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The family \( F\left( {\varepsilon, w}\right) ,\varepsilon \in (0,1\rbrack \) defined above, satisfies (9.32) if and only if \( \left( {{A}^{ij}\left( x\right) }\right) \) defined by \[ {A}^{ij}\left( x\right) = \mathop{\sum }\limits_{{\alpha = 1}}^{r}{V}_{\alpha }^{i}\left( x\right) {V}_{\alpha }^{j}\left( x\right) \]... | Proof. Since \[ F\left( {\varepsilon, w}\right) \sim {f}_{1} + \varepsilon {f}_{2} + {\varepsilon }^{2}{f}_{3} + \cdots \;\text{ in }{D}_{\infty }\left( {R}^{d}\right) \] where \( {f}_{t} \) are given in Theorem 10.4 and the Malliavin covariance of \( {f}_{1} \) coincides with \( A\left( x\right) = \left( {{A}^{ij}\lef... | Yes |
Theorem 10.6. Suppose that \( \det \left( {A\left( x\right) }\right) > 0 \) . Then \( {\delta }_{x}\left( {{X}^{\varepsilon }\left( {1, x, w}\right) }\right) \) has the following asymptotic expansion in \( {\overline{\mathbf{D}}}_{-\infty } \) as \( \varepsilon \downarrow 0 \) :\n\n(10.51)\n\n\[ \n{\delta }_{x}\left( {... | can be obtained explicitly by Theorem 9.4:\n\n(10.52)\n\n\[ \n{\Phi }_{0} = {\delta }_{0}\left( {f}_{1}\right) \n\] \n\nand\n\n(10.53)\n\n\[ \n{\Phi }_{k}\left( w\right) = \mathop{\sum }\limits_{\alpha }\mathop{\sum }\limits_{n}\frac{1}{\left| \alpha \right| !}{D}_{\alpha }{\delta }_{0}\left( {f}_{1}\right) {f}_{{n}_{1... | Yes |
Lemma 2.1. Let \( \left( {{X}_{t},{B}_{t}}\right) \) be a pair of \( d \) -dimensional continuous \( \left( {\mathcal{F}}_{t}\right) \) - adapted processes defined on a probability space \( \left( {\Omega ,\mathcal{F}, P}\right) \) with a reference family \( \left( {\mathcal{F}}_{t}\right) \) such that \( \left\{ {B}_{... | The proof follows in exactly the same way as in Theorem IV-1.1. | No |
Theorem 3.1. (1) If \( s\left( {l + }\right) = - \infty \) and \( s\left( {r - }\right) = \infty \), then\n\n\[ \n{P}_{x}\left( {e = \infty }\right) = {P}_{x}\left( {\mathop{\lim }\limits_{{t \uparrow \infty }}X\left( t\right) = r}\right) = {P}_{x}\left( {\mathop{\lim }\limits_{{t \uparrow \infty }}X\left( t\right) = l... | Proof. Let \( l < a < x < b < r \) and \( \tau = \inf \left\{ {t;{X}_{t} \in \left\lbrack {a, b}\right\rbrack }\right\} \) . By Itô’s formula and (3.4),\n\n\[ \ns\left( {{X}^{x}\left( {t \land \tau }\right) }\right) - s\left( x\right) = {\int }_{0}^{t \land \tau }{s}^{\prime }\left( {{X}^{x}\left( s\right) }\right) \si... | Yes |
Lemma 3.1. Let \( u\left( x\right) \) be the unique solution of\n\n(3.11)\n\n\[ \begin{cases} {Lu}\left( x\right) & = u\left( x\right) \\ u\left( c\right) & = 1 \\ {u}^{\prime }\left( c\right) & = 0. \end{cases} \]\n\nThen\n\n(3.12)\n\n\[ 1 + \kappa \left( x\right) \leq u\left( x\right) \leq \exp \{ \kappa \left( x\rig... | Proof. \( u\left( x\right) \) is the solution of (3.11) if and only if it satisfies\n\n\[ u\left( x\right) = 1 + {\int }_{c}^{x}{ds}\left( y\right) {\int }_{c}^{y}u\left( z\right) {dm}\left( z\right) ,\]\n\nwhere\n\n\[ {ds}\left( y\right) = \exp \left\lbrack {-{\int }_{c}^{y}\frac{{2b}\left( z\right) }{{\sigma }^{2}\le... | Yes |
Theorem 3.2. (1) If \( \kappa \left( {r - }\right) = \kappa \left( {l + }\right) = \infty \), then\n\n\[{P}_{x}\left( {e = \infty }\right) = 1\;\text{ for all }x \in I.\] | Proof. Let \( u\left( x\right) \) be defined by (3.11). Let \( l < a < x < b < r \) and \( \tau = \inf \{ t;X\left( t\right) \in \left( {a, b}\right) \} \) . By Itô’s formula\n\n\[d{e}^{-t}u\left( {X\left( t\right) }\right) = {e}^{-t}{u}^{\prime }\left( {X\left( t\right) }\right) \sigma \left( {X\left( t\right) }\right... | Yes |
Theorem 4.1 ([54]). Let \( {x}_{0} \in {\mathbf{R}}^{d} \smallsetminus S \) be fixed and \( {\xi }_{0} = p\left( {x}_{0}\right) \in {I}^{0} \) . Let \( X = \left( {X}_{t}\right) \) be the above diffusion starting at \( {x}_{0} \) . Then we can construct \( \bar{I} \) - valued continuous stochastic processes \( {\eta }_... | Proof. For simplicity we assume that \( \zeta = \infty \) a.s. and that \( \left( {\xi }_{t}^{\pm \pm }\right) \) are all conservative diffusion processes on \( {I}^{0} \) ; the general case can be proved with a slight modification.\n\nSet\n\n\[ \n{\phi }^{ + }\left( t\right) = {\int }_{0}^{t}\left\lbrack {a\left( {X}_... | Yes |
Lemma 5.1. Let the Ricci curvature \( \rho \left( X\right) \) satisfy that\n\n\[ \left( {d - 1}\right) b \leq \rho \left( X\right) ,\;X \in {T}_{x}\left( M\right) ,\;\parallel X\parallel = 1, \]\n\nand the sectional curvature \( K\left( \xi \right) \) satisfy that \( K\left( \xi \right) \leq a \) for every plane sectio... | For a proof, see [7] (pp. \( {253} \sim {256} \) ) and [42]. | No |
Theorem 5.1. Let the Ricci curvature \( \rho \left( X\right) \) satisfy that \( \left( {d - 1}\right) b \leq \rho \left( X\right) \) for all \( X \in {T}_{x}\left( M\right) ,\parallel X\parallel = 1 \) and \( x \in M \) . (Let the sectional curvature \( K\left( \xi \right) \) satisfy that \( K\left( \xi \right) \leq a ... | \[ \overset{*1}{}0 < {r}_{0} < {r}^{ * } \] \n\n*2 We have automatically that \( {r}^{ * } \leq {r}_{0}\left( b\right) \) (respectively \( {r}^{ * } \leq {r}_{{x}_{0}}\left( M\right) \) ).\n\n(respectively \( {\xi }_{0}^{\left( \omega \right) } = d\left( {{x}_{0},{X}_{0}}\right) \) ) and the following is satisfied: if ... | No |
Corollary 2. Let \( M \) be a connected, complete \( d \) -dimensional Riemannian manifold \( \left( {d \geqq 2}\right) \) with non-positive sectional curvature for all plane sections. Furthermore, we assume that the Ricci curvature \( \rho \) satisfies the condition\n\n\[ 0 \geq \rho \left( X\right) \geq \left( {d - 1... | Proof. Without loss of generality, we may assume that \( M \) is simply connected because the Brownian motion on \( M \) is obtained as the projection of the Brownian motion on the universal covering space \( \widetilde{M} \) of \( M \) . Then \( {r}^{ * } = \infty \) by the Cartan-Hadamard theorem ([7]) and the assert... | Yes |
\[ {\int }_{X\left\lbrack {0, t}\right\rbrack }\alpha = {\int }_{0}^{t}{\widetilde{\alpha }}_{k}\left( {r\left( s\right) }\right) d{w}^{k}\left( s\right) + {\int }_{0}^{t}\left( {\alpha \left( b\right) - \frac{1}{2}{\delta \alpha }}\right) \left( {X\left( s\right) }\right) {ds} \] | Proof. We have \[ {\int }_{X\left\lbrack {0, t}\right\rbrack }\alpha = {\int }_{0}^{t}{\bar{\alpha }}_{k}\left( {r\left( s\right) }\right) \circ d{w}^{k}\left( s\right) + {\int }_{0}^{t}\alpha \left( b\right) \left( {X\left( s\right) }\right) {ds} \] \[ = {\int }_{0}^{t}{\bar{\alpha }}_{k}\left( {r\left( s\right) }\rig... | Yes |
Let \( M \) be a Riemannian manifold. The horizontal Brownian motion \( r = \{ r\left( t\right) \} \) is a diffusion process on \( O\left( M\right) \) determined by the equation (6.1) with \( {\widetilde{L}}_{0} = 0 \) : (6.13) \[ \left\{ \begin{array}{l} {dr}\left( t\right) = {\widetilde{L}}_{k}\left( {r\left( t\right... | Suppose that \( {\bar{b}}^{t}\left( r\right) = {\overline{\omega \left( b\right) }}_{t}\left( r\right) \) is bounded on \( O\left( M\right) \) for \( i = 1,2,\ldots, d \) . Then (6.14) \[ M\left( t\right) = \exp \left\{ {\mathop{\sum }\limits_{{i = 1}}^{d}{\int }_{0}^{t}{\bar{b}}^{t}\left( {r\left( s\right) }\right) d{... | Yes |
Choose \( {\phi }^{i} \in \Phi, i = 1,2,\ldots, r \), and set\n\n\[ \n{B}_{\delta }^{i}\left( {t, w}\right) = {w}^{i}\left( {k\delta }\right) + {\phi }^{i}\left( {\left( {t - {k\delta }}\right) /\delta }\right) {\Delta }_{k}{w}^{i}, \]\n\n\[ \n\text{if}{k\delta } \leq t < \left( {k + 1}\right) \delta, k = 0,1,\ldots \t... | For example,\n\n\[ \nE\left\lbrack {\left\{ {\int }_{0}^{\delta }\left| {\dot{B}}_{\delta }^{i}\left( s, w\right) \right| ds\right\} }^{6}\right\rbrack = E\left\lbrack {\left| {\Delta }_{0}{w}^{i}\right| }^{6}\right\rbrack {\left( {\int }_{0}^{1}\left| {\dot{\phi }}^{i}\left( s\right) \right| ds\right) }^{6} \]\n\n\[ \... | Yes |
Let \( r = 2 \) and choose \( {\phi }^{l} \in \Phi, i = 1,2 \) . Set\n\n(7.9)\n\n\[ \n{B}_{\delta }^{l}\left( {t, w}\right) = \left\{ \begin{array}{ll} {w}^{l}\left( {k\delta }\right) + {\phi }^{l}\left( {\left( {t - {k\delta }}\right) /\delta }\right) {\Delta }_{k}{w}^{l}, & {\Delta }_{k}{w}^{1}{\Delta }_{k}{w}^{2} \g... | It is easy to see that \( {\left\{ {B}_{\delta }\left( t, w\right) \right\} }_{\delta > 0} \) is an approximation of a Wiener process. In this case, however, \n\n\[ \n\frac{1}{2}{\int }_{0}^{\delta }\left\{ {{B}_{\delta }^{1}\left( {s, w}\right) {\dot{B}}_{\delta }^{2}\left( {s, w}\right) - {B}_{\delta }^{2}\left( {s, ... | Yes |
Example 7.3 (mollifiers). Let \( \\rho \) be a non-negative \( {C}^{\\infty } \) -function whose support is contained in \( \\left\\lbrack {0,1}\\right\\rbrack \) and \( {\\int }_{0}^{1}\\rho \\left( s\\right) {ds} = 1 \) . Set\n\n\[ \n{\\rho }_{\\delta }\\left( s\\right) = \\frac{1}{\\delta }\\rho \\left( \\frac{s}{\\... | Indeed, (i) - (iv) in Definition 7.1 are obvious and (v) and (vi) are verified as follows. We have\n\n\[ \n{\\left| {B}_{\\delta }^{l}\\left( 0, w\\right) \\right| }^{2m} = {\\left( {\\int }_{0}^{1}\\frac{1}{\\sqrt{\\delta }}{w}^{l}\\left( \\delta s\\right) \\rho \\left( s\\right) ds\\right) }^{2m}{\\delta }^{m} \n\]\n... | Yes |
Theorem 8.1. \( \mathcal{S}\left( {P}_{x}\right) = \overline{{\mathcal{S}}^{x}}\; \) for every \( x \in {R}^{d} \) . | Proof. First we shall prove the inclusion \( \mathcal{S}\left( {P}_{x}\right) \subset \overline{{\mathcal{S}}^{x}} \) . For each \( n = 1,2,\ldots \) and \( t \geq 0 \), we set \( {t}_{n} = \left\lbrack {{2}^{n}t}\right\rbrack /{2}^{n} \) and \( {\bar{t}}_{n} = \left( {\left\lbrack {{2}^{n}t}\right\rbrack + 1}\right) /... | Yes |
Lemma 8.1. There exist positive constants \( {c}_{1} \) and \( {c}_{2} \) such that\n\n\[ {P}^{W}\left( {\parallel w{\parallel }_{T} < \varepsilon }\right) \sim {c}_{1}\exp \left( {-\frac{{c}_{2}}{{\varepsilon }^{2}}}\right) \;\text{ as }\;\varepsilon \downarrow 0. \] | Proof. Let \( {P}_{x} \) be the \( r \) -dimensional Wiener measure starting at \( x \in {\mathbf{R}}^{r} \) ; so, in particular, \( {P}_{0} = {P}^{w} \) . Let \( D = \left\{ {x \in {\mathbf{R}}^{r};\left| x\right| < 1}\right\} \) and set\n\n\[ \sigma \left( w\right) = \inf \{ t;w\left( t\right) \notin D\} \;\text{ for... | Yes |
Lemma 8.2. Set\n\n(8.11)\n\n\[ \n{\eta }^{ij}\left( t\right) = \frac{1}{2}{\int }_{0}^{t}\left\lbrack {{w}^{i}\left( s\right) d{w}^{j}\left( s\right) - {w}^{j}\left( s\right) d{w}^{i}\left( s\right) }\right\rbrack \n\]\n\n\[ \n\text{for}i, j = 1,2,\ldots, r\text{.} \n\]\n\nThen we have\n\n(8.12)\n\n\[ \n\mathop{\lim }\... | Proof. Let \( i \neq j \) be fixed and set\n\n\[ \na\left( t\right) = \frac{1}{4}{\int }_{0}^{t}\left\lbrack {{\left( {w}^{i}\right) }^{2}\left( s\right) + {\left( {w}^{j}\right) }^{2}\left( s\right) }\right\rbrack {ds}. \n\]\n\nWe have remarked in Section 6 that \( B\left( t\right) \mathrel{\text{:=}} {\eta }^{t\prime... | Yes |
Let \( {L}_{0},{L}_{1},\ldots ,{L}_{r} \) be vector fields on \( {\mathbf{R}}^{d} \) whose coefficients (in the Euclidean coordinates) are bounded and smooth with bounded derivatives. Let \( {X}^{x} = \left( {X\left( {t, w}\right) }\right) \) be the solution of\n\n\[ \left\{ \begin{array}{l} {dX}\left( t\right) = \math... | \[ \xi \left( {x,\phi }\right) \) is the solution of the dynamical system\n\n\[ \left\{ \begin{array}{l} \frac{d{\xi }_{t}}{dt} = \mathop{\sum }\limits_{{i = 1}}^{r}{L}_{t}\left( {\xi }_{t}\right) {\dot{\phi }}^{i}\left( t\right) + {L}_{0}\left( {\xi }_{t}\right) . \\ {\xi }_{0} = x. \end{array}\right. \] \n\nIt is kno... | Yes |
Lemma 8.4. If \( f \) is a nonpositive continuous function in \( D \) with compact support, then the function \( u \) defined by\n\n\[ \nu\left( y\right) = {E}_{y}\left\lbrack {{\int }_{0}^{\tau }f\left( {w\left( s\right) }\right) {ds}}\right\rbrack \]\n\nis a bounded \( A \) -subharmonic function in \( D \) . | Proof of Lemma 8.4. \( u \) is bounded because of the above assumption on \( D \) . First we shall prove that it is upper semicontinuous. We saw in the proof of Proposition V-2.1 that if \( X\left( {t, x}\right) \) is the solution of (8.1), then for every \( t > 0,{\mathbf{R}}^{d} \ni x \mapsto X\left( {t, x}\right) \i... | Yes |
Theorem 9.2. The diffusion \( \left\{ {P}_{x}\right\} \) is symmetrizable if and only if\n\n(9.16)\n\n\[ \mathop{\lim }\limits_{{\varepsilon \downarrow 0}}\frac{{P}_{x}\left( {\parallel w - \phi {\parallel }_{T} < \varepsilon }\right) }{{P}_{x}\left( {\parallel w - {\phi }_{ - }{\parallel }_{T} < \varepsilon }\right) }... | Proof. By Theorem 9.1, the limit in (9.16) is equal to\n\n\[ \exp \left( {2{\int }_{0}^{T}b\left( {\phi \left( s\right) }\right) \dot{\phi }\left( s\right) {ds}}\right) = \exp \left( {2{\int }_{\phi }\omega }\right) .\n\nThus (9.16) holds if and only if \( {\int }_{\phi }\omega = 0 \) . | Yes |
Real projective \( n \) -space \( {\mathbb{{RP}}}^{n} \) is defined to be the space of all lines through the origin in \( {\mathbb{R}}^{n + 1} \) . Each such line is determined by a nonzero vector in \( {\mathbb{R}}^{n + 1} \) , unique up to scalar multiplication, and \( {\mathbb{{RP}}}^{n} \) is topologized as the quo... | Since \( \partial {D}^{n} \) with antipodal points identified is just \( {\mathbb{{RP}}}^{n - 1} \), we see that \( {\mathbb{{RP}}}^{n} \) is obtained from \( {\mathbb{{RP}}}^{n - 1} \) by attaching an \( n \) -cell, with the quotient projection \( {S}^{n - 1} \rightarrow {\mathbb{{RP}}}^{n - 1} \) as the attaching map... | Yes |
Complex projective \( \mathbf{n} \) -space \( {\mathbb{{CP}}}^{n} \) is the space of complex lines through the origin in \( {\mathbb{C}}^{n + 1} \), that is,1-dimensional vector subspaces of \( {\mathbb{C}}^{n + 1} \) . As in the case of \( {\mathbb{{RP}}}^{n} \), each line is determined by a nonzero vector in \( {\mat... | From this description of \( {\mathbb{{CP}}}^{n} \) as the quotient of \( {D}_{ + }^{2n} \) under the identifications \( v \sim {\lambda v} \) for \( v \in {S}^{{2n} - 1} \) it follows that \( {\mathbb{{CP}}}^{n} \) is obtained from \( {\mathbb{{CP}}}^{n - 1} \) by attaching a cell \( {e}^{2n} \) via the quotient map \(... | Yes |
Let \( X \) be the union of a torus with \( n \) meridional disks. To obtain a CW structure on \( X \), choose a longitudinal circle in the torus, intersecting each of the meridional disks in one point. These intersection points are then the 0 -cells, the 1-cells are the rest of the longitudinal circle and the boundary... | \n\nequivalent space \( Y \) consisting of \( {n2} \) -spheres, each tangent to its two neighbors, a ’necklace with \( n \) beads.’ The third space \( Z \) in the figure, a strand of \( n \) beads with a string joining... | No |
Let us rederive the result in Example 0.8 that a sphere with two points identified is homotopy equivalent to \( {S}^{1} \vee {S}^{2} \) . | The sphere with two points identified can be obtained by attaching \( {S}^{2} \) to \( {S}^{1} \) by a map that wraps a closed arc \( A \) in \( {S}^{2} \) around \( {S}^{1} \) , as shown in the figure. Since \( A \) is contractible, this attaching map is homotopic to a constant map, and attaching \( {S}^{2} \) to \( {... | Yes |
In similar fashion we can see that the necklace in Example 0.9 is homotopy equivalent to the wedge sum of a circle with \( n \) 2-spheres. | The necklace can be obtained from a circle by attaching \( n \) 2-spheres along arcs, so the necklace is homotopy equivalent to the space obtained by attaching \( n \) 2-spheres to a circle at points. Then we can slide these attaching points around the circle until they all coincide, producing the wedge sum. | Yes |
Here is an application of the earlier fact that collapsing a contractible subcomplex is a homotopy equivalence: If \( \left( {X, A}\right) \) is a CW pair, consisting of a cell complex \( X \) and a subcomplex \( A \), then \( X/A \simeq X \cup {CA} \), the mapping cone of the inclusion \( A \hookrightarrow X \) . | For we have \( X/A = \left( {X \cup {CA}}\right) /{CA} \simeq X \cup {CA} \) since \( {CA} \) is a contractible subcomplex of \( X \cup {CA} \) . | Yes |
If \( \left( {X, A}\right) \) is a CW pair and \( A \) is contractible in \( X \), that is, the inclusion \( A \hookrightarrow X \) is homotopic to a constant map, then \( X/A \simeq X \vee {SA} \). | Namely, by the previous example we have \( X/A \simeq X \cup {CA} \), and then since \( A \) is contractible in \( X \), the mapping cone \( X \cup {CA} \) of the inclusion \( A \hookrightarrow X \) is homotopy equivalent to the mapping cone of a constant map, which is \( X \vee {SA} \). | Yes |
A pair \( \left( {X, A}\right) \) has the homotopy extension property if \( A \) has a mapping cylinder neighborhood in \( X \), by which we mean a closed neighborhood \( N \) containing a subspace \( B \), thought of as the boundary of \( N \), with \( N - B \) an open neighborhood of \( A \) , such that there exists ... | To verify the homotopy extension property, notice first that \( I \times I \) retracts onto \( I \times \{ 0\} \cup \partial I \times I \), hence \( B \times I \times I \) retracts onto \( B \times I \times \{ 0\} \cup B \times \partial I \times I \), and this retraction induces a retraction of \( {M}_{f} \times I \) o... | Yes |
Proposition 0.16. If \( \left( {X, A}\right) \) is a CW pair, then \( X \times \{ 0\} \cup A \times I \) is a deformation retract of \( X \times I \), hence \( \left( {X, A}\right) \) has the homotopy extension property. | Proof: There is a retraction \( r : {D}^{n} \times I \rightarrow {D}^{n} \times \{ 0\} \cup \partial {D}^{n} \times I \), for example the radial projection from the point \( \left( {0,2}\right) \in {D}^{n} \times \mathbb{R} \). Then setting \( {r}_{t} = {tr} + \left( {1 - t}\right) \mathbb{1} \) gives a deformation ret... | Yes |
Proposition 0.18. If \( \left( {{X}_{1}, A}\right) \) is a CW pair and we have attaching maps \( f, g : A \rightarrow {X}_{0} \) that are homotopic, then \( {X}_{0}{ \sqcup }_{f}{X}_{1} \simeq {X}_{0}{ \sqcup }_{g}{X}_{1} \) rel \( {X}_{0} \) . | Proof: If \( F : A \times I \rightarrow {X}_{0} \) is a homotopy from \( f \) to \( g \), consider the space \( {X}_{0}{ \sqcup }_{F}\left( {{X}_{1} \times I}\right) \) . This contains both \( {X}_{0}{ \sqcup }_{f}{X}_{1} \) and \( {X}_{0}{ \sqcup }_{g}{X}_{1} \) as subspaces. A deformation retraction of \( {X}_{1} \ti... | Yes |
Corollary 0.21. A map \( f : X \rightarrow Y \) is a homotopy equivalence iff \( X \) is a deformation retract of the mapping cylinder \( {M}_{f} \) . Hence, two spaces \( X \) and \( Y \) are homotopy \( \parallel \) equivalent iff there is a third space containing both \( X \) and \( Y \) as deformation retracts. | Proof: In the diagram at the right the maps \( i \) and \( j \) are the inclusions and \( r \) is the canonical retraction, so \( f = {ri} \) and \( i \simeq {jf} \) . Since \( j \) and \( r \) are homotopy equivalences, it follows that \( f \) is a homotopy equivalence iff \( i \) is a homotopy equivalence, since the ... | Yes |
Proposition 1.5. The map \( {\beta }_{h} : {\pi }_{1}\left( {X,{x}_{1}}\right) \rightarrow {\pi }_{1}\left( {X,{x}_{0}}\right) \) defined by \( {\beta }_{h}\left\lbrack f\right\rbrack = \left\lbrack {h \cdot f \cdot \bar{h}}\right\rbrack \) is an isomorphism. | Proof: If \( {f}_{t} \) is a homotopy of loops based at \( {x}_{1} \) then \( h \cdot {f}_{t} \cdot \bar{h} \) is a homotopy of loops based at \( {x}_{0} \), so \( {\beta }_{h} \) is well-defined. Further, \( {\beta }_{h} \) is a homomorphism since \( {\beta }_{h}\left\lbrack {f \cdot g}\right\rbrack = \left\lbrack {h ... | Yes |
Proposition 1.6. A space \( X \) is simply-connected iff there is a unique homotopy class | Proof: Path-connectedness is the existence of paths connecting every pair of points, so we need be concerned only with the uniqueness of connecting paths. Suppose \( {\pi }_{1}\left( X\right) = 0 \) . If \( f \) and \( g \) are two paths from \( {x}_{0} \) to \( {x}_{1} \), then \( f \simeq f \cdot \bar{g} \cdot g \sim... | Yes |
Theorem 1.9. Every continuous map \( h : {D}^{2} \rightarrow {D}^{2} \) has a fixed point, that is, a point \( x \) with \( h\left( x\right) = x \) . | Proof: Suppose on the contrary that \( h\left( x\right) \neq x \) for all \( x \in {D}^{2} \) . Then we can define a map \( r : {D}^{2} \rightarrow {S}^{1} \) by letting \( r\left( x\right) \) be the point of \( {S}^{1} \) where the ray in \( {\mathbb{R}}^{2} \) starting at \( h\left( x\right) \) and passing through \(... | Yes |
Proposition 1.12. \( {\pi }_{1}\left( {X \times Y}\right) \) is isomorphic to \( {\pi }_{1}\left( X\right) \times {\pi }_{1}\left( Y\right) \) if \( X \) and \( Y \) are path-connected. | Proof: A basic property of the product topology is that a map \( f : Z \rightarrow X \times Y \) is continuous iff the maps \( g : Z \rightarrow X \) and \( h : Z \rightarrow Y \) defined by \( f\left( z\right) = \left( {g\left( z\right), h\left( z\right) }\right) \) are both continuous. Hence a loop \( f \) in \( X \t... | Yes |
For a point \( x \) in \( {\mathbb{R}}^{n} \), the complement \( {\mathbb{R}}^{n} - \{ x\} \) is homeomorphic to \( {S}^{n - 1} \times \mathbb{R} \) | so by Proposition 1.12 \( {\pi }_{1}\left( {{\mathbb{R}}^{n}-\{ x\} }\right) \) is isomorphic to \( {\pi }_{1}\left( {S}^{n - 1}\right) \times {\pi }_{1}\left( \mathbb{R}\right) \) . Hence \( {\pi }_{1}\left( {{\mathbb{R}}^{n}-\{ x\} }\right) \) is \( \mathbb{Z} \) for \( n = 2 \) and trivial for \( n > 2 \) | No |
Lemma 1.19. If \( {\varphi }_{t} : X \rightarrow Y \) is a homotopy and \( h \) is the path \( {\varphi }_{t}\left( {x}_{0}\right) \) formed by the images of a basepoint \( {x}_{0} \in X \), then the three maps in the diagram at the right satisfy \( {\varphi }_{0 * } = {\beta }_{h}{\varphi }_{1 * } \) . | Proof: Let \( {h}_{t} \) be the restriction of \( h \) to the interval \( \left\lbrack {0, t}\right\rbrack \) , with a reparametrization so that the domain of \( {h}_{t} \) is still \( \left\lbrack {0,1}\right\rbrack \) . Explicitly, we can take \( {h}_{t}\left( s\right) = h\left( {ts}\right) \) . Then if \( f \) is a ... | Yes |
Example 1.22. Let \( X \) be the graph shown in the figure, consisting of the twelve edges of a cube. The seven heavily shaded edges form a maximal tree \( T \subset X \), a contractible subgraph containing all the vertices of \( X \). We claim that \( {\pi }_{1}\left( X\right) \) is the free product of five copies of ... | To deduce this from van Kampen’s theorem, choose for each edge \( {e}_{\alpha } \) of \( X - T \) an open neighborhood \( {A}_{\alpha } \) of \( T \cup {e}_{\alpha } \) in \( X \) that deformation retracts onto \( T \cup {e}_{\alpha } \). The intersection of two or more \( {A}_{\alpha } \)’s deformation retracts onto \... | Yes |
Proposition 1.26. The inclusion \( X \hookrightarrow Y \) induces a surjection \( {\pi }_{1}\left( {X,{x}_{0}}\right) \rightarrow {\pi }_{1}\left( {Y,{x}_{0}}\right) \) whose kernel is \( N \) . Thus \( {\pi }_{1}\left( Y\right) \approx {\pi }_{1}\left( X\right) /N \) . | Proof: Let us expand \( Y \) to a slightly larger space \( Z \) that deformation retracts onto \( Y \) and is more convenient for applying van Kampen’s theorem. The space \( Z \) is obtained from \( Y \) by attaching rectangular strips \( {S}_{\alpha } = I \times I \), with the lower edge \( I \times \{ 0\} \) attached... | Yes |
Corollary 1.27. The surface \( {M}_{g} \) is not homeomorphic, or even homotopy equivalent, to \( {M}_{h} \) if \( g \neq h \) . | Proof: The abelianization of \( {\pi }_{1}\left( {M}_{g}\right) \) is the direct sum of \( {2g} \) copies of \( \mathbb{Z} \) . So if \( {M}_{g} \simeq {M}_{h} \) then \( {\pi }_{1}\left( {M}_{g}\right) \approx {\pi }_{1}\left( {M}_{h}\right) \), hence the abelianizations of these groups are isomorphic, which implies \... | Yes |
Corollary 1.28. For every group \( G \) there is a 2-dimensional cell complex \( {X}_{G} \) with | Proof: Choose a presentation \( G = \left\langle {{g}_{\alpha } \mid {r}_{\beta }}\right\rangle \) . This exists since every group is a quotient of a free group, so the \( {g}_{\alpha } \) ’s can be taken to be the generators of this free group with the \( {r}_{\beta } \) ’s generators of the kernel of the map from the... | Yes |
Proposition 1.30. Given a covering space \( p : \widetilde{X} \rightarrow X \), a homotopy \( {f}_{t} : Y \rightarrow X \), and a map \( {\widetilde{f}}_{0} : Y \rightarrow \widetilde{X} \) lifting \( {f}_{0} \), then there exists a unique homotopy \( {\widetilde{f}}_{t} : Y \rightarrow \widetilde{X} \) of \( {\widetil... | Proof: For the covering space \( p : \mathbb{R} \rightarrow {S}^{1} \) this is property (c) in the proof of Theorem 1.7 , and the proof there applies to any covering space. | No |
Proposition 1.31. The map \( {p}_{ * } : {\pi }_{1}\left( {\widetilde{X},{\widetilde{x}}_{0}}\right) \rightarrow {\pi }_{1}\left( {X,{x}_{0}}\right) \) induced by a covering space \( p : \left( {\widetilde{X},{\widetilde{x}}_{0}}\right) \rightarrow \left( {X,{x}_{0}}\right) \) is injective. The image subgroup \( {p}_{ ... | Proof: An element of the kernel of \( {p}_{ * } \) is represented by a loop \( {\widetilde{f}}_{0} : I \rightarrow \widetilde{X} \) with a homotopy \( {f}_{t} : I \rightarrow X \) of \( {f}_{0} = p{\widetilde{f}}_{0} \) to the trivial loop \( {f}_{1} \) . By the remarks preceding the proposition, there is a lifted homo... | Yes |
Proposition 1.32. The number of sheets of a covering space \( p : \left( {\widetilde{X},{\widetilde{x}}_{0}}\right) \rightarrow \left( {X,{x}_{0}}\right) \) \( \parallel \) with \( X \) and \( \widetilde{X} \) path-connected equals the index of \( {p}_{ * }\left( {{\pi }_{1}\left( {\widetilde{X},{\widetilde{x}}_{0}}\ri... | Proof: For a loop \( g \) in \( X \) based at \( {x}_{0} \), let \( \widetilde{g} \) be its lift to \( \widetilde{X} \) starting at \( {\widetilde{x}}_{0} \) . A product \( h \cdot g \) with \( \left\lbrack h\right\rbrack \in H = {p}_{ * }\left( {{\pi }_{1}\left( {\widetilde{X},{\widetilde{x}}_{0}}\right) }\right) \) h... | Yes |
Proposition 1.33. Suppose given a covering space \( p : \left( {\widetilde{X},{\widetilde{x}}_{0}}\right) \rightarrow \left( {X,{x}_{0}}\right) \) and a map \( f : \left( {Y,{y}_{0}}\right) \rightarrow \left( {X,{x}_{0}}\right) \) with \( Y \) path-connected and locally path-connected. Then a lift \( f : \left( {Y,{y}_... | Proof: The ’only if’ statement is obvious since \( {f}_{ * } = {p}_{ * }{\widetilde{f}}_{ * } \) . For the converse, let \( y \in Y \) and let \( y \) be a path in \( Y \) from \( {y}_{0} \) to \( y \) . The path \( {fy} \) in \( X \) starting at \( {x}_{0} \) has a unique lift \( \widetilde{fy} \) starting at \( {\wid... | Yes |
Proposition 1.34. Given a covering space \( p : \widetilde{X} \rightarrow X \) and a map \( f : Y \rightarrow X \) with two lifts \( {\widetilde{f}}_{1},{\widetilde{f}}_{2} : Y \rightarrow \widetilde{X} \) that agree at one point of \( Y \), then if \( Y \) is connected, these two lifts must agree on all of \( Y \) . | Proof: For a point \( y \in Y \), let \( U \) be an open neighborhood of \( f\left( y\right) \) in \( X \) for which \( {p}^{-1}\left( U\right) \) is a disjoint union of open sets \( {\widetilde{U}}_{\alpha } \) each mapped homeomorphically to \( U \) by \( p \), and let \( {\widetilde{U}}_{1} \) and \( {\widetilde{U}}... | Yes |
For integers \( m, n \geq 2 \), let \( {X}_{m, n} \) be the quotient space of a cylinder \( {S}^{1} \times I \) under the identifications \( \left( {z,0}\right) \sim \left( {{e}^{{2\pi i}/m}z,0}\right) \) and \( \left( {z,1}\right) \sim \left( {{e}^{{2\pi i}/n}z,1}\right) \) . Let \( A \subset X \) and \( B \subset X \... | The figure for Example 1.29 at the end of the preceding section\n\n\n\nshows what \( A \) looks like in the typical case \( m = 3 \) . We have \( {\pi }_{1}\left( A\right) \approx \mathbb{Z} \) , and the universal cove... | Yes |
Theorem 1.38. Let \( X \) be path-connected, locally path-connected, and semilocally simply-connected. Then there is a bijection between the set of basepoint-preserving isomorphism classes of path-connected covering spaces \( p : \left( {\widetilde{X},{\widetilde{x}}_{0}}\right) \rightarrow \left( {X,{x}_{0}}\right) \)... | Proof: It remains only to prove the last statement. We show that for a covering space \( p : \left( {\widetilde{X},{\widetilde{x}}_{0}}\right) \rightarrow \left( {X,{x}_{0}}\right) \), changing the basepoint \( {\widetilde{x}}_{0} \) within \( {p}^{-1}\left( {x}_{0}\right) \) corresponds exactly to changing \( {p}_{ * ... | Yes |
Proposition 1.39. Let \( p : \left( {\widetilde{X},{\widetilde{x}}_{0}}\right) \rightarrow \left( {X,{x}_{0}}\right) \) be a path-connected covering space of the path-connected, locally path-connected space \( X \), and let \( H \) be the subgroup \( {p}_{ * }\left( {{\pi }_{1}\left( {\widetilde{X},{\widetilde{x}}_{0}}... | Proof: We observed earlier in the proof of the classification theorem that changing the basepoint \( {\widetilde{x}}_{0} \in {p}^{-1}\left( {x}_{0}\right) \) to \( {\widetilde{x}}_{1} \in {p}^{-1}\left( {x}_{0}\right) \) corresponds precisely to conjugating \( H \) by an element \( \left\lbrack y\right\rbrack \in {\pi ... | Yes |
Proposition 1.40. If an action of a group \( G \) on a space \( Y \) satisfies \( \left( *\right) \), then:\n\n(a) The quotient map \( p : Y \rightarrow Y/G, p\left( y\right) = {Gy} \), is a normal covering space.\n\n(b) \( G \) is the group of deck transformations of this covering space \( Y \rightarrow Y/G \) if \( Y... | Proof: Given an open set \( U \subset Y \) as in condition \( \left( *\right) \), the quotient map \( p \) simply identifies all the disjoint homeomorphic sets \( \{ g\left( U\right) \mid g \in G\} \) to a single open set \( p\left( U\right) \) in \( Y/G \) . By the definition of the quotient topology on \( Y/G, p \) r... | Yes |
In Example 1.35 we constructed a contractible 2-complex \( {\widetilde{X}}_{m, n} = \) \( {T}_{m, n} \times \mathbb{R} \) as the universal cover of a finite 2-complex \( {X}_{m, n} \) that was the union of the mapping cylinders of the two maps \( {S}^{1} \rightarrow {S}^{1}, z \mapsto {z}^{m} \) and \( z \mapsto {z}^{n... | We described a decomposition of \( {\widetilde{X}}_{m, n} \) into rectangles, with \( {X}_{m, n} \) the quotient of one rectangle. These rectangles in fact define a cell structure on \( {\widetilde{X}}_{m, n} \) lifting a cell structure on \( {X}_{m, n} \) with two vertices, three edges, and one 2-cell. The group \( {G... | Yes |
For \( G = {\mathbb{Z}}_{2} = \left\langle {x \mid {x}^{2}}\right\rangle ,{X}_{G} \) is \( {\mathbb{{RP}}}^{2} \) and \( {\widetilde{X}}_{G} = {S}^{2} \). | More generally, for \( {\mathbb{Z}}_{n} = \left\langle {x \mid {x}^{n}}\right\rangle ,{X}_{G} \) is \( {S}^{1} \) with a disk attached by the map \( z \mapsto {z}^{n} \) and \( {\widetilde{X}}_{G} \) consists of \( n \) disks \( {D}_{1},\cdots ,{D}_{n} \) with their boundary circles identified. A generator of \( {\math... | Yes |
Example 1.48. If \( G = {\mathbb{Z}}_{2} * {\mathbb{Z}}_{2} = \left\langle {a, b \mid {a}^{2},{b}^{2}}\right\rangle \) then the Cayley graph is a union of an infinite sequence of circles each tangent to its two neighbors. | We obtain \( {\widetilde{X}}_{G} \) from this graph by making each circle the equator of a 2-sphere, yielding an infinite sequence of tangent 2-spheres. Elements of the index-two normal subgroup \( \mathbb{Z} \subset {\mathbb{Z}}_{2} * {\mathbb{Z}}_{2} \) generated by \( {ab} \) act on \( {\widetilde{X}}_{G} \) as tran... | Yes |
Lemma 2.1. The composition \( {\Delta }_{n}\left( X\right) \overset{{\partial }_{n}}{ \rightarrow }{\Delta }_{n - 1}\left( X\right) \overset{{\partial }_{n - 1}}{ \rightarrow }{\Delta }_{n - 2}\left( X\right) \) is zero. | Proof: We have \( {\partial }_{n}\left( \sigma \right) = \mathop{\sum }\limits_{i}{\left( -1\right) }^{i}\sigma \mid \left\lbrack {{v}_{0},\cdots ,{\widehat{v}}_{i},\cdots ,{v}_{n}}\right\rbrack \), and hence\n\n\[{\partial }_{n - 1}{\partial }_{n}\left( \sigma \right) = \mathop{\sum }\limits_{{j < i}}{\left( -1\right)... | Yes |
Example 2.2. \( X = {S}^{1} \), with one vertex \( v \) and one edge \( e \) . Then \( {\Delta }_{0}\left( {S}^{1}\right) \) | and \( {\Delta }_{1}\left( {S}^{1}\right) \) are both \( \mathbb{Z} \) and the boundary map \( {\partial }_{1} \) is zero since \( \partial e = v - v \) . The groups \( {\Delta }_{n}\left( {S}^{1}\right) \) are 0 for \( n \geq 2 \) since there are no simplices in these dimensions. Hence \[ {H}_{n}^{\Delta }\left( {S}^{... | Yes |
Example 2.3. \( X = T \), the torus with the \( \Delta \) -complex structure pictured earlier, having one vertex, three edges \( a, b \), and \( c \), and two 2-simplices \( U \) and \( L \) . As in the previous example, \( {\partial }_{1} = 0 \) so \( {H}_{0}^{\Delta }\left( T\right) \approx \mathbb{Z} \) . Since \( {... | Thus \[ {H}_{n}^{\Delta }\left( T\right) \approx \left\{ \begin{array}{ll} \mathbb{Z} \oplus \mathbb{Z} & \text{ for }n = 1 \\ \mathbb{Z} & \text{ for }n = 0,2 \\ 0 & \text{ for }n \geq 3 \end{array}\right. \] | Yes |
Example 2.4. \( X = {\mathbb{{RP}}}^{2} \), as pictured earlier, with two vertices \( v \) and \( w \), three edges \( a, b \), and \( c \), and two 2-simplices \( U \) and \( L \) . Then \( \operatorname{Im}{\partial }_{1} \) is generated by \( w - v \), so \( {H}_{0}^{\Delta }\left( X\right) \approx \mathbb{Z} \) wit... | Further, \( \operatorname{Ker}{\partial }_{1} \approx \mathbb{Z} \oplus \mathbb{Z} \) with basis \( a - b \) and \( c \), and \( \operatorname{Im}{\partial }_{2} \) is an index-two subgroup of \( \operatorname{Ker}{\partial }_{1} \) since we can choose \( c \) and \( a - b + c \) as a basis for \( \operatorname{Ker}{\p... | Yes |
Proposition 2.6. Corresponding to the decomposition of a space \( X \) into its path- \( \parallel \) components \( {X}_{\alpha } \) there is an isomorphism of \( {H}_{n}\left( X\right) \) with the direct sum \( { \oplus }_{\alpha }{H}_{n}\left( {X}_{\alpha }\right) \) . | Proof: Since a singular simplex always has path-connected image, \( {C}_{n}\left( X\right) \) splits as the direct sum of its subgroups \( {C}_{n}\left( {X}_{\alpha }\right) \) . The boundary maps \( {\partial }_{n} \) preserve this direct sum decomposition, taking \( {C}_{n}\left( {X}_{\alpha }\right) \) to \( {C}_{n ... | Yes |
Proposition 2.7. If \( X \) is nonempty and path-connected, then \( {H}_{0}\left( X\right) \approx \mathbb{Z} \) . Hence for \( \parallel \) any space \( X,{H}_{0}\left( X\right) \) is a direct sum of \( \mathbb{Z} \) ’s, one for each path-component of \( X \) . | Proof: By definition, \( {H}_{0}\left( X\right) = {C}_{0}\left( X\right) /\operatorname{Im}{\partial }_{1} \) since \( {\partial }_{0} = 0 \) . Define a homomorphism \( \varepsilon : {C}_{0}\left( X\right) \rightarrow \mathbb{Z} \) by \( \varepsilon \left( {\mathop{\sum }\limits_{i}{n}_{i}{\sigma }_{i}}\right) = \matho... | Yes |
Proposition 2.8. If \( X \) is a point, then \( {H}_{n}\left( X\right) = 0 \) for \( n > 0 \) and \( {H}_{0}\left( X\right) \approx \mathbb{Z} \) . | Proof: In this case there is a unique singular \( n \) -simplex \( {\sigma }_{n} \) for each \( n \), and \( \partial \left( {\sigma }_{n}\right) = \) \( \mathop{\sum }\limits_{i}{\left( -1\right) }^{i}{\sigma }_{n - 1} \), a sum of \( n + 1 \) terms, which is therefore 0 for \( n \) odd and \( {\sigma }_{n - 1} \) for... | Yes |
Theorem 2.13. If \( X \) is a space and \( A \) is a nonempty closed subspace that is a deformation retract of some neighborhood in \( X \), then there is an exact sequence\n\n\[ \cdots \rightarrow {\widetilde{H}}_{n}\left( A\right) \overset{{i}_{ * }}{ \rightarrow }{\widetilde{H}}_{n}\left( X\right) \overset{{j}_{ * }... | The map \( \partial \) will be constructed in the course of the proof. The idea is that an element \( x \in {\widetilde{H}}_{n}\left( {X/A}\right) \) can be represented by a chain \( \alpha \) in \( X \) with \( \partial \alpha \) a cycle in \( A \) whose homology class is \( \partial x \in {\widetilde{H}}_{n - 1}\left... | Yes |
In the long exact sequence of reduced homology groups for the pair \( \left( {{D}^{n},\partial {D}^{n}}\right) \), the maps \( {H}_{i}\left( {{D}^{n},\partial {D}^{n}}\right) \overset{\partial }{ \rightarrow }{\widetilde{H}}_{i - 1}\left( {S}^{n - 1}\right) \) are isomorphisms for all \( i > 0 \) since the remaining te... | Thus we obtain the calculation\n\n\[ \n{H}_{i}\left( {{D}^{n},\partial {D}^{n}}\right) \approx \left\{ \begin{array}{ll} \mathbb{Z} & \text{ for }i = n \\ 0 & \text{ otherwise } \end{array}\right.\n\] | Yes |
Proposition 2.22. For good pairs \( \left( {X, A}\right) \), the quotient map \( q : \left( {X, A}\right) \rightarrow \left( {X/A, A/A}\right) \) induces isomorphisms \( {q}_{ * } : {H}_{n}\left( {X, A}\right) \rightarrow {H}_{n}\left( {X/A, A/A}\right) \approx {\widetilde{H}}_{n}\left( {X/A}\right) \) for all \( n \) ... | Proof: Let \( V \) be a neighborhood of \( A \) in \( X \) that deformation retracts onto \( A \) . We have a commutative diagram\n\n\[ \n{H}_{n}\left( {X/A, A/A}\right) \rightarrow {H}_{n}\left( {X/A, V/A}\right) \leftarrow {H}_{n}\left( {X/A - A/A, V/A - A/A}\right) \]\n\nThe upper left horizontal map is an isomorphi... | Yes |
Corollary 2.24. If the CW complex \( X \) is the union of subcomplexes \( A \) and \( B \), then the inclusion \( \left( {B, A \cap B}\right) \hookrightarrow \left( {X, A}\right) \) induces isomorphisms \( {H}_{n}\left( {B, A \cap B}\right) \rightarrow {H}_{n}\left( {X, A}\right) \) for all \( n \) . | Proof: Since CW pairs are good, Proposition 2.22 allows us to pass to the quotient spaces \( B/\left( {A \cap B}\right) \) and \( X/A \) which are homeomorphic, assuming we are not in the trivial case \( A \cap B = \varnothing \) . | No |
Corollary 2.25. For a wedge sum \( { \vee }_{\alpha }{X}_{\alpha } \), the inclusions \( {i}_{\alpha } : {X}_{\alpha } \hookrightarrow { \vee }_{\alpha }{X}_{\alpha } \) induce an isomorphism \( { \oplus }_{\alpha }{i}_{\alpha * } : { \oplus }_{\alpha }{\widetilde{H}}_{n}\left( {X}_{\alpha }\right) \rightarrow {\wideti... | Proof: Since reduced homology is the same as homology relative to a basepoint, this follows from the proposition by taking \( \left( {X, A}\right) = \left( {\mathop{\coprod }\limits_{\alpha }{X}_{\alpha },\mathop{\coprod }\limits_{\alpha }\left\{ {x}_{\alpha }\right\} }\right) \) . | Yes |
Theorem 2.26. If nonempty open sets \( U \subset {\mathbb{R}}^{m} \) and \( V \subset {\mathbb{R}}^{n} \) are homeomorphic, I then \( m = n \) . | Proof: For \( x \in U \) we have \( {H}_{k}\left( {U, U-\{ x\} }\right) \approx {H}_{k}\left( {{\mathbb{R}}^{m},{\mathbb{R}}^{m}-\{ x\} }\right) \) by excision. From the long exact sequence for the pair \( \left( {{\mathbb{R}}^{m},{\mathbb{R}}^{m}-\{ x\} }\right) \) we get \( {H}_{k}\left( {{\mathbb{R}}^{m},{\mathbb{R}... | Yes |
Proposition 2.29. \( {\mathbb{Z}}_{2} \) is the only nontrivial group that can act freely on \( {S}^{n} \) if \( n \) is even. | Proof: Since the degree of a homeomorphism must be \( \pm 1 \), an action of a group \( G \) on \( {S}^{n} \) determines a degree function \( d : G \rightarrow \{ \pm 1\} \) . This is a homomorphism since \( \deg {fg} = \deg f\deg g \) . If the action is free, then \( d \) sends every nontrivial element of \( G \) to \... | Yes |
We can use this result to construct a map \( {S}^{n} \rightarrow {S}^{n} \) of any given degree, for each \( n \geq 1 \) . | Let \( q : {S}^{n} \rightarrow { \vee }_{k}{S}^{n} \) be the quotient map obtained by collapsing the complement of \( k \) disjoint open balls \( {B}_{i} \) in \( {S}^{n} \) to a point, and let \( p : {\bigvee }_{k}{S}^{n} \rightarrow {S}^{n} \) identify all the summands to a single sphere. Consider the composition \( ... | Yes |
In the case of \( {S}^{1} \), the map \( f\left( z\right) = {z}^{k} \), where we view \( {S}^{1} \) as the unit circle in \( \mathbb{C} \), has degree \( k \). | This is evident in the case \( k = 0 \) since \( f \) is then constant. The case \( k < 0 \) reduces to the case \( k > 0 \) by composing with \( z \mapsto {z}^{-1} \), which is a reflection, of degree -1 . To compute the degree when \( k > 0 \), observe first that for any \( y \in {S}^{1},{f}^{-1}\left( y\right) \) co... | Yes |
Proposition 2.33. \( \deg {Sf} = \deg f \), where \( {Sf} : {S}^{n + 1} \rightarrow {S}^{n + 1} \) is the suspension of the \( \parallel \operatorname{map}f : {S}^{n} \rightarrow {S}^{n} \) . | Proof: Let \( C{S}^{n} \) denote the cone \( \left( {{S}^{n} \times I}\right) /\left( {{S}^{n} \times 1}\right) \) with base \( {S}^{n} = {S}^{n} \times 0 \subset C{S}^{n} \) , so \( C{S}^{n}/{S}^{n} \) is the suspension of \( {S}^{n} \) . The map \( f \) induces \( {Cf} : \left( {C{S}^{n},{S}^{n}}\right) \rightarrow \... | Yes |
Lemma 2.34. If \( X \) is a \( {CW} \) complex, then:\n\n(a) \( {H}_{k}\left( {{X}^{n},{X}^{n - 1}}\right) \) is zero for \( k \neq n \) and is free abelian for \( k = n \), with a basis in one-to-one correspondence with the \( n \) -cells of \( X \) .\n\n(b) \( {H}_{k}\left( {X}^{n}\right) = 0 \) for \( k > n \) . In ... | Proof: Statement (a) follows immediately from the observation that \( \left( {{X}^{n},{X}^{n - 1}}\right) \) is a good pair and \( {X}^{n}/{X}^{n - 1} \) is a wedge sum of \( n \) -spheres, one for each \( n \) -cell of \( X \) . Here we are using Proposition 2.22 and Corollary 2.25.\n\nTo prove (b), consider the long ... | Yes |
Theorem 2.35. \( {H}_{n}^{CW}\left( X\right) \approx {H}_{n}\left( X\right) \) . | Proof: From the diagram above, \( {H}_{n}\left( X\right) \) can be identified with \( {H}_{n}\left( {X}^{n}\right) /\operatorname{Im}{\partial }_{n + 1} \) . Since \( {j}_{n} \) is injective, it maps \( \operatorname{Im}{\partial }_{n + 1} \) isomorphically onto \( \operatorname{Im}\left( {{j}_{n}{\partial }_{n + 1}}\r... | Yes |
Let \( {M}_{g} \) be the closed orientable surface of genus \( g \) with its usual CW structure consisting of one \( 0 - \mathrm{{cell}},{2g1} - \mathrm{{cells}} \), and one 2-cell attached by the product of commutators \( \left\lbrack {{a}_{1},{b}_{1}}\right\rbrack \cdots \left\lbrack {{a}_{g},{b}_{g}}\right\rbrack \)... | As observed above, \( {d}_{1} \) must be 0 since there is only one 0 -cell. Also, \( {d}_{2} \) is 0 because each \( {a}_{i} \) or \( {b}_{i} \) appears with its inverse in \( \left\lbrack {{a}_{1},{b}_{1}}\right\rbrack \cdots \left\lbrack {{a}_{g},{b}_{g}}\right\rbrack \), so the maps \( {\Delta }_{\alpha \beta } \) a... | Yes |
The closed nonorientable surface \( {N}_{g} \) of genus \( g \) has a cell structure with one \( 0 - \mathrm{{cell}},{g1} - \mathrm{{cells}} \), and one 2-cell attached by the word \( {a}_{1}^{2}{a}_{2}^{2}\cdots {a}_{g}^{2} \). | Again \( {d}_{1} = 0 \), and \( {d}_{2} : \mathbb{Z} \rightarrow {\mathbb{Z}}^{g} \) is specified by the equation \( {d}_{2}\left( 1\right) = \left( {2,\cdots ,2}\right) \) since each \( {a}_{i} \) appears in the attaching word of the 2-cell with total exponent 2 , which means that each \( {\Delta }_{\alpha \beta } \) ... | Yes |
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 three 2-cells have degree zero. | Each \( {\Delta }_{\alpha \beta } \) maps the interiors of two opposite faces of the cube homeomorphically onto the complement of a point in the target \( {S}^{2} \) and sends the remaining four faces to this point. Computing local degrees at the center points of the two opposite faces, we see that the local degree is ... | 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\[ 0 \rightarrow {C}_{k}\overset{{d}_{k}}{ \rightarrow }{C}_{k - 1} \rightarrow \cdots \rightarrow {C}_{1}\overset{{d}_{1}}{ \rightarrow }{C}_{0} \rightarrow 0 \]\n\nbe a chain complex of finitely generated abelian groups, with cycles \( {Z}_{n} = \operatorname{Ker}{d}_{n... | 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}{ \rig... | Hence if \( F \) has characteristic 2, for example if \( F = {\mathbb{Z}}_{2} \), then \( {H}_{k}\left( {{\mathbb{{RP}}}^{n};F}\right) \approx F \) for \( 0 \leq k \leq n \), a more uniform answer than with \( \mathbb{Z} \) coefficients. On the other hand, if \( F \) has characteristic different from 2 then the boundar... | Yes |
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 \( \mathbb{Z} \) coefficients sin... | \[ 0 = {\widetilde{H}}_{n + 1}\left( {{S}^{n};{\mathbb{Z}}_{m}}\right) \rightarrow {\widetilde{H}}_{n + 1}\left( {X;{\mathbb{Z}}_{m}}\right) \overset{{f}_{ * }}{ \rightarrow }{\widetilde{H}}_{n + 1}\left( {X/{S}^{n};{\mathbb{Z}}_{m}}\right) \] 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 \( {H}^{n}\left( {X;\mathbb{Z}}\right) \) consis... | 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... | Here we take reduced cohomology to be cohomology relative to a basepoint, and we use relative cup products. We should assume the basepoints \( {x}_{\alpha } \in {X}_{\alpha } \) are deformation retracts of neighborhoods, to be sure that the claimed isomorphism indeed holds. | 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 |
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 If \( {H}^{k}\left( ... | 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 a point... | 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 |
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