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Lemma 13.2. Let \( \mathcal{C} \) and \( {\mathcal{C}}^{\prime } \) be non-negative chain complexes. Let \( \phi : \mathcal{C} \rightarrow \) \( {\mathcal{C}}^{\prime } \) be a chain equivalence with chain-homotopy inverse \( {\phi }^{\prime } \) . Suppose \( \mathcal{C} \) and \( {\mathcal{C}}^{\prime } \) are augment... | Proof. If \( {D}^{\prime } \) is a chain homotopy between \( \phi \circ {\phi }^{\prime } \) and the identity, then in dimension 0 ,\n\n\[ \n{\partial }^{\prime }{D}^{\prime }{c}_{0}^{\prime } = \phi {\phi }^{\prime }\left( {c}_{0}^{\prime }\right) - {c}_{0}^{\prime }, \n\]\n\nso\n\n\[ \n0 = {\epsilon }^{\prime }\left(... | Yes |
Theorem 13.4 (Acyclic carrier theorem, algebraic version). Let \( \mathcal{O} \) and \( {\mathcal{O}}^{\prime } \) be augmented chain complexes; let \( \mathcal{O} \) be free. Let \( \Phi \) be an acyclic carrier from \( \mathcal{C} \) to \( {\mathcal{C}}^{\prime } \), relative to some set of preferred bases for \( \ma... | Proof. The proof of this theorem is just a jazzed-up version of the preceding proof. The requirement that the restriction of \( {\epsilon }^{\prime } \) give an augmentation for \( \Phi \left( {\sigma }_{p}^{\alpha }\right) \) means that \( {\epsilon }^{\prime } \) must map the 0 -dimensional group of this chain comple... | No |
Lemma 13.5. If \( w * K \) is a cone over the complex \( K \), then \( w * K \) is acyclic in ordered homology. | Proof. Define\n\n\[ D : {C}_{p}^{\prime }\left( {w * K}\right) \rightarrow {C}_{p + 1}^{\prime }\left( {w * K}\right) \]\n\nfor \( p \geq 0 \) by the equation\n\n\[ D\left( \left( {{v}_{0},\ldots ,{v}_{p}}\right) \right) = \left( {w,{v}_{0},\ldots ,{v}_{p}}\right) . \]\n\nNote that it is irrelevant here whether any of ... | Yes |
Theorem 13.6. Choose a partial ordering of the vertices of \( K \) that induces a linear ordering on the vertices of each simplex of \( K \) . Define \( \phi : {C}_{p}\left( K\right) \rightarrow \) \( {C}_{p}^{\prime }\left( K\right) \) by letting\n\n\[ \n\phi \left( \left\lbrack {{v}_{0},\ldots ,{v}_{p}}\right\rbrack ... | The proof is an application of the acyclic carrier theorem; it is left as an exercise. | No |
Theorem 13.7. Let \( f : \left( {K,{K}_{0}}\right) \rightarrow \left( {L,{L}_{0}}\right) \) be a simplicial map. Let \( \phi \) and \( \psi \) be as in the preceding theorem. Then the following diagram commutes: \n\nSi... | Proof. One checks directly from the definition that \( {f}_{y} \circ \psi = \psi \circ {f}_{y}^{\prime } \) . Thus the diagram already commutes on the chain level, so it commutes on the homology level as well. It is not true that \( \phi \circ {f}_{\# } = {f}_{\# }^{\prime } \circ \phi \), since \( \phi \) depends on a... | Yes |
Lemma 14.1. Let \( h : \left| K\right| \rightarrow \left| L\right| \) satisfy the star condition with respect to \( K \) and \( L \) . Choose \( f : {K}^{\left( 0\right) } \rightarrow {L}^{\left( 0\right) } \) so that for each vertex \( v \) of \( K \), \[ h\left( {\operatorname{St}v}\right) \subset \operatorname{St}f\... | Proof. (a) Let \( \sigma = {v}_{0}\ldots {v}_{p} \) . Then \( x \in \) St \( {v}_{i} \) for each \( i \), so \[ h\left( x\right) \in h\left( {\operatorname{St}{v}_{i}}\right) \subset \operatorname{St}f\left( {v}_{i}\right) . \] This means that \( h\left( x\right) \) has a positive barycentric coordinate with respect to... | Yes |
Lemma 14.2. Let \( f : K \rightarrow L \) be a simplicial approximation to \( h : \left| K\right| \rightarrow \) \( \left| L\right| \) . Given \( x \in \left| K\right| \), there is a simplex \( \tau \) of \( L \) such that \( h\left( x\right) \in \operatorname{Int}\tau \) and \( f\left( x\right) \in \tau \) . | Proof. This follows immediately from (a) of the preceding lemma. | No |
Theorem 14.3. Let \( h : \left| K\right| \rightarrow \left| L\right| \) and \( k : \left| L\right| \rightarrow \left| M\right| \) have simplicial approximations \( f : K \rightarrow L \) and \( g : L \rightarrow M \), respectively. Then \( g \circ f \) is a simplicial approximation to \( k \circ h \) . | Proof. We know \( g \circ f \) is a simplicial map. If \( v \) is a vertex of \( K \), then\n\n\[ h\left( {\operatorname{St}v}\right) \subset \operatorname{St}f\left( v\right) \]\n\n because \( f \) is a simplicial approximation to \( h \) . It follows that\n\n\[ k\left( {h\left( {\mathrm{{St}}v}\right) }\right) \subse... | Yes |
Let \( K \) and \( L \) be the complexes pictured in Figure 14.1, whose underlying spaces are homeomorphic to the circle and to the annulus, respectively. Let \( {K}^{\prime } \) be the complex obtained from \( K \) by inserting extra vertices, as pictured. Let \( h \) be the indicated continuous map, where we denote \... | Now \( h \) does not satisfy the star condition relative to \( K \) and \( L \), but it does satisfy the star condition relative to \( {K}^{\prime } \) and \( L \) . Hence \( h \) has a simplicial approximation \( f : {K}^{\prime } \rightarrow L \) . One such is pictured; we denote \( f\left( a\right) \) by \( {A}^{\pr... | Yes |
Lemma 14.4. Let \( h : \left| K\right| \rightarrow \left| L\right| \) satisfy the star condition relative to \( K \) and \( L \) ; suppose \( h \) maps \( \left| {K}_{0}\right| \) into \( \left| {L}_{0}\right| \) .\n\n(a) Any simplicial approximation \( f : K \rightarrow L \) to \( h \) also maps \( \left| {K}_{0}\righ... | Proof. Let \( f, g \) be simplicial approximations to \( h \) . Given \( \sigma \in {K}_{0} \), choose \( x \in \operatorname{Int}\sigma \), and let \( \tau \) be the simplex of \( L \) such that \( h\left( x\right) \in \operatorname{Int}\tau \) . Because \( h \) maps \( \left| {K}_{0}\right| \) into \( \left| {L}_{0}\... | Yes |
Lemma 15.1. Let \( {K}^{\prime } \) be a subdivision of \( K \) . Then for each vertex \( w \) of \( {K}^{\prime } \) , there is a vertex \( v \) of \( K \) such that\n\n\[ \operatorname{St}\left( {w,{K}^{\prime }}\right) \subset \operatorname{St}\left( {v, K}\right) . \]\n\nIndeed, if \( \sigma \) is the simplex of \(... | Proof. If this inclusion holds, then since \( w \) belongs to \( \operatorname{St}\left( {w,{K}^{\prime }}\right), w \) must lie in some open simplex of \( K \) that has \( v \) as a vertex.\n\nConversely, suppose \( w \in \) Int \( \sigma \) and \( v \) is a vertex of \( \sigma \) . It suffices to show that\n\n\[ \lef... | Yes |
Lemma 15.2. If \( K \) is a complex, then the intersection of any collection of subcomplexes of \( K \) is a subcomplex of \( K \) . Conversely, if \( \left\{ {K}_{\alpha }\right\} \) is a collection of complexes in \( {\mathbf{E}}^{J} \), and if the intersection of every pair \( \left| {K}_{\alpha }\right| \cap \left|... | To verify that \( {L}_{p + 1} \) is a complex, we note that\n\n\[ \left| {{w}_{\sigma } * {L}_{\sigma }}\right| \cap \left| {L}_{p}\right| = \operatorname{Bd}\sigma \]\n\nwhich is the polytope of the subcomplex \( {L}_{a} \) of both \( {w}_{a} * {L}_{a} \) and \( {L}_{p} \) . Similarly, if \( \tau \) is another \( p + ... | No |
Lemma 15.3. The complex \( \operatorname{sd}K \) equals the collection of all simplices’ of the form\n\n\[ \n{\widehat{\sigma }}_{1}{\widehat{\sigma }}_{2}\ldots {\widehat{\sigma }}_{n} \n\]\n\nwhere \( {\sigma }_{1} \succ {\sigma }_{2} \succ \ldots \succ {\sigma }_{n} \) . | Proof. We prove this fact by induction. It is immediate that the simplices of sd \( K \) lying in the subdivision of \( {K}^{\left( 0\right) } \) are of this form. (Each such simplex is a vertex of \( K \), and \( \widehat{v} = v \) for a vertex.)\n\nSuppose now that each simplex of sd \( K \) lying in \( \left| {K}^{\... | Yes |
Theorem 16.1 (The finite simplicial approximation theorem). Let \( K \) and \( L \) be complexes; let \( K \) be finite. Given a continuous map \( h : \left| K\right| \rightarrow \left| L\right| \), there is an \( N \) such that \( h \) has a simplicial approximation \( f : {\operatorname{sd}}^{N}K \rightarrow L \) . | Proof. Cover \( \left| K\right| \) by the open sets \( {h}^{-1}\left( {\mathrm{{St}}w}\right) \), as \( w \) ranges over the vertices of \( L \) . Now given this open covering \( \mathcal{A} \) of the compact metric space \( K \), there is a number \( \lambda \) such that any set of diameter less than \( \lambda \) lie... | Yes |
Lemma 16.2. Let \( {K}_{0} \) be a subcomplex of \( K \) . (a) If \( \tau \) is a simplex of \( \operatorname{sd}\left( {K/{K}_{0}}\right) \), then \( \tau \) is of the form \[ \tau = {\widehat{\sigma }}_{1}\ldots {\widehat{\sigma }}_{q}{v}_{0}\ldots {v}_{p} \] where \( s = {v}_{0}\ldots {v}_{p} \) is a simplex of \( {... | Proof. (a) The result is true if \( \tau \) is in \( {J}_{0} \) . In general, let \( \tau \) be a simplex of \( {J}_{p + 1} \) not in \( {J}_{p} \) . Then either \( \tau \) belongs to \( {K}_{0} \), in which case \( \tau \) is of the form \( {v}_{0}\ldots {v}_{r} \), or \( \tau \) belongs to one of the cones \( \wideha... | Yes |
Theorem 16.5 (The general simplicial approximation theorem). Let \( K \) and \( L \) be complexes; let \( h : \\left| K\\right| \\rightarrow \\left| L\\right| \) be a continuous map. There exists a subdivision \( {K}^{\\prime } \) of \( K \) such that \( h \) has a simplicial approximation \( f : {K}^{\\prime } \\right... | Proof. Let \( \\mathcal{A} \) be the covering of \( \\left| K\\right| \) by the open sets \( {h}^{-1}\\left( {\\operatorname{St}\\left( {w, L}\\right) }\\right) \), as \( w \) ranges over the vertices of \( L \) . Choose a subdivision \( {K}^{\\prime } \) of \( K \) whose closed stars refine \( \\mathcal{A} \) . Then \... | Yes |
Lemma 17.1. Let \( {K}^{\prime } \) be a subdivision of \( K \) . Then the identity map \( i : \left| K\right| \rightarrow \left| K\right| \) has a simplicial approximation\n\n\[ g : {K}^{\prime } \rightarrow K\text{.} \]\n\nLet \( \tau \) be a simplex of \( {K}^{\prime } \) and let \( \sigma \) be a simplex of \( K \)... | Proof. By Lemma 15.1, the map \( i \) has a simplicial approximation \( g \) . Given \( \tau \subset \sigma \), let \( w \) be a vertex of \( \tau \) . Then \( w \) lies interior to \( \sigma \) or to a face of \( \sigma \) . Then \( g \) maps \( w \) to a vertex of \( \sigma \), by Lemma 14.1. | No |
Theorem 17.3. Let \( {K}_{0} \) be a subcomplex of \( K \) . Given the subdivision \( {K}^{\prime } \) of \( K \), let \( {K}_{0}^{\prime } \) denote the induced subdivision of \( {K}_{0} \) . The subdivision operator \( \lambda \) induces a chain map\n\n\[ \lambda : {C}_{p}\left( {K,{K}_{0}}\right) \rightarrow {C}_{p}... | Proof. We check that each of the acyclic carriers defined in Step 1 of the preceding proof preserves the subcomplexes involved. Certainly if \( \sigma \in {K}_{0} \), then \( \Psi \left( \sigma \right) = K\left( \sigma \right) \) and \( \Lambda \left( \sigma \right) = {K}^{\prime }\left( \sigma \right) \) are subcomple... | Yes |
Theorem 18.1 (The functorial properties). The identity map \( i : \left| K\right| \rightarrow \left| K\right| \) induces the identity homomorphism \( {i}_{ * } : {H}_{p}\left( K\right) \rightarrow {H}_{p}\left( K\right) \) . If \( h : \left| K\right| \rightarrow \left| L\right| \) and \( k : \left| L\right| \rightarrow... | Proof. That \( {i}_{ * } \) is the identity is immediate from the definition. To check the second statement, choose \( {f}_{0} : {L}^{\prime } \rightarrow M \) and \( {g}_{0} : {L}^{\prime } \rightarrow L \) as simplicial approximations to \( k \) and \( {i}_{\left| l\right| } \), respectively. Then choose \( {f}_{1} :... | Yes |
Corollary 18.2 (Topological invariance of homology groups). If \( h : \left| K\right| \rightarrow \left| L\right| \) is a homeomorphism, then \( {h}_{ * } : {H}_{p}\left( K\right) \rightarrow {H}_{p}\left( L\right) \) is an isomorphism. The same result holds for reduced homology. | Proof. Let \( k : \left| L\right| \rightarrow \left| K\right| \) be the inverse of \( h \) . Then \( {h}_{ * } \circ {k}_{ * } \) equals \( {\left( {i}_{\left| L\right| }\right) }_{ * } \) and \( {k}_{ * } \circ {h}_{ * } \) equals \( {\left( {i}_{\left| K\right| }\right) }_{ * } \) . Thus \( {h}_{ * } \circ {k}_{ * } ... | Yes |
If \( K \) is the complex consisting of a 1-simplex and its faces, then \( \left| K\right| \times I \) is by the procedure of the preceding lemma subdivided into the complex pictured in Figure 19.2. |  | No |
Theorem 19.2. If \( h, k : \left| K\right| \rightarrow \left| L\right| \) are homotopic, then \( {h}_{ * },{k}_{ * } : {H}_{p}\left( K\right) \rightarrow \) \( {H}_{p}\left( L\right) \) are equal. The same holds for reduced homology. | Proof. Let \( K \) be a complex. Let \( M \) be a complex whose underlying space is \( \left| K\right| \times I \), such that for each \( \sigma \in K \), both \( \sigma \times 0 \) and \( \sigma \times 1 \) are simplices of \( M \) , and \( \sigma \times I \) is the polytope of a subcomplex of \( M \) . Let \( F : \le... | Yes |
Theorem 19.3. If \( h \) and \( k \) are homotopic as maps of pairs of spaces, then \( {h}_{ * } = {k}_{ * } \) as maps of relative homology groups. | Proof. The proof of Theorem 19.2 goes through without difficulty. Both \( i \) and \( j \) carry \( \left| {K}_{0}\right| \) into \( \left| {K}_{0}\right| \times I \), and so does the chain homotopy connecting \( {i}_{ji} \) and \( {j}_{ * } \) . Then \( {i}_{ * } = {j}_{ * } \) as maps of relative homology, and the pr... | No |
Theorem 19.4. If \( f : K \rightarrow L \) is a simplicial approximation to the continuous map \( h : \left| K\right| \rightarrow \left| L\right| \), then \( f \) is homotopic to \( h \) . | Proof. For each \( x \) in \( \left| K\right| \), we know from Lemma 14.2 that \( f\left( x\right) \) and \( h\left( x\right) \) lie in a single simplex of \( L \) . Therefore, the \ | No |
Theorem 19.5. If \( f : \left| K\right| \rightarrow \left| L\right| \) is a homotopy equivalence, then \( {f}_{ * } \) is an isomorphism. In particular, if \( \left| K\right| \) is contractible, then \( K \) is acyclic. | Proof. The proof is immediate. If \( g \) is a homotopy inverse for \( f \), then \( {g}_{ * } \) is an inverse for \( {f}_{ * } \). | No |
Theorem 19.6. The unit sphere \( {S}^{n - 1} \) is a deformation retract of punctured euclidean space \( {\mathbf{R}}^{n} - \mathbf{0} \) . | Proof. Let \( X = {\mathbf{R}}^{n} - \mathbf{0} \) . We define \( F : X \times I \rightarrow X \) by the equation\n\n\[ F\left( {x, t}\right) = \left( {1 - t}\right) x + {tx}/\parallel x\parallel . \]\n\nThe map \( F \) gradually shrinks each open ray emanating from the origin to the point where it intersects the unit ... | Yes |
The euclidean spaces \( {\mathbf{R}}^{n} \) and \( {\mathbf{R}}^{m} \) are not homeomorphic if \( n \neq m \) . | Proof. Suppose that \( h \) is a homeomorphism of \( {\mathbf{R}}^{n} \) with \( {\mathbf{R}}^{m} \) . Then \( h \) is a homeomorphism of \( {\mathbf{R}}^{n} - \mathbf{0} \) with \( {\mathbf{R}}^{m} - p \) for some \( p \in {\mathbf{R}}^{m} \) . The latter space is homeomorphic with \( {\mathbf{R}}^{m} - \mathbf{0} \) ... | Yes |
Let \( X \) be the subspace of \( {\mathbf{R}}^{3} \) obtained by rotating the unit circle in the \( x - z \) plane centered at \( \left( {2,0,0}\right) \) about the \( z \) -axis. Using cylindrical coordinates \( \left( {r,\theta, z}\right) \) in \( {\mathbf{R}}^{3} \), we can express \( X \) as the set of points sati... | You can check that \( p \) maps \( {I}^{2} \) onto \( X \) and is a closed quotient map. | No |
Theorem 20.1. Let \( p : X \rightarrow Y \) be a quotient map. If \( C \) is a locally compact Hausdorff space, then\n\n\[ p \times {i}_{C} : X \times C \rightarrow Y \times C \]\n\nis a quotient map. | Proof: Let \( \pi = p \times {i}_{C} \) . Let \( A \) be a subset of \( Y \times C \) such that \( {\pi }^{-1}\left( A\right) \) is open in \( X \times C \) . We show \( A \) is open in \( Y \times C \) . That is, given \( \left( {{y}_{0},{c}_{0}}\right) \) in \( A \), we find an open set about \( \left( {{y}_{0},{c}_{... | Yes |
If \( p : A \rightarrow B \) and \( q : C \rightarrow D \) are quotient maps, and if the domain of \( p \) and the range of \( q \) are locally compact Hausdorff spaces, then \( p \times q : A \times C \rightarrow B \times D \) is a quotient map. | We can write \( p \times q \) as the composite \( A \times C\xrightarrow[]{{i}_{A} \times q}A \times D\xrightarrow[]{p \times {i}_{D}}B \times D \). Since each of these maps is a quotient map, so is \( p \times q \) . | No |
Theorem 20.4. If the topology of \( X \) is coherent with the subspaces \( {X}_{\alpha } \) , and if \( Y \) is a locally compact Hausdorff space, then the topology of \( X \times Y \) is coherent with the subspaces \( {X}_{\alpha } \times Y \) . | Proof. Let \( E = \sum \left( {{X}_{\alpha }\times \{ \alpha \} }\right) \) ; let \( p : E \rightarrow X \) be the projection map. Because \( Y \) is locally compact Hausdorff, the map\n\n\[ p \times {i}_{Y} : E \times Y \rightarrow X \times Y \]\n\n is also a quotient map. Now \( E \) is the topological sum of the sub... | Yes |
Corollary 20.5. The topology of \( \left| K\right| \times I \) is coherent with the subspaces \( \sigma \times I \), for \( \sigma \in K \) . | Proof. By definition, the topology of \( \left| K\right| \) is coherent with the subspaces \( \sigma \) , for \( \sigma \in K \) . Since \( I \) is locally compact Hausdorff (in fact, compact Hausdorff), the preceding theorem applies. | Yes |
Corollary 20.6. Let \( w * K \) be a cone over the complex \( K \) . The map \( \pi : \left| K\right| \times I \rightarrow \left| {w * K}\right| \) defined by\n\n\[ \pi \left( {x, t}\right) = \left( {1 - t}\right) x + {tw} \]\n\n is a quotient map; it collapses \( \left| K\right| \times 1 \) to the point \( w \) and is... | Proof. If \( \sigma = {v}_{0}\ldots {v}_{n} \) is a simplex of \( K \), let \( w * \sigma \) denote the simplex \( w{v}_{0}\ldots {v}_{n} \) of \( w * K \) . A set \( B \) is closed in \( \left| {w * K}\right| \) if and only if its intersection with each simplex \( w * \sigma \) is closed in that simplex. A set \( A \)... | Yes |
Theorem 21.1. There is no retraction \( r : {B}^{n + 1} \rightarrow {S}^{n} \) . | Proof. Such a map \( r \) would be an extension of the identity map \( i : {S}^{n} \rightarrow \) \( {S}^{n} \) . Since \( i \) has degree \( 1 \neq 0 \), there is no such extension. | Yes |
Theorem 21.2 (Brouwer fixed-point theorem). Every continuous map \( \phi : {B}^{n} \rightarrow {B}^{n} \) has a fixed point. | Proof. If \( \phi : {B}^{n} \rightarrow {B}^{n} \) has no fixed point, we can define a map \( h : {B}^{n} \rightarrow \) \( {S}^{n - 1} \) by the equation\n\n\[ h\left( x\right) = \frac{x - \phi \left( x\right) }{\parallel x - \phi \left( x\right) \parallel }, \]\n\nsince \( x - \phi \left( x\right) \neq 0 \) . Let \( ... | Yes |
Theorem 21.3. Let \( n \geq 1 \) . The degree of the antipodal map \( a : {S}^{n} \rightarrow {S}^{n} \) is \( {\left( -1\right) }^{n + 1} \) . | Proof. We show in fact that the reflection map\n\n\[ \rho \left( {{x}_{1},\ldots ,{x}_{n + 1}}\right) = \left( {{x}_{1},\ldots ,{x}_{n}, - {x}_{n + 1}}\right) \]\n\nhas degree -1 . It then follows that any reflection map\n\n\[ {\rho }_{i}\left( {{x}_{1},\ldots ,{x}_{i},\ldots ,{x}_{n + 1}}\right) = \left( {{x}_{1},\ldo... | Yes |
Theorem 21.4. If \( h : {S}^{n} \rightarrow {S}^{n} \) has degree different from \( {\left( -1\right) }^{n + 1} \), then \( h \) has a fixed point. | Proof. We shall suppose that \( h : {S}^{n} \rightarrow {S}^{n} \) has no fixed point and prove that \( h \simeq a \) . The theorem follows. Intuitively, we construct the homotopy by simply moving the point \( h\left( x\right) \) to the point \( - x \), along the shorter great circle arc joining these two points; becau... | Yes |
Theorem 21.5. If \( h : {S}^{n} \rightarrow {S}^{n} \) has degree different from 1, then \( h \) carries some point \( x \) to its antipode \( - x \) . | Proof. If \( a \) is the antipodal map, then \( a \circ h \) has degree different from \( {\left( -1\right) }^{n + 1} \), so it has a fixed point \( x \) . Thus \( a\left( {h\left( x\right) }\right) = x \), so \( - h\left( x\right) = x \) as desired. | Yes |
Corollary 21.6. \( {S}^{n} \) has a non-zero tangent vector field if and only if \( n \) is odd. | Proof. If \( n \) is odd, let \( n = {2k} - 1 \) . Then for \( x \in {S}^{n} \), we define\n\n\[ \overrightarrow{v}\left( x\right) = \left( {-{x}_{2},{x}_{1}, - {x}_{4},{x}_{3},\ldots , - {x}_{2k},{x}_{{2k} - 1}}\right) .\n\]\n\nNote that \( \overrightarrow{v}\left( x\right) \) is perpendicular to \( x \), so that \( \... | Yes |
Theorem 22.2. Let \( K \) be a finite complex. Let \( {\beta }_{p} = \operatorname{rank}{H}_{p}\left( K\right) /{T}_{p}\left( K\right) \) ; it is the betti number of \( K \) in dimension \( p \) . Then\n\n\[ \chi \left( K\right) = {\sum }_{p}{\left( -1\right) }^{p}{\beta }_{p} \] | Proof. If \( \phi : {C}_{p}\left( K\right) \rightarrow {C}_{p}\left( K\right) \) is the identity chain map, then the matrix of \( \phi \) relative to any basis is the identity matrix. We conclude that \( \operatorname{tr}\left( {\phi ,{C}_{p}}\right) = \) rank \( {C}_{p} \) . Similarly, because \( {\phi }_{ * } \) is t... | Yes |
Lemma 22.4. Let \( K \) be a finite complex; let \( h : \left| K\right| \rightarrow \left| K\right| \) be a continuous map. If \( \left| K\right| \) is connected, then \( {h}_{ * } : {H}_{0}\left( K\right) \rightarrow {H}_{0}\left( K\right) \) is the identity. | Proof. Let \( f : {K}^{\prime } \rightarrow K \) be a simplicial approximation to \( h \) . If \( v \) is a vertex of \( K \), the subdivision operator \( \lambda \) carries \( v \) to a 0 -chain carried by the subdivision of \( v \), which is just \( v \) itself. Thus \( \lambda \left( v\right) \) is a multiple of \( ... | Yes |
Theorem 22.5. Let \( K \) be a finite complex; let \( h : \left| K\right| \rightarrow \left| K\right| \) be a continuous map. If \( \left| K\right| \) is acyclic, then \( h \) has a fixed point. | Proof. The group \( {H}_{0}\left( K\right) \) is infinite cyclic, and \( {h}_{ * } \) is the identity on \( {H}_{0}\left( K\right) \) . Thus \( \operatorname{tr}\left( {{h}_{ * },{H}_{0}\left( K\right) }\right) = 1 \) . Since all the higher dimensional homology vanishes, \( \Lambda \left( h\right) = 1 \) . Therefore, \... | Yes |
Theorem 22.6. The antipodal map of \( {S}^{n} \) has degree \( {\left( -1\right) }^{n + 1} \) . | Proof. Let \( h : {S}^{n} \rightarrow {S}^{n} \) be a map of degree \( d \) . We compute \( \Lambda \left( h\right) \) . Now \( {h}_{ * } \) is the identity on 0-dimensional homology. On \( n \) -dimensional homology, its matrix is a one by one matrix with single entry \( d = \) degree \( f \) . Therefore,\n\n\[ \Lambd... | Yes |
Theorem 23.1. Let \( 0 \rightarrow {A}_{1}\xrightarrow[]{\phi }{A}_{2}\xrightarrow[]{\psi }{A}_{3} \rightarrow 0 \) be exact. The following are equivalent:\n\n(1) The sequence splits.\n\n(2) There is a map \( p : {A}_{2} \rightarrow {A}_{1} \) such that \( p \circ \phi = {i}_{{A}_{1}} \) .\n\n(3) There is a map \( j : ... | Proof. We show that (1) implies (2) and (3). It suffices to prove (2) and (3) for the sequence\n\n\[ 0 \rightarrow {A}_{1}\overset{i}{ \rightarrow }{A}_{1} \oplus {A}_{3}\overset{\pi }{ \rightarrow }{A}_{3} \rightarrow 0. \]\n\nAnd this is easy; we define \( p : {A}_{1} \oplus {A}_{3} \rightarrow {A}_{1} \) as projecti... | Yes |
Corollary 23.2. Let \( 0 \rightarrow {A}_{1}\xrightarrow[]{\phi }{A}_{2}\xrightarrow[]{\psi }{A}_{3} \rightarrow 0 \) be exact. If \( {A}_{3} \) is free abelian, the sequence splits. | Proof. We choose a basis for \( {A}_{3} \), and define the value of \( j : {A}_{3} \rightarrow {A}_{2} \) on the basis element \( e \) to be any element of the nonempty set \( {\psi }^{-1}\left( e\right) \) . | No |
Theorem 23.3 (The exact homology sequence of a pair). Let \( K \) be a complex; let \( {K}_{0} \) be a subcomplex. Then there is a long exact sequence\n\n\[ \cdots \rightarrow {H}_{p}\left( {K}_{0}\right) \overset{{i}_{ * }}{ \rightarrow }{H}_{p}\left( K\right) \overset{{\pi }_{ * }}{ \rightarrow }{H}_{p}\left( {K,{K}_... | The proof of this theorem is basically algebraic in nature. We shall formulate it in a purely algebraic fashion and prove it in the next section. | No |
Let \( K \) be the complex pictured in Figure 23.1, whose polytope is a square. Let \( {K}_{0} \) be the subcomplex whose polytope is the boundary of the square. We know from Example 3 of \( §9 \) that \( {H}_{2}\left( {K,{K}_{0}}\right) \) is infinite cyclic and is generated by the 2-chain \( \gamma \) that is the sum... | This fact can also be proved by considering the exact homology sequence of the pair \( \left( {K,{K}_{0}}\right) \) . A portion of this sequence is \[ {H}_{2}\left( K\right) \rightarrow {H}_{2}\left( {K,{K}_{0}}\right) \overset{{\partial }_{ * }}{ \rightarrow }{H}_{1}\left( {K}_{0}\right) \rightarrow {H}_{1}\left( K\ri... | Yes |
Let \( K \) be the complex pictured in Figure 23.2, whose underlying space is an annulus. Let \( {K}_{0} \) be the subcomplex of \( K \) whose polytope equals the union of the inner and outer edges of the square. In Example 4 of \( §9 \), we computed the homology of \( \left( {K,{K}_{0}}\right) \) . We recompute it her... | \[ 0 \rightarrow {H}_{2}\left( {K,{K}_{0}}\right) \overset{{\partial }_{ * }}{ \rightarrow }{H}_{1}\left( {K}_{0}\right) \overset{{\widehat{i}}_{ * }}{ \rightarrow }{H}_{1}\left( K\right) \overset{{\pi }_{ * }}{ \rightarrow }{H}_{1}\left( {K,{K}_{0}}\right) \overset{{\partial }_{ * }}{ \rightarrow }{\widetilde{H}}_{0}\... | No |
We consider the next two examples together. Let \( \left( {K,{K}_{0}}\right) \) denote either the cylinder and its top edge, or the Möbius band and its edge. In each case, \( \left| {K}_{0}\right| \) is a circle. Furthermore, \( \left| K\right| \) has the homotopy type of a circle; the central circle \( C \) indicated ... | Everything depends on computing the homomorphism \( {i}_{ * } \) . Since the retraction \( r : \left| K\right| \rightarrow C \) is a homotopy equivalence, it suffices to compute the homomorphism induced by the composite map \( r \circ i : \left| {K}_{0}\right| \rightarrow C \), which collapses the edge \( \left| {K}_{0... | No |
Lemma 24.1 (The zig-zag lemma). Suppose one is given chain complexes \( \mathcal{C} = \left\{ {{C}_{p},{\partial }_{C}}\right\} ,\mathcal{D} = \left\{ {{D}_{p},{\partial }_{D}}\right\} \), and \( \mathcal{E} = \left\{ {{E}_{p},{\partial }_{E}}\right\} \), and chain maps \( \phi ,\psi \) such that the sequence\n\n\[ 0 \... | Proof. The proof is of a type now commonly known as \ | No |
Theorem 24.2. Suppose one is given the commutative diagram\n\n\n\nwhere the horizontal sequences are exact sequences of chain complexes, and \( \alpha \) , \( \beta ,\gamma \) are chain maps. Then the following diagr... | Proof. Commutativity of the first two squares is immediate, because commutativity holds already on the chain level. Commutativity of the last square involves examining the definitions of \( {\partial }_{ * } \) and \( {\partial }_{ * }^{\prime } \).\n\nGiven \( \left\{ {e}_{p}\right\} \in {H}_{p}\left( \mathcal{E}\righ... | Yes |
Lemma 24.4. Let \( h : \left( {K,{K}_{0}}\right) \rightarrow \left( {L,{L}_{0}}\right) \) be a simplicial map.\n\n(a) The induced homology homomorphisms \( {h}_{ * } \) give a homomorphism of the exact homology sequence of \( \left( {K,{K}_{0}}\right) \) with that of \( \left( {L,{L}_{0}}\right) \) . | Proof. We know \( {h}_{\mu } \) is a chain map, and the following diagram commutes: \n\nThen (a) follows. | No |
Lemma 25.2. Let \( h : \left( {K,{K}_{0},{K}_{1}}\right) \rightarrow \left( {L,{L}_{0},{L}_{1}}\right) \) be a simplicial map, where \( K = {K}_{0} \cup {K}_{1} \) and \( L = {L}_{0} \cup {L}_{1} \) . Then \( h \) induces a homomorphism of Mayer-Vietoris sequences. | Proof. One checks immediately that the chain maps \( {h}_{\mu } \) induced by \( h \) commute with the chain maps \( \phi \) and \( \psi \) defined in the preceding proof. Naturality then follows from Theorem 24.2. | No |
Theorem 25.4. If \( K \) is a complex, then for all \( p \), there is an isomorphism\n\n\[ \n{\widetilde{H}}_{p}\left( {S\left( K\right) }\right) \rightarrow {\widetilde{H}}_{p - 1}\left( K\right) .\n\] | Proof. Let \( {K}_{0} = {w}_{0} * K \) and \( {K}_{1} = {w}_{1} * K \) . Then \( {K}_{0} \cup {K}_{1} = S\left( K\right) \) and \( {K}_{0} \cap {K}_{1} = K \) . In the reduced Mayer-Vietoris sequence\n\n\[ \n{\widetilde{H}}_{p}\left( {K}_{0}\right) \oplus {\widetilde{H}}_{p}\left( {K}_{1}\right) \rightarrow {\widetilde... | Yes |
Theorem 27.1. Simplicial homology theory on the class of triangulable pairs satisfies the Eilenberg-Steenrod axioms. | Proof. Axioms \( 1 - 5 \) and 7 express familiar properties of the homology of simplicial complexes that carry over at once to the homology of triangulable pairs. Only Axioms 6 and 8, the excision axiom and the axiom of compact support, require comment.\n\nTo check the axiom of compact support, it suffices to show that... | Yes |
Theorem 27.3. Let \( i : \left( {{X}_{0},{A}_{0}}\right) \rightarrow \left( {X, A}\right) \) be an inclusion of triangulable pairs, where \( \left( {{X}_{0},{A}_{0}}\right) \) is a compact pair. If \( \alpha \in {H}_{p}\left( {{X}_{0},{A}_{0}}\right) \) and \( {i}_{ * }\left( \alpha \right) = 0 \), then there are a com... | Proof. We may assume that \( \left( {X, A}\right) \) is the polytope of a simplicial pair \( \left( {K, C}\right) \) . Because \( {X}_{0} \) is compact, it is contained in the polytope of a finite sub-complex \( {K}_{0} \) of \( K \) . Then \( {A}_{0} \) is contained in the polytope of \( C \cap {K}_{0} = {C}_{0} \) . ... | Yes |
Given a homology theory, let \( p \) be fixed, and consider the following two functors, defined on admissible pairs:\n\n\[ G\left( {X, A}\right) = {H}_{p}\left( {X, A}\right) ;\;G\left( f\right) = {f}_{ * }.\]\n\n\[ H\left( {X, A}\right) = {H}_{p - 1}\left( A\right) ;\;H\left( f\right) = {\left( f \mid A\right) }_{ * }... | tells us that \( {\partial }_{ * } \) is a natural transformation of the functor \( G \) to the functor \( H \) . This is precisely the third of the Eilenberg-Steenrod axioms. | Yes |
Consider the category of pairs of spaces and pairs of maps. Let \( G \) and \( H \) be the functors\n\n\[ G\left( {X, Y}\right) = X \times Y;\;G\left( {f, g}\right) = f \times g. \]\n\n\[ H\left( {X, Y}\right) = Y \times X;\;H\left( {f, g}\right) = g \times f. \]\n\nGiven \( \left( {X, Y}\right) \), let \( {T}_{\left( ... | Given \( \left( {X, Y}\right) \), let \( {T}_{\left( X, Y\right) } \) be the homeomorphism of \( X \times Y \) with \( Y \times X \) that switches coordinates. Then \( T \) is a natural equivalence of \( G \) with \( H \) . | No |
If \( V \) is a vector space over \( \mathbf{R} \), consider the space \( \mathcal{L}\left( {V,\mathbf{R}}\right) \) of linear functionals on \( V \) (linear transformations of \( V \) into \( \mathbf{R} \) ). It is often called the dual space to \( V \) . The space \( \mathcal{L}\left( {V,\mathbf{R}}\right) \) has the... | The assignment\n\n\[ \nV \rightarrow \mathcal{L}\left( {V,\mathbf{R}}\right) \;\text{ and }\;f \rightarrow {f}^{tr}\n\]\n\nis a contravariant functor from the category of vector spaces and linear transformations to itself. | Yes |
Theorem 29.1. The homomorphism \( {f}_{y} \) commutes with \( \partial \) . Furthermore, \( {\partial }^{2} = 0 \) . | Proof. The first statement follows by direct computation:\n\n\[ \partial {f}_{\# }\left( T\right) = \mathop{\sum }\limits_{{i = 0}}^{p}{\left( -1\right) }^{i}\left( {f \circ T}\right) \circ l\left( {{\epsilon }_{0},\ldots ,{\widehat{\epsilon }}_{l},\ldots ,{\epsilon }_{p}}\right) ,\] \n\n\[ {f}_{\# }\left( {\partial T}... | Yes |
Theorem 29.2. If \( i : X : \rightarrow X \) is the identity, then \( {i}_{ * } : {H}_{p}\left( X\right) \rightarrow {H}_{p}\left( X\right) \) is the identity. If \( f : X \rightarrow Y \) and \( g : Y \rightarrow Z \), then \( {\left( g \circ f\right) }_{ * } = {g}_{ * } \circ {f}_{ * } \) . The same holds in reduced ... | Proof. Both equations in fact hold on the chain level. For \( {i}_{\# }\left( T\right) = \) \( i \circ T = T \) . And \( {\left( g \circ f\right) }_{\# }\left( T\right) = \left( {g \circ f}\right) \circ T = g \circ \left( {f \circ T}\right) = {g}_{\# }\left( {{f}_{\# }\left( T\right) }\right) \) . | Yes |
Theorem 29.4. Let \( X \) be a topological space. Then \( {H}_{0}\left( X\right) \) is free abelian. If \( \left\{ {X}_{\alpha }\right\} \) is the collection of path components of \( X \), and if \( {T}_{\alpha } \) is a singular 0 -simplex with image in \( {X}_{\alpha } \), for each \( \alpha \), then the homology cla... | Proof. Let \( {x}_{\alpha } \) be the point \( {T}_{\alpha }\left( {\Delta }_{0}\right) \) . If \( T : {\Delta }_{0} \rightarrow X \) is any singular 0 -simplex of \( X \), then there is a path \( f : \left\lbrack {0,1}\right\rbrack \rightarrow X \) from the point \( T\left( {\Delta }_{0}\right) \) to some point \( {x}... | Yes |
Theorem 29.6. Let \( X \) be a subspace of \( {\mathbf{E}}^{\prime } \) that is star convex relative to \( w \) . Then \( X \) is acyclic in singular homology. | Proof. To show that \( {\widetilde{H}}_{0}\left( X\right) = 0 \), let \( c \) be a singular 0 -chain on \( X \) such that \( \epsilon \left( c\right) = 0 \) . Then by the preceding lemma,\n\n\[ \partial \left\lbrack {c, w}\right\rbrack = \epsilon \left( c\right) {T}_{w} - c = - c, \]\n\nso \( c \) bounds a 1-chain.\n\n... | Yes |
Theorem 30.2. There is a homomorphism \( {\partial }_{ * } : {H}_{p}\left( {X, A}\right) \rightarrow {H}_{p - 1}\left( A\right) \), defined for \( A \subset X \) and all \( p \), such that the sequence\n\n\[ \cdots \rightarrow {H}_{p}\left( A\right) \overset{{\dot{i}}_{ * }}{ \rightarrow }{H}_{p}\left( X\right) \overse... | Proof. For the existence of \( {\partial }_{ * } \) and the exact sequence, we apply the \ | No |
Theorem 30.3. If \( P \) is a one-point space, then \( {H}_{p}\left( P\right) = 0 \) for \( p \neq 0 \) and \( {H}_{0}\left( P\right) \simeq \mathbf{Z} \) . | Proof. This follows from Theorem 29.6, once one notes that a one-point space in \( {\mathbf{R}}^{N} \) is star convex! For a more direct proof, we compute the chain complex \( \mathcal{S}\left( P\right) \) . There is exactly one singular simplex \( {T}_{p} : {\Delta }_{p} \rightarrow P \) in each non-negative dimension... | No |
Theorem 30.4. Given \( \alpha \in {H}_{p}\left( {X, A}\right) \), there is a compact pair \( \left( {{X}_{0},{A}_{0}}\right) \subset \) \( \left( {X, A}\right) \), such that \( \alpha \) is in the image of the homomorphism induced by inclusion \[ {i}_{ * } : {H}_{p}\left( {{X}_{0},{A}_{0}}\right) \rightarrow {H}_{p}\le... | Proof. If \( T : {\Delta }_{p} \rightarrow X \) is a singular simplex, its minimal carrier is defined to be the image set \( T\left( {\Delta }_{p}\right) \) . The minimal carrier of a singular \( p \) -chain \( \sum {n}_{i}{\bar{T}}_{i} \) (where each \( {n}_{i} \neq 0 \) ) is the union of the minimal carriers of the \... | Yes |
Theorem 30.5. Let \( i : \left( {{X}_{0},{A}_{0}}\right) \rightarrow \left( {X, A}\right) \) be inclusion, where \( \left( {{X}_{0},{A}_{0}}\right) \) is a compact pair. If \( \alpha \in {H}_{p}\left( {{X}_{0},{A}_{0}}\right) \) and \( {i}_{ * }\left( \alpha \right) = 0 \), then there are a compact pair \( \left( {{X}_... | Proof. Let \( {c}_{p} \) be a singular \( p \) -chain of \( {X}_{0} \) representing \( \alpha \) ; then \( \partial {c}_{p} \) is carried by \( {A}_{0} \) . By hypothesis, there is a chain \( {d}_{p + 1} \) of \( X \) such that \( {c}_{p} - \partial {d}_{p + 1} \) is carried by \( A \) . Let \( {X}_{1} \) be the union ... | Yes |
Lemma 30.6. There exists, for each space \( X \) and each non-negative integer \( p \), a homomorphism\n\n\[ \n{D}_{X} : {S}_{p}\left( X\right) \rightarrow {S}_{p + 1}\left( {X \times I}\right) \n\]\n\nhaving the following properties:\n\n(a) If \( T : {\Delta }_{p} \rightarrow X \) is a singular simplex, then\n\n\[ \n\... | Proof. We proceed by induction on \( p \) . The case \( p = 0 \) is easy. Given \( T : {\Delta }_{0} \rightarrow X \), let \( {x}_{0} \) denote the point \( T\left( {\Delta }_{0}\right) \) . Define \( {D}_{X}T : {\Delta }_{1} \rightarrow X \times I \) by the equation\n\n\[ \n{D}_{x}T\left( {t,0,\ldots }\right) = \left(... | No |
Theorem 30.7. If \( f, g : \left( {X, A}\right) \rightarrow \left( {Y, B}\right) \) are homotopic, then \( {f}_{ * } = {g}_{ * } \) . The same holds in reduced homology if \( A = B = \varnothing \) . | Proof. Let \( F : \left( {X \times I, A \times I}\right) \rightarrow \left( {Y, B}\right) \) be the homotopy between \( f, g : \left( {X, A}\right) \rightarrow \left( {Y, B}\right) \) . Let \( i, j : \left( {X, A}\right) \rightarrow \left( {X \times I, A \times I}\right) \) be given by \( i\left( x\right) = \) \( \left... | Yes |
Theorem 30.8. Let \( f : \left( {X, A}\right) \rightarrow \left( {Y, B}\right) \). (a) If \( f \) is a homotopy equivalence, then \( {f}_{ * } \) is an isomorphism in relative homology. | Proof. If \( f \) is a homotopy equivalence, it is immediate that \( {f}_{ * } \) is an isomorphism. To prove (b), we examine the long exact homology sequences of \( \left( {X, A}\right) \) and of \( \left( {Y, B}\right) \), and the homomorphism \( {f}_{ * } \) carrying the one exact sequence to the other. The hypothes... | Yes |
Consider the inclusion map \( j : \left( {{B}^{n},{S}^{n - 1}}\right) \rightarrow \left( {{\mathbf{R}}^{n},{\mathbf{R}}^{n} - \mathbf{0}}\right) \). Since \( {B}^{n} \) is a deformation retract of \( {\mathbf{R}}^{n} \), and \( {S}^{n - 1} \) is a deformation retract of \( {\mathbf{R}}^{n} - \mathbf{0} \), the map \( {... | \[ g \circ j : \left( {{B}^{n},{S}^{n - 1}}\right) \rightarrow \left( {{B}^{n},{S}^{n - 1}}\right) \] carries all of \( {B}^{n} \) into \( {S}^{n - 1} \), so it induces the trivial homomorphism in homology. On the other hand, this map is by hypothesis homotopic to the identity, so it induces the identity homomorphism o... | Yes |
Lemma 31.1. The homomorphism \( {\operatorname{sd}}_{x} \) is an augmentation-preserving chain map, and it is natural in the sense that for any continuous map \( f \) : \( X \rightarrow Y \), we have\n\n\[ \n{f}_{\# } \circ {\operatorname{sd}}_{x} = {\operatorname{sd}}_{Y} \circ {f}_{\# }\n\] | Proof. The map \( {\operatorname{sd}}_{X} \) preserves augmentation because it is the identity in dimension 0 . Naturality holds in dimension 0 for the same reason. Naturality holds in positive dimensions by direct computation:\n\n\[ \n{f}_{\# }\left( {{\operatorname{sd}}_{X}T}\right) = {f}_{\# }{T}_{\# }\left( {{\oper... | Yes |
Lemma 31.2. Let \( T : {\Delta }_{p} \rightarrow \sigma \) be a linear homeomorphism of \( {\Delta }_{p} \) with the p-simplex \( \sigma \) . Then each term of \( \operatorname{sd}T \) is a linear homeomorphism of \( {\Delta }_{p} \) with a simplex in the first barycentric subdivision of \( \sigma \) . | Proof. The lemma is trivial for \( p = 0 \) ; suppose it is true in dimensions less than \( p \) . Consider first the identity linear homeomorphism \( {i}_{p} : {\Delta }_{p} \rightarrow {\Delta }_{p} \) . Now\n\n\[ \operatorname{sd}{i}_{p} = \left\lbrack {\operatorname{sd}\partial {i}_{p},{\widehat{\Delta }}_{p}}\righ... | Yes |
Theorem 31.3. Let \( \mathcal{A} \) be a collection of subsets of \( X \) whose interiors cover \( X \) . Given \( T : {\Delta }_{p} \rightarrow X \), there is an \( m \) such that each term of \( {\operatorname{sd}}^{m}T \) is A-small. | Proof. It follows from the preceding lemma that if \( L \) is a linear homeomorphism of \( {\Delta }_{p} \) with the \( p \) -simplex \( \sigma \), then each term of \( {\operatorname{sd}}^{m}L \) is a linear homeomorphism of \( {\Delta }_{p} \) with a simplex in the \( m \) th barycentric subdivision of \( \sigma \) .... | Yes |
Lemma 31.4. Let \( m \) be given. For each space \( X \), there is a homomorphism \( {D}_{X} : {S}_{p}\left( X\right) \rightarrow {S}_{p + 1}\left( X\right) \) such that for each singular p-simplex \( T \) of \( X \) ,\n\n(*) \n\n\[ \partial {D}_{x}T + {D}_{x}\partial T = {\operatorname{sd}}^{m}T - T. \] \n\nFurthermor... | Proof. If \( T : {\Delta }_{0} \rightarrow X \) is a singular 0 -simplex, define \( {D}_{X}T = 0 \) . Formula (*) and naturality follow trivially. Now let \( p > 0 \) . Suppose \( {D}_{x} \) is defined, satisfying \( \left( *\right) \) and naturality, in dimensions less than \( p \) . We proceed by a method similar to ... | No |
Theorem 31.5. Let \( X \) be a space; let \( \mathcal{A} \) be a collection of subsets of \( X \) whose interiors cover \( X \) . Then the inclusion map \( {\mathcal{S}}^{\mathcal{A}}\left( X\right) \rightarrow \mathcal{S}\left( X\right) \) induces an isomorphism in homology, both ordinary and reduced. | Proof. The obvious way to proceed is to attempt to define a chain map \( \lambda : \mathcal{S}\left( X\right) \rightarrow {\mathcal{S}}^{\mathcal{A}}\left( X\right) \) that is a chain-homotopy inverse for the inclusion map. This is not as easy as it looks. For any particular singular chain, there is an \( m \) such tha... | Yes |
Corollary 31.6. Let \( X \) and \( \mathcal{A} \) be as in the preceding theorem. If \( B \subset X \) , let \( {S}_{p}^{A}\left( B\right) \) be generated by those singular simplices \( T : {\Delta }_{p} \rightarrow B \) whose image sets lie in elements of \( \mathcal{A} \) . Let \( {S}_{p}^{\mathcal{A}}\left( {X, B}\r... | Proof. The inclusions \( {\mathcal{S}}^{\mathcal{A}}\left( B\right) \rightarrow \mathcal{S}\left( B\right) \) and \( {\mathcal{S}}^{\mathcal{A}}\left( X\right) \rightarrow \mathcal{S}\left( X\right) \) give rise to a homomorphism of the long exact homology sequence derived from\n\n\[ 0 \rightarrow \mathcal{S}\left( B\r... | Yes |
Theorem 31.7 (Excision for singular theory). Let \( A \subset X \) . If \( U \) is a subset of \( X \) such that \( \bar{U} \subset \operatorname{Int}{A}_{i} \) then inclusion \[ j : \left( {X - U, A - U}\right) \rightarrow \left( {X, A}\right) \] induces an isomorphism in singular homology. | Proof. Let \( \mathcal{A} \) denote the collection \( \{ X - U, A\} \) . Now \( X - U \) contains the open set \( X - \bar{U} \) . Since \( \bar{U} \subset \operatorname{Int}A \), the interiors of the sets \( X - U \) and \( A \) cover \( X \) . Consider the homomorphisms \[ \frac{{S}_{p}\left( {X - U}\right) }{{S}_{p}... | Yes |
Let \( n \geq 0 \) . The group \( {H}_{i}\left( {{B}^{n},{S}^{n - 1}}\right) \) is infinite cyclic for \( i = n \) and vanishes otherwise. The group \( {\widetilde{H}}_{i}\left( {S}^{n}\right) \) is infinite cyclic for \( i = n \) and vanishes otherwise. The homomorphism of \( {\widetilde{H}}_{n}\left( {S}^{n}\right) \... | Proof. We verify the theorem for \( n = 0 \) . It is trivial that \( {H}_{p}\left( {{B}^{0},\varnothing }\right) \) is infinite cyclic for \( p = 0 \) and vanishes otherwise, since \( {B}^{0} \) is a single point.\n\nIt is similarly easy to see that \( {H}_{p}\left( {S}^{0}\right) = 0 \) for \( p \neq 0 \), since \( {S... | Yes |
Consider the singular chain complex functor, from the topological category to \( \mathbf{A} \) . Let \( \mathcal{M} \) be the collection \( \left\{ {{\Delta }_{p} \mid p = 0,1,\ldots }\right\} \) . This functor is acyclic relative to \( \mathcal{M} \) . We show it is free relative to \( \mathcal{M} \) : for each \( p \... | It is immediate that; as \( T \) ranges over all continuous maps \( {\Delta }_{p} \rightarrow X \), the elements \( {T}_{p}\left( {i}_{p}\right) = T \) form a basis. for \( {S}_{p}\left( X\right) \) . | Yes |
Consider the following functor \( G \), defined on the category of topological pairs:\n\n\[ \left( {X, Y}\right) \rightarrow \mathcal{S}\left( {X \times Y}\right) \;\text{ and }\;\left( {f, g}\right) \rightarrow {\left( f \times g\right) }_{i}. \]\n\nLet \( \mathcal{M} = \left\{ {\left( {{\Delta }_{p},{\Delta }_{q}}\ri... | For each index \( p \), let \( {J}_{p} \) consist of a single element; let the corresponding family consist of \( \left( {{\Delta }_{p},{\Delta }_{p}}\right) \) alone; and let the corresponding element of \( {S}_{p}\left( {{\Delta }_{p} \times {\Delta }_{p}}\right) \) be the diagonal map \( {d}_{p}\left( x\right) = \le... | Yes |
Let \( G \) be the functor\n\n\[ \nX \rightarrow \mathcal{S}\left( {X \times I}\right) \;\text{ and }\;f \rightarrow {\left( f \times {i}_{t}\right) }_{\mu }.\n\]\n\nLet \( \mathcal{M} = \left\{ {{\Delta }_{p} \mid p = 0,1,\ldots }\right\} \) . Then \( G \) is acyclic relative to \( \mathcal{M} \) . | It is also true that \( G \) is free relative to \( \mathcal{M} \), but the proof is not obvious. Let \( {J}_{p} \) be the set of all continuous functions \( \alpha : {\Delta }_{p} \rightarrow I \) . Let the family \( {\left\{ {M}_{\alpha }\right\} }_{\alpha \in {J}_{p}} \) be defined by setting \( {M}_{\alpha } = {\De... | Yes |
Theorem 32.2. Let \( \mathbf{C} \) be a category; let \( G \) and \( {G}^{\prime } \) be functors from \( \mathbf{C} \) to \( \mathbf{A} \) . If \( G \) and \( {G}^{\prime } \) are free and acyclic relative to the collection \( \mathcal{M} \) of objects of \( \mathbf{C} \) , then there is a natural transformation \( {T... | Proof. We apply the preceding theorem four times. Because \( G \) is free and \( {G}^{\prime } \) is acyclic, \( {T}_{X} \) exists. Because \( {G}^{\prime } \) is free and \( G \) is acyclic, there is a natural transformation \( {S}_{X} \) of \( {G}^{\prime } \) to \( G \) . Now \( {S}_{X} \circ {T}_{X} \) and the iden... | Yes |
Theorem 33.1. Let \( X = {X}_{1} \cup {X}_{2} \) ; suppose \( \left\{ {{X}_{1},{X}_{2}}\right\} \) is an excisive couple. Let \( A = {X}_{1} \cap {X}_{2} \) . Then there is an exact sequence\n\n\[ \cdots \rightarrow {H}_{p}\left( A\right) \overset{{\phi }_{ * }}{ \rightarrow }{H}_{p}\left( {X}_{1}\right) \oplus {H}_{p}... | Proof. We define a short exact sequence of chain complexes\n\n(*)\n\n\[ 0 \rightarrow {S}_{p}\left( A\right) \overset{\phi }{ \rightarrow }{S}_{p}\left( {X}_{1}\right) \oplus {S}_{p}\left( {X}_{2}\right) \overset{\psi }{ \rightarrow }{S}_{p}\left( {X}_{1}\right) + {S}_{p}\left( {X}_{2}\right) \rightarrow 0 \]\n\nby the... | Yes |
Theorem 33.2. Let \( X \) be a space. There is for all \( p \) an isomorphism\n\n\[ \n{\widetilde{H}}_{p}\left( {S\left( X\right) }\right) \rightarrow {\widetilde{H}}_{p - 1}\left( X\right) .\n\] | Proof. Let \( \pi : X \times \left\lbrack {-1,1}\right\rbrack \rightarrow S\left( X\right) \) be the quotient map. Let \( v = \) \( \pi \left( {X \times 1}\right) \) and \( w = \pi \left( {X \times \left( {-1}\right) }\right) \) ; these points are called the \ | No |
Lemma 34.1. Let \( \psi : \mathcal{O} \rightarrow {\mathcal{C}}^{\prime } \) be a chain map of augmented chain complexes. Then \( {\psi }_{ * } \) is an isomorphism in reduced homology if and only if it is an isomorphism in ordinary homology. | Proof. We recall the proof that \( {H}_{0}\left( \mathcal{C}\right) \cong {\widetilde{H}}_{0}\left( \mathcal{C}\right) \oplus \mathbf{Z} \) . (See the exercises of §7.) Begin with the exact sequences\n\n\n\nChoose \(... | No |
Theorem 34.4. The isomorphism \( {\eta }_{ * } \) commutes with homomorphisms induced by simplicial maps. | Proof. Let \( f : \left( {K,{K}_{0}}\right) \rightarrow \left( {L,{L}_{0}}\right) \) be a simplicial map. We have already proven that \( {f}_{ * } \) commutes with \( {\phi }_{ * } \) . (See Theorem 13.7.) We show it commutes with \( {\theta }_{ * } \) . In fact, \( {f}_{\theta } \) commutes with \( \theta \) on the ch... | Yes |
Theorem 34.5. The isomorphism \( {\eta }_{ * } \) commutes with homomorphisms induced by continuous maps. | Proof. Let\n\n\[ h : \left( {\left| K\right| ,\left| {K}_{0}\right| }\right) \rightarrow \left( {\left| L\right| ,\left| {L}_{0}\right| }\right) \]\n\nbe a continuous map. Let\n\n\[ f : \left( {{K}^{\prime },{K}_{0}^{\prime }}\right) \rightarrow \left( {L,{L}_{0}}\right) \;\text{ and }\; g : \left( {{K}^{\prime },{K}_{... | Yes |
Lemma 35.1. Let \( A \subset X \) . If \( A \) contains a neighborhood of the point \( x \) , then\n\n\[ \n{H}_{p}\left( {X, X - x}\right) \simeq {H}_{p}\left( {A, A - x}\right) .\n\] | Proof. Let \( U \) denote the set \( X - A \) . Because \( A \) contains a neighborhood of \( x \) ,\n\n\[ \n\bar{U} \subset X - x = \operatorname{Int}\left( {X - x}\right) .\n\]\n\nIt follows from the excision property that\n\n\[ \n{H}_{p}\left( {X, X - x}\right) \simeq {H}_{p}\left( {X - U, X - x - U}\right) = {H}_{p... | Yes |
If \( x \in {\mathbf{R}}^{m} \), we show that \( {H}_{i}\left( {{\mathbf{R}}^{m},{\mathbf{R}}^{m} - x}\right) \) is infinite cyclic for \( i = m \) and vanishes otherwise. | Let \( B \) denote a ball centered at \( x \) . By the preceding lemma,\n\n\[ \n{H}_{i}\left( {{\mathbf{R}}^{m},{\mathbf{R}}^{m} - x}\right) \simeq {H}_{i}\left( {B, B - x}\right) \simeq {H}_{i}\left( {{B}^{m},{B}^{m} - \mathbf{0}}\right) .\n\]\n\nNow \( {S}^{m - 1} \) is a deformation retract of \( {B}^{m} - \mathbf{0... | Yes |
If \( x \in \mathrm{{Bd}}{\mathbf{H}}^{m} \), then the group \( {H}_{i}\left( {{\mathbf{H}}^{m},{\mathbf{H}}^{m} - x}\right) \) vanishes for all \( i \) . If \( x \in {\mathbf{H}}^{m} \) and \( x \notin \operatorname{Bd}{\mathbf{H}}^{m} \), then this group is infinite cyclic for \( i = m \) and vanishes otherwise. | If \( x \notin \mathrm{{Bd}}{\mathbf{H}}^{m} \), this result follows from the preceding example, once we note that \( x \) has a neighborhood that is an open set of \( {\mathbf{R}}^{m} \) . So suppose \( x \in \operatorname{Bd}{\mathbf{H}}^{m} \) ; we can assume without loss of generality that \( x = \mathbf{0} \) . Le... | Yes |
The unit ball \( {B}^{n} \) in \( {\mathbf{R}}^{n} \) is an \( n \) -manifold with boundary, and Bd \( {B}^{n} = \) \( {S}^{n - 1} \). | If \( p \in {B}^{n} - {S}^{n - 1} \), then the set of all \( x \) with \( \parallel x\parallel < 1 \) is an open set of \( {\mathbf{R}}^{n} \) ; thus there is a coordinate patch about \( p \) . Now let \( p \in {S}^{n - 1} \) ; we find a coordinate patch about \( p \) . Some coordinate of \( p \) is non-zero; suppose f... | Yes |
Lemma 35.2. Let \( s \) be a simplex of the complex \( K \) . If \( x \) and \( y \) are points of Int \( s \), then the local homology groups of \( \left| K\right| \) at \( x \) and at \( y \) are isomorphic. | Proof. It suffices to prove the theorem when \( x = \widehat{s} \), the barycenter of \( s \) . Let sd \( K \) be the first barycentric subdivision of \( K \) . Let \( {K}^{\prime } \) be a subdivision of \( K \) defined exactly as sd \( K \) was, except that \( y \) is used instead of \( \widehat{s} \) when \ | No |
Theorem 35.3. Let \( M \) be an m-manifold with boundary; suppose \( K \) is a complex and \( h : \left| K\right| \rightarrow M \) is a homeomorphism. Then \( {h}^{-1}\left( {\operatorname{Bd}M}\right) \) is the polytope of a subcomplex of \( K \) . | Proof. If an open simplex Int \( s \) of \( K \) intersects the set \( {h}^{-1}\left( {\operatorname{Bd}M}\right) \), it lies in this set, by the preceding lemma; since this set is closed, it must contain \( s \) . | Yes |
Lemma 35.4. Let \( v \) be a vertex of the simplicial complex \( K \) . Then\n\n\[ \n{H}_{i}\left( {\left| K\right| ,\left| K\right| - v}\right) \simeq {H}_{i}\left( {\overline{\mathrm{{St}}}v,\mathrm{{Lk}}v}\right) .\n\] | Proof. The set \( \overline{\mathrm{{St}}}v \) contains a neighborhood of \( v \) ; therefore, it follows from Lemma 35.1 that\n\n\[ \n{H}_{i}\left( {\left| K\right| ,\left| K\right| - v}\right) \simeq {H}_{i}\left( {\overline{\mathrm{{St}}}v,\overline{\mathrm{{St}}}v - v}\right) .\n\]\n\nLet \( L \) denote the subcomp... | Yes |
Lemma 35.5. Let \( v * L \) be a cone over \( L \) . Then \( \left| L\right| \) is a deformation retract of \( \left| {v * L}\right| - v \) . | Proof. Consider the quotient map\n\n\[ \pi : \left| L\right| \times I \rightarrow \left| {v * L}\right| \]\n\ndefined by \( \pi \left( {x, t}\right) = \left( {1 - t}\right) x + {tv} \) . (See Corollary 20.6.) Since \( \left| L\right| \times \lbrack 0,1) \) is open in \( \left| L\right| \times I \) and is saturated with... | Yes |
Theorem 35.6. Let \( K \) be a complex of dimension \( n \) ; let \( X = \left| K\right| \) . For \( p > n \) the local homology groups \( {H}_{p}\left( {X, X - x}\right) \) vanish, while for \( p = n \) at least one of the groups \( {H}_{n}\left( {X, X - x}\right) \) is non-trivial. | Proof. Let \( \sigma \) be an \( n \) -simplex of \( K \) . Then \( \sigma \) is a face of no other simplex of \( K \), so the set Int \( \sigma \) is in fact an open set of \( \left| K\right| \) . (Its complement is the union of all simplices of \( K \) different from \( \sigma \) .) If \( x \) is the barycenter \( \w... | Yes |
Theorem 36.1. Let \( B \) be a \( k \) -cell in \( {S}^{n} \) . Then \( {S}^{n} - B \) is acyclic. In particular, \( B \) does not separate \( {S}^{n} \) . | Proof. Let \( n \) be fixed. We proceed by induction on \( k \) . First take the case \( k = 0 \) . Then \( B \) is a single point. The space \( {S}^{n} - B \) is a single point if \( n = 0 \) , while if \( n > 0 \), it is homeomorphic to \( {\mathbf{R}}^{n} \) . In either case, \( {S}^{n} - B \) is acyclic.\n\nWe now ... | Yes |
Theorem 36.2. Let \( n > k \geq 0 \) . Let \( h : {S}^{k} \rightarrow {S}^{n} \) be an imbedding. Then\n\n\[ \n{\widetilde{H}}_{i}\left( {{S}^{n} - h\left( {S}^{k}\right) }\right) \simeq \left\{ \begin{array}{ll} \mathbf{Z} & \text{ if }i = n - k - 1, \\ 0 & \text{ otherwise. } \end{array}\right. \n\] | Proof. Let \( n \) be fixed. We prove the theorem by induction on \( k \) . First take the case \( k = 0 \) . Then \( h\left( {S}^{0}\right) \) consists of two points \( p \) and \( q \) . Since \( {S}^{n} - p - q \approx \) \( {\mathbf{R}}^{n} - \mathbf{0} \), and \( {\mathbf{R}}^{n} - \mathbf{0} \) has the homotopy t... | Yes |
Theorem 36.3 (The generalized Jordan curve theorem). Let \( n > 0 \) . Let \( C \) be a subset of \( {S}^{n} \) homeomorphic to the \( n - 1 \) sphere. Then \( {S}^{n} - C \) has precisely two components. of which \( C \) is the common (topological) boundary. | Proof. Applying the preceding theorem to the case \( k = n - 1 \), we see that \( {\widetilde{H}}_{0}\left( {{S}^{n} - C}\right) \cong \mathbf{Z} \) . Thus \( {S}^{n} - C \) has precisely two path components (which are the same as its components, as noted earlier). Let \( {W}_{1} \) and \( {W}_{2} \) be these path comp... | Yes |
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