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A 3-dimensional torus \( {T}^{3} = {S}^{1} \times {S}^{1} \times {S}^{1} \) can be constructed from a cube by identifying each pair of opposite square faces as in the first of the two figures. The second figure shows a slightly different pattern of identifications of opposite faces, with the front and back faces now id... | In the case of the 3-torus \( {T}^{3} \) the cellular boundary map \( {d}_{2} \) is zero by the same calculation as for the 2-dimensional torus. We claim that \( {d}_{3} \) is zero as well. This amounts to saying that the three maps \( {\Delta }_{\alpha \beta } : {S}^{2} \rightarrow {S}^{2} \) corresponding to the thre... | Yes |
Example 2.42: Real Projective Space \( {\mathbb{{RP}}}^{n} \) . As we saw in Example 0.4, \( {\mathbb{{RP}}}^{n} \) has a CW structure with one cell \( {e}^{k} \) in each dimension \( k \leq n \), and the attaching map for \( {e}^{k} \) is the 2-sheeted covering projection \( \varphi : {S}^{k - 1} \rightarrow {\mathbb{... | \[ 0 \rightarrow \mathbb{Z}\overset{2}{ \rightarrow }\mathbb{Z}\overset{0}{ \rightarrow }\cdots \overset{2}{ \rightarrow }\mathbb{Z}\overset{0}{ \rightarrow }\mathbb{Z}\overset{2}{ \rightarrow }\mathbb{Z}\overset{0}{ \rightarrow }\mathbb{Z} \rightarrow 0\;\text{ if }n\text{ is even } \] \[ 0 \rightarrow \mathbb{Z}\over... | Yes |
Theorem 2.44. \( \chi \left( X\right) = \mathop{\sum }\limits_{n}{\left( -1\right) }^{n}\operatorname{rank}{H}_{n}\left( X\right) \) . | Proof of 2.44: This is purely algebraic. Let\n\n\[ \n0 \rightarrow {C}_{k}\overset{{d}_{k}}{ \rightarrow }{C}_{k - 1} \rightarrow \cdots \rightarrow {C}_{1}\overset{{d}_{1}}{ \rightarrow }{C}_{0} \rightarrow 0 \]\n\n\nbe a chain complex of finitely generated abelian groups, with cycles \( {Z}_{n} = \operatorname{Ker}{d... | Yes |
Take \( X = {S}^{n} \) with \( A \) and \( B \) the northern and southern hemispheres, so that \( A \cap B = {S}^{n - 1} \). Then in the reduced Mayer-Vietoris sequence the terms \( {\widetilde{H}}_{i}\left( A\right) \oplus {\widetilde{H}}_{i}\left( B\right) \) are zero, so we obtain isomorphisms \( {\widetilde{H}}_{i}... | This gives another way of calculating the homology groups of \( {S}^{n} \) by induction. | No |
We can decompose the Klein bottle \( K \) as the union of two Möbius bands \( A \) and \( B \) glued together by a homeomorphism between their boundary circles. | Then \( A, B \), and \( A \cap B \) are homotopy equivalent to circles, so the interesting part of the reduced Mayer-Vietoris sequence for the decomposition \( K = A \cup B \) is the segment\n\n\[ 0 \rightarrow {H}_{2}\left( K\right) \rightarrow {H}_{1}\left( {A \cap B}\right) \overset{\Phi }{ \rightarrow }{H}_{1}\left... | Yes |
Let us describe an exact sequence which is somewhat similar to the Mayer-Vietoris sequence and which in some cases generalizes it. If we are given two maps \( f, g : X \rightarrow Y \) then we can form a quotient space \( Z \) of the disjoint union of \( X \times I \) and \( Y \) via the identifications \( \left( {x,0}... | The exact sequence we want has the form\n\n\( \left( *\right) \)\n\n\[ \cdots \rightarrow {H}_{n}\left( X\right) \xrightarrow[]{{f}_{ * } - {g}_{ * }}{H}_{n}\left( Y\right) \overset{{i}_{ * }}{ \rightarrow }{H}_{n}\left( Z\right) \rightarrow {H}_{n - 1}\left( X\right) \overset{{f}_{ * } - {g}_{ * }}{ \rightarrow }{H}_{... | Yes |
Lemma 2.49. If \( f : {S}^{k} \rightarrow {S}^{k} \) has degree \( m \), then \( {f}_{ * } : {H}_{k}\left( {{S}^{k};G}\right) \rightarrow {H}_{k}\left( {{S}^{k};G}\right) \) is multiplication by \( m \) . | Proof: As a preliminary observation, note that a homomorphism \( \varphi : {G}_{1} \rightarrow {G}_{2} \) induces maps \( {\varphi }_{\sharp } : {C}_{n}\left( {X, A;{G}_{1}}\right) \rightarrow {C}_{n}\left( {X, A;{G}_{2}}\right) \) commuting with boundary maps, so there are induced homomorphisms \( {\varphi }_{ * } : {... | Yes |
It is instructive to see what happens to the homology of \( {\mathbb{{RP}}}^{n} \) when the coefficient group \( G \) is chosen to be a field \( F \). | The cellular chain complex is\n\n\[ \cdots \overset{0}{ \rightarrow }F\overset{2}{ \rightarrow }F\overset{0}{ \rightarrow }F\overset{2}{ \rightarrow }F\overset{0}{ \rightarrow }F \rightarrow 0 \]\n\nHence if \( F \) has characteristic 2, for example if \( F = {\mathbb{Z}}_{2} \), then \( {H}_{k}\left( {{\mathbb{{RP}}}^... | Yes |
Example 2.51. Let \( X \) be a Moore space \( M\left( {{\mathbb{Z}}_{m}, n}\right) \) obtained from \( {S}^{n} \) by attaching a cell \( {e}^{n + 1} \) by a map of degree \( m \) . The quotient map \( f : X \rightarrow X/{S}^{n} = {S}^{n + 1} \) induces trivial homomorphisms on reduced homology with \( \mathbb{Z} \) co... | \[ 0 = {\widetilde{H}}_{n + 1}\left( {{S}^{n};{\mathbb{Z}}_{m}}\right) \rightarrow {\widetilde{H}}_{n + 1}\left( {X;{\mathbb{Z}}_{m}}\right) \overset{{f}_{ * }}{ \rightarrow }{\widetilde{H}}_{n + 1}\left( {X/{S}^{n};{\mathbb{Z}}_{m}}\right) \] Exactness says that \( {f}_{ * } \) is injective, hence nonzero since \( {\w... | Yes |
Theorem 3.2. If a chain complex \( C \) of free abelian groups has homology groups \( {H}_{n}\left( C\right) \), then the cohomology groups \( {H}^{n}\left( {C;G}\right) \) of the cochain complex \( \operatorname{Hom}\left( {{C}_{n}, G}\right) \) are determined by split exact sequences | \[ 0 \rightarrow \operatorname{Ext}\left( {{H}_{n - 1}\left( C\right), G}\right) \rightarrow {H}^{n}\left( {C;G}\right) \overset{h}{ \rightarrow }\operatorname{Hom}\left( {{H}_{n}\left( C\right), G}\right) \rightarrow 0 \] | Yes |
Theorem 3.5. \( {H}^{n}\left( {X;G}\right) \approx \operatorname{Ker}{d}_{n}/\operatorname{Im}{d}_{n - 1} \) . Furthermore, the cellular cochain complex \( \left\{ {{H}^{n}\left( {{X}^{n},{X}^{n - 1};G}\right) ,{d}_{n}}\right\} \) is isomorphic to the dual of the cellular chain complex, obtained by applying \( \operato... | Proof: The universal coefficient theorem implies that \( {H}^{k}\left( {{X}^{n},{X}^{n - 1};G}\right) = 0 \) for \( k \neq n \) . The long exact sequence of the pair \( \left( {{X}^{n},{X}^{n - 1}}\right) \) then gives isomorphisms \( {H}^{k}\left( {{X}^{n};G}\right) \approx \) \( {H}^{k}\left( {{X}^{n - 1};G}\right) \... | Yes |
Let \( M \) be the closed orientable surface of genus \( g \geq 1 \) with the \( \Delta \) -complex structure shown in the figure for the case \( g = 2 \). The cup product of interest is \( {H}^{1}\left( M\right) \times {H}^{1}\left( M\right) \rightarrow {H}^{2}\left( M\right) \). Taking \( \mathbb{Z} \) coefficients, ... | To represent \( {\alpha }_{i} \) by a simplicial cocycle \( {\varphi }_{i} \) we need to choose values for \( {\varphi }_{i} \) on the edges radiating out from the central vertex in such a way that \( \delta {\varphi }_{i} = 0 \). This is the 'cocycle condition' discussed in the introduction to this chapter, where we s... | Yes |
The closed nonorientable surface \( N \) of genus \( g \) can be treated in similar fashion if we use \( {\mathbb{Z}}_{2} \) coefficients. Using the \( \Delta \) -complex structure shown, the edges \( {a}_{i} \) give a basis for \( {H}_{1}\left( {N;{\mathbb{Z}}_{2}}\right) \), and the dual basis elements \( {\alpha }_{... | The remarks in the paragraph preceding this example apply here also, but with the following difference: When one tries to deform a second copy of the loop \( {\alpha }_{i} \) in the present example to be disjoint from the original copy, the best one can do is make it intersect the original in one point. This reflects t... | Yes |
Let \( X \) be the 2-dimensional CW complex obtained by attaching a 2-cell to \( {S}^{1} \) by the degree \( m \) map \( {S}^{1} \rightarrow {S}^{1}, z \mapsto {z}^{m} \). Using cellular cohomology, or cellular homology and the universal coefficient theorem, we see that \( {H}^{n}\left( {X;\mathbb{Z}}\right) \) consist... | To obtain a \( \Delta \) -complex structure on \( X \), take a regular \( m \) -gon subdivided into \( m \) triangles \( {T}_{i} \) around a central vertex \( v \), as shown in the figure for the case \( m = 4 \), then identify all the outer edges by rotations of the \( m \) -gon. This gives \( X \) a \( \Delta \) -com... | Yes |
For a map \( f : X \rightarrow Y \), the induced maps \( {f}^{ * } : {H}^{n}\left( {Y;R}\right) \rightarrow {H}^{n}\left( {X;R}\right) \) satisfy \( {f}^{ * }\left( {\alpha \smile \beta }\right) = {f}^{ * }\left( \alpha \right) \smile {f}^{ * }\left( \beta \right) \), and similarly in the relative case. | This comes from the cochain formula \( {f}^{\sharp }\left( \varphi \right) \smile {f}^{\sharp }\left( \psi \right) = {f}^{\sharp }\left( {\varphi \smile \psi }\right) \) :\n\n\[ \left( {{f}^{\sharp }\varphi \smile {f}^{\sharp }\psi }\right) \left( \sigma \right) = {f}^{\sharp }\varphi \left( {\sigma \mid \left\lbrack {... | Yes |
Theorem 3.12. \( {H}^{ * }\left( {{\mathbb{{RP}}}^{n};{\mathbb{Z}}_{2}}\right) \approx {\mathbb{Z}}_{2}\left\lbrack \alpha \right\rbrack /\left( {\alpha }^{n + 1}\right) \) and \( {H}^{ * }\left( {{\mathbb{{RP}}}^{\infty };{\mathbb{Z}}_{2}}\right) \approx {\mathbb{Z}}_{2}\left\lbrack \alpha \right\rbrack \), where \( \... | Proof: Let us do the case of \( {\mathbb{{RP}}}^{n} \) first. To simplify notation we abbreviate \( {\mathbb{{RP}}}^{n} \) to \( {P}^{n} \) and we let the coefficient group \( {\mathbb{Z}}_{2} \) be implicit. Since the inclusion \( {P}^{n - 1} \hookrightarrow {P}^{n} \) induces an isomorphism on \( {H}^{i} \) for \( i ... | Yes |
The isomorphism \( {H}^{ * }\left( {\mathop{\coprod }\limits_{\alpha }{X}_{\alpha };R}\right) \overset{ \approx }{ \rightarrow }\mathop{\prod }\limits_{\alpha }{H}^{ * }\left( {{X}_{\alpha };R}\right) \) whose coordinates are induced by the inclusions \( {i}_{\alpha } : {X}_{\alpha } \hookrightarrow \mathop{\coprod }\l... | Similarly for a wedge sum the isomorphism \( {\widetilde{H}}^{ * }\left( {\mathop{\bigvee }\limits_{\alpha }{X}_{\alpha };R}\right) \approx \mathop{\prod }\limits_{\alpha }{\widetilde{H}}^{ * }\left( {{X}_{\alpha };R}\right) \) is a ring isomorphism. Here we take reduced cohomology to be cohomology relative to a basepo... | No |
Theorem 3.14. The identity \( \alpha \smile \beta = {\left( -1\right) }^{k\ell }\beta \smile \alpha \) holds for all \( \alpha \in {H}^{k}\left( {X, A;R}\right) \) and \( \parallel \beta \in {H}^{\ell }\left( {X, A;R}\right) \), when \( R \) is commutative. | Proof: Consider first the case \( A = \varnothing \) . For cochains \( \varphi \in {C}^{k}\left( {X;R}\right) \) and \( \psi \in {C}^{\ell }\left( {X, R}\right) \) one can see from the definition that the cup products \( \varphi \smile \psi \) and \( \psi \smile \varphi \) differ only by a permutation of the vertices o... | Yes |
The theorem says that \( {H}^{ * }\left( {{\mathbb{{RP}}}^{\infty } \times {\mathbb{{RP}}}^{\infty };{\mathbb{Z}}_{2}}\right) \) is isomorphic as a ring to \( {H}^{ * }\left( {{\mathbb{{RP}}}^{\infty };{\mathbb{Z}}_{2}}\right) \otimes {H}^{ * }\left( {{\mathbb{{RP}}}^{\infty };{\mathbb{Z}}_{2}}\right) \) | By Theorem 3.12 this is \( {\mathbb{Z}}_{2}\left\lbrack \alpha \right\rbrack \otimes {\mathbb{Z}}_{2}\left\lbrack \beta \right\rbrack \), which is just the polynomial ring \( {\mathbb{Z}}_{2}\left\lbrack {\alpha ,\beta }\right\rbrack \) | Yes |
Proposition 3.19. If a natural transformation between unreduced cohomology theories on the category of \( {CW} \) pairs is an isomorphism when the \( {CW} \) pair is \( \left( {\text{point},\varnothing }\right) \) , then it is an isomorphism for all CW pairs. | Proof: Let \( \mu : {h}^{ * }\left( {X, A}\right) \rightarrow {k}^{ * }\left( {X, A}\right) \) be the natural transformation. By the five-lemma it will suffice to show that \( \mu \) is an isomorphism when \( A = \varnothing \) .\n\nFirst we do the case of finite-dimensional \( X \) by induction on dimension. The induc... | Yes |
Theorem 3.21. For \( {CW} \) pairs \( \left( {X, A}\right) \) and \( \left( {Y, B}\right) \) the cross product homomorphism \( {H}^{ * }\left( {X, A;R}\right) { \otimes }_{R}{H}^{ * }\left( {Y, B;R}\right) \rightarrow {H}^{ * }\left( {X \times Y, A \times Y \cup X \times B;R}\right) \) is an isomorphism of rings \( \pa... | Proof: The case \( B = \varnothing \) was covered in the course of the proof of the absolute case, so it suffices to deduce the case \( B \neq \varnothing \) from the case \( B = \varnothing \) .\n\nThe following commutative diagram shows that collapsing \( B \) to a point reduces the proof to the case that \( B \) is ... | Yes |
Proposition 3.22. For \( n > 0,{H}^{ * }\left( {J\left( {S}^{n}\right) ;\mathbb{Z}}\right) \) consists of a \( \mathbb{Z} \) in each dimension a multiple of \( n \) . If \( n \) is even, the \( {i}^{\text{th }} \) power of a generator of \( {H}^{n}\left( {J\left( {S}^{n}\right) ;\mathbb{Z}}\right) \) is \( i \) ! times... | Proof: Giving \( {S}^{n} \) its usual CW structure, the resulting CW structure on \( J\left( {S}^{n}\right) \) consists of exactly one cell in each dimension a multiple of \( n \) . Thus if \( n > 1 \) we deduce immediately from cellular cohomology that \( {H}^{ * }\left( {J\left( {S}^{n}\right) ;\mathbb{Z}}\right) \) ... | Yes |
Lemma 3.27. Let \( M \) be a manifold of dimension \( n \) and let \( A \subset M \) be a compact subset. Then:\n\n(a) If \( x \mapsto {\alpha }_{x} \) is a section of the covering space \( {M}_{R} \rightarrow M \), then there is a unique class \( {\alpha }_{A} \in {H}_{n}\left( {M \mid A;R}\right) \) whose image in \(... | Proof of 3.27: The coefficient ring \( R \) will play no special role in the argument so we shall omit it from the notation. We break the proof up into four steps. (1) First we observe that if the lemma is true for compact sets \( A, B \), and \( A \cap B \), then it is true for \( A \cup B \) . To see this, consider t... | Yes |
Corollary 3.28. If \( M \) is a closed connected \( n \) -manifold, the torsion subgroup of \( {H}_{n - 1}\left( {M;\mathbb{Z}}\right) \) is trivial if \( M \) is orientable and \( {\mathbb{Z}}_{2} \) if \( M \) is nonorientable. | Proof: This is an application of the universal coefficient theorem for homology, using the fact that the homology groups of \( M \) are finitely generated, from Corollaries A. 8 and A. 9 in the Appendix. In the orientable case, if \( {H}_{n - 1}\left( {M;\mathbb{Z}}\right) \) contained torsion, then for some prime \( p... | Yes |
Proposition 3.29. If \( M \) is a connected noncompact \( n \) -manifold, then \( {H}_{i}\left( {M;R}\right) = 0 \) for \( i \geq n \) . | Proof: Represent an element of \( {H}_{i}\left( {M;R}\right) \) by a cycle \( z \) . This has compact image in \( M \) , so there is an open set \( U \subset M \) containing the image of \( z \) and having compact closure \( \bar{U} \subset M \) . Let \( V = M - \bar{U} \) . Part of the long exact sequence of the tripl... | Yes |
Let us compute these cohomology groups when \( X = \mathbb{R} \) with the \( \Delta \) -complex structure having vertices at the integer points. For a simplicial 0 -cochain to be a cocycle it must take the same value on all vertices, but then if the cochain lies in \( {\Delta }_{c}^{0}\left( X\right) \) it must be iden... | Namely, consider the map \( \sum : {\Delta }_{c}^{1}\left( {\mathbb{R};G}\right) \rightarrow G \) sending each cochain to the sum of its values on all the 1-simplices. Note that \( \sum \) is not defined on all of \( {\Delta }^{1}\left( X\right) \), just on \( {\Delta }_{c}^{1}\left( X\right) \) . The map \( \sum \) va... | No |
Proposition 3.33. If a space \( X \) is the union of a directed set of subspaces \( {X}_{\alpha } \) with the property that each compact set in \( X \) is contained in some \( {X}_{\alpha } \), then the natural map \( \mathop{\lim }\limits_{ \rightarrow }{H}_{i}\left( {{X}_{\alpha };G}\right) \rightarrow {H}_{i}\left( ... | Proof: For surjectivity, represent a cycle in \( X \) by a finite sum of singular simplices. The union of the images of these singular simplices is compact in \( X \), hence lies in some \( {X}_{\alpha } \), so the map \( \mathop{\lim }\limits_{ \rightarrow }{H}_{i}\left( {{X}_{\alpha };G}\right) \rightarrow {H}_{i}\le... | Yes |
Theorem 3.35. The duality map \( {D}_{M} : {H}_{c}^{k}\left( {M;R}\right) \rightarrow {H}_{n - k}\left( {M;R}\right) \) is an isomorphism \( \parallel \) for all \( k \) whenever \( M \) is an \( R \) -oriented \( n \) -manifold. | The proof will not be difficult once we establish a technical result stated in the next lemma, concerning the commutativity of a certain diagram. Commutativity statements of this sort are usually routine to prove, but this one seems to be an exception. The reader who consults other books for alternative expositions wil... | No |
Lemma 3.36. If \( M \) is the union of two open sets \( U \) and \( V \), then there is a diagram of Mayer-Vietoris sequences, commutative up to sign: | Proof: Compact sets \( K \subset U \) and \( L \subset V \) give rise to the Mayer-Vietoris sequence in the upper row of the following diagram, whose lower row is also a Mayer-Vietoris sequence. | No |
Proposition 3.38. The cup product pairing is nonsingular for closed \( R \) -orientable manifolds when \( R = \mathbb{Z} \) and torsion in \( {H}^{ * }\left( {M;\mathbb{Z}}\right) \) is factored out. | Proof: Consider the composition\n\n\[ \n{H}^{n - k}\left( {M;R}\right) \overset{h}{ \rightarrow }{\operatorname{Hom}}_{R}\left( {{H}_{n - k}\left( {M;R}\right), R}\right) \overset{{D}^{ * }}{ \rightarrow }{\operatorname{Hom}}_{R}\left( {{H}^{k}\left( {M;R}\right), R}\right) \]\n\nwhere \( h \) is the map appearing in t... | Yes |
Corollary 3.39. If \( M \) is a closed connected orientable \( n \) -manifold, then for each element \( \alpha \in {H}^{k}\left( {M;\mathbb{Z}}\right) \) of infinite order that is not a proper multiple of another element, there exists an element \( \beta \in {H}^{n - k}\left( {M;\mathbb{Z}}\right) \) such that \( \alph... | Proof: The hypotheses on \( \alpha \) mean that it generates a \( \mathbb{Z} \) summand of \( {H}^{k}\left( {M;\mathbb{Z}}\right) \) . There is then a homomorphism \( \varphi : {H}^{k}\left( {M;\mathbb{Z}}\right) \rightarrow \mathbb{Z} \) with \( \varphi \left( \alpha \right) = 1 \) . By the nonsingularity of the cup p... | Yes |
Proposition 3.42. If \( M \) is a compact manifold with boundary, then \( \partial M \) has a collar neighborhood. | Proof: Let \( {M}^{\prime } \) be \( M \) with an external collar attached, the quotient of the disjoint union of \( M \) and \( \partial M \times \left\lbrack {0,1}\right\rbrack \) in which \( x \in \partial M \) is identified with \( \left( {x,0}\right) \in \partial M \times \left\lbrack {0,1}\right\rbrack \) . It wi... | Yes |
Theorem 3.43. Suppose \( M \) is a compact \( R \) -orientable \( n \) -manifold whose boundary \( \partial M \) is decomposed as the union of two compact \( \left( {n - 1}\right) \) -dimensional manifolds \( A \) and \( B \) with a common boundary \( \partial A = \partial B = A \cap B \) . Then cap product with a fund... | Proof: The cap product map \( {D}_{M} : {H}^{k}\left( {M, A;R}\right) \rightarrow {H}_{n - k}\left( {M, B;R}\right) \) is defined since the existence of collar neighborhoods of \( A \cap B \) in \( A \) and \( B \) and \( \partial M \) in \( M \) implies that \( A \) and \( B \) are deformation retracts of open neighbo... | No |
Theorem 3.44. If \( K \) is a compact, locally contractible, nonempty, proper subspace of \( {S}^{n} \), then \( {\widetilde{H}}_{i}\left( {{S}^{n} - K;\mathbb{Z}}\right) \approx {\widetilde{H}}^{n - i - 1}\left( {K;\mathbb{Z}}\right) \) for all \( i \) . | Proof: We will obtain the desired isomorphism when \( i \neq 0 \) as the composition of five isomorphisms\n\n\[ \n{H}_{i}\left( {{S}^{n} - K}\right) \approx {H}_{c}^{n - i}\left( {{S}^{n} - K}\right) \n\] \n\n\[ \n\approx \mathop{\lim }\limits_{ \rightarrow }{H}^{n - i}\left( {{S}^{n} - K, U - K}\right) \n\] \n\n\[ \n\... | Yes |
Proposition 4.1. A covering space projection \( p : \left( {\widetilde{X},{\widetilde{x}}_{0}}\right) \rightarrow \left( {X,{x}_{0}}\right) \) induces isomor- \( \parallel \) phisms \( {p}_{ * } : {\pi }_{n}\left( {\widetilde{X},{\widetilde{x}}_{0}}\right) \rightarrow {\pi }_{n}\left( {X,{x}_{0}}\right) \) for all \( n... | Proof: For surjectivity of \( {p}_{ * } \) we apply the lifting criterion in Proposition 1.33, which implies that every map \( \left( {{S}^{n},{s}_{0}}\right) \rightarrow \left( {X,{x}_{0}}\right) \) lifts to \( \left( {\widetilde{X},{\widetilde{x}}_{0}}\right) \) provided that \( n \geq 2 \) so that \( {S}^{n} \) is s... | Yes |
For a product \( {\Pi }_{\alpha }{X}_{\alpha } \) of an arbitrary collection of path-connected spaces \( {X}_{\alpha } \) there are isomorphisms \( {\pi }_{n}\left( {{\Pi }_{\alpha }{X}_{\alpha }}\right) \approx {\Pi }_{\alpha }{\pi }_{n}\left( {X}_{\alpha }\right) \) for all \( n \) . | A map \( f : Y \rightarrow \mathop{\prod }\limits_{\alpha }{X}_{\alpha } \) is the same thing as a collection of maps \( {f}_{\alpha } : Y \rightarrow {X}_{\alpha } \) . Taking \( Y \) to be \( {S}^{n} \) and \( {S}^{n} \times I \) gives the result. | No |
Let \( {CX} \) be the cone on a path-connected space \( X \), the quotient space of \( X \times I \) obtained by collapsing \( X \times \{ 0\} \) to a point. We can view \( X \) as the subspace \( X \times \{ 1\} \subset {CX} \) . Since \( {CX} \) is contractible, the long exact sequence of homotopy groups for the pair... | The long exact sequence of homotopy groups is clearly natural: A map of base-pointed triples \( \left( {X, A, B,{x}_{0}}\right) \rightarrow \left( {Y, C, D,{y}_{0}}\right) \) induces a map between the associated long exact sequences, with commuting squares. | No |
Lemma 4.6. Let \( \left( {X, A}\right) \) be a \( {CW} \) pair and let \( \left( {Y, B}\right) \) be any pair with \( B \neq \varnothing \) . For each \( n \) such that \( X - A \) has cells of dimension \( n \), assume that \( {\pi }_{n}\left( {Y, B,{y}_{0}}\right) = 0 \) for all \( {y}_{0} \in B \) . Then every map \... | Proof: Assume inductively that \( f \) has already been homotoped to take the skeleton \( {X}^{k - 1} \) to \( B \) . If \( \Phi \) is the characteristic map of a cell \( {e}^{k} \) of \( X - A \), the composition \( {f\Phi } : \left( {{D}^{k},\partial {D}^{k}}\right) \rightarrow \left( {Y, B}\right) \) can be homotope... | Yes |
Theorem 4.8. Every map \( f : X \rightarrow Y \) of CW complexes is homotopic to a cellular map. If \( f \) is already cellular on a subcomplex \( A \subset X \), the homotopy may be taken to be stationary on \( A \) . | Proof of 4.8: Suppose inductively that \( f : X \rightarrow Y \) is already cellular on the skeleton \( {X}^{n - 1} \), and let \( {e}^{n} \) be an \( n \) -cell of \( X \) . The closure of \( {e}^{n} \) in \( X \) is compact, being the image of a characteristic map for \( {e}^{n} \), so \( f \) takes the closure of \(... | Yes |
Lemma 4.10. Let \( f : {I}^{n} \rightarrow Z \) be a map, where \( Z \) is obtained from a subspace \( W \) by attaching a cell \( {e}^{k} \) . Then \( f \) is homotopic \( \operatorname{rel}{f}^{-1}\left( W\right) \) to a map \( {f}_{1} \) for which there is a simplex \( {\Delta }^{k} \subset {e}^{k} \) with \( {f}_{1... | Proof of 4.10: Identifying \( {e}^{k} \) with \( {\mathbb{R}}^{k} \), let \( {B}_{1},{B}_{2} \subset {e}^{k} \) be the closed balls of radius 1 and 2 centered at the origin. Since \( {f}^{-1}\left( {B}_{2}\right) \) is closed and therefore compact in \( {I}^{n} \), it follows that \( f \) is uniformly continuous on \( ... | Yes |
Corollary 4.12. A CW pair \( \left( {X, A}\right) \) is \( n \) -connected if all the cells in \( X - A \) have dimension greater than \( n \) . In particular the pair \( \left( {X,{X}^{n}}\right) \) is \( n \) -connected, hence the inclusion \( {X}^{n} \hookrightarrow X \) induces isomorphisms on \( {\pi }_{i} \) for ... | Proof: Applying cellular approximation to maps \( \left( {{D}^{i},\partial {D}^{i}}\right) \rightarrow \left( {X, A}\right) \) with \( i \leq n \) gives the first statement. The last statement comes from the long exact sequence of the pair \( \left( {X,{X}^{n}}\right) \) . | No |
When \( X \) is path-connected and \( A \) is a point, the construction of a 0 -connected CW model for \( \left( {X, A}\right) \) gives a CW approximation to \( X \) with a single 0 -cell and all higher cells attached by basepoint-preserving maps. | In particular, any connected CW complex is homotopy equivalent to a CW complex with these properties. | No |
Corollary 4.16. If \(\left( {X, A}\right)\) is an \(n\)-connected CW pair, then there exists a CW pair \(\left( {Z, A}\right) \simeq \left( {X, A}\right)\) rel \(A\) such that all cells of \(Z - A\) have dimension greater than \(n\). | Proof: An \( n \) -connected CW approximation \( f : \left( {Z, A}\right) \rightarrow \left( {X, A}\right) \) given by the preceding proposition will do the trick. First we check that \( f \) induces isomorphisms \( {\pi }_{i}\left( Z\right) \approx \) \( {\pi }_{i}\left( X\right) \) for all \( i \) . This is true for ... | Yes |
Proposition 4.18. Suppose we are given:\n\n(i) an \( n \) -connected \( {CW} \) model \( f : \left( {Z, A}\right) \rightarrow \left( {X, A}\right) \) ,\n\n(ii) an \( {n}^{\prime } \) -connected CW model \( {f}^{\prime } : \left( {{Z}^{\prime },{A}^{\prime }}\right) \rightarrow \left( {{X}^{\prime },{A}^{\prime }}\right... | Proof: By Corollary 4.16 we may assume all cells of \( Z - A \) have dimension greater than \( n \) . Let \( W \) be the quotient space of the mapping cylinder of \( {f}^{\prime } \) obtained by collapsing each line segment \( \left\{ {a}^{\prime }\right\} \times I \) to a point, for \( {a}^{\prime } \in {A}^{\prime } ... | Yes |
Corollary 4.19. An \( n \) -connected \( {CW} \) model for \( \left( {X, A}\right) \) is unique up to homotopy equivalence \( \operatorname{rel}A \) . In particular, CW approximations to spaces are unique up to homotopy equivalence. | Proof: Given two \( n \) -connected CW models \( \left( {Z, A}\right) \) and \( \left( {{Z}^{\prime }, A}\right) \) for \( \left( {X, A}\right) \), we apply the proposition twice with \( g \) the identity map to obtain maps \( h : Z \rightarrow {Z}^{\prime } \) and \( {h}^{\prime } : {Z}^{\prime } \rightarrow Z \) . Th... | Yes |
Proposition 4.21. A weak homotopy equivalence \( f : X \rightarrow Y \) induces isomorphisms \( {f}_{ * } : {H}_{n}\left( {X;G}\right) \rightarrow {H}_{n}\left( {Y;G}\right) \) and \( {f}^{ * } : {H}^{n}\left( {Y;G}\right) \rightarrow {H}^{n}\left( {X;G}\right) \) for all \( n \) and all coefficient groups \( G \) . | Proof: Replacing \( Y \) by the mapping cylinder \( {M}_{f} \) and looking at the long exact sequences of homotopy, homology, and cohomology groups for \( \left( {{M}_{f}, X}\right) \), we see that it suffices to show:\n\n- If \( \left( {Z, X}\right) \) is an \( n \) -connected pair of path-connected spaces, then \( {H... | Yes |
Theorem 4.23. Let \( X \) be a CW complex decomposed as the union of subcomplexes A and B with nonempty connected intersection \( C = A \cap B \) . If \( \left( {A, C}\right) \) is \( m \) -connected and \( \left( {B, C}\right) \) is \( n \) -connected, \( m, n \geq 0 \), then the map \( {\pi }_{i}\left( {A, C}\right) ... | This yields the Freudenthal suspension theorem: | No |
Corollary 4.24. The suspension map \( {\pi }_{i}\left( {S}^{n}\right) \rightarrow {\pi }_{i + 1}\left( {S}^{n + 1}\right) \) is an isomorphism for \( i < {2n} - 1 \) and a surjection for \( i = {2n} - 1 \) . More generally this holds for the suspension \( {\pi }_{i}\left( X\right) \rightarrow {\pi }_{i + 1}\left( {SX}\... | Proof: Decompose the suspension \( {SX} \) as the union of two cones \( {C}_{ + }X \) and \( {C}_{ - }X \) intersecting in a copy of \( X \) . The suspension map is the same as the map\n\n\[ \n{\pi }_{i}\left( X\right) \approx {\pi }_{i + 1}\left( {{C}_{ + }X, X}\right) \rightarrow {\pi }_{i + 1}\left( {{SX},{C}_{ - }X... | Yes |
Corollary 4.25. \( {\pi }_{n}\left( {S}^{n}\right) \approx \mathbb{Z} \), generated by the identity map, for all \( n \geq 1 \) . In \( \parallel \) particular, the degree map \( {\pi }_{n}\left( {S}^{n}\right) \rightarrow \mathbb{Z} \) is an isomorphism. | Proof: From the preceding corollary we know that in the suspension sequence\n\n\[ \n{\pi }_{1}\left( {S}^{1}\right) \rightarrow {\pi }_{2}\left( {S}^{2}\right) \rightarrow {\pi }_{3}\left( {S}^{3}\right) \rightarrow \cdots \n\]\n\nthe first map is surjective and all the subsequent maps are isomorphisms. Since \( {\pi }... | Yes |
Let us show that \( {\pi }_{n}\left( {{S}^{1} \vee {S}^{n}}\right) \) for \( n \geq 2 \) is free abelian on a countably infinite number of generators. | By Proposition 4.1 we may compute \( {\pi }_{i}\left( {{S}^{1} \vee {S}^{n}}\right) \) for \( i \geq 2 \) by passing to the universal cover. This consists of a copy of \( \mathbb{R} \) with a sphere \( {S}_{k}^{n} \) attached at each integer point \( k \in \mathbb{R} \), so it is homotopy equivalent to \( { \vee }_{k}{... | Yes |
Proposition 4.28. If a CW pair \( \left( {X, A}\right) \) is \( r \) -connected and \( A \) is \( s \) -connected, with \( \parallel r, s\parallel \geq 0 \), then the map \( {\pi }_{i}\left( {X, A}\right) \rightarrow {\pi }_{i}\left( {X/A}\right) \) induced by the quotient map \( X \rightarrow X/A \) . I is an isomorph... | Proof: Consider \( X \cup {CA} \), the complex obtained from \( X \) by attaching a cone \( {CA} \) along \( A \subset X \) . Since \( {CA} \) is a contractible subcomplex of \( X \cup {CA} \), the quotient map \( X \cup {CA} \rightarrow \left( {X \cup {CA}}\right) /{CA} = X/A \) is a homotopy equivalence by Propositio... | Yes |
Suppose \( X \) is obtained from a wedge of spheres \( \mathop{\bigvee }\limits_{\alpha }{S}_{\alpha }^{n} \) by attaching cells \( {e}_{\beta }^{n + 1} \) via basepoint-preserving maps \( {\varphi }_{\beta } : {S}^{n} \rightarrow { \vee }_{\alpha }{S}_{\alpha }^{n} \), with \( n \geq 2 \). By cellular approximation we... | To see that \( {\pi }_{n}\left( X\right) \) is as claimed, consider the following portion of the long exact sequence of the pair \( \left( {X,\mathop{\bigvee }\limits_{\alpha }{S}_{\alpha }^{n}}\right) \):\n\n\[ \n{\pi }_{n + 1}\left( {X,\mathop{\bigvee }\limits_{\alpha }{S}_{\alpha }^{n}}\right) \overset{\partial }{ \... | Yes |
Theorem 4.32. If a space \( X \) is \( \left( {n - 1}\right) \) -connected, \( n \geq 2 \), then \( {\widetilde{H}}_{i}\left( X\right) = 0 \) for \( i < n \) and \( {\pi }_{n}\left( X\right) \approx {H}_{n}\left( X\right) \) . If a pair \( \left( {X, A}\right) \) is \( \left( {n - 1}\right) \) -connected, \( n \geq 2 \... | Proof: We may assume \( X \) is a CW complex and \( \left( {X, A}\right) \) is a CW pair by taking CW approximations to \( X \) and \( \left( {X, A}\right) \) . For CW pairs the relative case then reduces to the absolute case since \( {\pi }_{i}\left( {X, A}\right) \approx {\pi }_{i}\left( {X/A}\right) \) for \( i \leq... | Yes |
Corollary 4.33. A map \( f : X \rightarrow Y \) between simply-connected CW complexes is a ho- \( \parallel \) motopy equivalence if \( {f}_{ * } : {H}_{n}\left( X\right) \rightarrow {H}_{n}\left( Y\right) \) is an isomorphism for each \( n \) . | Proof: After replacing \( Y \) by the mapping cylinder \( {M}_{f} \) we may take \( f \) to be an inclusion \( X \hookrightarrow Y \) . Since \( X \) and \( Y \) are simply-connected, we have \( {\pi }_{1}\left( {Y, X}\right) = 0 \) . The relative Hurewicz theorem then says that the first nonzero \( {\pi }_{n}\left( {Y... | Yes |
We construct a space \( X = \left( {{S}^{1} \vee {S}^{n}}\right) \cup {e}^{n + 1} \), for arbitrary \( n > 1 \), such that the inclusion \( {S}^{1} \hookrightarrow X \) induces an isomorphism on all homology groups and on \( {\pi }_{i} \) for \( i < n \), but not on \( {\pi }_{n} \) . | From Example 4.27 we have \( {\pi }_{n}\left( {{S}^{1} \vee {S}^{n}}\right) \approx \mathbb{Z}\left\lbrack {t,{t}^{-1}}\right\rbrack \) . Let \( X \) be obtained from \( {S}^{1} \vee {S}^{n} \) by attaching a cell \( {e}^{n + 1} \) via a map \( {S}^{n} \rightarrow {S}^{1} \vee {S}^{n} \) corresponding to \( {2t} - 1 \i... | Yes |
Proposition 4.36. The Hurewicz map \( h : {\pi }_{n}\left( {X, A,{x}_{0}}\right) \rightarrow {H}_{n}\left( {X, A}\right) \) is a homomor- 1 phism, assuming \( n > 1 \) so that \( {\pi }_{n}\left( {X, A,{x}_{0}}\right) \) is a group. | Proof: It suffices to show that for maps \( f, g : \left( {{D}^{n},\partial {D}^{n}}\right) \rightarrow \left( {X, A}\right) \), the induced maps on homology satisfy \( {\left( f + g\right) }_{ * } = {f}_{ * } + {g}_{ * } \), for if this is the case then \( h\left( \left\lbrack {f + g}\right\rbrack \right) = \) \( {\le... | Yes |
Lemma 4.38. If \( X \) is a connected \( {CW} \) complex to which cells \( {e}_{\alpha }^{n} \) are attached for a fixed \( n \geq 2 \), forming a \( {CW} \) complex \( W = X\mathop{\bigcup }\limits_{\alpha }{e}_{\alpha }^{n} \), then \( {\pi }_{n}\left( {W, X}\right) \) is a free \( {\pi }_{1}\left( X\right) \) -modul... | Proof: Since \( W/X = \mathop{\bigvee }\limits_{\alpha }{S}_{\alpha }^{n} \), we have \( {\pi }_{n}\left( {W, X}\right) \approx {\pi }_{n}\left( {\mathop{\bigvee }\limits_{\alpha }{S}_{\alpha }^{n}}\right) \) when \( X \) is simply-connected, by Proposition 4.28. The conclusion of the lemma in this case is then immedia... | Yes |
Lemma 4.39. For any \( \left( {X, A,{x}_{0}}\right) \), the formula \( a + b - a = \left( {\partial a}\right) b \) holds for all \( \parallel a, b\parallel \in {\pi }_{2}\left( {X, A,{x}_{0}}\right) \), where \( \partial : {\pi }_{2}\left( {X, A,{x}_{0}}\right) \rightarrow {\pi }_{1}\left( {A,{x}_{0}}\right) \) is the ... | Proof: The formula is obtained by constructing a homotopy from \( a + b - a \) to \( \left( {\partial a}\right) b \) as indicated in the pictures below. | No |
Proposition 4.40. Let \( X \) be a connected \( {CW} \) complex with \( {H}_{1}\left( X\right) = 0 \) . Then there is a simply-connected \( {CW} \) complex \( {X}^{ + } \) and a map \( X \rightarrow {X}^{ + } \) inducing isomorphisms on all homology groups. | Proof: Choose loops \( {\varphi }_{\alpha } : {S}^{1} \rightarrow {X}^{1} \) generating \( {\pi }_{1}\left( X\right) \) and use these to attach cells \( {e}_{\alpha }^{2} \) to \( X \) to form a simply-connected CW complex \( {X}^{\prime } \) . The homology exact sequence\n\n\[ 0 \rightarrow {H}_{2}\left( X\right) \rig... | Yes |
Theorem 4.41. Suppose \( p : E \rightarrow B \) has the homotopy lifting property with respect to disks \( {D}^{k} \) for all \( k \geq 0 \) . Choose basepoints \( {b}_{0} \in B \) and \( {x}_{0} \in F = {p}^{-1}\left( {b}_{0}\right) \) . Then the map \( {p}_{ * } : {\pi }_{n}\left( {E, F,{x}_{0}}\right) \rightarrow {\... | Proof: First we show that \( {p}_{ * } \) is onto. Represent an element of \( {\pi }_{n}\left( {B,{b}_{0}}\right) \) by a map \( f : \left( {{I}^{n},\partial {I}^{n}}\right) \rightarrow \left( {B,{b}_{0}}\right) \) . The constant map to \( {x}_{0} \) provides a lift of \( f \) to \( E \) over the subspace \( {J}^{n - 1... | Yes |
One of the simplest nontrivial fiber bundles is the Möbius band, which is a bundle over \( {S}^{1} \) with fiber an interval. Specifically, take \( E \) to be the quotient of \( I \times \left\lbrack {-1,1}\right\rbrack \) under the identifications \( \left( {0, v}\right) \sim \left( {1, - v}\right) \), with \( p : E \... | Glueing two copies of \( E \) together by the identity map between their boundary circles produces a Klein bottle, a bundle over \( {S}^{1} \) with fiber \( {S}^{1} \) . | No |
Projective spaces yield interesting fiber bundles. In the real case we have the familiar covering spaces \( {S}^{n} \rightarrow {\mathbb{{RP}}}^{n} \) with fiber \( {S}^{0} \). Over the complex numbers the analog of this is a fiber bundle \( {S}^{1} \rightarrow {S}^{{2n} + 1} \rightarrow {\mathbb{{CP}}}^{n} \). Here \(... | To see that the local triviality condition for fiber bundles is satisfied, let \( {U}_{i} \subset {\mathbb{{CP}}}^{n} \) be the open set of equivalence classes \( \left\lbrack {{z}_{0},\cdots ,{z}_{n}}\right\rbrack \) with \( {z}_{i} \neq 0 \). Define \( {h}_{i} : {p}^{-1}\left( {U}_{i}\right) \rightarrow {U}_{i} \time... | Yes |
The case \( n = 1 \) is particularly interesting since \( {\mathbb{{CP}}}^{1} = {S}^{2} \) and the bundle becomes \( {S}^{1} \rightarrow {S}^{3} \rightarrow {S}^{2} \) with fiber, total space, and base all spheres. This is known as the Hopf bundle, and is of low enough dimension to be seen explicitly. The projection \(... | In polar coordinates we have \( p\left( {{r}_{0}{e}^{i{\theta }_{0}},{r}_{1}{e}^{i{\theta }_{1}}}\right) = \left( {{r}_{0}/{r}_{1}}\right) {e}^{i\left( {{\theta }_{0} - {\theta }_{1}}\right) } \) where \( {r}_{0}^{2} + {r}_{1}^{2} = 1 \) . For a fixed ratio \( \rho = {r}_{0}/{r}_{1} \in \left( {0,\infty }\right) \) the... | Yes |
Another Hopf bundle \( {S}^{7} \rightarrow {S}^{15} \rightarrow {S}^{8} \) can be defined using the octonion algebra \( \mathbb{O} \) . Elements of \( \mathbb{O} \) are pairs of quaternions \( \left( {{a}_{1},{a}_{2}}\right) \) with multiplication given by \( \left( {{a}_{1},{a}_{2}}\right) \left( {{b}_{1},{b}_{2}}\rig... | Let \( {U}_{0} \) and \( {U}_{1} \) be the complements of \( \infty \) and 0 in the base space \( \mathbb{O} \cup \{ \infty \} \) . Define \( {h}_{i} : {p}^{-1}\left( {U}_{i}\right) \rightarrow {U}_{i} \times {S}^{7} \) and \( {g}_{i} : {U}_{i} \times {S}^{7} \rightarrow {p}^{-1}\left( {U}_{i}\right) \) by\n\n\[ \n{h}_... | Yes |
Proposition 4.48. A fiber bundle \( p : E \rightarrow B \) has the homotopy lifting property with \( \parallel \) respect to all \( {CW} \) pairs \( \left( {X, A}\right) \) . | Proof: As noted earlier, the homotopy lifting property for CW pairs is equivalent to the homotopy lifting property for disks, or equivalently, cubes. Let \( G : {I}^{n} \times I \rightarrow B \) , \( G\left( {x, t}\right) = {g}_{t}\left( x\right) \), be a homotopy we wish to lift, starting with a given lift \( {\wideti... | Yes |
Applying this theorem to a covering space \( p : E \rightarrow B \) with \( E \) and \( B \) path-connected, and discrete fiber \( F \), the resulting long exact sequence of homotopy groups yields Proposition 4.1 that \( {p}_{ * } : {\pi }_{n}\left( E\right) \rightarrow {\pi }_{n}\left( B\right) \) is an isomorphism fo... | We also obtain a short exact sequence \( 0 \rightarrow {\pi }_{1}\left( E\right) \rightarrow {\pi }_{1}\left( B\right) \rightarrow {\pi }_{0}\left( F\right) \rightarrow 0 \), consistent with the covering space theory facts that \( {p}_{ * } : {\pi }_{1}\left( E\right) \rightarrow {\pi }_{1}\left( B\right) \) is injecti... | No |
For \( m < n \leq k \) there are fiber bundles\n\n\[ \n{V}_{n - m}\left( {\mathbb{R}}^{k - m}\right) \rightarrow {V}_{n}\left( {\mathbb{R}}^{k}\right) \overset{p}{ \rightarrow }{V}_{m}\left( {\mathbb{R}}^{k}\right) \n\]\n\nwhere the projection \( p \) sends an \( n \) -frame onto the \( m \) -frame formed by its first ... | Local trivializations can be constructed as follows. For an \( m \) -frame \( F \) , choose an orthonormal basis for the \( \left( {k - m}\right) \) -plane orthogonal to \( F \) . This determines orthonormal bases for the \( \left( {k - m}\right) \) -planes orthogonal to all nearby \( m \) -frames by orthogonal project... | Yes |
Theorem 4.57. There are natural bijections \( T : \langle X, K\left( {G, n}\right) \rangle \rightarrow {H}^{n}\left( {X;G}\right) \) for all \( {CW} \) complexes \( X \) and all \( n > 0 \), with \( G \) any abelian group. Such a \( T \) has the form \( T\left( \left\lbrack f\right\rbrack \right) = {f}^{ * }\left( \alp... | In the course of the proof we will define a natural group structure on \( \langle X, K\left( {G, n}\right) \rangle \) such that the transformation \( T \) is an isomorphism. | No |
Theorem 4.58. If \( \left\{ {K}_{n}\right\} \) is an \( \Omega \) -spectrum, then the functors \( X \mapsto {h}^{n}\left( X\right) = \left\langle {X,{K}_{n}}\right\rangle \) , \( \parallel n \in \mathbb{Z} \), define a reduced cohomology theory on the category of basepointed CW complexes and basepoint-preserving maps. | Proof: Two of the three axioms for a cohomology theory, the homotopy axiom and the wedge sum axiom, are quite easy to check. For the homotopy axiom, a basepoint-preserving map \( f : X \rightarrow Y \) induces \( {f}^{ * } : \left\langle {Y,{K}_{n}}\right\rangle \rightarrow \left\langle {X,{K}_{n}}\right\rangle \) by c... | Yes |
Theorem 4.59. If \( {h}^{ * } \) is an unreduced cohomology theory on the category of \( {CW} \) pairs and \( {h}^{n}\left( \text{point}\right) = 0 \) for \( n \neq 0 \), then there are natural isomorphisms \( {h}^{n}\left( {X, A}\right) \approx \) \( {H}^{n}\left( {X, A;{h}^{0}\left( \text{point }\right) }\right) \) f... | Proof: The case of homology is slightly simpler, so let us consider this first. For CW complexes, relative homology groups reduce to absolute groups, so it suffices to deal with the latter. For a CW complex \( X \) the long exact sequences of \( {h}_{ * } \) homology groups for the pairs \( \left( {{X}^{n},{X}^{n - 1}}... | No |
For a fibration \( p : E \rightarrow B \), the fibers \( {F}_{b} = {p}^{-1}\left( b\right) \) over each path component of \( B \) are all homotopy equivalent. | Proof: A path \( y : I \rightarrow B \) gives rise to a homotopy \( {g}_{t} : {F}_{y\left( 0\right) } \rightarrow B \) with \( {g}_{t}\left( {F}_{y\left( 0\right) }\right) = y\left( t\right) \) . The inclusion \( {F}_{y\left( 0\right) } \hookrightarrow E \) provides a lift \( {\widetilde{g}}_{0} \), so by the homotopy ... | Yes |
Proposition 4.64. The map \( p : {E}_{f} \rightarrow B, p\left( {a,\gamma }\right) = \gamma \left( 1\right) \), is a fibration. | Proof: Continuity of \( p \) follows from (a) of Proposition A. 14 in the Appendix which says that the evaluation map \( {B}^{I} \times I \rightarrow B,\left( {y, s}\right) \mapsto y\left( s\right) \), is continuous.\n\nTo verify the fibration property, let a homotopy \( {g}_{t} : X \rightarrow B \) and a lift \( {\wid... | Yes |
Proposition 4.65. If \( p : E \rightarrow B \) is a fibration, then the inclusion \( E \hookrightarrow {E}_{p} \) is a fiber homotopy equivalence. In particular, the homotopy fibers of \( p \) are homotopy equivalent to the actual fibers. | Proof: We apply the homotopy lifting property to the homotopy \( {g}_{t} : {E}_{p} \rightarrow B,{g}_{t}\left( {e, y}\right) = \) \( y\left( t\right) \), with initial lift \( {\widetilde{g}}_{0} : {E}_{p} \rightarrow E,{\widetilde{g}}_{0}\left( {e, y}\right) = e \) . The lifting \( {\widetilde{g}}_{t} : {E}_{p} \righta... | Yes |
Proposition 4.66. If \( F \rightarrow E \rightarrow B \) is a fibration or fiber bundle with \( E \) contractible, then there is a weak homotopy equivalence \( F \rightarrow {\Omega B} \) . | Proof: If we compose a contraction of \( E \) with the projection \( p : E \rightarrow B \) then we have for each point \( x \in E \) a path \( {y}_{x} \) in \( B \) from \( p\left( x\right) \) to a basepoint \( {b}_{0} = p\left( {x}_{0}\right) \), where \( {x}_{0} \) is the point to which \( E \) contracts. This yield... | Yes |
For the Postnikov tower of a connected CW complex \( X \) the natural map \( X \rightarrow \lim {X}_{n} \) is a weak homotopy equivalence, so \( X \) is a CW approximation to \( \mathop{\lim }\limits_{ \leftarrow }{X}_{n} \) | The composition \( {\pi }_{i}\left( X\right) \rightarrow {\pi }_{i}\left( {\mathop{\lim }\limits_{ \rightarrow }{X}_{n}}\right) \overset{\lambda }{ \rightarrow }\mathop{\lim }\limits_{ \leftarrow }{\pi }_{i}\left( {X}_{n}\right) \) is an isomorphism since \( {\pi }_{i}\left( X\right) \rightarrow {\pi }_{i}\left( {X}_{n... | Yes |
Lemma 4.70. Let \( \left( {X, A}\right) \) be a CW pair with both \( X \) and \( A \) connected, such that the homotopy fiber of the inclusion \( A \hookrightarrow X \) is a \( K\left( {\pi, n}\right), n \geq 1 \) . Then there exists a fibration \( F \rightarrow E \rightarrow B \) and a map \( \left( {X, A}\right) \rig... | Proof: It remains only to prove the ’if’ implication. As we noted just before the statement of the lemma, the groups \( {\pi }_{i}\left( {X, A}\right) \) are zero except for \( {\pi }_{n + 1}\left( {X, A}\right) \approx \pi \) . If the action of \( {\pi }_{1}\left( A\right) \) on \( {\pi }_{n + 1}\left( {X, A}\right) \... | Yes |
Theorem 4.71. Every map \( f : X \rightarrow Y \) between connected CW complexes has a Moore-Postnikov tower, which is unique up to homotopy equivalence. A Moore-Postnikov tower of principal fibrations exists iff \( {\pi }_{1}\left( X\right) \) acts trivially on \( {\pi }_{n}\left( {{M}_{f}, X}\right) \) for all \( n >... | Proof: The existence and uniqueness of a diagram satisfying (1) and (2) and commutative at least up to homotopy follows from Propositions 4.13 and 4.18 applied to the pair \( \left( {{M}_{f}, X}\right) \) with \( {M}_{f} \) the mapping cylinder of \( f \) . Having such a diagram, we proceed as in the earlier case of Po... | Yes |
Proposition 1.1. Let \( \mathcal{L} \) be an ordinary fractal string with sequence of lengths \( {l}_{1},{l}_{2},{l}_{3},\ldots \) . Then\n\n\( {N}_{\mathcal{L}}\left( x\right) = O\left( {x}^{D}\right) , \) as \( x \rightarrow \infty \; \) if and only if \( \;{l}_{j} = O\left( {j}^{-1/D}\right) , \) as \( j \rightarrow... | Proof. Suppose we have the estimate\n\n\[ \n{N}_{\mathcal{L}}\left( x\right) \leq C \cdot {x}^{D} \n\]\n\n---\n\n\( {}^{1} \) Beginning in Chapter 4, Definition 4.1, we adopt the convention that the integers \( j \) such that \( {l}_{j}^{-1} = x \) must be counted with the weight \( 1/2 \) . A similar convention will b... | Yes |
Theorem 1.10. Suppose \( \mathcal{L} \) has infinitely many lengths. Then the abscissa of convergence of the geometric zeta function of \( \mathcal{L} \) coincides with \( D \), the Minkowski dimension of \( \partial \mathcal{L} \) . | Proof. We write \( \sigma \) for the abscissa of convergence of \( {\zeta }_{\mathcal{L}} \) . Let \( d > D \) . In view of definition (1.4) of the Minkowski dimension \( D \), there exists a constant \( {C}_{1} \) such that \( V\left( \varepsilon \right) \leq {C}_{1}{\varepsilon }^{1 - d} \) . For \( n \geq 1 \) we ch... | Yes |
Theorem 1.16. Let \( \mathcal{L} \) be a fractal string of dimension \( D \), and assume that \( {\zeta }_{\mathcal{L}} \) has a meromorphic extension to a neighborhood of \( D \) . If\n\n\[ \n{N}_{\mathcal{L}}\left( x\right) = O\left( {x}^{D}\right) ,\;\text{ as }x \rightarrow \infty ,\n\]\n\nor if the volume of the t... | Proof. First, recall from Remark 1.14 or from [Pos] or [Wid] that \( D \) is a singularity of \( {\zeta }_{\mathcal{L}} \) . Suppose \( {N}_{\mathcal{L}}\left( x\right) \leq C \cdot {x}^{D} \) and \( {N}_{\mathcal{L}}\left( x\right) = 0 \) for \( x \leq {x}_{0} \) . Then for \( s > D \) ,\n\n\[ \n{\zeta }_{\mathcal{L}}... | Yes |
Let\n\n\[ \n{\zeta }_{\mathcal{L}}\left( s\right) = \mathop{\sum }\limits_{{j = 1}}^{\infty }j{e}^{D{j}^{2}}{e}^{-{j}^{2}s}.\n\]\n\nThat is, we consider the generalized fractal string with lengths \( {e}^{-{j}^{2}} \), each repeated with multiplicity \( j{e}^{D{j}^{2}} \), for \( 0 < D < 1 \) (we obtain an ordinary fra... | On the other hand, \( \left( {s - D}\right) {\zeta }_{\mathcal{L}}\left( s\right) \) has limit \( 1/2 \) as \( s \rightarrow {D}^{ + } \), as we now show. For \( t > 0 \), we have\n\n\[ \nt{\zeta }_{\mathcal{L}}\left( {D + t}\right) = t\mathop{\sum }\limits_{{j = 1}}^{\infty }j{e}^{-{j}^{2}t} = \mathop{\sum }\limits_{{... | Yes |
Theorem 1.19. The spectral counting function of \( \mathcal{L} \) is given by\n\n\[ \n{N}_{\nu }\left( x\right) = {N}_{\mathcal{L}}\left( x\right) + {N}_{\mathcal{L}}\left( \frac{x}{2}\right) + {N}_{\mathcal{L}}\left( \frac{x}{3}\right) + \ldots \n\]\n\n(1.36)\n\n\[ \n= \mathop{\sum }\limits_{{j = 1}}^{\infty }\left\lb... | Proof. For the spectral counting function, this follows from the following computation:\n\n\[ \n{N}_{\nu }\left( x\right) = \mathop{\sum }\limits_{{k = 1}}^{\infty }\mathop{\sum }\limits_{{j : k \cdot {l}_{j}^{-1} \leq x}}1 = \mathop{\sum }\limits_{{k = 1}}^{\infty }\# \left\{ {j : {l}_{j}^{-1} \leq x/k}\right\} = \mat... | Yes |
Theorem 1.20 (Weyl’s Asymptotic Law). Let \( \mathcal{L} \) be a fractal string of dimension \( D \) and of total length\n\n\[{\operatorname{vol}}_{1}\left( \mathcal{L}\right) = \mathop{\sum }\limits_{{j = 1}}^{\infty }{l}_{j} = {\zeta }_{\mathcal{L}}\left( 1\right)\]\n\n(as defined in (1.19)). Then, for every \( \delt... | Proof. We write \( \{ x\} \) for the fractional part of \( x \) . By formula (1.37),\n\n\[{N}_{\nu }\left( x\right) = \mathop{\sum }\limits_{{j = 1}}^{\infty }{l}_{j}x - \mathop{\sum }\limits_{{j = 1}}^{\infty }\left\{ {{l}_{j}x}\right\}\]\n\nBoth sums are convergent, and the first sum equals \( {W}_{\mathcal{L}}\left(... | Yes |
Theorem 2.4. Let \( \\mathcal{L} \) be a self-similar string as in Section 2.1. Then the geometric zeta function of this string has a meromorphic continuation to the whole complex plane, given by\n\n\[ \n{\\zeta }_{\\mathcal{L}}\\left( s\\right) = \\frac{{L}^{s}\\mathop{\\sum }\\limits_{{k = 1}}^{K}{g}_{k}^{s}}{1 - \\m... | Proof. Indeed, we have\n\n\[ \n\\mathop{\\sum }\\limits_{{{\\nu }_{1} = 1}}^{N}\\cdots \\mathop{\\sum }\\limits_{{{\\nu }_{q} = 1}}^{N}{\\left( {r}_{{\\nu }_{1}}\\cdots {r}_{{\\nu }_{q}}\\right) }^{s} = \\mathop{\\sum }\\limits_{{{\\nu }_{1} = 1}}^{N}\\cdots \\mathop{\\sum }\\limits_{{{\\nu }_{q} = 1}}^{N}{r}_{{\\nu }_... | Yes |
Theorem 2.17. Let \( \mathcal{L} \) be a self-similar string of dimension \( D \), with scaling ratios \( {r}_{1},\ldots ,{r}_{N} \) and gaps \( {g}_{1},\ldots ,{g}_{K} \), as defined in Section 2.1. Then all the complex dimensions of \( \mathcal{L} \) lie to the left of or on the line \( \operatorname{Re}s = D \) : | \[ \mathcal{D} = {\mathcal{D}}_{\mathcal{L}}\left( \mathbb{C}\right) \subset \{ s \in \mathbb{C} : \operatorname{Re}s \leq D\} . \] (2.36) The value \( s = D \) is the only pole of \( {\zeta }_{\mathcal{L}} \) on the real line. (In particular, Equation (1.27) holds, with \( D = {D}_{\mathcal{L}} \) .) Moreover, \( 0 < ... | Yes |
Corollary 2.21. Every self-similar string has infinitely many complex dimensions with positive real part. | Proof. In the lattice case, this follows from the fact that \( \mathcal{D} \) always contains the vertical line of complex dimensions \( D + {in}\mathbf{p} \), for \( n \in \mathbb{Z} \) (see formula (2.39) in Theorem 2.17).\n\nIn the nonlattice case, this follows from the fact that \( D > 0 \), combined with the state... | Yes |
Consider the self-similar string with scaling ratios \( {r}_{1} = 1/2,{r}_{2} = 1/4,{r}_{3} = {2}^{-1 - \sqrt{2}} \) and one gap \( g = 1/4 - {r}_{3} \) . | For example, the first approximation gives the lattice string with scaling ratios \( {\widetilde{r}}_{1} = {2}^{-1},{\widetilde{r}}_{2} = {2}^{-2},{\widetilde{r}}_{3} = {2}^{-5/2} \), the complex dimensions of which are the solutions to the equation \( {z}^{2} + {z}^{4} + {z}^{5} = 1,{2}^{-\omega /2} = z \) . The oscil... | Yes |
Problem 3.4 (Transition in the Nongeneric Nonlattice Case). A nongeneric nonlattice string has a vertical line of transition inside the vertical strip \( {D}_{l} \leq \operatorname{Re}s \leq D \), to the left of which the density of the real parts is infinitely higher than to the right. Such a transition does not occur... | Thus, for the nongeneric nonlattice string of Example 3.3, this transition occurs at \( \operatorname{Re}s = 0 \), as indicated by the corner of the density graph at this point, and the vertical part of the graph to the left of \( \operatorname{Re}s = 0 \) . Moreover, from other numerical evidence, it seems that this l... | No |
The complex dimensions of \( \mathcal{L} \) are found by solving the Dirichlet polynomial equation\n\n\[ {2}^{-s} + {2}^{-{\phi s}} + {2}^{-{2s}} = 1 \] | For the real parts, we observe the same phenomenon of phase transition in the complex dimensions as discussed in Example 3.3 and Problem 3.4. The complex dimensions with positive real part again correspond to those of the golden string of Section 2.3.5, like in the 2-3-4 equation discussed at the beginning of Section 3... | No |
Theorem 3.6. Let \( f \) be a Dirichlet polynomial with \( M \) different scaling ratios \( 1 > {r}_{1} > \cdots > {r}_{M} > 0 \) and complex multiplicities \( {m}_{j} \) as in Equation (3.5). Then, both in the lattice and the nonlattice case, the set \( {\mathcal{D}}_{f} \) of complex roots of \( f \) is contained in ... | \[ {\mathcal{D}}_{f} = {\mathcal{D}}_{f}\left( \mathbb{C}\right) \subseteq \left\{ {s \in \mathbb{C} : {D}_{l} \leq \operatorname{Re}s \leq {D}_{r}}\right\} . \] (3.9) It has density \( \frac{{w}_{M}}{2\pi } \) (with \( {w}_{M} = \log {r}_{M}^{-1} \) ): \[ \# \left( {{\mathcal{D}}_{f}\cap \{ \omega \in \mathbb{C} : 0 \... | Yes |
Every integral positive \( {}^{3} \) Dirichlet polynomial has infinitely many complex roots with positive real part. | Proof of Theorem 3.6. The case of positive and integral weights was proved in Section 2.5. There, the real numbers \( {D}_{l} \) and \( {D}_{r} \) were not defined, but their main property (3.9) can be deduced by an argument similar to that used for \( D \) . Indeed, let \( s \) be a complex number with real part \( \s... | Yes |
Theorem 3.14. Let \( f \) be a meromorphic function and let \( {\left\{ {f}_{n}\right\} }_{n = 1}^{\infty } \) be a sequence of meromorphic functions such that \( {f}_{n} \rightarrow f \) . Then \( {\mathfrak{D}}_{n} \rightarrow \widehat{\mathfrak{D}} \) . | Proof. Let \( C \subset W \) be compact and choose circles \( {T}_{0} \) and \( {T}_{\infty } \) around 0 and \( \infty \), respectively, so small that the pre-image \( {f}^{-1}\left( {{T}_{0} \cup {T}_{\infty }}\right) \cap C \) is the union of disjoint small circles (really, closed Jordan curves) around each point \(... | Yes |
Theorem 3.18. Let weights \( {w}_{0} = 0 < {w}_{1} < \cdots < {w}_{M} \) and multiplicities \( {m}_{0} = - 1 \) and \( {m}_{1},\ldots ,{m}_{M} \in \mathbb{C} \) be given as in (3.5), such that at least one ratio \( {w}_{j}/{w}_{1} \) is irrational, for some \( j = 2,\ldots, M \) . Let\n\n\[ f\left( s\right) = 1 - \math... | Proof. Let \( {r}_{j} = {e}^{-{w}_{j}} \) for \( j = 1,\ldots, M \), and \( \widetilde{r} = {e}^{-\widetilde{w}} \) . To show that \( f \) is well approximated by \( \widetilde{f} \), we consider the expression\n\n\[ {r}_{j}^{s} - {\widetilde{r}}^{{k}_{j}s} = - s{\int }_{{k}_{j}\widetilde{w}}^{{w}_{j}}{e}^{-{sx}}{dx}. ... | Yes |
Theorem 3.19. Let \( f \) be a Dirichlet polynomial, with scaling ratios\n\n\[ \n{r}_{0} = 1 > {r}_{1} > \cdots > {r}_{M} > 0 \n\]\n\nand multiplicities \( {m}_{j} \in \mathbb{C} \) . Let \( {\left\{ {f}_{n}\right\} }_{n = 1}^{\infty } \) be a sequence of Dirichlet polynomials, with scaling ratios\n\n\[ \n1 > {r}_{1}^{... | Proof of Theorem 3.19. Suppose first that \( {f}_{n} \) and \( f \) are lattice polynomials with the same multiplicative generator \( r \) . Since, then, both functions are polynomials in \( {r}^{s} \), the theorem follows from [Ahl,§5.5]. In general, we choose lattice strings approximating everything on the level of t... | Yes |
Theorem 3.23. Let \( \mathcal{L} \) be a nonlattice self-similar string. Then there exists a sequence of simple complex dimensions of \( \mathcal{L} \) approaching the line \( \operatorname{Re}s = D \) from the left. | Proof. Note that \( \mathop{\lim }\limits_{{\sigma \rightarrow + \infty }}f\left( {\sigma + {it}}\right) = 1 \) and \( \mathop{\lim }\limits_{{\sigma \rightarrow - \infty }}\left| {f\left( {\sigma + {it}}\right) }\right| = \infty \), for any given real value of \( t \) . Given \( t \in \mathbb{R} \), let\n\n\[ l\left( ... | Yes |
Theorem 3.25. Let \( \mathcal{L} \) be a nonlattice self-similar string with scaling ratios \( {r}_{1},\ldots ,{r}_{N} \) and gaps \( {g}_{1},\ldots ,{g}_{K} \). Then there exists a screen \( S \) such that \( {\zeta }_{\mathcal{L}} \) is bounded on \( S \) and all complex dimensions to the right of \( S \) are simple ... | Proof. Let \( \widetilde{\mathcal{L}} \) be a lattice string with scaling ratios \( {\widetilde{r}}_{1},\ldots ,{\widetilde{r}}_{N} \), approximating \( \mathcal{L} \). Let \( D \) be the dimension of \( \mathcal{L} \), and assume \( {}^{5} \) that \( D \) is also the dimension of \( \widetilde{\mathcal{L} \). Since th... | Yes |
Theorem 3.26. Let \( \mathcal{L} \) be a self-similar string as in Theorem 3.25. Then there exists a sequence of positive numbers \( {T}_{1},{T}_{2},\ldots \), tending to infinity, such that \( \left| {{\zeta }_{\mathcal{L}}\left( s\right) }\right| \) is uniformly bounded from above on each horizontal line \( \operator... | Proof. If \( \mathcal{L} \) is a lattice string, then the existence of such a sequence \( {\left\{ {T}_{n}\right\} }_{n = 1}^{\infty } \) follows easily from the fact that the denominator of \( {\zeta }_{\mathcal{L}} \) is periodic (with period \( i\mathbf{p} \), where \( \mathbf{p} \) is the oscillatory period).\n\nIf... | Yes |
Lemma 3.28. Let \( n \) be given by formula (3.27). Suppose that the last \( k \) digits of \( n \) vanish: \( k \geq 0 \) is such that \( {d}_{k} \neq 0 \) and \( {d}_{k - 1} = \cdots = {d}_{0} = 0 \) . Put \( m = \mathop{\sum }\limits_{{\nu = k}}^{\infty }{d}_{\nu }{p}_{\nu } \) . Then \( {n\alpha } - m \) lies stric... | Proof. We have \( {n\alpha } - m = \mathop{\sum }\limits_{{\nu = k}}^{\infty }{d}_{\nu }\left( {\alpha {q}_{\nu } - {p}_{\nu }}\right) \), which is close to the first term \( {d}_{k}{\left( -1\right) }^{k}/{q}_{k + 1}^{\prime } \) by Equation (3.26). Again by this equation, the terms in this sum are alternately positiv... | Yes |
Lemma 3.29. Let \( {w}_{1},{w}_{2} > 0 \) and \( \alpha = {w}_{2}/{w}_{1} > 1 \) . Let \( \Delta = \Delta \left( x\right) \) be the function of \( x \), defined implicitly by \[ {m}_{1}{e}^{-{w}_{1}D}{e}^{-{w}_{1}\Delta } + {m}_{2}{e}^{-{w}_{2}D}{e}^{-x}{e}^{-{w}_{2}\Delta } = 1, \] and \( \Delta \left( 0\right) = 0 \)... | Proof. Write \( {e}^{-{w}_{1}\Delta } = y\left( x\right) \), so that \( y \) is defined by \[ {m}_{1}{e}^{-{w}_{1}D}y + {m}_{2}{e}^{-{w}_{2}D}{e}^{-x}{y}^{\alpha } = 1\;\text{ and }\;y\left( 0\right) = 1. \] Since \( y = 0 \) is not a solution of this equation, it follows that if \( y\left( x\right) \) is analytic in a... | Yes |
Theorem 3.30. Let \( {w}_{2} = \alpha {w}_{1} \), with \( \alpha > 1 \) and irrational. Then the complex roots of the Dirichlet polynomial equation\n\n\[ \n{m}_{1}{e}^{-{w}_{1}s} + {m}_{2}{e}^{-{w}_{2}s} = 1 \n\]\n\nare simple. | Proof. Writing \( A = {m}_{1}{e}^{-{w}_{1}s} \) and \( B = {m}_{2}{e}^{-{w}_{2}s} \), the assumption that \( s \) is a double root of this Dirichlet polynomial leads to the equations \( A + B = 1 \) and \( A + {\alpha B} = 0 \) . Thus \( A = \frac{\alpha }{\alpha - 1} \) and \( B = - \frac{1}{\alpha - 1} \) . It follow... | Yes |
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