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Proposition 119.12. Let \( X \) be a topological space. If \( X \) is dominated by a CW-complex, then \( X \) is homotopy equivalent to a CW-complex. | Proof. We will prove Proposition 119.12 in Section 119.4, using the Whitehead Theorem 119.9. Alternatively see the proofs of Theorems 85.17, 85.20 and 85.21 for alternative proofs. | No |
Theorem 85.14. (Dugundji Extension Theorem) Let \( V \) be a real vector space and \( \parallel - \parallel : V \rightarrow {\mathbb{R}}_{ \geq 0} \) a norm on \( V \) . We equip \( V \) with the corresponding metric and we use this metric to view \( V \) as a topological space. With respect to this topology every conv... | Proof. This theorem was proved by James Dugundji [Dug51, Corollary 4.2] in 1952. A proof is also given in [MaS82, Theorem I.3.1], [Maye89, Theorem I.5.7] and [Saka13], Theorem 6.1.1]. | Yes |
Theorem 85.15. (Hanner’s Theorem) Let \( X \) be a topological space which is second-countable and metrizable. If \( X \) is the union of a family \( {\left\{ {U}_{i}\right\} }_{i \in I} \) of open subsets such that each \( {U}_{i} \) is an ANR, then \( X \) itself is an ANR. | Proof. This theorem was proved by Olof Hanner [Han51, Theorem 3.3] in 1950. An exposition of the proof is also given in Fera04, Theorem 2.77] and in [Hu65, Theorem III.8.1]. | No |
Corollary 85.16. Every topological manifold is an ANR. | Proof. Let \( X \) be an \( n \) -dimensional topological manifold. By definition \( X \) is second-countable. By Corollary 9.2 we know that \( X \) is metrizable. In Lemma 6.9 (1) we showed that \( X \) is locally homeomorphic to some non-empty convex subset of \( {\mathbb{R}}^{n} \) . By the Dugundji Extension Theore... | Yes |
Theorem 85.17. Let \( X \) be a topological space. The following statements are equivalent:\n\n(1) \( X \) is dominated by a countable CW-complex.\n\n(2) \( X \) is homotopy equivalent to a countable \( \mathrm{{CW}} \) -complex.\n\n(3) \( X \) is homotopy equivalent to a countable simplicial complex.\n\n(3’) \( X \) i... | Proof. The fact that all the statements are equivalent goes well beyond the scope of these notes. Thus we just give the necessary references. The implications (3’) \( \Rightarrow \left( 3\right) \Rightarrow \left( 2\right) \Rightarrow \left( 1\right) \) are of course trivial. Next note that the equivalence of (2) and (... | Yes |
Every topological manifold is homotopy equivalent to a countable locally finite simplicial complex. | This statement follows from Corollary \( {85.16} \) together with Theorem 85.17 (4) \( \Rightarrow \) (3’) and the fact that topological manifolds are by definition second-countable. | Yes |
Proposition 85.19. If \( X \) is a CW-complex with countably many cells, then for any \( n \in \mathbb{N} \) and any \( {x}_{0} \in X \) the group \( {\pi }_{n}\left( {X,{x}_{0}}\right) \) is countable. | Proof. This follows from the combination of Theorem 85.17 \( \left( 1\right) \Rightarrow \left( 4\right) \) together with Proposition 40.7 and Proposition 62.9. | No |
Theorem 85.20. If \( X \) is a topological space, then the following statements are equivalent:\n\n(1) \( X \) is dominated by a CW-complex.\n\n(2) \( X \) is homotopy equivalent to a CW-complex.\n\n(3) \( X \) is homotopy equivalent to a simplicial complex.\n\n(4) \( X \) is homotopy equivalent to an ANR. | Proof. The implication \ | No |
Theorem 85.21. Let \( X \) be a topological space. The following statements are equivalent:\n\n(2) \( X \) is homotopy equivalent to a finite \( \mathrm{{CW}} \) -complex.\n\n(3) \( X \) is homotopy equivalent to a finite simplicial complex.\n\n(4) \( X \) is homotopy equivalent to a compact ANR.\n\nFurthermore the fol... | Proof. The fact that (2) and (3) are equivalent is a consequence of Lemma 61.24 and Proposition 62.12. Next note that one can show, with some effort, that it is a consequence of the Dugundji Extension Theorem 85.14 that every finite simplicial complex is an ANR, we refer to [LW69, p. 210] and HNV04, p. 472] for details... | Yes |
Theorem 85.22. There exists a topological space that is dominated by a finite CW-complex but that is not homotopy equivalent to a finite CW-complex. | Proof. The statement is proved in [Wal165a, p. 66]. Alternatively see [Var89, p. 163] and FR01, Theorem 3.1]. More precisely, in [Wal165a, Var89] it is shown that to a topological space \( X \) that is dominated by a finite CW-complex one can associate an invariant in the \ | No |
Corollary 85.23. Every compact topological manifold is homotopy equivalent to a finite simplicial complex. | Proof. This statement follows immediately from Corollary 85.16 together with Theorem 85.21 (4) \( \Rightarrow \) (3). A very different proof for the corollary was given by Rob Kirby and Larry Siebenmann in [KSi69, Chapter III.2] and [KSi77, p. 744]. | No |
Every closed \( n \) -dimensional topological manifold is homotopy equivalent to a finite \( n \) -dimensional simplicial complex. | Sketch of PROOF. First note that the 1-dimensional case follows immediately from the classification result proved in Theorem 7.1. Furthermore the 2-dimensional case and the 3-dimensional case follow from Radó’s Theorem 85.27 and Moise’s Theorem [85.28] below. Thus we can now assume that \( n \geq 4 \) .\n\nFirst note t... | No |
Proposition 85.25. Every closed connected non-empty n-dimensional topological manifold is homotopy equivalent to a finite \( n \) -dimensional CW-complex which has a single 0-cell and which has a single n-cell. | Proof. Let \( X \) be a closed connected non-empty \( n \) -dimensional topological manifold. We pick an embedding \( \varphi : {\bar{B}}^{n} \rightarrow X \) . It follows from Proposition 44.3 that \( Y \mathrel{\text{:=}} X \smallsetminus \varphi \left( {B}^{n}\right) \) is a compact connected non-empty \( n \) -dime... | Yes |
Theorem 85.27. (Radó's Theorem) Every 2-dimensional topological manifold admits a PL-structure, in particular it admits a simplicial and a CW-structure. | Proof. In 1926 it was shown by Tibor Radó Rad26 that every 2-dimensional topological manifold admits a simplicial structure. It is elementary to see that the simplicial structure of any 1-dimensional topological manifold is in fact a PL-structure. It follows from Proposition 64.11 that the simplicial structure of any 2... | No |
Theorem 85.30. (Kirby-Quinn-Siebenmann Theorem) Every closed topological manifold of dimension \( \geq 5 \) has a CW-structure. | Proof. For dimensions \( \geq 6 \) this theorem was proved in the 1970s by Rob Kirby and Larry Siebenmann KSi77, Essay III.2. In 1982 this result was extended to the 5-dimensional case by Frank Quinn Qu82, Theorem 2.3.1]. | Yes |
Theorem 85.32. There exists a closed 4-dimensional topological manifold that does not admit a simplicial structure. | Proof. This theorem was proved by Andrew Casson, see AkM90, page xvi], in the 1980s building on the work of Mike Freedman Fre82. An account of the proof is also given in Sav12, Theorem 18.3]. | Yes |
Theorem 85.33. (Manolescu’s Theorem) For every \( n \in {\mathbb{N}}_{ \geq 5} \) there exists a closed \( n \) -dimensional topological manifold that does not admit a simplicial structure. | Proof. The theorem is proved in Man16a. A more relaxed exposition of the key ideas of the proof is also given in [Man16b]. Note that, as is explained in [Man16a, Man16b], in dimension 5 the examples are necessarily non-orientable, whereas for every \( n \geq 5 \) there exists a closed orientable \( n \) -dimensional to... | No |
Let \( X \) and \( Y \) be homotopy equivalent topological spaces. If the Euler characteristic is defined for one of the two topological spaces, then it is defined for the other topological space and we have\n\n\[ \chi \left( X\right) = \chi \left( Y\right) \] | This statement follows immediately from Corollary 42.8. | No |
Lemma 86.1. Let \( n \in \mathbb{N} \), let \( U \) be an open subset of \( {\mathbb{R}}^{n} \), let \( A \) be a convex bounded subset of \( U \) with \( \bar{A} \subset U \) and let \( x \in A \) . Then for every abelian group \( G \) and every \( i \in {\mathbb{N}}_{0} \) the following inclusion induced maps are iso... | Proof. Let \( U \) be an open subset of \( {\mathbb{R}}^{n} \), let \( A \) be a convex bounded subset of \( U \) with \( \bar{A} \subset U \) and let \( x \in A \) . To simplify the notation we drop the coefficients \( G \) from the notation and without loss of generality we assume that \( x = 0 \) . Now let \( i \in ... | Yes |
For any \( k \in {\mathbb{N}}_{0} \) we have\n\n\[
{\mathrm{H}}_{k}\left( {{\Delta }^{n},\partial {\Delta }^{n};R}\right) = \left\{ \begin{array}{ll} 0, & \text{ if }k \neq n, \\ R \cdot \left\lbrack {{\operatorname{id}}_{{\Delta }^{n}} \otimes 1}\right\rbrack , & \text{ if }k = n. \end{array}\right.
\] | (1) For \( k \neq n \) the statement is proved the same way as Lemma 45.1. Next recall that in Lemma 45.2 (2) we saw that the identity map id: \( {\Delta }^{n} \rightarrow {\Delta }^{n} \) represents a generator of \( {\mathrm{H}}_{n}\left( {{\Delta }^{n},\partial {\Delta }^{n};\mathbb{Z}}\right) \cong \mathbb{Z} \) . ... | Yes |
Lemma 86.3. Let \( R \) be a commutative ring, let \( U \) be an open subset of \( {\mathbb{R}}^{n} \), let \( P \in U \) and let \( \sigma : {\Delta }^{n} \rightarrow U \) be a singular \( n \) -simplex with the following properties:\n\n(1) the point \( P \) lies in the image of \( {\overset{ \circ }{\Delta }}^{n} \),... | Proof. We denote by \( Q \in {\overset{ \circ }{\Delta }}^{n} \) the point with \( \sigma \left( Q\right) = P \) . To simplify the notation we work throughout the proof with the coefficients \( R = \mathbb{Z} \) . Using the fact that the inclusion induced map\n\n\[ \n{\mathrm{H}}_{n}\left( {U, U\smallsetminus \{ P\} }\... | No |
Lemma 86.4. Let \( M \) be an \( n \) -dimensional topological manifold and let \( R \) be a commutative ring. Then for every \( x \in M \smallsetminus \partial M \) and any \( k \in {\mathbb{N}}_{0} \) we have\n\n\[ \n{\mathrm{H}}_{k}\left( {M, M\smallsetminus \{ x\} ;R}\right) \cong \left\{ \begin{array}{ll} R, & \te... | Proof. Let \( M \) be an \( n \) -dimensional topological manifold, let \( R \) be a commutative ring and let \( x \in M \smallsetminus \partial M \) . Since \( x \notin \partial M \) we can pick a chart \( \Phi : U \rightarrow V \) such that \( V \) is an open subset of \( {\mathbb{R}}^{n} \) . We write \( y = \Phi \l... | Yes |
Lemma 86.5. Let \( M \) be an \( n \) -dimensional topological manifold and let \( R \) be a commutative ring.\n\n(1) Given any subset \( A \subset M \smallsetminus \partial M \) the set \( {\Gamma }_{R}^{A}\left( M\right) \) admits a unique \( R \) -module structure such that for each \( y \in A \) the map \( {\Gamma ... | Proof (*).\n\n(1) We will prove this statement in Exercise 86.2.\n\n(2) Let \( A \subset M \smallsetminus \partial M \) be a subset and let \( \alpha \in {\mathrm{H}}_{n}\left( {M, M \smallsetminus A;R}\right) \) . Let \( x \in A \) . We need to show that there exists an open neighborhood \( U \) of \( x \) in \( A \) ... | No |
Lemma 86.9. Let \( M \) be an-dimensional topological manifold.\n\n(1) Let \( R \) be a commutative ring and let \( A \subset M \smallsetminus \partial M \) be a connected subset. If \( M \) is\n\n\( R \) -orientable, then for every \( x \in M \smallsetminus \partial M \) the map\n\n\[ \Phi : {\Gamma }_{R}^{A}\left( M\... | Proof. Let \( M \) be a connected \( n \) -dimensional topological manifold.\n\n(1) Let \( R \) be a commutative ring such that \( M \) is \( R \) -orientable. We pick an \( R \) -orientation \( {\left\{ {\mu }_{y}\right\} }_{y \in M \smallsetminus \partial M} \) for \( M \) . Next let \( A \subset M \smallsetminus \pa... | Yes |
Lemma 86.10. Let \( M \) be a topological manifold.\n\n(1) For any point \( x \in M \smallsetminus \partial M \) there exist precisely two \( \mathbb{Z} \) -orientations for \( M \) at \( x \) .\n\n(2) If \( M \) is connected and orientable, then \( M \) admits precisely two \( \mathbb{Z} \) -orientations. | Proof. The first statement is an immediate consequence of the fact, proved in Lemma 86.4, that for any \( x \in M \smallsetminus \partial M \) we have \( {\mathrm{H}}_{n}\left( {M, M\smallsetminus \{ x\} ;\mathbb{Z}}\right) \cong \mathbb{Z} \) . The second statement follows easily from Proposition 86.6 and the definiti... | Yes |
Lemma 86.12. Let \( \Psi : V \rightarrow {V}^{\prime } \) be a diffeomorphism between two open subsets of \( {\mathbb{R}}^{n} \) . Then for any \( x \in V \) we have\n\n\( \Psi \) preserves the standard \( \Psi \) preserves the orientation of \( V \) in the \( \mathbb{Z} \) -orientation of \( V \) at the point \( x \) ... | Proof (*). Let \( \Psi : V \rightarrow {V}^{\prime } \) be a diffeomorphism between two open subsets of \( {\mathbb{R}}^{n} \) and let \( x \in V \) . We write \( {x}^{\prime } = \Psi \left( x\right) \) . Given \( w \in {\mathbb{R}}^{n} \) we denote by \( {t}_{w} \) the translation \( {\mathbb{R}}^{n} \rightarrow {\mat... | Yes |
Proposition 86.13. Every topological manifold is \( {\mathbb{F}}_{2} \) -orientable. | Proof. Let \( M \) be an \( n \) -dimensional topological manifold. Let \( x \in M \smallsetminus \partial M \) . By Lemma 86.4 we know that \( {\mathrm{H}}_{n}\left( {M, M\smallsetminus \{ x\} ;{\mathbb{F}}_{2}}\right) \cong {\mathbb{F}}_{2} \) . We define \( {\mu }_{x} \) to be the unique non-trivial element in \( {\... | Yes |
Lemma 86.14. Let \( M \) be a topological manifold. If \( M \) is orientable, then it is \( R \) -orientable for any commutative ring \( R \) . | Proof. Let \( M \) be an \( n \) -dimensional orientable topological manifold and let \( R \) be a commutative ring. Let \( x \in M \smallsetminus \partial M \) . It follows from the Universal Coefficient Theorem 57.19, together with the fact that \( {\mathrm{H}}_{n - 1}\left( {M, M\smallsetminus \{ x\} ;\mathbb{Z}}\ri... | Yes |
Corollary 86.16. Let \( M \) be a connected topological manifold without boundary. If \( M \) is non-orientable, then there exists an epimorphism \( {\pi }_{1}\left( M\right) \rightarrow {\mathbb{Z}}_{2} \) . | Proof. The proof of the corollary is basically identical to the proof of Corollary 17.4, we just need to replace Proposition 17.3 by Proposition 86.15. | No |
Lemma 86.17. Let \( R \) be a commutative ring and let \( M \) be a topological manifold that is equipped with an \( R \)-orientation.\n\n(1) Let \( N \) be a topological manifold and let \( f : N \rightarrow M \) be a map. If \( f : N \smallsetminus \partial N \rightarrow M \) is a local homeomorphism, then there exis... | Proof. To simplify the notation we suppress the ring \( R \) from the notation. Thus let \( M \) be an \( n \)-dimensional topological manifold that is equipped with an orientation \( {\left\{ {\alpha }_{x}\right\} }_{x \in M \smallsetminus \partial M} \).\n\n(1) Let \( N \) be a topological manifold and let \( f : N \... | No |
Corollary 87.4. Let \( M \) be a compact \( n \) -dimensional topological manifold.\n\n(1) If \( \left\lbrack M\right\rbrack \) is a fundamental class, then \( {\left\{ {\left\lbrack M\right\rbrack }_{x}\right\} }_{x \in M \smallsetminus \partial M} \) defines an orientation for \( M \) .\n\n(2) If \( {\left\{ {\alpha ... | Proof. Let \( M \) be a compact non-empty \( n \) -dimensional topological manifold. By considering the components of \( M \) separately we see that without loss of generality we can assume that \( M \) is connected. It is also clear that we only have to study the case that \( M \) is non-empty.\n\n(1) Let \( \left\lbr... | Yes |
Corollary 87.5. Let \( M \) be a compact, non-empty \( n \) -dimensional topological manifold.\n\n(1) If \( M \) is connected, then\n\n\[ \n{\left\lbrack M\right\rbrack }_{{\mathbb{F}}_{2}} \mathrel{\text{:=}} \text{the unique non-zero element of}{\mathrm{H}}_{n}\left( {M,\partial M;{\mathbb{F}}_{2}}\right) \cong {\mat... | Proof. It follows immediately from Theorem 87.2 (2) that the class \( {\left\lbrack M\right\rbrack }_{{\mathbb{F}}_{2}} \) that we had just defined in (1) is indeed an \( {\mathbb{F}}_{2} \) -fundamental class for \( M \) and that it is unique. As we pointed out on page 2121, the second statement is an immediate conseq... | Yes |
Lemma 87.6. The two definitions of the fundamental class of a compact oriented nonempty smooth manifold agree. | Proof (*). Let \( M \) be a compact \( n \) -dimensional smooth manifold that is oriented in the sense of Section 6.11. The case \( n = 0 \) is basically trivial. Hence we now only consider the case \( n \geq 1 \) . As we will see, the lemma follows almost immediately from the definitions.\n\nNow let us first recall th... | Yes |
Proposition 87.8. Let \( M \) be a compact connected \( n \) -dimensional topological manifold. Let \( \left( {{\sigma }_{1} + \cdots + {\sigma }_{m}}\right) \otimes 1 \in {\mathrm{C}}_{n}\left( {M,\partial M;{\mathbb{F}}_{2}}\right) = {\mathrm{C}}_{n}\left( {M,\partial M}\right) \otimes {\mathbb{F}}_{2} \) be a cycle.... | Proof. We leave it to the reader to modify the proofs of Propositions [68.4] and [68.18] to obtain the above result. | No |
Corollary 87.11. Let \( M \) be a n-dimensional topological manifold with empty boundary, let \( R \) be a commutative ring and let \( A \subset M \) be a compact connected subset. If \( M \) is \( R \) -orientable, then for every \( x \in A \) the map\n\n\[ \n{\mathrm{H}}_{n}\left( {M, M \smallsetminus A;R}\right) \ri... | Proof. We consider the following diagram:\n\n\n\nThe diagram commutes by definition. Since \( A \) is compact we know by Theorem 87.10 (i) that the left diagonal map is an isomorphism. Furthermore, since \( A \) is... | Yes |
Let \( M \) be an oriented \( n \) -dimensional topological manifold with empty boundary and let \( R \) be a commutative ring. We suppose that \( M \) is equipped with an \( R \) - orientation. We use the notation from Theorem 87.10.\n\n(2) Let \( U \) be an open subset of \( M \) and let \( A \subset U \) be a compac... | Proof. The lemma follows easily from the functoriality of homology groups and the uniqueness statement of Theorem 87.10. | No |
Lemma 87.14. Let \( M \) be an \( n \) -dimensional topological manifold, let \( R \) be a commutative ring. Let \( \mathcal{A} \) be a family of compact subsets of \( M \) with the following two properties: (1) Statements (i) and (ii) from Theorem 87.10 hold for each \( A \in \mathcal{A} \). (2) The family \( \mathcal... | Proof. Given \( k \in \mathbb{N} \) we write \[ {\mathcal{U}}_{k} \mathrel{\text{:=}} \{ \text{all subsets of}M\text{that are the union of at most}k\text{subsets in}\mathcal{A}\} \text{.} \] We claim that for every \( k \in \mathbb{N} \) statements (i) and (ii) hold for all elements in \( {\mathcal{U}}_{k} \). We prove... | Yes |
Lemma 87.15. Let \( R \) be a commutative ring and let \( {\left\{ {\mu }_{x}\right\} }_{x \in {\mathbb{R}}^{n}} \) be an \( R \) -section on \( {\mathbb{R}}^{n} \), let \( A \subset {\mathbb{R}}^{n} \) be a convex bounded subset and let \( \beta \in {\mathrm{H}}_{n}\left( {{\mathbb{R}}^{n} \mid A;R}\right) \) . If \( ... | Proof. The proof is very similar to the proof of Proposition 86.6. The following are the two key observations:\n\n(1) a convex set is path-connected and thus connected,\n\n(2) since \( A \) is convex, for any \( x \in A \) and any \( \epsilon > 0 \) the intersection \( A \cap {B}_{\epsilon }\left( x\right) \) is the in... | No |
Lemma 87.16. Statements (i) and (ii) from Theorem 87.10 hold for \( M = {\mathbb{R}}^{n} \), any commutative ring \( R \) and any convex compact subset \( A \) of \( {\mathbb{R}}^{n} \) . | Proof. Let \( A \) be a convex compact subset of \( {\mathbb{R}}^{n} \) and let \( R \) be a commutative ring. As pointed out above, if \( A = \varnothing \), then there is nothing to prove. Thus we can assume that \( A \neq \varnothing \) .\n\nNow we can prove that \( A \) satisfies the statements (i) and (ii).\n\n(i)... | Yes |
Lemma 87.17. Every subset of \( {\mathbb{R}}^{n} \) that is the union of finitely many convex compact sets satisfies the statements (i) and (ii) from Theorem 87.10. | Proof. We consider\n\n\[ \mathcal{A} \mathrel{\text{:=}} \left\{ {\text{ all convex compact subsets of }{\mathbb{R}}^{n}}\right\} .\n\]\n\nThis family of subsets of \( {\mathbb{R}}^{n} \) is closed under intersections \( {}_{1}^{1253} \) By Lemma 87.16 the two statements hold for each \( A \in \mathcal{A} \) . The lemm... | No |
Lemma 87.19. Let \( M \) be an \( n \) -dimensional topological manifold with empty boundary and let \( A \subset M \) be a compact subset. If there exists a chart \( \Phi : U \rightarrow {\mathbb{R}}^{n} \) such that \( A \) is contained in \( U \), then statements (i) and (ii) hold for \( A \) . | Proof. The lemma is a fairly straightforward consequence of Lemma 87.18 and the fact that we have isomorphisms\n\n\[ \n{\mathrm{H}}_{n}\left( {M, M \smallsetminus A;R}\right) \underset{ \uparrow }{\overset{ \cong }{ \leftarrow }}{\mathrm{H}}_{n}\left( {U, U \smallsetminus A;R}\right) \underset{ \uparrow }{\overset{\Phi... | No |
Proposition 87.20. Let \( M \) be a connected \( n \) -dimensional topological manifold and let \( R \) be a commutative ring. If \( M \) is non-compact and if \( \partial M = \varnothing \), then \( {\mathrm{H}}_{k}\left( {M;R}\right) = 0 \) for every \( k \geq n \). | Proof. Let \( M \) be a connected non-compact \( n \) -dimensional topological manifold such that \( \partial M = \varnothing \), let \( R \) be a commutative ring and let \( k \geq n \). Let \( \alpha \in {\mathrm{H}}_{k}\left( {M;R}\right) \). We need to show that \( \alpha = 0 \). We represent \( \alpha \) by a cycl... | Yes |
Lemma 87.23. Let \( A \) be a finitely generated abelian group. Then the following holds:\n\n\( A \) is a free abelian group \( \Leftrightarrow \operatorname{Tor}\left( {A,{\mathbb{Z}}_{p}}\right) = 0 \) for all primes \( p \) . | Proof of Proposition 87.22 (1) I. Let \( M \) be a closed connected \( n \) -dimensional topological manifold. By Proposition 85.13 (4) we know that \( {\mathrm{H}}_{n - 1}\left( {M;\mathbb{Z}}\right) \) is a finitely generated abelian group.\n\nWe first consider the case that \( M \) is orientable. We want to show tha... | No |
Lemma 87.24. Let \( M \) be a compact oriented \( n \) -dimensional topological manifold. Furthermore let \( W \subset M \) be a compact non-empty codimension-zero submanifold. We write \( \overset{ \circ }{W} = W \smallsetminus \partial W \) . The following two statements hold:\n\n(1) The inclusion induced map \( {\ma... | Proof \( \left( *\right) \) .\n\n(1) The first statement is an immediate consequence of the Excision Theorem 44.10.\n\n(2) We denote by \( \Phi : {\mathrm{H}}_{n}\left( {M,\partial M;\mathbb{Z}}\right) \rightarrow {\mathrm{H}}_{n}\left( {W,\partial W;\mathbb{Z}}\right) \) the map given in statement (2). We have to show... | No |
Lemma 87.26. Let \( M \) be a compact oriented \( n \) -dimensional topological manifold with non-empty boundary. If we equip \( \mathrm{D}M \) with the orientation coming from Lemma 86.18, then\n\n\[ \left\lbrack {\mathrm{D}M}\right\rbrack = D\left( \left\lbrack M\right\rbrack \right) \in {\mathrm{H}}_{n}\left( {\math... | Proof. Let \( {\left\{ {\mu }_{x}\right\} }_{x \in M \smallsetminus \partial M} \) be an orientation for \( M \) . We denote by \( {\left\{ {\alpha }_{x}\right\} }_{x \in \mathrm{D}M} \) the corresponding orientation for D \( M \) that we had constructed in Lemma 86.18. To simplify the notation we write \( \mathrm{D}M ... | Yes |
Proposition 87.27. Let \( M \) be a compact \( n \) -dimensional topological manifold with nonempty boundary.\n\n(1) We denote by \( {\partial }_{n} : {\mathrm{H}}_{n}\left( {M,\partial M;\mathbb{Z}}\right) \rightarrow {\mathrm{H}}_{n - 1}\left( {\partial M;\mathbb{Z}}\right) \) the connecting homomorphism of the long ... | Proof (*). As a health warning we point out that in this proof we make heavy use of the notation introduced in Proposition 86.19. We only provide the proof of Statement (1); the proof of Statement (2) is almost identical.\n\nLet \( {\left\{ {\mu }_{y}\right\} }_{y \in M \smallsetminus \partial M} \) be the orientation ... | Yes |
Corollary 87.28. Let \( M \) be a compact \( n \) -dimensional topological manifold. Then the following two statements hold:\n\n(1) We have\n\n\[ \ker \left( {{\mathrm{H}}_{n - 1}\left( {\partial M;{\mathbb{F}}_{2}}\right) \rightarrow {\mathrm{H}}_{n - 1}\left( {M;{\mathbb{F}}_{2}}\right) }\right) = {\mathbb{F}}_{2} \c... | Proof. The proof is verbatim the same as the proof of Corollary [68.11, the only difference in the proof is that we need to replace Proposition 68.9 by Proposition 87.27,\n\nAs a reminder, the first statement is proved using the long exact sequence in homology of the pair \( \left( {M,\partial M}\right) \) together wit... | Yes |
Lemma 87.31. Let \( M \) be an oriented \( n \) -dimensional topological manifold and let \( R \) be a commutative ring. We suppose that \( M \) is equipped with an \( R \) -orientation. We use the notation from Theorem 87.30.\n\n(1) Let \( K \subset L \subset M \) be compact subsets that are products close to the boun... | Proof. The lemma follows easily from the functoriality of homology groups and the uniqueness statement of Theorem 87.30. | No |
Proposition 87.32. Let \( f : M \rightarrow N \) be a map between two closed oriented connected \( n \) -dimensional topological manifolds \( M \) and \( N \) and let \( x \in M \) . Let \( y \in N \) such that \( {f}^{-1}\left( {\{ y\} }\right) \) consists of finitely many points \( {x}_{1},\ldots ,{x}_{m} \) . Then\n... | Proof of Proposition 87.32. The proof of the proposition is similar to the proofs of Propositions 45.23 and 69.7. Thus, in a vein attempt to keep these notes concise, we will\n\nnot provide a proof. Instead we hope that the reader will take the baton and carry out the proof. | No |
The following equalities hold:\n\n\[ \n{\mathrm{{PD}}}_{M \times N}\left( {{i}_{ * }\left( \left\lbrack M\right\rbrack \right) }\right) = {\left( -1\right) }^{mn} \cdot {q}^{ * }\left( {\left\lbrack N\right\rbrack }^{ * }\right) \in {\mathrm{H}}^{m}\left( {M \times N, M \times \partial N;\mathbb{Z}}\right) \]\n\n\[ \n{... | Proof. By Proposition 84.2 (2) we have\n\n\[ \n{p}^{ * }\left( {\left\lbrack M\right\rbrack }^{ * }\right) \cap \left\lbrack {M \times N}\right\rbrack = {i}_{ * }\left( \left\lbrack M\right\rbrack \right) \;\text{ and }\;{q}^{ * }\left( {\left\lbrack N\right\rbrack }^{ * }\right) \cap \left\lbrack {M \times N}\right\rb... | Yes |
Proposition 88.3. Let \( M \) be a compact orientable \( n \) -dimensional topological manifold and suppose that we are given a decomposition \( \partial M = A \cup B \) where \( A \) and \( B \) are compact \( \left( {n - 1}\right) \) -dimensional submanifolds of \( \partial M \) such that \( A \cap B = \partial A = \... | Proof. We have the following isomorphisms\n\nPoincaré Duality Theorem 88.1 Universal Coefficient Theorem 75.13\n\n\[ \n\begin{aligned} {\mathrm{H}}_{n - k}\left( {M, A;\mathbb{Z}}\right) & \overset{ \downarrow }{ \cong }{\mathrm{H}}^{k}\left( {M, B;\mathbb{Z}}\right) & \overset{ \downarrow }{ \cong }\operatorname{Hom}\... | Yes |
Corollary 88.4. Let \( M \) be a compact orientable \( n \) -dimensional topological manifold and suppose that we are given a decomposition \( \partial M = A \cup B \) where \( A \) and \( B \) are compact \( \left( {n - 1}\right) \) -dimensional submanifolds of \( \partial M \) such that \( A \cap B = \partial A = \pa... | Proof. We have\n\n\[ \n{\mathrm{H}}_{n - 1}\left( {M, A;\mathbb{Z}}\right) \overset{\begin{matrix} \text{ Proposition } \\ \downarrow \end{matrix}}{ \cong }\underset{\text{free abelian group }}{\underbrace{{\mathrm{{FH}}}_{1}\left( {M, B}\right) }} \oplus \overset{\text{free abelian group }}{\overbrace{\underset{ = 0\t... | Yes |
Theorem 88.6. Let \( M \) be a compact \( n \) -dimensional topological manifold and let \( k \in {\mathbb{N}}_{0} \). (1) We have \[ {\dim }_{{\mathbb{F}}_{2}}\left( {{\mathrm{H}}_{k}\left( {M;{\mathbb{F}}_{2}}\right) }\right) = {\dim }_{{\mathbb{F}}_{2}}\left( {{\mathrm{H}}_{n - k}\left( {M,\partial M;{\mathbb{F}}_{2... | Proof. Let \( M \) be a compact \( n \) -dimensional topological manifold. We have \[ {\dim }_{{\mathbb{F}}_{2}}\left( {{\mathrm{H}}_{k}\left( {M;{\mathbb{F}}_{2}}\right) }\right) \underset{ \uparrow }{ = }{\dim }_{{\mathbb{F}}_{2}}\left( {{\mathrm{H}}^{n - k}\left( {M,\partial M;{\mathbb{F}}_{2}}\right) }\right) \unde... | Yes |
Proposition 88.7. For every compact odd-dimensional topological manifold \( M \) we have\n\n\[ \chi \left( M\right) = \frac{1}{2}\chi \left( {\partial M}\right) \]\n\nIn particular if \( M \) is closed, then \( \chi \left( M\right) = 0 \) . | Proof. Let \( M \) be a compact topological manifold of dimension \( {2n} + 1 \) . Then\n\n\[ \chi \left( M\right) = \mathop{\sum }\limits_{{i = 0}}^{{{2n} + 1}}{\left( -1\right) }^{i} \cdot {\dim }_{{\mathbb{F}}_{2}}\left( {{\mathrm{H}}_{i}\left( {M;{\mathbb{F}}_{2}}\right) }\right) \; = \mathop{\sum }\limits_{{i = 0}... | Yes |
Proposition 85.34 (4)\n\n\[ \n\begin{array}{l} = - \mathop{\sum }\limits_{{j = 0}}^{{{2n} + 1}}{\left( -1\right) }^{j} \cdot {\dim }_{{\mathbb{F}}_{2}}\left( {{\mathrm{H}}_{j}\left( {M,\partial M;{\mathbb{F}}_{2}}\right) }\right) = - \chi \left( {\overline{M,\partial }M}\right) = - \chi \left( M\right) + \chi \left( {\... | follows from the substitution \( j = {2n} + 1 - i\; \) Proposition 8.34 \( \left( 4\right) \; \) Proposition 8.54 \( \left( 4\right) \) \n\ntogether with the fact that \( {\left( -1\right) }^{{2n} + 1} = - 1 \n\nSummarizing we have shown that \( \chi \left( M\right) = - \chi \left( M\right) + \chi \left( {\partial M}\r... | No |
Corollary 88.9. The 2-dimensional topological manifold \( {\mathbb{{RP}}}^{2} \) is not the boundary of a compact 3-dimensional topological manifold. | Proof. Suppose there was a compact 3-dimensional topological manifold \( M \) with \( \partial M = \) \( {\mathbb{{RP}}}^{2} \) . We would obtain that\n\n\[ \chi \left( M\right) = \frac{1}{2}\chi \left( {\partial M}\right) = \frac{1}{2}\chi \left( {\mathbb{{RP}}}^{2}\right) = \frac{1}{2}. \]\n\nBut the Euler characteri... | Yes |
Lemma 88.11. Let \( M \) be a connected closed 3-dimensional topological manifold. If \( M \) is non-orientable, then \( {\mathrm{H}}_{1}\left( {M;\mathbb{Z}}\right) \) is infinite. | Proof. Let \( M \) be a connected closed 3-dimensional topological manifold. We have\n\n\[ \begin{aligned} 0 & = \chi \left( M\right) & = \mathop{\sum }\limits_{{n \in {\mathbb{N}}_{0}}}{\left( -1\right) }^{n} \cdot {b}_{n}\left( M\right) & = {b}_{0}\left( M\right) - {b}_{1}\left( M\right) + {b}_{2}\left( M\right) & = ... | Yes |
Lemma 88.12. Let \( p : \widetilde{N} \rightarrow N \) be a finite cover of a closed, oriented, non-empty \( n \) - dimensional topological manifold. If we equip \( \widetilde{N} \) with the orientation given by Lemma 86.17 (3), then for any \( k \in {\mathbb{N}}_{0} \) the Umkehr map \( {p}^{!} : {\mathrm{H}}_{k}\left... | Proof. We start our proof with the following claim.\n\nClaim. Let \( r \leq s \) . Given any singular simplex \( \sigma : {\Delta }^{s} \rightarrow N \) and any cochain \( \phi \in {\mathrm{C}}^{r}\left( {N;\mathbb{Z}}\right) \) we have\n\n\[ {p}^{ * }\left( \phi \right) \cap {p}^{ * }\left( \sigma \right) = {p}^{ * }\... | Yes |
Proposition 88.13. Let \( n \geq 2 \) . Give a closed connected \( n \) -dimensional topological manifold \( X \) the following statements are equivalent:\n\n(1) \( X \) is \( \left\lfloor \frac{n}{2}\right\rfloor \) -connected,\n\n(2) \( X \) is \( \left( {n - 1}\right) \) -connected,\n\n(3) \( {\pi }_{1}\left( X\righ... | Proof. We have \( \left( 1\right) \Rightarrow \left( 4\right) \) by Corollary 53.7. We can easily show that \( \left( 4\right) \Rightarrow \left( 3\right) \) by applying the Poincaré Duality Theorem 88.1 (here we use that \( X \) is orientable by Corol- \( \operatorname{lary}\left\lbrack \overline{86.16}\right\rbrack \... | No |
Proposition 88.14. Let \( f : M \rightarrow N \) be a map between closed oriented non-empty \( n \) -dimensional topological manifolds. For every \( k \in {\mathbb{N}}_{0} \) the following statements hold:\n\n(1) The map \( {f}_{ * } \circ {f}^{!} \) is multiplication by \( \deg \left( f\right) \) . | Proof. Let \( f : M \rightarrow N \) be a between closed oriented non-empty \( n \) -dimensional topological manifolds. We write \( d = \deg \left( f\right) \) . Let \( \sigma \in {\mathrm{H}}_{k}\left( {N;\mathbb{Z}}\right) \) . We have\n\n\[ \begin{aligned} \left( {{f}_{ * } \circ {f}^{!}}\right) \left( \sigma \right... | Yes |
Corollary 88.15. There is no degree-one map from \( {S}^{1} \times {S}^{2} \) to \( {\mathbb{{RP}}}^{3} \) . | Proof. The corollary follows immediately from Proposition 88.14 and the observation that the group \( {\mathrm{H}}_{1}\left( {{\mathbb{{RP}}}^{3};\mathbb{Z}}\right) \cong {\mathbb{Z}}_{2} \) is not a subsummand of \( {\mathrm{H}}_{1}\left( {{S}^{1} \times {S}^{2};\mathbb{Z}}\right) \cong \mathbb{Z} \) . | Yes |
Corollary 88.16. Let \( n \in {\mathbb{N}}_{ \geq 2} \) and let \( M \) be a closed oriented connected \( n \) -dimensional topological manifold. If there exists a degree one map \( f : {S}^{n} \rightarrow M \), then \( M \) is a homotopy \( n \) -sphere. | Proof. Let \( f : {S}^{n} \rightarrow M \) be a degree one map. Recall that by Proposition 69.9 we know that \( {f}_{ * } : {\pi }_{1}\left( {S}^{n}\right) \rightarrow {\pi }_{1}\left( M\right) \) is an epimorphism. Thus we see that \( M \) is simply connected. Furthermore it follows from Proposition 88.14 that for \( ... | Yes |
Theorem 88.17. Let \( M \) be a closed, non-empty \( n \) -dimensional topological manifold and let \( R \) be a commutative ring. We suppose that \( M \) is \( R \) -oriented. We denote by \( \left\lbrack M\right\rbrack \in \) \( {\mathrm{H}}_{n}\left( {M;R}\right) \) the \( R \) -fundamental class. Then for each \( k... | \[ {\mathrm{{PD}}}_{M} : {\mathrm{H}}^{k}\left( {M;R}\right) \overset{ \cong }{ \rightarrow }{\mathrm{H}}_{n - k}\left( {M;R}\right) \] \[ \sigma \mapsto \sigma \cap \left\lbrack M\right\rbrack \] is an isomorphism. | Yes |
Lemma 88.18. Let \( M \) be an oriented \( n \) -dimensional topological manifold with empty boundary.\n\n(1) Let \( K \subset L \subset M \) be compact subsets. We denote by \( {f}_{KL} : \left( {M, M \smallsetminus L}\right) \rightarrow \left( {M, M \smallsetminus K}\right) \) the inclusion map of pairs.\n\n(a) The m... | Proof. Let \( M \) be an oriented \( n \) -dimensional topological manifold with empty boundary.\n\n(1) (a) This is precisely Corollary 87.12 (1).\n\n(b) Let \( \varphi \in {\mathrm{H}}^{k}\left( {M, M \smallsetminus K}\right) \) . We see that\n\n\[ \n\varphi \cap {\mu }_{K}\overset{!}{ = }\varphi \cap {f}_{{KL} * }\le... | Yes |
Lemma 88.21. Let \( M \) be a topological manifold and let \( U, V \) be open subsets such that \( M = U \cup V \) . Then given any compact subset \( C \subset M \) there exist compact subsets \( K \subset U \) and \( L \subset V \) such that \( C = K \cup L \) . | Proof (*). Let \( M \) be an \( n \) -dimensional topological manifold and let \( U, V \) be open subsets with \( M = U \cup V \) . Furthermore let \( C \) be a compact subset of \( M \) . Using charts one can easily show that given any point \( x \in C \) there exists a compact set \( {A}_{x} \) with the following pro... | Yes |
Lemma 88.22. Let \( M \) be a compact oriented \( n \) -dimensional topological manifold and let \( U, V \) be open subsets such that \( M = U \cup V \) and let \( K \subset U \) and \( L \subset V \) be compact subsets. Then there exist singular \( n \) -chains \( {\alpha }_{U \smallsetminus L},{\alpha }_{U \cap V} \)... | Proof. We pick a representative \( \alpha \in {\mathrm{C}}_{n}\left( M\right) \) for \( {\mu }_{K \cup L}^{M} \) . Note that the open sets \( U \smallsetminus L, U \cap V \) and \( V \smallsetminus K \) cover \( M \) . It follows from Lemmas 43.28 and 43.31 that there exist singular chains \( {\alpha }_{U \smallsetminu... | Yes |
Theorem 88.23. (The inverted Mayer-Vietoris Theorem for Cohomology groups) Let \( X \) be a topological space and let \( A \) and \( B \) be open subsets.\n\n(1) For every \( n \in {\mathbb{N}}_{0} \) and every cochain \( \varphi \in {\mathrm{C}}^{n}\left( {X, A \cup B}\right) \) there exist \( {\varphi }_{A} \in {\mat... | Proof. We start out with the following two observations:\n\n(1) It is straightforward to verify that the following sequence of cochain maps between cochain complexes is exact:\n\n\[ 0 \rightarrow {\mathrm{C}}^{ * }\left( {X,\{ A, B\} }\right) \xrightarrow[]{\;{i}^{ * } \oplus - {i}^{ * }}{\mathrm{C}}^{ * }\left( {X, A}... | No |
Lemma 88.24. Let \( M \) be an oriented \( n \) -dimensional topological manifold with empty boundary that is the union of two open sets \( U \) and \( V \) . Then there exists a diagram\n\n\[ \ldots \rightarrow {\mathrm{H}}_{\mathrm{c}}^{k}\left( {U \cap V}\right) \xrightarrow[]{\;i \oplus - i\;}{\mathrm{H}}_{\mathrm{... | Proof. Let \( M \) be an oriented \( n \) -dimensional topological manifold with empty boundary. Throughout the proof, given any two subsets \( X \subset Y \) and \( i \in {\mathbb{N}}_{0} \) we write as before \( {\mathrm{H}}^{i}\left( {Y \mid X}\right) \mathrel{\text{:=}} {\mathrm{H}}^{i}\left( {Y, Y \smallsetminus X... | Yes |
Lemma 88.25. Let \( M \) be an oriented \( n \) -dimensional topological manifold with empty boundary and let \( U \) be an open subset of \( M \) . We denote by \( i : U \rightarrow M \) the inclusion. For any \( k \in {\mathbb{N}}_{0} \) the following two diagrams commute:\n\n\[ \n\\begin{matrix} {\\mathrm{H}}_{\\mat... | Proof. The statement that the diagram on the left commutes for any \( k \) is contained in Lemma 88.24 (1). Since the maps \( {\\mathrm{{PD}}}_{U} \) and \( {\\mathrm{{PD}}}_{M} \) are isomorphisms the statement that the diagram on the right commutes is just a reformulation of the fact that the diagrams on the left com... | Yes |
Theorem 88.1. (Poincaré Duality Theorem) Let \( M \) be a compact, non-empty \( n \) -dimensional topological manifold and let \( R \) be a commutative ring. We suppose that \( M \) is \( R \) -oriented. We denote by \( \left\lbrack M\right\rbrack \in {\mathrm{H}}_{n}\left( {M,\partial M;R}\right) \) the \( R \) -funda... | Proof of Theorem 88.1 for \( A = \partial M \) and \( B = \varnothing \) . In this proof we consider the case that \( A = \partial M \) and \( B = \varnothing \) . To simplify the notation we will once again only consider the case that \( R = \mathbb{Z} \) . The proof of the general case is basically the same. We intro... | No |
Proposition 88.26. Let \( M \) be a compact \( n \) -dimensional topological manifold with nonempty boundary. Let \( R \) be a commutative ring. We suppose that \( M \) is \( R \) -oriented. We equip the boundary \( \partial M \) with the R-orientation given by Proposition 86.19, As usual we denote by \( \left\lbrack M... | Proof. We had implicitly proved the proposition in the above proof of the Poincaré Duality Theorem 88.1 applied to the special case that \( A = \varnothing \) and \( B = \partial M \) . | Yes |
Proposition 88.27. Let \( M \) be a compact, non-empty \( n \) -dimensional topological manifold and let \( R \) be a commutative ring. We suppose that \( M \) is \( R \) -oriented. We denote by \( \left\lbrack M\right\rbrack \in {\mathrm{H}}_{n}\left( {M,\partial M;R}\right) \) the \( R \) -fundamental class. Furtherm... | Proof (*). We already showed on page 2178 that the first diagram commutes up to multiplication by \( {\left( -1\right) }^{k} \) . The fact that the second diagram commutes is a fairly straightforward consequence of Lemma 83.10. We leave it to the reader to fill in the details. | No |
Lemma 89.3. Let \( R \) be a commutative ring and let \( M \) be a topological manifold that is equipped with an \( R \)-orientation. If \( W \) is a codimension-zero submanifold of \( M \) with corner, then \( W \) admits a unique \( R \)-orientation such that the inclusion map \( N \rightarrow M \) is orientation-pre... | Proof. This statement follows immediately from Lemma 86.17 (1). | Yes |
Lemma 89.4. Let \( M \) be a compact oriented \( n \) -dimensional topological manifold. Furthermore let \( W \subset M \) be a compact non-empty codimension-zero submanifold with corner. We write \( \overset{ \circ }{W} = W \smallsetminus \partial W \) . The following two statements hold:\n\n(1) The inclusion induced ... | Proof. The proof of Lemma 89.4 is basically identical to the proof of Lemma 87.24. | No |
Lemma 89.6. Let \( M \) be a compact oriented \( n \) -dimensional topological manifold and let \( W \) be a compact codimension-zero submanifold with corner. Furthermore let \( A \) and \( B \) be disjoint unions of components of \( \partial M \) with \( \partial M = A \cup B \) . We fix the following notation:\n\n(1)... | Proof. We set \( \widehat{W} \mathrel{\text{:=}} W \smallsetminus \left( {{\partial }_{0}W \cup C}\right) \) . Note that \( M \smallsetminus \widehat{W} = \left( {M \smallsetminus \overset{ \circ }{W}}\right) \cup B \) . By definition of the isomorphisms \( {\mathrm{{PD}}}_{W} \) and \( {\mathrm{{PD}}}_{M} \), proving ... | Yes |
Lemma 90.1. Let \( R \) be a commutative ring. Let \( V \) and \( W \) be free \( R \)-modules of the same finite rank and let\n\n\[ \langle \rangle : \rangle : V \times W \rightarrow R \]\n\nbe a pairing.\n\n(1) We denote by \( n \) the rank of \( V \). Let \( {v}_{1},\ldots ,{v}_{n} \) be a basis for \( V \) and let ... | Proof. The second statement is evidently just a special case of the first statement. So it suffices to prove the first statement. Let \( {v}_{1},\ldots ,{v}_{n} \) be a basis for \( V \) and let \( {w}_{1},\ldots ,{w}_{n} \) be a basis for \( W \). We denote by \( {v}_{1}^{ * },\ldots ,{v}_{n}^{ * } \in \operatorname{H... | Yes |
Proposition 90.2. Let \( M \) be a closed \( n \) -dimensional topological manifold and let \( k \in {\mathbb{N}}_{0} \) .\n\n(1) Suppose \( M \) is oriented with fundamental class \( \left\lbrack M\right\rbrack \) . If \( {\mathrm{H}}_{k - 1}\left( {M;\mathbb{Z}}\right) \) and \( {\mathrm{H}}_{n - k - 1}\left( {M;\mat... | Proof. Let \( M \) be a closed \( n \) -dimensional topological manifold and let \( k \in {\mathbb{N}}_{0} \) .\n\n(1) We suppose that \( M \) is oriented with fundamental class \( \left\lbrack M\right\rbrack \) and we suppose that \( {\mathrm{H}}_{k - 1}\left( {M;\mathbb{Z}}\right) \) and \( {\mathrm{H}}_{n - k - 1}\l... | No |
Corollary 90.3. Let \( M \) be a closed connected \( n \) -dimensional topological manifold and let \( k \in {\mathbb{N}}_{0} \) .\n\n(1) Suppose that \( M \) is oriented and that \( {\mathrm{H}}_{k - 1}\left( {M;\mathbb{Z}}\right) \) and \( {\mathrm{H}}_{n - k - 1}\left( {M;\mathbb{Z}}\right) \) are torsion-free. Furt... | Proof. Let \( M \) be a closed connected \( n \) -dimensional topological manifold and let \( k \in {\mathbb{N}}_{0} \) .\n\n(1) By our hypothesis and by Proposition 90.2 (1) the cup product pairing\n\n\[ {\mathrm{H}}^{k}\left( {M;\mathbb{Z}}\right) \times {\mathrm{H}}^{n - k}\left( {M;\mathbb{Z}}\right) \rightarrow \m... | Yes |
Lemma 90.4. The topological space \( {S}^{2} \vee {S}^{4} \) is not homotopy equivalent to any closed topological manifold. | Proof. Let \( X \) be a closed topological manifold whose homology and cohomology groups are isomorphic to the corresponding groups of \( {S}^{2} \vee {S}^{4} \) . It follows from the above calculation of homology groups, our hypothesis that \( X \) is compact, from Propositions 41.5 and 87.22 and Theorems [87,1] and [... | Yes |
Corollary 90.5. If \( M \) is a closed oriented \( n \) -dimensional smooth manifold, then for any \( k \in {\mathbb{N}}_{0} \) the pairing\n\n\[ \n{\mathrm{H}}_{\mathrm{{dR}}}^{k}\left( M\right) \times {\mathrm{H}}_{\mathrm{{dR}}}^{n - k}\left( M\right) \rightarrow \mathbb{R} \n\]\n\n\[ \n\left( {\left\lbrack \varphi ... | Proof. It is straightforward to see that it suffices to deal with the case that \( M \) is connected. Theorem 8.1.19 and Proposition 79.7 give us the following commutative diagram with vertical isomorphisms:\n\n\n\... | Yes |
The topological spaces \( {S}^{2} \times {S}^{4} \) and \( {\mathbb{{CP}}}^{3} \) are not homotopy equivalent. | In Lemma 84.3 we saw that the cup product\n\n\[ \n{\mathrm{H}}^{2}\left( {{S}^{2} \times {S}^{4};\mathbb{Z}}\right) \times {\mathrm{H}}^{2}\left( {{S}^{2} \times {S}^{4};\mathbb{Z}}\right) \rightarrow {\mathrm{H}}^{4}\left( {{S}^{2} \times {S}^{4};\mathbb{Z}}\right) \n\]\n\nis zero, whereas we saw in Proposition 90.7 t... | Yes |
The smooth manifold \( {\mathbb{{CP}}}^{2} \) is chiral, i.e. there does not exist a self-homeomorphism \( f : {\mathbb{{CP}}}^{2} \rightarrow {\mathbb{{CP}}}^{2} \) with \( {f}_{ * }\left( \left\lbrack {\mathbb{{CP}}}^{2}\right\rbrack \right) = - \left\lbrack {\mathbb{{CP}}}^{2}\right\rbrack \) . | As usual we denote by \( x \in {\mathrm{H}}^{2}\left( {{\mathbb{{CP}}}^{2};\mathbb{Z}}\right) \) the standard generator. By Proposition 90.7 we know that \( x \cup x \in {\mathrm{H}}^{4}\left( {{\mathbb{{CP}}}^{2};\mathbb{Z}}\right) \) is a generator. This implies that \( x \cup x = \epsilon \cdot {\left\lbrack {\mathb... | Yes |
Lemma 81.10 \( \;{f}^{ * }\left( x\right) \) is also a generator of \( {\mathrm{H}}^{2}\left( {{\mathbb{{CP}}}^{4};\mathbb{Z}}\right) \) | \[ = \epsilon \cdot {\eta }^{2} \cdot \left( {x \cup x}\right) = \epsilon \cdot \left( {x \cup x}\right) = {\left\lbrack {\mathbb{{CP}}}^{2}\right\rbrack }^{ * }.\] | No |
For every \( k \in \mathbb{N} \) we have\n\n\[ \n{\mathrm{H}}^{k}\left( {{\mathbb{{CP}}}^{\infty };\mathbb{Z}}\right) \cong {\mathrm{H}}_{k}\left( {{\mathbb{{CP}}}^{\infty };\mathbb{Z}}\right) \cong \left\{ \begin{array}{ll} \mathbb{Z}, & \text{ if }k\text{ is even,} \\ 0, & \text{ if }k\text{ is odd. } \end{array}\rig... | The statements in (1) and (2) regarding homology groups follow from the discussion on page 1263. The statements in (1) and (2) regarding cohomology groups then follow from the Universal Coefficient Theorem 75.13 for Cohomology Groups. | No |
Proposition 90.12. (1) For every \( k \in \mathbb{N} \) the \( k \) -th power \( {x}^{k} \in {\mathrm{H}}^{2k}\left( {{\mathbb{{CP}}}^{\infty };\mathbb{Z}}\right) \cong \mathbb{Z} \) is a generator. (2) If we view \( \mathbb{Z}\left\lbrack x\right\rbrack \) as a superalgebra by equipping the ring with the grading deter... | Proof. Exactly as in the proof of Proposition 90.7 we see that (2) is a consequence of (1). Thus it remains to prove (1) So let \( k \in \mathbb{N} \) . We consider the inclusion map \( j : {\mathbb{{CP}}}^{k} \rightarrow {\mathbb{{CP}}}^{\infty } \) . By Proposition 90.7 and Lemma 81.10 we know that \( {j}^{ * }\left(... | Yes |
Proposition 90.13. Let \( n \in \mathbb{N} \) . (1) For every every \( k \in {\mathbb{N}}_{0} \) we have \[ {\mathrm{H}}_{k}\left( {{\mathbb{{CP}}}^{n};{\mathbb{Z}}_{2}}\right) \cong {\mathrm{H}}^{k}\left( {{\mathbb{{CP}}}^{n};{\mathbb{Z}}_{2}}\right) \cong \left\{ \begin{array}{ll} {\mathbb{Z}}_{2}, & \text{ if }k = 0... | Proof. This proposition can be proved in an almost identical way as Proposition 9.12. Alternatively we can deduce the proposition from Proposition 9.1.12, the Universal Coefficient Theorem 75.26, Lemma 75.30 and Proposition 82.4 (2). We leave it to the impeccably conscientious reader to fill in the details. | No |
Proposition 90.14. If \( 1 \leq m < n \), then \( {\mathbb{{CP}}}^{m} \subset {\mathbb{{CP}}}^{n} \) is not a retract of \( {\mathbb{{CP}}}^{n} \) . | Proof. Let \( 1 \leq m < n \) . We denote by \( i : {\mathbb{{CP}}}^{m} \rightarrow {\mathbb{{CP}}}^{n} \) the inclusion map. Suppose there exists a retraction \( r : {\mathbb{{CP}}}^{n} \rightarrow {\mathbb{{CP}}}^{m} \) . We denote by \( x \in {\mathrm{H}}^{2}\left( {{\mathbb{{CP}}}^{n};\mathbb{Z}}\right) \) the stan... | Yes |
Lemma 90.15. (1) For every \( n \in \mathbb{N} \) we have \[ {\mathrm{H}}^{k}\left( {{\mathbb{{RP}}}^{n};{\mathbb{F}}_{2}}\right) \cong {\mathrm{H}}_{k}\left( {{\mathbb{{RP}}}^{n};{\mathbb{F}}_{2}}\right) \cong \left\{ \begin{array}{ll} {\mathbb{F}}_{2}, & \text{ if }k = 0,1,\ldots, n, \\ 0, & \text{ otherwise. } \end{... | Proof. The calculation of the isomorphism types of the (co-) homology groups and of the induced maps on homology is basically the same as in the proof of Lemma 90.15. The argument that determines the induced map on \( {\mathbb{F}}_{2} \) -cohomology is slightly different. This time we use Proposition 75.19 which gives ... | Yes |
Proposition 90.16. Let \( n \in \mathbb{N} \) .\n\n(1) For every \( k \in \{ 0,\ldots, n\} \) the \( k \) -th power \( {x}^{k} \in {\mathrm{H}}^{k}\left( {{\mathbb{{RP}}}^{n};{\mathbb{F}}_{2}}\right) \cong {\mathbb{F}}_{2} \) is non-trivial.\n\n(2) If we view \( {\mathbb{F}}_{2}\left\lbrack x\right\rbrack /\left( {x}^{... | Proof. First recall that Proposition 86.13 implies that any topological manifold, in particular \( {\mathbb{{RP}}}^{n} \), is \( {\mathbb{F}}_{2} \) -orientable. With this observation the proof of the proposition is basically identical to the proofs of Proposition 90.7 and 90.12. We just need to replace Lemma 90.6 \n\nThe same argument also works for \( {\mathbb{{CP}}}^{n} \), we just need to replace Propositions 90.16, by Propositions 90.7 | No |
Theorem 59.3. (Borsuk-Ulam) For every map \( f : {S}^{n} \rightarrow {\mathbb{R}}^{n} \) there exists a pair of antipodal points \( x \) and \( - x \) on \( {S}^{n} \) with \( f\left( x\right) = f\left( {-x}\right) \) . | Proof of the Borsuk-Ulam 59.3 For \( n \geq 2 \) . Let us suppose that there exists a map \( f : {S}^{n} \rightarrow {\mathbb{R}}^{n} \) such that for any \( x \in {S}^{n} \) we have that \( f\left( x\right) \neq f\left( {-x}\right) \) . We consider the map\n\n\[ g : {S}^{n} \rightarrow {S}^{n - 1} \]\n\n\[ x \mapsto \... | Yes |
Lemma 90.24. Let \( k \in \mathbb{N} \) . We denote by \( \varphi \in {\mathrm{H}}^{1}\left( {{\mathbb{{RP}}}^{k};{\mathbb{F}}_{2}}\right) \) the unique non-trivial element. We denote by \( p \) and \( q \) the two obvious projection maps \( {\mathbb{{RP}}}^{k} \times {\mathbb{{RP}}}^{k} \rightarrow {\mathbb{{RP}}}^{k}... | Proof. The lemma is an immediate consequence of Propositions [90.16] and 84.24 and the discussion on page 2055. | No |
Theorem 90.25. (Bott-Kervaire-Milnor 1958) Every finite-dimensional division algebra over \( \mathbb{R} \) is of dimension 1,2,4 or 8 . | Proof. The theorem was first proved using topological methods by Michel Kervaire Kerv58 and John Milnor [Miln58a, Corollary 1], building on the \ | No |
Lemma 91.1. If \( f : {S}^{k} \rightarrow X \) is a constant map, then there exists a homeomorphism\n\n\[ \operatorname{Cone}\left( {f : {S}^{k} \rightarrow X}\right) \overset{ \cong }{ \rightarrow }{S}^{k + 1} \vee X \]\n\nwhere we perform the wedge for suitable points in \( {S}^{k + 1} \) and \( X \) . | Proof. Suppose \( f : {S}^{k} \rightarrow X \) is a constant map. It follows easily from the example on page 2213 and the homeomorphism \( {\bar{B}}^{k + 1}/{S}^{k} \cong {S}^{k + 1} \) from page 182 that \( \operatorname{Cone}\left( {f : {S}^{k} \rightarrow X}\right) \) is homeomorphic to a wedge \( {S}^{k + 1} \vee X... | No |
Lemma 91.2. There exist homeomorphisms\n\n(1)\n\n\[ \operatorname{Cone}\left( {H : {S}^{3} \rightarrow {S}^{2}}\right) \cong {\mathbb{{CP}}}^{2} \]\n\n(2)\n\[ \operatorname{Cone}\left( {{H}_{\mathbb{H}} : {S}^{7} \rightarrow {S}^{4}}\right) \cong {\mathbb{{HP}}}^{2} \]\n\n(3)\n\[ \operatorname{Cone}\left( {{\mathrm{H}}... | Proof.\n\n(1) In Lemma 36.1 we saw that \( {\mathrm{{CP}}}^{2} \) is a CW-complex that is obtained from the CW-complex \( \mathbb{C}{\mathrm{P}}^{1} = {S}^{2} \) by attaching a 4-cell. The attaching map \( {S}^{3} \rightarrow \mathbb{C}{\mathrm{P}}^{1} \) of the 4-cell is precisely the Hopf map. This observation, toget... | Yes |
(1) The Hopf map \( H : {S}^{3} \rightarrow {S}^{2} \) represents a non-trivial element in the third homotopy group \( {\pi }_{3}\left( {S}^{2}\right) \) . | Proof. First we prove (1). Recall that for two topological spaces \( X \) and \( Y \) we write \( X \simeq Y \) if they are homotopy equivalent. We have\n\n\[ \text{Cone(Hopf map}H\text{)} \cong {\mathbb{{CP}}}^{2} \notin {S}^{2} \vee {S}^{4} \cong \text{Cone(constant map).} \]\n\n\[ \text{Lemma 9.1.2 (1) Lemma 9.4 Lem... | Yes |
Lemma 91.4. Let \( R \) be a commutative ring.\n\n(1) Let \( f : A \rightarrow X \) be a map between topological spaces.\n\n(a) We denote by \( j\left( f\right) : X \rightarrow \operatorname{Cone}\left( f\right) = \left( {\operatorname{Cone}\left( A\right) \sqcup X}\right) / \sim \) the obvious inclusion map.\n\n(b) We... | Proof. The proof is basically identical to the proof of Lemma 46.18. The meticulous reader who filled in the details of the proof of Lemma 46.18 will have no troubles working out the details of the present proof. | No |
Lemma 91.5. Let \( f : {S}^{m} \rightarrow {S}^{n} \) be a map. If \( m \geq n + 1 \) and \( n \geq 1 \), then the maps\n\n\[ \n{\mathrm{H}}^{n}\left( {\operatorname{Cone}\left( f\right) ;\mathbb{Z}}\right) \;\xrightarrow[]{j{\left( f\right) }^{ * }}\;{\mathrm{H}}^{n}\left( {{S}^{n};\mathbb{Z}}\right) \n\]\n\nand\n\n\[... | Proof. The lemma is an almost immediate consequence of Lemma 91.4 and the calculation of the cohomology groups of spheres, see page 1844. For \( n = 1 \) one also needs to invoke Exercise 73.6. | No |
Lemma 91.6. Let \( n \geq 2 \) and let \( {f}_{0},{f}_{1} : {S}^{{2n} - 1} \rightarrow {S}^{n} \) be two maps. If \( {f}_{0} \) and \( {f}_{1} \) are homotopic, then \( \operatorname{Hopf}\left( {f}_{0}\right) = \operatorname{Hopf}\left( {f}_{1}\right) \) . | Proof (*). Let \( H : {S}^{{2n} - 1} \times \left\lbrack {0,1}\right\rbrack \rightarrow {S}^{n} \) be a homotopy between \( {f}_{0} \) and \( {f}_{1} \) . We denote by \( {i}_{0},{i}_{1} : {S}^{{2n} - 1} \rightarrow {S}^{{2n} - 1} \times \left\lbrack {0,1}\right\rbrack \) the two obvious inclusion maps. We obtain the f... | Yes |
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