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Theorem 6 (Jacobi). \( - \Delta = {\left( 2\pi \right) }^{12}q\mathop{\prod }\limits_{{n = 1}}^{\infty }{\left( 1 - {q}^{n}\right) }^{24} \) . | [This formula is proved in the most natural way by using elliptic functions. Since this method would take us too far afield, we sketch below a different proof, which is \ | No |
Proposition 2.1. The following sequence of groups and homomorphisms\n\n\[ 0 \rightarrow {\widetilde{H}}_{0}\left( X\right) \overset{\xi }{ \rightarrow }{H}_{0}\left( X\right) \overset{{\varepsilon }_{ * }}{ \rightarrow }\mathbf{Z} \rightarrow 0 \]\nis exact. Thus we may identify \( {\widetilde{H}}_{0}\left( X\right) \)... | The proof is easy. It follows that \( {H}_{0}\left( X\right) \) is the direct sum of \( {\widetilde{H}}_{0}\left( X\right) \) and an infinite cyclic subgroup; however, this direct sum decomposition is not natural or canonical; the infinite cyclic summand can often be chosen in many different ways. | No |
Theorem 4.1. Let \( f \) and \( g \) be continuous maps of \( X \) into \( Y \) . If \( f \) and \( g \) are homotopic, then the induced homomorphisms, \( {f}_{ * } \) and \( {g}_{ * } \), of \( {H}_{n}\left( X\right) \) into \( {H}_{n}\left( Y\right) \) are the same. Also, \( {f}_{ * } = {g}_{ * } : {\widetilde{H}}_{0... | Proof: Let \( F : I \times X \rightarrow Y \) be a continuous map such that \( F\left( {0, x}\right) = f\left( x\right) \) and \( F\left( {1, x}\right) = g\left( x\right) \) . We will use the continuous map \( F \) to construct a sequence of homomorphisms\n\n\[ \n{\varphi }_{n} : {C}_{n}\left( X\right) \rightarrow {C}_... | No |
Theorem 4.2. If \( f : X \rightarrow Y \) is a homotopy equivalence, then \( {f}_{ * } : {H}_{n}\left( X\right) \rightarrow {H}_{n}\left( Y\right) \) , \( n = 0,1,2,\ldots \), and \( {f}_{ * } : {\widetilde{H}}_{0}\left( X\right) \rightarrow {\widetilde{H}}_{0}\left( Y\right) \) are isomorphisms. | The proof, which is simple, is left to the reader. | No |
Theorem 5.1. The homology sequence of any pair \( \left( {X, A}\right) \) is exact. | In order to prove this theorem, it obviously suffices to prove the following six inclusion relations:\n\n\[ \text{ image }{i}_{ * } \subset \operatorname{kernel}{j}_{ * },\;\text{ image }{i}_{ * } \supset \operatorname{kernel}{j}_{ * },\]\n\n\[ \text{image}{j}_{ * } \subset \text{kernel}{\partial }_{ * },\;\text{image}... | No |
Theorem 6.1. Let \( f, g : \left( {X, A}\right) \rightarrow \left( {Y, B}\right) \) be maps of pairs. If \( f \) and \( g \) are homotopic (as maps of pairs), then the induced homomorphisms \( {f}_{ * } \) and \( {g}_{ * } \) of \( {H}_{n}\left( {X, A}\right) \) into \( {H}_{n}\left( {Y, B}\right) \) are the same. | The proof proceeds along the same lines as that of Theorem 4.1. Because of the stronger hypothesis on the homotopy \( F \), it follows that the homomorphisms \( {\varphi }_{n} \) constructed in the proof of 4.1 satisfy the following condition:\n\n\[ \n{\varphi }_{n}\left( {{C}_{n}\left( A\right) }\right) \subset {C}_{n... | No |
Theorem 6.2. Let \( \\left( {X, A}\\right) \) be a pair, and let \( W \) be a subset of \( A \) such that \( \\bar{W} \) is contained in the interior of \( A \) . Then the inclusion map \( \\left( {X - W, A - W}\\right) \\rightarrow \) \( \\left( {X, A}\\right) \) induces an isomorphism of relative homology groups: | The proof of this theorem depends on the fact that in the definition of homology groups we can restrict our consideration to singular cubes which are arbitrarily small, and this will not change anything. For example, if \( X \) is a metric space, and \( \\varepsilon \) is a small positive number, we can insist that onl... | No |
Theorem 6.3. Assume that \( \mathcal{U} \) satisfies the above hypotheses. Then the induced homomorphisms \( {\sigma }_{ * } : {H}_{n}\left( {X, A,\mathcal{U}}\right) \rightarrow {H}_{n}\left( {X, A}\right) \) are isomorphisms for all \( n \) . | This theorem is the precise formulation of the assertion made earlier that we can restrict our consideration to singular cubes which are small of order \( \mathcal{U} \) in defining \( {H}_{n}\left( {X, A}\right) \) . The proof, which is rather long, is given in the next section. | No |
Lemma 7.1. Consider the following diagram of abelian groups and homomorphisms.\n\n\n\nAssume that each row is exact, that each square is commutative, that \( {f}_{1} \) is an epimorphism, \( {f}_{2} \) and \( {f}_{4} \... | Proof: It suffices to prove the following two assertions:\n\n(a) For any \( x \in {A}_{3} \), if \( {f}_{3}\left( x\right) = 0 \) then \( x = 0 \) .\n\n(b) Given any \( x \in {B}_{3} \), there exists an element \( y \in {A}_{3} \) such that \( {f}_{3}\left( y\right) = x \) .\n\nThe proof of each of these two assertions... | No |
Theorem 2.1. For any integer \( n \geq 0 \) , \[ {\widetilde{H}}_{i}\left( {S}^{n}\right) = \left\{ \begin{array}{ll} \mathbf{Z} & \text{ if }i = n \\ \{ 0\} & \text{ if }i \neq n \end{array}\right. \] | Proof of THEOREM 2.1. The proof is by induction on \( n \) . The theorem is true for \( n = 0 \), because \( {S}^{0} \) is a space consisting of exactly two points. In order to make the inductive step, we will identify \( {S}^{n} \) with the \ | No |
Proposition 2.5 (Brouwer fixed point theorem). Any continuous map \( f : {E}^{n} \rightarrow {E}^{n} \) has at least one fixed point, i.e., a point \( x \) such that \( f\left( x\right) = x \) . | Proof: Assume to the contrary that \( f\left( x\right) \neq x \) for all \( x \in {E}^{n} \) . Then the two distinct points \( x \) and \( f\left( x\right) \) determine a unique straight line which intersects \( {S}^{n - 1} \) in two points. Let \( v\left( x\right) \) denote that point of the intersection which is such... | No |
Theorem 2.6. There exists a continuous nonzero tangent vector field on \( {S}^{n} \) if and only if \( n \) is odd. | It is easy to give an example of a continuous nonzero tangent vector field on \( {S}^{n} \) for \( n \) odd: One defines\n\n\[ v\left( {{x}_{1},\ldots ,{x}_{n + 1}}\right) = \left( {-{x}_{2},{x}_{1}, - {x}_{4},{x}_{3},\ldots , - {x}_{n + 1},{x}_{n}}\right) .\n\nTo prove that such a vector field does not exist on \( {S}... | Yes |
Theorem 3.1. Let \( \left( {X,{X}^{0}}\right) \) be a finite, regular graph with edges \( {e}_{1},{e}_{2},\ldots ,{e}_{k} \) . Then the inclusion map \( \left( {{\bar{e}}_{i},{\dot{e}}_{i}}\right) \rightarrow \left( {X,{X}^{0}}\right) \) induces a monomorphism \( {H}_{q}\left( {{\bar{e}}_{i},{\dot{e}}_{i}}\right) \righ... | Proof: The third sentence of the theorem is a consequence of the two preceding sentences, in view of Equation (3.1) above. Therefore we will concentrate our attention on the first two sentences of the theorem.\n\nAccording to the definition of a graph, the set \( {\bar{e}}_{i} \) is homeomorphic to the unit interval \(... | Yes |
Theorem 3.2. Let \( \left( {X,{X}^{0}}\right) \) be a finite, regular graph. Then \( {H}_{q}\left( X\right) = 0 \) for \( q > 1 \) , \( {H}_{1}\left( X\right) \) is a free abelian group, and | We leave it to the reader to prove this theorem, using the homology sequence of the pair \( \left( {X,{X}^{0}}\right) \) and the two results from linear algebra stated above. | No |
We wish to determine the induced homomorphism \( {f}_{ * } : {H}_{1}\left( {S}^{1}\right) \rightarrow {H}_{1}\left( {S}^{1}\right) \) . | In order to solve this problem, we need to subdivide \( {S}^{1} \) into a regular graph in two different ways. The first subdivision is into 6 equal arcs by means of the vertices\n\n\[ \n{v}_{j} = \exp \left( {j\frac{\pi \sqrt{-1}}{3}}\right) ,\;j = 0,1,\ldots ,5.\n\]\n\nThe corresponding (oriented) edges \( {e}_{0},{e... | Yes |
Proposition 4.1. The identification map \( f : \left( {{E}^{2},{\dot{E}}^{2}}\right) \rightarrow \left( {X,{X}^{1}}\right) \) induces an isomorphism \( {f}_{ * } : {H}_{q}\left( {{E}^{2},{\dot{E}}^{2}}\right) \rightarrow {H}_{q}\left( {X,{X}^{1}}\right) \) of relative homology groups for all \( q \) . Hence \( {H}_{q}\... | Proof: The last sentence is a consequence of the first sentence and Proposition 2.4. The pattern of proof of the first sentence of the proposition, using the excision property, deformation retracts, etc., is one that we have used before a couple of times.\n\nLet \( x \) denote the center point of the rectangle \( {E}^{... | Yes |
The projective plane may be obtained from a circular disc by identifying diametrically opposite points on the boundary. It is harder to visualize than the surfaces we have considered so far because it can not be imbedded homeomorphically in Euclidean 3- space. It is a nonorientable surface, and this results in a somewh... | As in the previous cases, denote the disc by \( {E}^{2} \), the projective plane by \( X \) , and let \( f : \left( {{E}^{2},{\dot{E}}^{2}}\right) \rightarrow \left( {X,{X}^{1}}\right) \) be the identification map. Here \( {\dot{E}}^{2} \) denotes the boundary circle of \( {E}^{2} \), and \( {X}^{1} = f\left( {\dot{E}}... | Yes |
An arbitrary nonorientable surface \( X \) is the connected sum of \( n \) projective planes, \( n \geq 1 \). If \( n \) is odd, it can be considered as the connected sum of a projective plane and an orientable surface, while if \( n \) is even, it can be considered as the connected sum of a Klein bottle and an orienta... | The final result is that \( {H}_{2}\left( X\right) = 0 \), and \( {H}_{1}\left( X\right) \) is the direct sum of a free abelian group of rank \( n - 1 \) and a cyclic group of order 2 . | Yes |
Theorem 5.1. Let \( A \) and \( B \) be subsets of the topological space \( X \) such that \( X = \) (interior \( A \) ) \( \cup \) (interior \( B \) ). Then it is possible to define natural homomorphisms\n\n\[ \Delta : {H}_{n}\left( X\right) \rightarrow {H}_{n - 1}\left( {A \cap B}\right) \]\n\nfor all values of \( n ... | Proof of Theorem 5.1. Let \( \mathcal{U} = \{ A, B\} \) ; in view of the hypotheses assumed on \( A \) and \( B \) we can apply Theorem II.6.3 to conclude that the inclusion homomorphisms \( \;\sigma :{C}_{n}\left( {X,\mathcal{U}}\right) \rightarrow {C}_{n}\left( X\right) \; \) induces isomorphisms \( \;{\sigma }_{ * }... | Yes |
Proposition 6.1. Let \( \left( {X, A}\right) \) be a pair consisting of a topological space \( X \) and subspace \( A \) . (a) Given any homology class \( u \in {H}_{n}\left( {X, A}\right) \), there exists a compact pair \( \left( {C, D}\right) \subset \left( {X, A}\right) \) and a homology class \( {u}^{\prime } \in {... | The proof of this proposition depends on the following fact: If \( a \in {Q}_{n}\left( X\right) \) , then there exists a compact set \( C \subset X \) such that \( a \in {Q}_{n}\left( C\right) \) . In fact, if \( a \) is a linear combination of the singular \( n \) -cubes \( {T}_{1},{T}_{2},\ldots ,{T}_{k} \), then we ... | No |
Theorem 6.3. Let \( A \) be a subset of \( {S}^{n} \) which is homeomorphic to \( {S}^{k},0 \leq k \leq \) \( n - 1 \) . Then \( {\widetilde{H}}_{n - k - 1}\left( {{S}^{n} - A}\right) = \mathbf{Z} \), and \( {\widetilde{H}}_{i}\left( {{S}^{n} - A}\right) = 0 \) for \( i \neq n - k - 1 \) . | Proof: Once again the proof is by induction on \( k \), using the Mayer-Vietoris sequence. If \( k = 0 \), then \( A \) consists of two points and \( {S}^{n} - A \) is homeomorphic to \( {\mathbf{R}}^{n} \) with one point removed. Hence \( {S}^{n} - A \) has the homotopy type of \( {S}^{n - 1} \), and the theorem is tr... | Yes |
Corollary 6.4 (Jordan-Brouwer theorem). Let \( A \) be a subset of \( {S}^{n} \) which is homeomorphic to \( {S}^{n - 1} \). Then \( {S}^{n} - A \) has exactly two components. | Proof: Apply the case \( k = n - 1 \) of the preceding theorem to conclude that \( {H}_{0}\left( {{S}^{n} - A}\right) \) has rank 2; hence \( {S}^{n} - A \) has exactly two arc components. But it is readily seen that \( {S}^{n} - A \) is locally arcwise connected, hence the components and arc-components are the same. | Yes |
Proposition 6.5. Let \( A \) be a subset of \( {S}^{n} \) which is homeomorphic to \( {S}^{n - 1} \) . Then \( A \) is the boundary of each component of \( {S}^{n} - A \) . | Proof of Proposition 6.5. Since \( {S}^{n} - A \) is locally connected, each component of \( {S}^{n} - A \) is an open subset of \( {S}^{n} - A \), and hence an open subset of \( {S}^{n} \) . Therefore the boundary of each component must be a subset of \( A \) . To complete the proof of the proposition, we must show th... | Yes |
Theorem 6.6. Let \( U \) and \( V \) be homeomorphic subsets of \( {S}^{n} \) . If \( U \) is open, then so is \( V \) (and conversely). | Proof: Let \( h : U \rightarrow V \) be a homeomorphism. For any point \( x \in U \) we can find a closed neighborhood \( N \) of \( x \) in \( U \) such that \( N \) is homeomorphic to \( {I}^{n} \) and its boundary, \( \dot{N} \), is homeomorphic to \( {S}^{n - 1} \) . Let \( y = h\left( x\right) \) ; then \( {N}^{\p... | Yes |
Theorem 7.1. Let \( X \) be an arcwise connected space. Then \( h \) is an isomorphism of the abelianized fundamental group \( {\pi }^{\prime }\left( {X,{x}_{0}}\right) \) onto \( {H}_{1}\left( X\right) \) . | Proof: In order to carry out the proof, it is convenient to show that one can compute the singular homology groups of an arcwise connected space \( X \) using only those singular cubes which have all their vertices mapped into the basepoint \( {x}_{0} \) . There is a certain analogy here with Theorem II.6.3.\n\nLet \( ... | Yes |
Lemma 7.2. If the space \( X \) is arcwise connected, then the homomorphism \( {\tau }_{ * } \) is an isomorphism for all \( n \) . | Proof of Lemma. The strategy of the proof is to show that the system of subgroups \( {C}_{n}\left( {X/{x}_{0}}\right), n = 0,1,2,\ldots \) is a \ | No |
Lemma 4.1. \( {H}_{q}\left( {K}^{n}\right) = 0 \) for all \( q > n \) . | The proof is by induction on \( n \) . For \( n = 0 \), the lemma is trivial, since \( {K}^{0} \) is a discrete space (by definition). The inductive step is proved by using the homology sequence of the pair \( \left( {{K}^{n},{K}^{n - 1}}\right) \) . | No |
Lemma 4.3. Let \( K \) be a finite CW-complex on the space \( X \) . Then\n\n\[ \mathop{\sum }\limits_{n}{\left( -1\right) }^{n}r\left( {{C}_{n}\left( K\right) }\right) = \mathop{\sum }\limits_{n}{\left( -1\right) }^{n}r\left( {{H}_{n}\left( K\right) }\right) . \] | We leave the proof, which depends on Statements (a) and (b) above, to the reader. | No |
Theorem 4.5. The induced homomorphisms \( {f}_{ * } : {H}_{n}\left( X\right) \rightarrow {H}_{n}\left( Y\right) \) and \( {\varphi }_{ * } : {H}_{n}\left( K\right) \rightarrow \) \( {H}_{n}\left( L\right) \) correspond under the isomorphisms \( {\theta }_{n} \) of Theorem 4.2; i.e., the following diagram is commutative... | Proof. This follows immediately from the fact that the following diagram is commutative for all \( n \), together with the definition of \( {\theta }_{n} \) contained in Theorem 4.2: | Yes |
Lemma 5.1. The incidence numbers of a CW-complex have the following properties:\n\n(a) For any n-cell \( {e}_{\lambda }^{n},\left\lbrack {{b}_{\lambda }^{n} : {b}_{\mu }^{n - 1}}\right\rbrack = 0 \) for all but a finite number of \( \left( {n - 1}\right) \) - cells \( {e}_{\mu }^{n - 1} \) .\n\n(b) For any \( n \) -cel... | Proof: The proof of (a) is a direct consequence of the definition of incidence numbers, and the proof of (b) follows from the relation \( {d}_{n - 1}{d}_{n} = 0 \) . To prove (c), recall that \( {C}_{1}\left( K\right) = {H}_{1}\left( {{K}^{1},{K}^{0}}\right) ,{C}_{0}\left( K\right) = {H}_{0}\left( {{K}^{0},{K}^{-1}}\ri... | Yes |
Lemma 5.2. If the cell \( {e}_{\mu }^{n - 1} \) is not contained in the closure of the cell \( {e}_{\lambda }^{n} \) , then \( \left\lbrack {{b}_{\lambda }^{n} : {b}_{\mu }^{n - 1}}\right\rbrack = 0 \) . | Proof: Earlier in this section, it was pointed out that the canonical direct sum decomposition of the group \( {C}_{n}\left( K\right) = {H}_{n}\left( {{K}^{n},{K}^{n - 1}}\right) \) is determined by the monomorphisms\n\n\[ \n{l}_{\lambda * } : {H}_{n}\left( {{\bar{e}}_{\lambda }^{n},{\dot{e}}_{\lambda }^{n}}\right) \ri... | Yes |
Lemma 5.3. Let \( f : \left( {X, A}\right) \rightarrow \left( {Y, B}\right) \) be a map of pairs which is homotopic to a map \( g : \left( {X, A}\right) \rightarrow \left( {Y, B}\right) \) such that \( g\left( X\right) \subset B \) . Then the induced homomorphism\n\n\[ \n{f}_{ * } : {H}_{n}\left( {X, A}\right) \rightar... | Proof. By the homotopy property, \( {f}_{ * } = {g}_{ * } \), hence we must prove that \( {g}_{ * } = 0 \) . The hypotheses imply that \( g \) can be factored, as follows:\n\n\[ \n\left( {X, A}\right) \overset{{g}^{\prime }}{ \rightarrow }\left( {B, B}\right) \overset{i}{ \rightarrow }\left( {Y, B}\right) . \n\] \n\nPa... | Yes |
Lemma 7.1. The incidence numbers \( \left\lbrack {{b}_{\lambda }^{n} : {b}_{\mu }^{n - 1}}\right\rbrack \) in a regular cell complex \( K \) satisfy the following four conditions:\n\n(1) If \( {e}_{\mu }^{n - 1} \) is not a face of \( {e}_{\lambda }^{n} \), then \( \left\lbrack {{b}_{\lambda }^{n} : {b}_{\mu }^{n - 1}}... | Proof: Condition (1) is a consequence of Lemma 5.2 and the definition of the term face.\n\nIn order to prove Statement (2), we will make use of Statement (2) of §6. According to this statement, \( {\bar{e}}_{\lambda }^{n} \) is a subcomplex of \( K \) which contains the cell \( {e}_{\mu }^{n - 1} \), and it is easy to ... | Yes |
Theorem 7.2. Let \( K \) be a regular CW-complex on the topological space \( X \). For each pair \( \left( {{e}_{\lambda }^{n},{e}_{\mu }^{n - 1}}\right) \) consisting of an \( n \) -cell and an \( \left( {n - 1}\right) \) -cell of \( K \), let there be given an integer \( {\alpha }_{\lambda \mu }^{n} = 0 \) or \( \pm ... | Proof: We will prove the existence of the required orientation \( {b}_{\lambda }^{n} \) on the cell \( {e}_{\lambda }^{n} \) by induction on \( n \). For \( n = 0 \) there is no choice: a 0 -cell has a unique orientation, which we denote by \( {b}_{\lambda }^{0 \). \n\nNext, let \( {e}_{\lambda }^{1} \) be a 1-cell, an... | Yes |
Theorem 8.1. If \( K \) is an orientable n-dimensional pseudomanifold, then \( {H}_{n}\left( K\right) \) is infinite cyclic; if \( K \) is nonorientable then \( {H}_{n}\left( K\right) = 0 \) . | The details of the proof are left to the reader. Note that since \( K \) is an \( n \) -dimensional CW-complex, \( {H}_{n}\left( K\right) = {Z}_{n}\left( K\right) \) . If \( K \) is orientable, and the \( n \) - cells are oriented so that any pair having a common face of dimension \( n - 1 \) are coherently oriented, t... | No |
Theorem 2.3. Let \( K = \left\{ {{K}_{n},{\partial }_{n}}\right\} \) and \( {K}^{\prime } = \left\{ {{K}_{n}^{\prime },{\partial }_{n}^{\prime }}\right\} \) be chain complexes such that \( {K}_{n} \) and \( {K}_{n}^{\prime } \) are free abelian groups for all \( n \) . Then a chain map \( f : K \rightarrow {K}^{\prime ... | The only if part of this theorem is a triviality, hence we will be concerned only with the if part. First, we need a couple of lemmas. | No |
Lemma 2.4. Let \( K \) be a chain complex such that \( {Z}_{n}\left( K\right) \) is a direct summand of \( {K}_{n} \) for all \( n \), and \( {H}_{n}\left( K\right) = 0 \) for all \( n \) . Then the identity map and the zero map of \( K \) into itself are chain homotopic. | Proof: For each \( n \), choose a direct sum decomposition\n\n\[ \n{K}_{n} = {Z}_{n}\left( K\right) \oplus {A}_{n} \n\]\n\nSince \( {H}_{n}\left( K\right) = 0,{B}_{n}\left( K\right) = {Z}_{n}\left( K\right) \) for all \( n \) . It follows that \( {\partial }_{n} \) maps \( {A}_{n} \) isomorphically onto \( {Z}_{n - 1}\... | Yes |
Lemma 2.5. Let \( K \) be a chain complex such that \( {K}_{n} \) is a free abelian group. Then \( {Z}_{n + 1}\left( K\right) \) is a direct summand of \( {K}_{n + 1} \) . | Proof. Since \( {K}_{n} \) is free abelian, it follows by a standard theorem of algebra that the subgroup \( {B}_{n}\left( K\right) \) is also free abelian. Because \( {\partial }_{n + 1} \) is a homomorphism of \( {K}_{n + 1} \) onto the free group \( {B}_{n}\left( K\right) \), we can conclude that \( {Z}_{n + 1}\left... | Yes |
Lemma 3.1. If the sequence \( 0 \rightarrow {K}^{\prime }\overset{f}{ \rightarrow }K\overset{g}{ \rightarrow }{K}^{\prime \prime } \rightarrow 0 \) is split exact, then so is the sequence \( 0 \rightarrow {K}^{\prime } \otimes G\xrightarrow[]{f \otimes 1}K \otimes G\xrightarrow[]{g \otimes 1}{K}^{\prime \prime } \otime... | In fact, if \( \left\{ {s}_{n}\right\} \) is a sequence of splitting homomorphisms for the original short exact sequence, then \( \left\{ {{s}_{n} \otimes 1}\right\} \) is a sequence of splitting homomorphisms for the second sequence. | Yes |
Lemma 3.2. If \( {K}^{\prime \prime } \) is a chain complex of free abelian groups, then any short exact sequence \( 0 \rightarrow {K}^{\prime } \rightarrow K \rightarrow {K}^{\prime \prime } \rightarrow 0 \) is split exact. | The proof is easy. | No |
Lemma 3.3. Let \( K \) and \( {K}^{\prime } \) be chain complexes of free abelian groups, and let \( f : K \rightarrow {K}^{\prime } \) be a chain map such that the induced homomorphism \( {f}_{ * } : {H}_{n}\left( K\right) \rightarrow \) \( {H}_{n}\left( {K}^{\prime }\right) \) is an isomorphism for all \( n \) . Then... | Proof: By Theorem 2.3, \( f \) is a chain homotopy equivalence. It follows readily that \( f \otimes {1}_{G} : K \otimes G \rightarrow {K}^{\prime } \otimes G \) is also a chain homotopy equivalence. Hence \( {\left( f \otimes {1}_{G}\right) }_{ * } \) is an isomorphism, as required,\n\nQ.E.D | Yes |
Lemma 6.1. If \( G \) is a free abelian group, then the homomorphism \( \alpha : {H}_{n}\left( K\right) \otimes \) \( G \rightarrow {H}_{n}\left( {K \otimes G}\right) \) is an isomorphism. | Proof: First, one considers the case where \( G = \mathbf{Z} \), which is trivial. In the general case, \( G \) is a direct sum of infinite cyclic groups, and \( \alpha \) obviously \ | No |
Corollary 6.3. For any pair \( \\left( {X, A}\\right) \) and any abelian group \( G \) there exists a split short exact sequence:\n\n\[ 0 \\rightarrow {H}_{n}\\left( {X, A}\\right) \\otimes G\\overset{\\alpha }{ \\rightarrow }{H}_{n}\\left( {X, A;G}\\right) \\overset{\\beta }{ \\rightarrow }\\operatorname{Tor}\\left( {... | The homomorphisms \( \\alpha \) and \( \\beta \) are natural with respect to homomorphisms induced by continuous maps of pairs and coefficient homomorphisms. The splitting can be chosen to be natural with respect to coefficient homomorphisms, but not with respect to homomorphisms induced by continuous maps. | No |
Theorem 2.1. Let \( K \) be a regular CW-complex on \( X \) with cells \( {e}_{1}^{m} \), and let \( L \) be a regular CW-complex on \( Y \) with cells \( {\sigma }_{j}^{n} \) . Assume that the incidence numbers have been chosen for both \( K \) and \( L \) . Then incidence numbers are defined for the product cells on ... | To prove this theorem, we must verify that Statements (1)-(4) of Theorem IV.7.2 are true with the stated choices of incidence numbers. This we leave to the reader as a nontrivial exercise. | No |
Theorem 4.1. Let \( K \) and \( L \) be chain complexes, at least one of which consists of free abelian groups. Then there exists a split exact sequence:\n\n\[ 0 \rightarrow \mathop{\sum }\limits_{{i + j = n}}{H}_{i}\left( K\right) \otimes {H}_{j}\left( L\right) \overset{\alpha }{ \rightarrow }{H}_{n}\left( {K \otimes ... | The proof of this important theorem is not difficult; it may be found in various books on homological algebra and algebraic topology, e.g., Vick [9], Hilton and Stammbach [4], Mac Lane [6], Cartan and Eilenberg [1], or Dold [2]. Actually, the theorem can be proved under slightly more general hypotheses than we have sta... | No |
Proposition 5.1. The tensor product of two free acyclic chain complexes (with augmentations) is again acyclic. | This follows from the Corollary 4.2 to the Künneth theorem and the commutative diagram above. | No |
Lemma 5.2. There exist homomorphisms \( {v}_{n}^{X} : {C}_{n}\left( X\right) \rightarrow {Q}_{n}\left( X\right) \), defined for each space \( X \) and each integer \( n \geq 0 \) such that \( {\mu }_{n}{v}_{n}^{X} = \) identity, and for any continuous map \( f : X \rightarrow Y \), the following diagram is commutative: | Proof: In order to save words, in the rest of this section we will call a homomorphism, such as \( {v}_{n}^{X} \) or \( {\mu }_{n}^{X} \), which is defined for each space \( X \) and commutes with the homomorphism \( {f}_{\# } \) induced by any continuous map \( f \), a natural homomorphism. As examples, we have the fa... | Yes |
Lemma 5.4. Let \( {\varphi }^{X, Y},{\psi }^{X, Y} : C\left( X\right) \otimes C\left( Y\right) \rightarrow C\left( X\right) \otimes C\left( Y\right) \) be a natural collection of chain maps. Then there exists a natural collection of chain homotopies | \[ {D}^{X, Y} : C\left( X\right) \otimes C\left( Y\right) \rightarrow C\left( X\right) \otimes C\left( Y\right) \] such that \[ {\varphi }^{X, Y} - {\psi }^{X, Y} = \partial {D}^{X, Y} + {D}^{X, Y}\partial \] for every ordered pair \( \left( {X, Y}\right) \) of spaces. | Yes |
Theorem 6.1. Let \( \left( {X, A}\right) \) and \( \left( {Y, B}\right) \) be pairs such that \( \{ A \times Y, X \times B\} \) is an excisive couple in \( X \times Y \) . Then there exists a split exact sequence\n\n\[ 0 \rightarrow \mathop{\sum }\limits_{{p + q = n}}{H}_{p}\left( {X, A}\right) \otimes {H}_{q}\left( {Y... | The homomorphisms \( \alpha \) and \( \beta \) are natural, but the splitting is not. | Yes |
The additive group of rational numbers is divisible. | It is easily proved that any quotient group of a divisible group is divisible, and any direct sum of divisible groups is divisible. Thus we could construct many more examples. | No |
Theorem 4.1. An abelian group is injective if and only if it is divisible. | The proof that an injective group is divisible is easy, and is left to the reader. Assume that \( G \) is divisible; we will prove that it is injective. Let \( A, B, h \) , and \( g \) be as in the diagram above. We may as well assume that \( B \) is a subgroup of \( A \), and \( h \) is the inclusion map. Consider all... | No |
Proposition 4.2. Any group is isomorphic to a subgroup of a divisible group. | Proof. There are various ways to prove this. One way is to express the given group \( G \) as the quotient group of a free group \( F \) :\n\n\[ G \approx F/R.\]\n\nObviously \( F \) can be considered as a subgroup of a divisible group \( D \) ; for if \( \left\{ {b}_{i}\right\} \) is a basis for \( F \), then we may t... | Yes |
Theorem 4.3 (Universal coefficient theorem for cohomology). Let \( K \) be a chain complex of free abelian groups, and let \( G \) be an arbitrary abelian group. Then there exists a split exact sequence\n\n\[ 0 \rightarrow \operatorname{Ext}\left( {{H}_{n - 1}\left( K\right), G}\right) \overset{\beta }{ \rightarrow }{H... | Proof. The proof we present is dual to that given in \( § \) V.6. For the reader who has some feeling for this duality, it is a purely mechanical exercise to transpose the previous proof to the present one.\n\nFirst we need a lemma, which is the dual of Lemma V.6.1. | No |
Lemma 4.4. If \( G \) is a divisible group, then the homomorphism\n\n\[ \alpha : {H}^{n}\left( {\operatorname{Hom}\left( {K, G}\right) }\right) \rightarrow \operatorname{Hom}\left( {{H}_{n}\left( K\right), G}\right) \]\n\nis an isomorphism for any chain complex \( K \) . | The proof of this lemma is a nice exercise, involving the various definitions and the fact that divisible groups are injective. | No |
Lemma 11.1. Let \( \left( {X, A}\right) \) be a pair such that \( {H}_{q}\left( {X, A}\right) \) is finitely generated for all \( q \) . Then there exists a chain complex \( K = \left\{ {{K}_{q},{\partial }_{q}}\right\} \) such that each \( {K}_{q} \) is a free abelian group of finite rank, and a chain homotopy equival... | Proof. For each \( q \), choose an epimorphism \( {e}_{q} \) of a finitely generated free abelian group \( {F}_{q} \) onto \( {H}_{q}\left( {X, A}\right) \) ; denote the kernel by \( {R}_{q + 1} \), and let \( {d}_{q + 1} : {R}_{q + 1} \rightarrow {F}_{q} \) denote the inclusion homomorphism. Then\n\n\[ 0 \leftarrow {H... | No |
Theorem 11.2. Let \( \left( {X, A}\right) \) and \( \left( {Y, B}\right) \) be pairs such that the following two conditions hold: \( {H}_{q}\left( {X, A}\right) \) is finitely generated for all \( q \), and \( \{ X \times B, A \times Y\} \) is an excisive couple in \( X \times Y \) . Then the cross product defines a ho... | We will indicate the main steps in the proof, leaving the verification of details to the reader.\n\nBy Lemma 11.1, there exists a chain complex \( K \) of finitely generated free abelian groups and a chain homotopy equivalence \( f : K \rightarrow C\left( {X, A}\right) \) . It follows that \( \operatorname{Hom}\left( {... | No |
Corollary 11.5. Let \( X \) be a space such that \( {H}_{q}\left( {X;F}\right) \) has finite rank over \( F \) for all \( q \) . Then for any space \( Y \), the cohomology algebra \( {H}^{ * }\left( {X \times Y;F}\right) \) is naturally isomorphic to the tensor product: | \[ \alpha : {H}^{ * }\left( {X;F}\right) \underset{F}{ \otimes }{H}^{ * }\left( {Y;F}\right) \approx {H}^{ * }\left( {X \times Y;F}\right) . \] The proof of this theorem and corollary is actually somewhat simpler than the proof of Theorem 11.2 and Corollary 11.3 because one has to deal with vector spaces over \( F \) r... | No |
Theorem 2.1. Let \( M \) be an \( n \) -manifold with orientation \( \mu \) . Then for each compact set \( K \subset M \) there exists a unique homology class \( {\mu }_{K} \in {H}_{n}\left( {M, M - K}\right) \) such that \[ {\rho }_{x}\left( {\mu }_{K}\right) = {\mu }_{x} \] for each \( x \in K \) . | Proof. The uniqueness of \( {\mu }_{K} \) is a direct consequence of a more general lemma below (Lemma 2.2). Therefore we will concentrate on the existence proof. Obviously, if the compact set \( K \) is contained in a sufficiently small neighborhood of some point, the continuity condition in the definition of \( \mu \... | Yes |
Theorem 2.3. Let \( M \) be an arbitrary \( n \) -dimensional manifold (i.e., \( M \) need not be orientable). Then for each compact set \( K \subset M \) there exists a unique homology class \( {\mu }_{K} \in {H}_{n}\left( {M, M - K;{\mathbf{Z}}_{2}}\right) \) such that\n\n\[{\rho }_{x}\left( {\mu }_{K}\right) = {\mu ... | The proof may be patterned on that of Theorem 2.1; the details are left to the reader. | No |
Theorem 4.1 (Poincaré duality). Let \( M \) be an oriented \( n \) -dimensional manifold and \( G \) an arbitrary abelian group. Then the homomorphism\n\n\[ P : {H}_{c}^{q}\left( {M;G}\right) \rightarrow {H}_{n - q}\left( {M;G}\right) \]\n\nis an isomorphism for all \( q \) . | We will give the proof of this theorem now, postponing the discussion of examples, special cases, and applications to later. As in the proof of Lemma 2.2, there are several cases, starting with \( M = {\mathbf{R}}^{n} \), and ending with the general case.\n\nCase 1: \( M = {\mathbf{R}}^{n} \) . Let \( {B}_{k} \) denote... | Yes |
Theorem 4.2. For any n-dimensional manifold \( M \) and any \( {\mathbf{Z}}_{2} \) -vector space \( G \) , the mod 2 Poincaré duality homomorphism \( {P}_{2} \) is an isomorphism of \( {H}_{c}^{q}\left( {M;G}\right) \) onto \( {H}_{n - q}\left( {M;G}\right) \) . | The proof is almost word for word the same as that of Theorem 4.1; the necessary modifications are rather obvious. | No |
Theorem 5.1. Let \( M \) be a compact oriented \( n \) -manifold and \( F \) a field. Then the bilinear form\n\n\[ \n{H}^{q}\left( {M;F}\right) \otimes {H}^{n - q}\left( {M;F}\right) \rightarrow F \]\n\n defined by\n\n\[ \nu \otimes v \rightarrow \left\langle {u \cup v,{\mu }_{M}}\right\rangle \]\n\n for any \( u \in {... | Proof. The relation\n\n\[ \left\langle {u \cup v,{\mu }_{M}}\right\rangle = \left\langle {u, v \cap {\mu }_{M}}\right\rangle \]\n\n can be interpreted as a commutativity relation, as indicated by the following diagram:\n\n be a compact manifold; then the integral homology group \( {H}_{q}\left( M\right) \) is finitely generated for all \( q \) . | If \( M \) could be given the structure of a CW-complex, then compactness would imply that this CW-complex was finite, and the theorem would follow. However it is not known at present whether or not all compact manifolds are CW-complexes. Fortunately, there is a way to avoid this difficulty. By results in Chapter IV, §... | Yes |
Theorem 5.3. Let \( M \) be a compact, connected, oriented \( n \) -manifold. Then the bilinear form\n\n\[ \n{B}^{q}\left( M\right) \otimes {B}^{n - q}\left( M\right) \rightarrow \mathbf{Z} \]\n\ndefined above is nonsingular, and induces an isomorphism of \( {B}^{q}\left( M\right) \) onto \( \operatorname{Hom}\left( {{... | The proof is very similar to that of Theorem 5.1, and may be left to the reader. | No |
Proposition 5.4. Let \( M \) be a compact, orientable manifold of dimension \( n = \) \( {4k} + 2 \), and let \( F \) be a field of characteristic \( \neq 2 \) . Then \( {H}^{{2k} + 1}\left( {M;F}\right) \) is a vector space over \( F \) whose dimension is even. | Proof. By Theorem 5.1, the bilinear form\n\n\[ \n{H}^{{2k} + 1}\left( {M;F}\right) \otimes {H}^{{2k} + 1}\left( {M;F}\right) \rightarrow F \n\]\n\ndefined by\n\n\[ \nu \otimes v \rightarrow \left\langle {u \cup v,{\mu }_{M}}\right\rangle \n\]\n\nis nonsingular. Moreover, by the commutative law for cup products,\n\n\[ \... | Yes |
Theorem 6.1. In each of the following four cases \( A \) is taut in \( X \) with respect to the Alexander-Spanier cohomology theory:\n\n(1) \( A \) is compact and \( X \) is Hausdorff.\n\n(2) \( A \) is closed and \( X \) is paracompact Hausdorff.\n\n(3) \( A \) is arbitrary and every open subset of \( X \) is paracomp... | This theorem is due to Spanier [10]; for a proof, see Massey, [7], pp. 238-241. One case of this theorem is proved in Spanier [9], pp. 316-317. | No |
Proposition 6.3. Let \( M \) be a paracompact \( n \) -manifold, and let \( A \) be a closed subset of \( M \) . Then\n\n\[ \operatorname{dir}\lim {H}^{q}\left( {N;G}\right) \approx {\bar{H}}^{q}\left( {A;G}\right) \]\n\nwhere the direct limit is taken over all neighborhoods \( N \) of \( A \) in \( M \) . | The proof of both of these propositions depends on the naturality of the homomorphism \( \lambda \) . The open neighborhoods of \( A \) are cofinal in the family of all neighborhoods of \( A \) ; and every open neighborhood \( N \) of \( A \) is also a paracompact manifold. Therefore \( \lambda : {\bar{H}}^{q}\left( N\... | No |
Proposition 6.5. Let \( A \) be a closed, proper subset of a compact, connected, orientable n-manifold. Then \( {\bar{H}}^{q}\left( {A;G}\right) = 0 \) for all \( q \geq n \) and all coefficient groups \( G \) . | This is a direct consequence of Proposition 6.4. | No |
Theorem 6.6 (Alexander duality theorem). Let \( M \) be a compact, connected, orientable n-manifold and \( q \) an integer such that \( {H}_{q}\left( {M, G}\right) = {H}_{q + 1}\left( {M, G}\right) = 0 \) . Then for any closed subset \( A \subset M \) , \[ {\bar{H}}^{n - q - 1}\left( A\right) \approx {H}_{q}\left( {M -... | The proof of the Alexander duality theorem follows immediately from Diagram (6.1); the details are left to the reader. | No |
Theorem 7.1. Let \( M \) be a compact \( n \) -dimensional manifold with boundary \( B \) . Then there exists an open neighborhood \( V \) of \( B \) and a homeomorphism \( g \) of \( B \times \lbrack 0,1) \) onto \( V \) such that \( g\left( {b,0}\right) = b \) for any \( b \in B \) . | For a short proof of this theorem, see R. Connelly [3]. Connelly's proof is reproduced in the appendix to Vick [11]. | No |
Corollary 7.3. Let \( {V}_{t} = g\left( {B\times \lbrack 0, t}\right) ) \) for \( 0 < t < 1 \), and \( {K}_{t} = M - {V}_{t} \) . Then \( {V}_{t} \) is an open neighborhood of \( B \) in \( M, B \) is a deformation retract of \( {V}_{t} \), and the collection \( \left\{ {{K}_{t} \mid 0 < t < 1}\right\} \) is cofinal in... | Next, for \( 0 < t < 1 \) let \( {i}_{t} : \left( {M, B}\right) \rightarrow \left( {M, M - {K}_{t}}\right) = \left( {M,{V}_{t}}\right) \) denote the inclusion map. It follows that the induced homomorphisms\n\n\[ \n{i}_{{t}^{ * }} : {H}_{q}\left( {M, B}\right) \rightarrow {H}_{q}\left( {M, M - {K}_{t}}\right)\n\]\n\n\[ ... | No |
Corollary 7.4. \( {H}_{c}^{q}\left( {M - B;G}\right) \) is naturally isomorphic to \( {H}^{q}\left( {M, B;G}\right) \) . | This corollary follows from the definition of \( {H}_{c}^{q}\left( {M - B}\right) \) as a direct limit, the fact that \( {H}^{q}\left( {M - B,\left( {M - B}\right) - K}\right) \approx {H}^{q}\left( {M, M - K}\right) \) for any compact set \( K \subset M - B \), and the cofinality of the family \( \left\{ {K}_{t}\right\... | No |
Theorem 7.5. Let \( M \) be a compact orientable \( n \) -dimensional manifold with boundary B. Then the homomorphism \[ {H}^{q}\left( {M, B;G}\right) \rightarrow {H}_{n - q}\left( {M;G}\right) , \] (defined by \( x \rightarrow x \cap {\mu }_{M} \) for any \( x \in {H}^{q}\left( {M, B;G}\right) \) ) is an isomorphism. | Proof. We already know that \( {H}^{q}\left( {M, B;G}\right) \) is isomorphic to \( {H}_{n - q}\left( {M;G}\right) \) . For, by Corollary 7.4, \( {H}^{q}\left( {M, B}\right) \approx {H}_{c}^{q}\left( {M - B}\right) \) ; then we have the Poincare duality isomorphism \( P : {H}_{c}^{q}\left( {M - B}\right) \approx {H}_{n... | Yes |
Theorem 7.6. Let \( M \) be a compact, oriented, \( n \) -dimensional manifold with boundary \( B \), and let \( {\partial }_{ * } : {H}_{n}\left( {M, B;\mathbf{Z}}\right) \rightarrow {H}_{n - 1}\left( {B,\mathbf{Z}}\right) \) denote the boundary operator of the pair \( \left( {M, B}\right) \) . Then \( {\partial }_{ *... | Proof. In order to prove this theorem, it is necessary to show that for any \( b \in B, j{\partial }_{ * }\left( {\mu }_{M}\right) \) is a generator of the infinite cyclic group \( {H}_{n - 1}\left( {B, B-\{ b\} ;\mathbf{Z}}\right) \) . Here \( j \) denotes the homomorphism \( {H}_{n - 1}\left( B\right) \rightarrow {H}... | Yes |
Lemma 8.3. There exists a homology class \( y \in {H}_{n + 1}\left( {X,{Y}_{1} \cup {Y}_{2}}\right) \) such that for any integer \( q \) and any \( w \in {H}^{q}\left( {X,{Y}_{1} \cap {Y}_{2}}\right) \) , \[ {\Delta }_{ * }\gamma \left( w\right) - \alpha {\Delta }^{ * }\left( w\right) = {\Delta }_{ * }\left( {\left( {{... | Proof of Lemma 8.3. The standard situation which leads to a commutative diagram of exact sequences is the following:  (8.3) In this diagram, the following two hypotheses are assumed: (i) The top and bottom lines are ... | Yes |
Theorem 2.4. There does not exist any continuous antipode preserving map \( f : {S}^{n} \rightarrow {S}^{n - 1}. \) | Proof. We will only give the proof for \( n > 2 \) ; the proof for \( n \leq 2 \) is contained in Algebraic Topology: An Introduction (loc. cit.) The proof is by contradiction. Assume that \( f : {S}^{n} \rightarrow {S}^{n - 1} \) is an antipode preserving map. Hence \( f \) induces a map \( g : R{P}^{n} \rightarrow R{... | Yes |
Lemma 3.1. Let \( p : M\left( f\right) \rightarrow C\left( f\right) \) denote the natural map which identifies the subset \( X = X \times \{ 1\} \) of \( M\left( f\right) \) to a single point \( P \) of \( C\left( f\right) \) . Then the induced homomorphism of relative cohomology groups\n\n\[ \left. {{p}^{ * } : {H}^{q... | Proof. Let \( \bar{X} \) denote the subset \( X \times \left\lbrack {\frac{1}{2},1}\right\rbrack \) of \( M\left( f\right) \), and let \( \bar{P} \) denote the image of \( \bar{X} \) under \( p \) . Consider the following commutative diagram:\n\n, we have the following isomorphisms of homology and cohomology groups:\n\n\[ \n{H}_{p}^{S}\left( {M;G}\right) \approx {H}_{p}\left( {M;G}\right) \]\n\n\[ \n{H}_{S}^{p}\left( {M;G}\right) \approx {H}^{p}\left( {M;G}\right) . \]\n | The corollary follows from the theorem by use of standard techniques (cf. Theorem V.2.3). | No |
Corollary 1. If \( \omega \) is a closed (in particular, holomorphic) 1-form, then (under the hypothesis of the theorem)\n\n\[{\int }_{\delta D}\omega = 0\] | Proof. Take \( f \) to be the constant function with value 1 . | No |
Corollary 2. Let \( f \) be \( {C}^{1} \) function and \( \omega \) a \( {C}^{1} \) -form on the Riemann surface M. If either \( f \) or \( \omega \) has compact support, then\n\n\[ \n{\iint }_{M}{fd\omega } - {\iint }_{M}\omega \land {df} = 0 \n\] | Proof. If \( M \) is not compact, then take \( D \) to be compact and have nice boundary so that either \( f \) or \( \omega \) vanishes on \( {\delta D} \) and use (4.1.1). If \( M \) is compact cover \( M \) by a finite number of disjoint triangles \( {\Delta }_{j}, j = 1,\ldots, n \) . Over each triangle (4.1.1) is ... | Yes |
Corollary 1. If the genus of \( M \) is zero, then \( M \) is conformally equivalent to the complex sphere \( \mathbb{C} \cup \{ \infty \} \) . | Proof. Consider the point divisor, \( P \in M \) . Then\n\n\[ r\left( {P}^{-1}\right) = 2\text{.} \]\n\nThus, there is a non-constant meromorphic function \( z \) in \( L\left( {P}^{-1}\right) \) . Such a function provides an isomorphism between \( M \) and \( \mathbb{C} \cup \{ \infty \} \) by Proposition I.1.6. | Yes |
Corollary 2. The degree of the canonical class \( Z \) is \( {2g} - 2 \) . | Proof. If \( g = 0 \), then compute the degree of the divisor \( {dz} \) (which is regular except for a double pole at \( \infty \) ). Thus we may assume \( g > 0 \) . Since the space of holomorphic abelian differentials has positive dimension, we may choose one such non-trivial differential; say \( \zeta \) . Since \(... | No |
Corollary 3. The Riemann-Roch theorem holds for the divisor \( \mathfrak{A} \) provided that\n\na. \( \mathfrak{A} \) is equivalent to an integral divisor, or\n\nb. \( Z/\mathfrak{A} \) is equivalent to an integral divisor for some canonical divisor \( Z \) . | Proof. Statement (a) follows from the trivial observation that all integers appearing in the Riemann-Roch theorem depend only on the divisor class of \( \mathfrak{A} \) (by Theorem III.4.6). Thus, to verify (b), we need verify Riemann-Roch for \( \mathfrak{A} \) provided we know it for \( Z/\mathfrak{A} \) . Now\n\n\[ ... | Yes |
Corollary 2. Under the hypothesis of Corollary 1, for an open dense set in D, the basis \( \left\{ {{\varphi }_{1},\ldots ,{\varphi }_{n}}\right\} \) of \( A \) adapted to \( z \) has the property | Proof. By the hypothesis \( {\operatorname{ord}}_{z}{\varphi }_{j} = {\mu }_{j} \) . It follows from Corollary 1 that \( \tau \left( z\right) = \mathop{\sum }\limits_{{j = 1}}^{n}\left( {{\mu }_{j} - j + 1}\right) = 0 \) for an open dense set. Since (as we have previously remarked) \( {\mu }_{j} \geq j - 1 \) we have \... | Yes |
Corollary 1. If \( M \) is of genus 1, then\n\n\[ \varphi : M \rightarrow J\left( M\right) \]\n\nis an isomorphism (conformal homeomorphism). | Proof. Clearly \( \varphi \) is surjective (since it is not constant). Let \( P, Q \in M, P \neq Q \) . If \( \varphi \left( P\right) = \varphi \left( Q\right) \), then \( P/Q \) is principal by Abel’s theorem. Thus there is a meromorphic function on \( M \) with a single simple pole. This contradiction shows that \( \... | Yes |
Theorem. Let \( D \in \operatorname{Div}\left( M\right) \) with \( D \geq 1 \) and \( \deg D = g \) . There is an integral divisor \( {D}^{\prime } \) of degree \( g \) close to \( D \) such that \( {D}^{\prime } \) is not special. Further \( {D}^{\prime } \) may be chosen to consist of \( g \) distinct points. | Proof of Theorem. Note that for divisors of degree \( g \) ,\n\n\[ r\left( {D}^{-1}\right) = 1 \Leftrightarrow i\left( D\right) = 0. \]\n\nAssume that \( D \) is given by (6.5.1) and define\n\n\[ {D}_{j} = {P}_{1}\cdots {P}_{j}\;\left( {j = 1,\ldots, g}\right) . \]\n\nThus \( {D}_{g} = D \) . Set \( {D}_{0} = 1 \) . We... | Yes |
Corollary 1. The \( g \) differentials\n\n\[ \frac{{z}^{j}{dz}}{w},\;j = 0,\ldots, g - 1, \]\n\n(7.5.1)\n\nform a basis for the abelian differentials of the first kind on \( M \) . | Proof. Without loss of generality \( z\left( {P}_{j}\right) \neq 0 \), and\n\n\[ \left( z\right) = \frac{{Q}_{3}{Q}_{4}}{{Q}_{1}{Q}_{2}} \]\n\nIt is clear of course that the differentials in (7.5.1) are linearly independent, and all we must show is that they are holomorphic. Since\n\n\[ \left( {dz}\right) = \frac{{P}_{... | Yes |
Corollary 2. On a hyperelliptic surface of genus \( g \geq 2 \) the products of the holomorphic abelian differentials (taken 2 at a time) form a \( \left( {{2g} - 1}\right) \) -dimensional subspace of the \( \left( {{3g} - 3}\right) \) -dimensional space of all holomorphic quadratic differentials. | Proof. Using the basis constructed in Corollary 1, the span of the products has a basis consisting of\n\n\[ \frac{{z}^{j}{\left( dz\right) }^{2}}{{w}^{2}},\;j = 0,\ldots ,{2g} - 2. \] | Yes |
Corollary 2. If \( g \geq 2 \), then the hyperelliptic involution is the unique involution with \( {2g} + 2 \) fixed points. | Proof. Let \( \widetilde{J} \) be another involution with \( {2g} + 2 \) fixed points. Then, as we have seen, the fixed points must be the Weierstrass points. Also if \( z \) is a function of degree 2 on \( M \), so is \( z \circ \widetilde{J} \) . Thus by Theorem III.7.3, there is a Möbius transformation \( A \) such ... | Yes |
Corollary 3. The hyperelliptic involution \( J \) on a (hyperelliptic) surface \( M \) of genus \( g \geq 2 \) is in the center of Aut \( M \) . | Proof. Let \( h \in \) Aut \( M \) . Then \( h \circ J \circ {h}^{-1} \) is an involution and fixes the \( {2g} + 2 \) points \( h\left( {P}_{j}\right) \) . Thus it is the hyperelliptic involution. Hence \( h \circ J \circ {h}^{-1} = J \) , or \( h \) commutes with \( J \) . | Yes |
Corollary 1. Under the hypothesis of the proposition,\n\n\[ c\left( {D}_{1}\right) + c\left( {D}_{2}\right) \geq c\left( D\right) + c\left( {{D}_{1}{D}_{2}{D}^{-1}}\right) . \]\n\n(8.3.4) | Proof. The proof is by direct computation. | No |
Corollary 2. If \( D \) is a special divisor with complementary divisor \( {D}^{ * } \), then\n\n\[ c\left( D\right) \geq c\left( \left( {D,{D}^{ * }}\right) \right) . \] | Proof. By Corollary 1 and Proposition III.8.2,\n\n\[ {2c}\left( D\right) = c\left( D\right) + c\left( {D}^{ * }\right) \geq c\left( \left( {D,{D}^{ * }}\right) \right) + c\left( \frac{D{D}^{ * }}{\left( D,{D}^{ * }\right) }\right) \]\n\n\[ = {2c}\left( \left( {D,{D}^{ * }}\right) \right) \text{.} \] | Yes |
Corollary 1. Let \( D \) be a divisor on \( M \) with \( 0 \leq \deg D \leq {2g} - 2 \) . Then \( c\left( D\right) \geq 0 \) . Equality occurs if and only if \( D \) is principal or canonical, unless \( M \) is a hyperelliptic Riemann surface. | Proof. The remarks preceding the statement of the corollary show that unless \( r\left( {D}^{-1}\right) \geq 2, c\left( D\right) \geq \deg D \) . If \( r\left( {D}^{-1}\right) \geq 2 \), we have \( D \) equivalent to an integral divisor \( {D}^{\prime } \) of the same degree. Proposition III.8.2 allows us to assume wit... | Yes |
Corollary 2. If \( D \) is a divisor on \( M \) with \( 0 \leq \deg D \leq {2g} - 2 \), then\n\n\[ i\left( D\right) \leq g - \frac{\deg D}{2}. \]\n\nEquality implies that \( D \) is principal or canonical, unless \( M \) is hyperelliptic. | Proof. In view of (8.2.1), this is a restatement of Corollary 1. | No |
Corollary 3. Let \( M \) be a compact Riemann surface of genus \( g > 4 \) . Let \( {D}_{1} \) and \( {D}_{2} \) be two inequivalent integral divisors of degree 3 such that \( r\left( {D}_{1}^{-1}\right) = \) \( 2 = r\left( {D}_{2}^{-1}\right) \) . Then \( M \) is hyperelliptic. | Proof. Choose non-constant functions \( {f}_{j} \in L\left( {D}_{j}^{-1}\right), j = 1,2 \) . We may assume that each \( {f}_{j} \) is of degree 3 as otherwise there is nothing to prove. Since \( {D}_{1} \) and \( {D}_{2} \) are inequivalent, \( {f}_{1} \neq c{f}_{2} \) for any \( c \in \mathbb{C} \) . As a matter of f... | Yes |
Corollary 2. Let \( M \) be a compact Riemann surface of genus \( g \geq 4 \) . Let \( A \) and B be inequivalent integral divisors with\n\n\[ 3 \leq \deg B \leq \deg A \leq g - 1 \]\n\nand\n\n\[ c\left( A\right) = 1 = c\left( B\right) . \]\n\nThen unless \( B = {A}^{ * } \) (a complementary divisor of \( A \) ) \( M \... | Proof. From the definition of Clifford index,\n\n\[ {2r}\left( {A}^{-1}\right) = 1 + \deg A \geq 4. \]\n\nThus \( r\left( {A}^{-1}\right) \geq 2 \) and \( r\left( {B}^{-1}\right) \geq 2 \) . There are now two possibilities.\n\nCase \( I : B \) is not the polar divisor of a function. Then there is at least one \( P \in ... | Yes |
Corollary 1. If \( f \neq 0 \) is a multiplicative function, then \( \deg \left( f\right) = 0 \) . | Proof. The order of \( f \) at \( P \), ord \( {}_{P}f \), just as in the ordinary case, is given by the residue at \( P \) of \( {df}/f \) . Since \( {df}/f \) is an abelian differential, the sum of its residues is zero. | No |
Corollary 2. If \( \omega \neq 0 \) is a multiplicative differential, then \( \deg \left( \omega \right) = {2g} - 2 \) . | Proof. Choose an abelian differential \( {\omega }_{1} \) on \( M,{\omega }_{1} \neq 0 \) . Then \( \omega /{\omega }_{1} \) is a multiplicative function belonging to the same character as \( \omega \) . Since \( \deg \left( {\omega }_{1}\right) = \) \( {2g} - 2 \), Corollary 1 yields Corollary 2. | No |
Corollary 1. If \( u \) is harmonic and bounded in \( \{ 0 < \left| z\right| < 1\} \), then \( \alpha = 0 \) . | Proof. If \( M = \sup \left| u\right| \) on \( 0 < \left| z\right| < 1 \), then\n\n\[ \left| {{\int }_{0}^{2\pi }u\left( {r{e}^{i\theta }}\right) {d\theta }}\right| \leq {M2\pi } \] | Yes |
Corollary 2. If \( u \) is harmonic and bounded in \( \{ 0 < \left| z\right| < 1\} \), then \( u \) can be extended as a harmonic function to \( \{ \left| z\right| < 1\} \) . | Proof. Since \( \alpha = 0,{\int }_{\left| z\right| = r} * {du} = 0 \) . Thus (we assume \( u \) is real, this involves no loss of generality) there exists an analytic function \( f \) on \( \{ 0 < \left| z\right| < 1\} \) , with \( u = \operatorname{Re}f \) on \( \{ 0 < \left| z\right| < 1\} \) . Set \( F = \exp f \) ... | Yes |
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