n}-\{ p\} }\right) \rightarrow \left( {{U}_{\a... | Yes |
Theorem 6.2.6. Every manifold is \( \mathbb{Z}/2\mathbb{Z} \) -orientable. | Proof. Following the diagram in Definition 6.1.3 all the way around from \( G \) to \( G \) gives an isomorphism from \( G \) to \( G \), and \( {\bar{\varphi }}_{x} \) and \( {\bar{\varphi }}_{y} \) are compatible if and only if this isomorphism is the identity. But the only isomorphism \( \bar{\varphi } : \mathbb{Z}/... | Yes |
Theorem 6.2.7. (1) Let \( M \) be the union of components \( M = {M}_{1} \cup {M}_{2} \cup \cdots \) . Then \( M \) is \( \mathbb{Z} \) -orientable if and only if each \( {M}_{i} \) is \( \mathbb{Z} \) -orientable. | Proof. The important thing to note is that if \( {\bar{\varphi }}_{x} : \mathbb{Z} \rightarrow {H}_{n}\left( {M, M - x;\mathbb{Z}}\right) \) is a local \( \mathbb{Z} \) -orientation, there is exactly one other local \( \mathbb{Z} \) -orientation at \( x \), namely \( - {\bar{\varphi }}_{x} \), where \( - {\bar{\varphi ... | Yes |
Theorem 6.2.10. Let \( M \) be a connected \( n \) -manifold. Then a system of local orientations \( \left\{ {\bar{\varphi }}_{x}\right\} \) is an orientation of \( M \) if and only if for every \( x, y \in M \) and every path \( f : I \rightarrow M \) with \( f\left( 0\right) = x \) and \( f\left( 1\right) = y,{\bar{\... | Proof. If \( M \) is orientable, let \( \left\{ {\bar{\varphi }}_{x}\right\} \) be an orientation, i.e., a compatible system of local orientations. Then for any path \( f,{f}_{y}\left( {\bar{\varphi }}_{x}\right) = {\bar{\varphi }}_{y} \) is independent of the choice of \( f \) .\n\nConversely, if \( {f}_{y}\left( {\ba... | Yes |
Lemma 6.2.11. (1) Let \( x \) and \( y \) be two points in \( M \) that are both contained in some coordinate patch \( {U}_{\alpha } \) . Then for any two paths \( f \) and \( g \) from \( x \) to \( y \) with \( f\left( I\right) \subset {U}_{\alpha } \) and \( g\left( I\right) \subset {U}_{\alpha },{f}_{y}\left( {\bar... | Proof. (1) Since \( I \) is compact, \( f\left( I\right) \) is a compact subset of \( {U}_{\alpha } \), and hence \( {\varphi }_{\alpha }^{-1}\left( {f\left( I\right) }\right) \) is a compact subset of \( {\mathbb{R}}^{n} \), as is \( {\varphi }_{\alpha }^{-1}\left( {g\left( I\right) }\right) \) . But then we may choos... | Yes |
Theorem 6.2.13. Let \( M \) be a connected \( n \) -manifold. If \( M \) is simply connected, then \( M \) is orientable. | Proof. By Theorem 6.2.10, we must show that \( \left\{ {{f}_{y}\left( {\bar{\varphi }}_{x}\right) }\right\} \) is independent of the choice of \( f \) . By Lemma 6.2.12(2), that will be the case if \( {h}_{x}\left( {\bar{\varphi }}_{x}\right) = {\bar{\varphi }}_{x} \) for any loop \( {h}_{x} \) based at \( x \) . But b... | Yes |
Theorem 6.2.15. A connected manifold \( M \) is orientable if and only if its orientation character \( w\left( f\right) = 0 \) for every loop \( f \) in \( M \) . | Proof. In light of Lemma 6.2.12, this is just a restatement of Theorem 6.2.10. | No |
Lemma 6.2.16. Let \( M \) be a connected manifold. The orientation character gives a homomorphism\n\n\[ w : {\pi }_{1}\left( {M, x}\right) \rightarrow \mathbb{Z}/2\mathbb{Z} \]\n\ndefined by \( w\left( \alpha \right) = w\left( f\right) \) where \( f \) is a loop in \( M \) representing \( \alpha \in {\pi }_{1}\left( {M... | Proof. By Lemma 6.2.12(2), \( w \) depends only on the homotopy class of \( f \), and by Lemma 6.2.12(1), \( w \) is a homomorphism. | Yes |
Corollary 6.2.17. Let \( M \) be a connected nonorientable manifold. Then \( M \) has a unique 2-fold cover \( N \) that is orientable. | Proof. \( N \) is the cover of \( M \) corresponding to the subgroup \( \operatorname{Ker}\left( w\right) \subset {\pi }_{1}\left( {M, x}\right) \) of index 2 as in Theorem 2.2.19. | No |
Lemma 6.2.18. Let \( M \) be a manifold. The orientation character gives a homomorphism\n\n\[ w : {H}_{1}\left( {M;\mathbb{Z}}\right) \rightarrow \mathbb{Z}/2\mathbb{Z} \] | Proof. The map \( w : {\pi }_{1}\left( {M, x}\right) \rightarrow \mathbb{Z}/2\mathbb{Z} \) is a map to an abelian group, so factors through the abelianization of \( {\pi }_{1}\left( {M, x}\right) \) . In case \( M \) is connected that is just \( {H}_{1}\left( {M;\mathbb{Z}}\right) \) by Theorem 5.2.4. In the general ca... | Yes |
Theorem 6.2.19. Let \( M \) be a manifold. The orientation character gives a homomorphism\n\n\[ w : {H}_{1}\left( {M;\mathbb{Z}/2\mathbb{Z}}\right) \rightarrow \mathbb{Z}/2\mathbb{Z} \] | Proof. The map \( w : {H}_{1}\left( {M;\mathbb{Z}}\right) \rightarrow \mathbb{Z}/2\mathbb{Z} \) factors through \( {H}_{1}\left( {M;\mathbb{Z}}\right) /2{H}_{1}\left( {M;\mathbb{Z}}\right) \) (i.e. \( w\left( {2\alpha }\right) = {2w}\left( \alpha \right) = 0 \) for any \( \alpha \in {H}_{1}\left( {M;\mathbb{Z}}\right) ... | No |
Corollary 6.2.20. Let \( M \) be a manifold. If \( {H}_{1}\left( {M;\mathbb{Z}/2\mathbb{Z}}\right) = 0 \), then \( M \) is orientable. | Recall we have the universal coefficient theorem, Theorem 5.5.12. Since \( {H}_{0}\left( {M;\mathbb{Z}}\right) = \mathbb{Z} \), that theorem gives an isomorphism\n\n\[ e : {H}^{1}\left( {M;\mathbb{Z}/2\mathbb{Z}}\right) \rightarrow \operatorname{Hom}\left( {{H}_{1}\left( M\right) ,\mathbb{Z}/2\mathbb{Z}}\right) . \] | No |
Theorem 6.2.26. Let \( M \) be an oriented manifold with boundary. Then \( \partial M \) has a well-defined induced orientation given by the construction in Definition 6.2.25. | Proof. This is simply a matter of checking that the local orientations \( \left\{ {\bar{\varphi }}_{x}\right\} \) are indeed compatible, and that they are independent of the choice of coordinate patches \( \left( {{U}_{\alpha },{\varphi }_{\alpha }}\right) \) used in the construction. | No |
Lemma 6.2.28. Let \( G = \mathbb{F} \) be a field of characteristic 0 or odd characteristic. Then a manifold \( M \) is \( G \) -orientable if and only if it is orientable. If \( G = \mathbb{F} \) is a field of characteristic 2, then every manifold \( M \) is \( G \) -orientable. | Proof. We do the more interesting case of a field \( \mathbb{F} \) of characteristic \( \neq 2 \) . Consider the diagram in Definition 6.2.3.\n\nIf we let \( V = {\varphi }_{x}\left( D\right) \) and replace \( G \) in that diagram by \( {H}_{n}\left( {M, M - V;G}\right) \) and the two vertical maps by the isomorphisms ... | Yes |
Corollary 6.2.31. Let \( M \) be a compact connected \( n \) -manifold with boundary.\n\n(1) For any such \( M,{H}_{n}\left( {M,\partial M;\mathbb{Z}/2\mathbb{Z}}\right) \cong \mathbb{Z}/2\mathbb{Z} \) and \( {H}^{n}\left( {M,\partial M;\mathbb{Z}/2\mathbb{Z}}\right) \cong \mathbb{Z}/2\mathbb{Z} \) . | Proof. The statements on homology are a direct consequence of Theorems 6.2.6 and 6.2.30.\n\nThe statements for cohomology then follow from the universal coefficient theorem and Theorem 6.1.12. | Yes |
Corollary 6.2.36. Let \( M \) be a \( G \) -oriented \( n \) -manifold with boundary with fundamental class \( \left\lbrack {M,\partial M}\right\rbrack \), and let \( \partial M \) have the induced \( G \) -orientation with fundamental class \( \left\lbrack {\partial M}\right\rbrack \) . If \( i : \partial M \rightarro... | Proof. We have the exact sequence of the pair \( \left( {M,\partial M}\right) \) :\n\n\[ \n{H}_{n}\left( {M,\partial M;G}\right) \overset{\partial }{ \rightarrow }{H}_{n - 1}\left( {\partial M;G}\right) \overset{{i}_{ * }}{ \rightarrow }{H}_{n - 1}\left( {M;G}\right) .\n\]\n\nBut \( \left\lbrack {\partial M}\right\rbra... | Yes |
We shall show that \( {S}^{1} \) is orientable. | We take a rather strange looking description and parameterization of \( {S}^{1} \), but we do so to use this as a \ | No |
Let \( G = \mathbb{Z} \). Choose an orientation of \( {S}^{p} \), and let \( {S}^{p} \) have fundamental homology class \( \left\lbrack {S}^{p}\right\rbrack \) and fundamental cohomology class \( \left\{ {S}^{p}\right\} \). Also choose an orientation of \( {S}^{q} \) and let \( {S}^{q} \) have fundamental homology clas... | By Poincaré duality, \( \cap \left\lbrack M\right\rbrack : {H}^{q}\left( M\right) \rightarrow {H}_{p}\left( M\right) \) is an isomorphism. Hence \( \widetilde{\beta } \cap \left\lbrack M\right\rbrack = \pm \widetilde{a} \), and we choose the orientation on \( M \) so that the sign is positive. Then\n\n\[ 1 = e\left( {\... | Yes |
To define an orientation of \( \mathbb{C}{P}^{1} \) it suffices to give a local orientation \( {\bar{\varphi }}_{{z}_{0}} \) at a single point \( {z}_{0} \), and we choose \( {z}_{0} \) to be the point with homogeneous coordinates \( \left\lbrack {0,1}\right\rbrack \) . We specify \( {\bar{\varphi }}_{{z}_{0}} \) by le... | \[ {H}_{1}\left( {S}^{1}\right) \rightarrow {H}_{1}\left( {\mathbb{C}-\{ 0\} }\right) \rightarrow {H}_{2}\left( {\mathbb{C},\mathbb{C}-\{ 0\} }\right) \rightarrow {H}_{2}\left( {\mathbb{C}{P}^{1},\mathbb{C}{P}^{1}-\{ \left\lbrack {0,1}\right\rbrack \} }\right) . \] Here the first isomorphism is induced by inclusion, th... | Yes |
Theorem 6.4.5. Let \( M \) be a compact \( n \) -dimensional manifold with \( n \) odd. Then the Euler characteristic \( \chi \left( M\right) = 0 \) . | Proof. We may use any coefficients to compute the Euler characteristic, so we choose \( \mathbb{Z}/2\mathbb{Z} \) . This means that \( M \) is orientable with these coefficients. Also, they form a field, so for any \( j,{H}_{j}\left( {M;\mathbb{Z}/2\mathbb{Z}}\right) \) and \( {H}^{j}\left( {M;\mathbb{Z}/2\mathbb{Z}}\r... | Yes |
Theorem 6.4.6. Let \( M \) be a compact \( n \) -manifold with odd Euler characteristic. Then \( M \) is not the boundary of a compact \( \left( {n + 1}\right) \) -manifold. | Proof. Suppose that \( M \) is the boundary of the compact \( \left( {n + 1}\right) \) -manifold \( X \) . Consider the exact sequence of the pair \( \left( {X, M}\right) \) :\n\n\[ 0 \rightarrow {H}_{n + 1}\left( {X;\mathbb{Z}/2\mathbb{Z}}\right) \rightarrow {H}_{n + 1}\left( {X, M;\mathbb{Z}/2\mathbb{Z}}\right) \righ... | Yes |
Corollary 6.4.11. Let \( M \) be a compact connected oriented manifold of dimension \( {2n}, n \) odd. Then for \( G = \mathbb{Z} \) or any field \( \mathbb{F} \) of characteristic not equal to 2, \( \operatorname{rank}\left( {{K}^{n}\left( {M;G}\right) }\right) \) is even. Also, the Euler characteristic \( \chi \left(... | Proof. By Theorem 6.4.8, \( \langle \) , \( \rangle {isanonsingularskew} - {symmetricbilinearformon} \) \( {K}^{n}\left( {M;G}\right) \), so by Theorem B.2.1, \( {K}^{n}\left( {M;G}\right) \) must have even rank.\n\nWe may use any field to compute Euler characteristic. Choosing \( \mathbb{F} = \mathbb{Q} \), say, and u... | Yes |
Theorem 6.4.15. Let \( M \) be a compact connected oriented manifold of dimension \( {2n} \) with \( n \) even. If the signature \( \sigma \left( M\right) \neq 0 \), then \( M \) is not the boundary of an oriented \( \left( {{2n} + 1}\right) \) -manifold. | Proof. Suppose that \( M \) is the boundary of \( {X}^{{2n} + 1} \) . Let \( V = {H}^{n}\left( {{M}^{2n};\mathbb{R}}\right) \), and let \( V \) have dimension \( t \) . We will use Lefschetz duality to find a subspace \( {V}_{0} \) of \( V \) of dimension \( t/2 \) with the restriction of the intersection form on \( M ... | Yes |
Theorem 6.4.17. Let \( M \) and \( N \) be compact connected oriented \( n \) -manifolds, \( n > 0 \) , with fundamental classes \( \left\lbrack M\right\rbrack \) and \( \left\lbrack N\right\rbrack \) respectively. Then\n\n\[ \n{H}_{n}\left( {M\# N}\right) \cong {H}^{n}\left( {M\# N}\right) \cong \mathbb{Z} \]\n\n\[ \n... | Proof. We work in cohomology as we wish to obtain the cup product structure. The argument in homology is very similar.\n\nConsider the disjoint union \( M \cup N \) of \( M \) and \( N \) . Then it is certainly true that \( {H}^{j}\left( {M \cup N}\right) \cong {H}^{j}\left( M\right) \oplus {H}^{j}\left( N\right) \) fo... | Yes |
Let \( n \) be even. We adopt the notation and language of Example 6.4.4.\n\nIn the basis of \( {H}^{n}\left( {\mathbb{C}{P}^{n}}\right) \) consisting of the element \( {\alpha }^{n/2} \), intersection form of the oriented manifold \( \mathbb{C}{P}^{n} \) has matrix [1], so \( H\left( {\mathbb{C}{P}^{n}}\right) \) has ... | \[ \diamond \] | No |
Lemma 7.1.4. For \( n \geq 2,{\pi }_{n}\left( {X,{x}_{0}}\right) \) is an abelian group. For \( n \geq 3,{\pi }_{n}\left( {X, A,{x}_{0}}\right) \) is an abelian group. | Proof. Here is a picture of a homotopy between \( {\alpha \beta } \) and \( {\beta \alpha } \) in case \( n = 2 \), for \( {\pi }_{n}\left( {X,{x}_{0}}\right) \) .  Similarly for \( n > 2 \) for \( {\pi }_{n}\left( {... | No |
Theorem 7.1.9. (1) If \( f : \left( {X, A,{x}_{0}}\right) \rightarrow \left( {X, A,{x}_{0}}\right) \) is the identity map, then \( {f}_{ * } \) : \( {\pi }_{n}\left( {X, A,{x}_{0}}\right) \rightarrow {\pi }_{n}\left( {X, A,{x}_{0}}\right) \) is the identity map. | Proof. Parts (1), (2), (3), and (5) are immediate. We leave the proof of (4) as an exercise. | No |
Theorem 7.1.11. Let \( X \) be a path-connected space and let \( {x}_{0},{x}_{1} \in X \) . Let \( \alpha : I \rightarrow X \) be a path from \( {x}_{0} \) to \( {x}_{1} \), i.e., \( \alpha \left( 0\right) = {x}_{0} \) and \( \alpha \left( 1\right) = {x}_{1} \) . Then \( \alpha \) induces an isomorphism \( {\alpha }_{ ... | Proof. Let \( f : \left( {{I}^{n},\partial {I}^{n}}\right) \rightarrow \left( {X,{x}_{0}}\right) \) represent an element of \( {\pi }_{n}\left( {X,{x}_{0}}\right) \) . The following picture shows how to obtain \( {\alpha }_{ * }\left( f\right) \in {\pi }_{n}\left( {X,{x}_{1}}\right) \) for \( n = 2 \), with the general... | No |
Corollary 7.1.12. Let \( X \) be a path-connected space and let \( {x}_{0} \in X \) . Then the construction of Theorem 7.1.11 gives an action of \( {\pi }_{1}\left( {X,{x}_{0}}\right) \) on \( {\pi }_{n}\left( {X,{x}_{0}}\right) \) for every \( n \) . The set of equivalence classes of elements of \( {\pi }_{n}\left( {X... | Proof. The only point to note is that we are considering homotopies \( F : {I}^{n} \times I \rightarrow X \) with the property that for every \( t \in I, F \mid \partial {I}^{n} \times \{ t\} \) is a map to a single point, so this gives us homotopies of maps \( f : {S}^{n} \rightarrow X \) where the point 1 is allowed ... | No |
Theorem 7.2.1. Let \( X \) and \( Y \) be path-connected spaces. Let \( {x}_{0} \in X \) and \( {y}_{0} \in Y \) . Let \( p : X \times Y \rightarrow X \) and \( q : X \times Y \rightarrow Y \) be projection on the first and second factors respectively. Then \( {p}_{ * } \times {q}_{ * } : {\pi }_{n}\left( {X \times Y,\... | Proof. First we show \( {p}_{ * } \times {q}_{ * } \) is onto. Let \( f : \left( {{S}^{n},1}\right) \rightarrow \left( {X,{x}_{0}}\right) \) represent an element \( \alpha \) of \( {\pi }_{n}\left( {X,{x}_{0}}\right) \) and let \( g : \left( {{S}^{n},1}\right) \rightarrow \left( {Y,{y}_{0}}\right) \) represent an eleme... | Yes |
Theorem 7.2.2. Let \( X \) be a path-connected space and let \( \widetilde{X} \) be a connected covering space of \( X \) . Let \( p : \widetilde{X} \rightarrow X \) be the covering projection. Let \( {\widetilde{x}}_{0} \in \widetilde{X} \) and let \( {x}_{0} \in X \) with \( p\left( {\widetilde{x}}_{0}\right) = {x}_{... | Proof. First we show \( {p}_{ * } \) is onto. Let \( f : \left( {{S}^{n},1}\right) \rightarrow \left( {X,{x}_{0}}\right) \) represent an element of \( {\pi }_{n}\left( {X,{x}_{0}}\right) \) . We wish to show there is an \( \widetilde{f} : \left( {{S}^{n},1}\right) \rightarrow \left( {\widetilde{X},{\widetilde{x}}_{0}}\... | Yes |
Corollary 7.2.3. For every \( n \geq 2,{\pi }_{n}\left( {{S}^{1},1}\right) = 0 \) . | Proof. We know from Example 2.2.3 that \( p : \mathbb{R} \rightarrow {S}^{1} \) by \( p\left( t\right) = \exp \left( {2\pi it}\right) \) is a covering map, so for \( n \geq 2,{\pi }_{n}\left( {{S}^{1},1}\right) \cong {\pi }_{n}\left( {\mathbb{R},0}\right) = 0 \) as \( \mathbb{R} \) is contractible. | Yes |
The projection \( p : X \times Y \rightarrow X \) is a locally trivial fiber bundle with fiber \( Y \) . | Indeed, we call this a globally trivial fiber bundle. | No |
Theorem 7.2.9. Let \( p : E \rightarrow B \) be a locally trivial fiber bundle. Let \( {b}_{0} \in B \) , \( F = {p}^{-1}\left( {b}_{0}\right) \), and \( {e}_{0} \in F \) . Then for every \( n \) , \[ {p}_{ * } : {\pi }_{n}\left( {E, F,{e}_{0}}\right) \rightarrow {\pi }_{n}\left( {B,{b}_{0}}\right) \] is an isomorphism... | Proof. First we show \( {p}_{ * } \) is onto. Let \( g : \left( {{I}^{n},\partial {I}^{n}}\right) \rightarrow \left( {B,{b}_{0}}\right) \) represent an element of \( {\pi }_{n}\left( {B,{b}_{0}}\right) \) . Regard \( {I}^{n} \) as \( I \times {I}^{n - 1} \) . Then \( \{ 0\} \times {I}^{n - 1} \subset \partial {I}^{n - ... | Yes |
Corollary 7.2.10. Let \( p : E \rightarrow B \) be a locally trivial fiber bundle with fiber \( F = \) \( {p}^{-1}\left( {b}_{0}\right) \) and let \( {f}_{0} \in F \) . Then there is an exact sequence\n\n\[ \cdots \rightarrow {\pi }_{n}\left( {F,{f}_{0}}\right) \rightarrow {\pi }_{n}\left( {E,{f}_{0}}\right) \overset{{... | Proof. The first claim follows immediately from Theorems 7.1.9 and 7.2.9.\n\nAs for the second claim, if \( s \) is a section, then \( {s}_{ * } : {\pi }_{n}\left( {B,{b}_{0}}\right) \rightarrow {\pi }_{n}\left( {E,{f}_{0}}\right) \) splits \( {p}_{ * } \) , so this long exact sequence breaks up into a series of split ... | Yes |
Theorem 7.2.14. \( {\pi }_{i}\left( {S}^{n}\right) = 0 \) for \( i < n \) . | Proof. Give \( {S}^{i} \) a CW-structure with one cell in dimension \( i \) and one cell in dimension 0, and give \( {S}^{n} \) a CW-structure with one cell in dimension \( n \) and one cell in dimension 0 . Let \( f : {S}^{i} \rightarrow {S}^{n} \) represent an element of \( {\pi }_{i}\left( {S}^{n}\right) \) . Then b... | Yes |
Corollary 7.2.16. For any \( n \geq 1,{\pi }_{n}\left( {S}^{n}\right) \cong \mathbb{Z} \) . | Proof. By Hopf's theorem, we have an isomorphism\n\n\[ \n{\pi }_{n}\left( {S}^{n}\right) \rightarrow \left\{ {\text{ degrees of maps from }{S}^{n}\text{ to }{S}^{n}}\right\} \n\] \n\nBut this latter set is \( \mathbb{Z} \) by Theorem 4.2.31. | Yes |
Lemma 7.2.18. The following diagram commutes: | \[ \cdots \rightarrow {\pi }_{n}\left( {A,{x}_{0}}\right) \rightarrow {\pi }_{n}\left( {X,{x}_{0}}\right) \rightarrow {\pi }_{n}\left( {X, A,{x}_{0}}\right) \overset{\partial }{ \rightarrow }{\pi }_{n - 1}\left( {A,{x}_{0}}\right) \rightarrow \cdots \] | No |
Theorem 7.2.21. (a) For any space \( X,{\sum X} \) is path-connected. | Proof. (a) is trivial | No |
Let \( A \) be a nonsingular \( n \times n \) rational matrix and \( b \) a rational \( n \) -vector. The encoding size of the unique solution of \( {Ax} = b \) is polynomially bounded by the encoding size of \( \left( {A, b}\right) \) . | By Remark 1.1, we may assume that \( \left( {A, b}\right) \) has integer entries. Let \( \theta \) be the largest absolute value of an entry in \( \left( {A, b}\right) \) . The absolute value of the determinant of any square \( n \times n \) submatrix of \( \left( {A, b}\right) \) is at most \( n!{\theta }^{n} \) (by u... | Yes |
Lemma 1.3. Let \( S \subset {\mathbb{R}}^{n} \) and \( c \in {\mathbb{R}}^{n} \) . Then \( \sup \{ {cx} : x \in S\} = \sup \{ {cx} : \) \( x \in \operatorname{conv}\left( S\right) \} \) . | Proof. Since \( S \subseteq \operatorname{conv}\left( S\right) \), we have \( \sup \{ {cx} : x \in S\} \leq \sup \{ {cx} : x \in \) \( \operatorname{conv}\left( S\right) \} \) . We prove \( \sup \{ {cx} : x \in S\} \geq \sup \{ {cx} : x \in \operatorname{conv}\left( S\right) \} \) . Let \( {z}^{ * } \mathrel{\text{:=}}... | Yes |
Lemma 1.4. Consider the 2-variable mixed integer linear set \( S : = \{ \left( {x, y}\right) \in \) \( \left. {\mathbb{Z} \times {\mathbb{R}}_{ + } : x - y \leq \beta }\right\} \) . Let \( f \mathrel{\text{:=}} \beta - \lfloor \beta \rfloor \) . Then\n\n\[ x - \frac{1}{1 - f}y \leq \lfloor \beta \rfloor \]\n\n(1.11)\ni... | Proof. We prove that the inequality (1.11) is satisfied by every \( \left( {x, y}\right) \in S \) . Note that, since \( x \in \mathbb{Z} \), either \( x \leq \lfloor \beta \rfloor \) or \( x \geq \lfloor \beta \rfloor + 1 \) . If \( x \leq \lfloor \beta \rfloor \), then adding this inequality to the inequality \( - y \... | Yes |
Proposition 1.6. Let \( a, b \in \mathbb{Z},\left( {a, b}\right) \neq \left( {0,0}\right) \) . Then\n\n\[ \gcd \left( {a, b}\right) = \min \{ {ax} + {by} : x, y \in \mathbb{Z},{ax} + {by} \geq 1\} . \] | Proof. Let \( m \mathrel{\text{:=}} \min \{ {ax} + {by} : x, y \in \mathbb{Z},{ax} + {by} \geq 1\} \) (it is clear that this set is not empty, as \( \left( {a, b}\right) \neq \left( {0,0}\right) \), thus this minimum exists). Clearly, if \( d\left| {a\text{and}d}\right| b \) then \( d \mid {ax} + {by} \) for every \( x... | Yes |
Proposition 1.7. Given \( a, b \in {\mathbb{Z}}_{ + } \) such that \( a \geq b \) and \( a > 0 \), the Euclidean algorithm runs in polynomial time and correctly returns \( \gcd \left( {a, b}\right) \) . | Proof. At each iteration the pair \( \left( {a, b}\right) \) is replaced by \( \left( {b, r}\right) \), where \( r = a - \) \( \lfloor a/b\rfloor b \) . Therefore \( r < a/2 \) . Furthermore, in the pair \( \left( {a, b}\right), r \) replaces \( a \) after two iterations. Therefore the Euclidean algorithm terminates af... | Yes |
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