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Theorem 117.18. For any \( n \geq 2 \) we have \( {\pi }_{n + 2}\left( {S}^{n}\right) \cong {\mathbb{Z}}_{2} \) . | ## Proof.\n\n(1) As mentioned before, the theorem was proved by Lev Pontryagin [Pon50b, Pon59]. See also [Scor05, p. 233] and [WX10, Chapter 2.5] for outlines of the proof. The most readable account of the Thom-Pontryagin approach to proving the theorem is surely Put.\n\n(2) At the same time as Pontryagin two very diff... | No |
Theorem 117.19.\n\n(1) We have \( {}^{1652}{\pi }_{5}\left( {S}^{3}\right) \cong {\mathbb{Z}}_{2},{\pi }_{6}\left( {S}^{3}\right) \cong {\mathbb{Z}}_{12},{\pi }_{7}\left( {S}^{4}\right) \cong \mathbb{Z} \oplus {\mathbb{Z}}_{12} \) .\n\n(2) For any \( \overrightarrow{n \geq }5 \) we have \( {\pi }_{n + 3}\left( {S}^{n}\... | Proof.\n\n(a) The theorem was proved by Vladimir Rokhlin Rok52 | Rok86a, p. 21] in 1952, using the above Thom-Pontryagin approach.\n\n(b) A somewhat different proof is sketched in [WX10, Chapter 2.6]. Some of the results can also be found in [Ser53, Proposition 10]. Other proofs are given in [Hat, Theorem 1.40 (b)] and... | Yes |
(1) If \( n \) is odd, then \( {\pi }_{k}\left( {S}^{n}\right) \) is finite for \( k \geq n + 1 \). | Proof. The theorem was first proved by Serre [Ser53] using \ | No |
Corollary 119.2. (*) Let \( A \) be a CW-complex and let \( f : A \rightarrow X \) be a map to a topological space. There exists a CW-complex \( B \) and a map \( g : B \rightarrow X \) such that the following statements hold:\n\n(1) \( A \) is a subcomplex of \( B \) ,\n\n(2) the restriction of \( g \) to \( A \) equa... | Proof of Corollary \( {119.2}\left( *\right) \) . Let \( A \) be a CW-complex and let \( f : A \rightarrow X \) be a map to a topological space. Without loss of generality we can and will assume that \( A \) and \( X \) are 0-connected. We set \( {B}_{0} \mathrel{\text{:=}} A \) and \( {g}_{0} \mathrel{\text{:=}} f \) ... | Yes |
Lemma 119.3. Let \( \left( {X, A}\right) \) be a pair of CW-complexes and let \( f : A \rightarrow Y \) be a map to a path-connected topological space. Suppose the following condition holds:\n\n(*) For every \( n \in \mathbb{N} \) such that \( X \smallsetminus A \) contains an \( n \) -cell we have the equality \( {}^{... | Proof. We start out with the following claim.\n\nClaim. Given any \( n \in {\mathbb{Z}}_{ \geq - 1} \) we can extend \( f : A \rightarrow Y \) to a map \( A \cup {X}^{n} \rightarrow Y \) .\n\nWe prove the claim by induction on \( n \in {\mathbb{Z}}_{ \geq - 1} \) . For \( n = - 1 \) there is nothing to prove. So suppos... | Yes |
Lemma 119.5. Let \( f : X \rightarrow Y \) be a map between topological spaces.\n\n(1) If \( f \) is a homotopy equivalence, then it is also a weak homotopy equivalence. | (1) This statement is an immediate consequence of Proposition 40.7 (2). | Yes |
Proposition 119.6. Let \( f : X \rightarrow Y \) be a map between topological spaces.\n\n(1) If \( f \) is a weak homotopy equivalence, then for any abelian group \( G \) and any \( n \in {\mathbb{N}}_{0} \) the induced maps \( {f}_{ * } : {\mathrm{H}}_{n}\left( {X;G}\right) \rightarrow {\mathrm{H}}_{n}\left( {Y;G}\rig... | (1) Let \( f : X \rightarrow Y \) be a weak homotopy equivalence. It follows easily from Lemmas 41.14 and Lemmas 7.14 that we can assume, without loss of generality, that \( X \) and \( Y \) are path-connected. Let \( {x}_{0} \in X \) . By our hypothesis we know that the map \( {f}_{ * } : {\pi }_{n}\left( {X,{x}_{0}}\... | Yes |
Lemma 119.7. (*) Let \( X \) and \( Y \) be two CW-complexes. We define the topological space \( X \otimes Y \), whose underlying set is \( X \times Y \), as on page [963. The identity map \( \mathrm{{id}} : X \otimes Y \rightarrow X \times Y \) is a CW-approximation. | Proof (*). By Proposition 36.23 we know that \( X \otimes Y \) is a CW-complex and we know that the identity map id: \( X \otimes Y \rightarrow X \times Y \) is continuous. In Proposition 36.26 we showed that the induced maps on fundamental groups are isomorphisms. Basically the same argument, with the obvious extensio... | No |
Theorem 119.8. (CW-Approximation Theorem)\n\n(1) Every topological space \( X \) admits a CW-approximation \( f : Z \rightarrow X \) .\n\n(2) Let \( k \in {\mathbb{N}}_{0} \) . If \( X \) is a \( k \) -connected topological space, then there exists a CW-approximation \( f : Z \rightarrow X \) such that \( Z \) has a si... | Proof of the CW-Approximation Theorem 119.8. Let \( X \) be a topological space.\n\n(1) We need to show that \( X \) admits a CW-approximation. By considering each path-component of \( X \) separately we can assume that \( X \) is path-connected.\n\nThus let \( X \) be a path-connected topological space. We pick a poin... | Yes |
Lemma 119.11. Let \( \left( {X, A}\right) \) be a pair of CW-complexes and let \( \left( {Y, B}\right) \) be a pair of topological spaces with \( B \neq \varnothing \) . We make the following assumption: For every \( n \in {\mathbb{N}}_{0} \) such that \( X \smallsetminus A \) has at least one n-dimensional cell we ass... | Proof. We start out with the following claim.\n\nClaim. Let \( k \in \mathbb{N} \) . Let \( f : \left( {X, A}\right) \rightarrow \left( {Y, B}\right) \) be a map. If \( f\left( {X}^{k - 1}\right) \subset B \), then there exists a homotopy rel \( A \cup {X}^{k - 1} \) from \( f \) to a map \( g \) with \( g\left( {X}^{k... | Yes |
Proposition 119.15. Let \( X \) be a CW-complex, let \( {x}_{0} \in X \) and let \( n \in {\mathbb{N}}_{0} \) . If \( X \) is \( n \) -connected, then there exists a CW-complex \( Y \) which has one 0-cell \( {y}_{0} \) and no cells in dimensions \( 1,\ldots, n \) such that the pair \( \left( {X,{x}_{0}}\right) \) is h... | Proof. Let \( X \) be an \( n \) -connected CW-complex and let \( {x}_{0} \in X \) . By the CW-Approximation Theorem 119.8 (2) there exists a CW-complex \( Y \) which has a single 0-cell \( \left\{ {y}_{0}\right\} \) and no cells in dimensions \( 1,\ldots, n \) and a map \( f : \left( {Y,{y}_{0}}\right) \rightarrow \le... | Yes |
Lemma 119.16. Let \( X \) and \( Y \) be CW-complexes. If \( X \) and \( Y \) are simple homotopy equivalent, then they are homotopy equivalent. | Proof (*). By Lemma 18.11 the notion of being homotopy equivalent is an equivalence relation. Thus it suffices to show that if a CW-complex \( X \) is an elementary collapse of a CW-complex \( Y \), then \( X \) is homotopy equivalent to \( Y \) . In fact, by Lemma 18.14 it suffices to show that \( X \) is a deformatio... | No |
Corollary 119.19. Let \( X \) be a CW-complex. If \( X \) is simply connected and if we have \( {\mathrm{H}}_{n}\left( {X;\mathbb{Z}}\right) = 0 \) for all \( n \geq 2 \), then \( X \) is contractible, i.e. \( X \) is homotopy equivalent to a point. | Proof. We let \( Y = \{ * \} \) be the topological space that consists of a single point and let \( f : X \rightarrow Y = \{ * \} \) be the only map there is. It follows immediately from Proposition 119.6 that \( f \) is a weak homotopy equivalence. Thus the desired statement follows from the Whitehead Theorem 119.9 (1... | Yes |
Proposition 47.11. Given any \( n \in \mathbb{N} \) and given any abelian group \( \pi \) there exists an \( \left( {n + 1}\right) \) -dimensional CW-complex that has no cells in dimensions \( 1,\ldots, n - 1 \) and that is a Moore space of type \( \mathrm{M}\left( {\pi, n}\right) \) . | Sketch of PROOF. Since \( \pi \) is abelian it follows immediately from Lemma 57.16 (1) that there exists a free resolution of \( \pi \) of the following form:\n\n\[ 0 \rightarrow {\mathbb{Z}}^{\left( A\right) }\overset{\varphi }{ \rightarrow }{\mathbb{Z}}^{\left( B\right) }\overset{\rho }{ \rightarrow }\pi \rightarrow... | No |
Lemma 120.1. Let \( m \in \mathbb{N} \) and let \( {r}_{1},{r}_{2},\cdots \in \mathbb{N} \) be natural numbers that are coprime to \( m \) .\n\n(1) The map\n\n\[ \n{\mathbb{Z}}_{m} \times {S}^{\infty } \rightarrow {S}^{\infty } \]\n\n\[ \n\left( {\left\lbrack k\right\rbrack ,\underset{ \in {\mathbb{C}}^{\infty }}{\unde... | (1) The proof of this statement is quite similar to the task performed in Exercise 36.4 We leave it to the motivated reader to prove the desired statement. As always it takes a little effort to work rigorously with the topology of \( {S}^{\infty } = \underline{\lim }{S}^{n} \) . | No |
(1) Let \( X \) be an Eilenberg-Maclane space of type \( \mathrm{K}\left( {A, n}\right) \) and let \( Y \) be an Eilenberg-Maclane space of type \( \mathrm{K}\left( {B, n}\right) \) .\n\n(a) The CW-complex \( X \otimes Y \), that is defined on page 963, is an Eilenberg-Maclane space of type \( \mathrm{K}\left( {A \time... | (1) Recall that by construction and by Proposition 36.23 we know \( X \otimes Y \) is a CW-complex whose underlying set is \( X \times Y \) . Let id: \( X \otimes Y \rightarrow X \times Y \) be the identity map. We pick \( x \in X \) and \( y \in Y \) . Given any \( k \in {\mathbb{N}}_{0} \) we have the following isomo... | Yes |
Lemma 120.4. Let \( n \in \mathbb{N} \) and let \( \pi \) be a group. If \( X \) is an Eilenberg-Maclane space of type \( \mathrm{K}\left( {\pi, n}\right) \), then given any abelian group \( G \) the following statements hold:\n\n(1) We have \( {\mathrm{H}}_{0}\left( {X;G}\right) \cong G \) and \( {\mathrm{H}}^{0}\left... | Proof. The statements follow easily from the Hurewicz Theorems 52.5 and 53.5 and from the Universal Coefficient Theorems 57.19 and 75.13 The experienced reader will have no troubles with filling in the details. | No |
Proposition 120.5. Let \( I \) be a set, let \( n \in \mathbb{N} \) and for each \( i \in I \) let \( {S}_{i}^{n} \) be a copy of \( {S}^{n} \) . For each \( i \in I \) we denote by \( {\varphi }_{i} : {S}^{n} \rightarrow {S}_{i}^{n} \) the obvious homeomorphism. By a slight abuse of notation we denote by \( {\varphi }... | (1) As we pointed out above, we can view \( \mathop{\bigvee }\limits_{{i \in I}}{S}_{i}^{n} \) as a CW-complex with one 0-cell and no other cells outside of dimension \( n \) . It follows from Proposition 40.9 that all homotopy groups in dimension \( 1,\ldots, n - 1 \) vanish.\n\n(2) In the case that \( I \) is a finit... | Yes |
Proposition 120.6. Let \( k \in \mathbb{N} \), let \( X \) be a connected \( \left( {k + 1}\right) \) -dimensional CW-complex \( {}^{1686} \) and let \( {x}_{0} \in X \) be a base point.\n\n(1) There exists a \( \left( {k + 2}\right) \) -dimensional CW-complex \( Y \) with the following properties:\n\n(a) the \( \left(... | Proof.\n\n(1) We pick a set \( {\left\{ {\varphi }_{j} : \left( {S}^{k + 1}, * \right) \rightarrow \left( X,{x}_{0}\right) \right\} }_{j \in J} \) of maps that represent a generating set for the group \( {\pi }_{k + 1}\left( {X,{x}_{0}}\right) \) . We set\n\n\[ Y = \left( {X \sqcup \mathop{\bigsqcup }\limits_{{j \in J}... | Yes |
Proposition 120.7. Let \( k \in \mathbb{N} \), let \( Y \) be a connected \( \left( {k + 1}\right) \) -dimensional CW-complex and let \( {y}_{0} \in Y \). (1) There exists a CW-complex \( Z \) with the following three properties: (a) the \( \left( {k + 1}\right) \) -skeleton of \( Z \) equals \( Y \), (b) we have \( {\... | Proof. Let \( k \in \mathbb{N} \) and let \( Y \) be a connected \( \left( {k + 1}\right) \) -dimensional CW-complex. Let \( {y}_{0} \in Y \). (1) We define \( {Z}_{k + 1} = Y \). For \( i = k + 1, k + 2,\ldots \) we iteratively apply Proposition 120.6 to \( {Z}_{i} \) to obtain a sequence of CW-complexes \( {Z}_{k + 1... | Yes |
Lemma 120.11. Given any \( n \in \mathbb{N} \) there exists an Eilenberg-Maclane space of type \( \mathrm{K}\left( {\mathbb{Z}, n}\right) \) with precisely one 0-cell, precisely one n-cell, no cell of dimension \( n + 1 \) and only countably many cells in dimensions \( \geq n + 2 \) . | Proof. Let \( n \in \mathbb{N} \) . We consider \( Y = {S}^{n} \) . Recall that by Corollary 53.6 we know that \( {\pi }_{n}\left( {S}^{n}\right) \cong \mathbb{Z} \) . By Proposition 120.7 (3) there exists a CW-complex \( Z \) with the following four properties:\n\n(1) the \( \left( {n + 1}\right) \) -skeleton of \( Z ... | Yes |
Corollary 120.14. There is no finite-dimensional Eilenberg-Maclane space of type \( \mathrm{K}\left( {{\mathbb{Z}}_{2},1}\right) \) . | Proof. Let \( K \) be an Eilenberg-Maclane space of type \( \mathrm{K}\left( {{\mathbb{Z}}_{2},1}\right) \) . Given any \( i \in \mathbb{N} \) we calculate that\n\n\( {\mathrm{H}}_{i}\left( {K;{\mathbb{Z}}_{2}}\right) \underset{ \uparrow }{ \cong }{\mathrm{H}}_{i}\left( {\text{ any Eilenberg-Maclane space of type }\mat... | Yes |
Proposition 120.18. (*) Let \( \pi \) be a group, let \( Y \) be an Eilenberg-Maclane space of type \( \mathrm{K}\left( {\pi ,1}\right) \) and let \( {y}_{0} \in Y \) . Furthermore let \( X \) be a 0-connected \( \mathrm{{CW}} \) -complex and let \( {x}_{0} \in X \) be a point in the 0 -skeleton \( {X}^{0} \) . Suppose... | Proof (*). The proof of the proposition is similar to the proof of Proposition 120.15 (2). In the following provide a sketch of the argument. We leave it to the reader to fill in the details. First of all we point out that by Proposition 119.15 we can assume that \( X \) has a single 0-cell \( \left\{ {x}_{0}\right\} \... | No |
Proposition 120.21. If \( M \) is a compact orientable connected \( n \) -dimensional smooth manifold, then the following two statements are equivalent:\n\n(1) there exists an epimorphism \( {\pi }_{1}\left( M\right) \rightarrow \left\langle {{x}_{1},\ldots ,{x}_{k}}\right\rangle \),\n\n(2) there exist disjoint connect... | SKETCH OF A PROOF. First note that the \ | No |
Theorem 120.23. (1) Any CW-complex that is homotopy equivalent to the product of a family of Eilenberg-Maclane spaces of type \( \mathrm{K}\left( {\mathbb{Q},{k}_{i}}\right), i \in I \) is social. (2) Conversely, if \( X \) is a connected social \( \mathrm{{CW}} \) -complex, then it is is homotopy equivalent to the pro... | Proof. This theorem is proved in [EGH62, p. 90] and [Weinb04, Theorem 1.3], see also Eckm04, Theorems 4 and 7]. For the most part the proofs do not go much beyond what we have covered so far, except that they build on a result of Alexander Grothendieck Groth57, Chapter V], see also [HZ74, p. 32] and [Hirz69, p. 254]. | Yes |
Lemma 120.25. (*) Let \( \left( {X,{x}_{0}}\right) \) be a pointed 0-connected topological space. There exists a natural map\n\n\[ \eta : {\mathrm{H}}_{2}\left( {X;\mathbb{Z}}\right) \rightarrow {\mathrm{H}}_{2}\left( {{\pi }_{1}\left( {X,{x}_{0}}\right) }\right) \]\n\nwhich has the following property: whenever \( \lef... | Proof (*). Let \( \left( {X,{x}_{0}}\right) \) be a pointed 0-connected topological space. We write \( \pi = {\pi }_{1}\left( {X,{x}_{0}}\right) \) . Let \( \left( {Z,{z}_{0},\varphi : {\pi }_{1}\left( {Z,{z}_{0}}\right) \rightarrow \pi }\right) \) be the canonical triple introduced in Theorem 120.8. By the CW-Approxim... | Yes |
Lemma 120.27. Let \( \pi \) be an abelian group and let \( n \in {\mathbb{N}}_{ \geq 2} \) . If \( Y \) is an Eilenberg-Maclane space of type \( \mathrm{K}\left( {\pi, n}\right) \), then \( {\mathrm{H}}_{n + 1}\left( {Y;\mathbb{Z}}\right) = 0 \) . | Proof of Lemma 120.27. Let \( n \in {\mathbb{N}}_{ \geq 2} \) and let \( \pi \) be an abelian group. It follows from Corollary 120.13 that it suffices to show that there exists a single Eilenberg-Maclane space \( Z \) of type \( \overline{\mathrm{K}\left( {\pi, n}\right) } \) with \( {\mathrm{H}}_{n + 1}\left( {Z;\math... | Yes |
Corollary 120.28. Let \( \left( {X,{x}_{0}}\right) \) be some pointed 0-connected topological space. If \( {\pi }_{1}\left( {X,{x}_{0}}\right) \cong \) \( \mathbb{Z} \), then the Hurewicz homomorphism \( {\Phi }_{\left( X,{x}_{0}\right) } : {\pi }_{2}\left( {X,{x}_{0}}\right) \rightarrow {\mathrm{H}}_{2}\left( {X;\math... | Proof. In the discussion on page 2836 we saw that \( {\mathrm{H}}_{2}\left( {{\pi }_{1}\left( {X,{x}_{0}}\right) }\right) \cong {\mathrm{H}}_{2}\left( \mathbb{Z}\right) \cong {\mathrm{H}}_{2}\left( {S}^{1}\right) = 0 \) . Thus it follows immediately from Theorem 120.26 (2) that the Hurewicz homomorphism \( {\pi }_{2}\l... | Yes |
In particular we see that \( {\mathrm{H}}_{2}\left( {X;\mathbb{Z}}\right) \) is non-trivial. | To simplify the discussion let us consider the case \( m = 2 \) . Thus let \( \left( {X,{x}_{0}}\right) \) be a pointed 0-connected topological space with \( {\pi }_{1}\left( {X,{x}_{0}}\right) \cong {\mathbb{Z}}^{2} \) . We proceed as follows:\n\n(1) We pick two loops \( \alpha ,\beta : \left( {{S}^{1}, * }\right) \ri... | No |
Lemma 120.30. We continue with the above notation. The map\n\n\[ \n\\begin{aligned} \\mu : \\overset{ = \\mathbb{Z} \\cdot \\left\\lbrack T\\right\\rbrack }{\\overbrace{{\\mathrm{H}}_{2}\\left( {T;\\mathbb{Z}}\\right) }} & \\rightarrow \\operatorname{coker}\\left( {{\\pi }_{2}\\left( {X,{x}_{0}}\\right) \\rightarrow {\... | Sketch of PROOF. As in the proof of Theorem 120.26 (2) we can assume that \( X \) is a 3-dimensional CW-complex. Furthermore, by the proof of Theorem 120.26 (2) we know that there exists an inclusion \( i : X \\rightarrow Y \) of \( X \) into an Eilenberg-Maclane space \( Y \) of type \( \\mathrm{K}\\left( {{\\mathbb{Z... | Yes |
Lemma 121.1. Let \( f : X \rightarrow Y \) be a map between topological spaces. The corresponding morphism \( \left\lbrack f\right\rbrack \in {\operatorname{Mor}}_{\mathcal{H}}\left( {X, Y}\right) \) is invertible in the sense of the definition on page 478 if and only if \( f \) is a homotopy equivalence. | Proof. Note that by definition \( \left\lbrack f\right\rbrack \in {\operatorname{Mor}}_{\mathcal{H}}\left( {X, Y}\right) \) is invertible if and only if there exists \( \left\lbrack g\right\rbrack \in {\operatorname{Mor}}_{\mathcal{H}}\left( {Y, X}\right) \) with \( \left\lbrack {f \circ g}\right\rbrack = \left\lbrack ... | Yes |
Lemma 121.2. Let \( X, Y \) and \( Z \) be topological spaces. The map\n\n\[ \chi : \left\lbrack {X, Y}\right\rbrack \times \left\lbrack {X, Z}\right\rbrack \rightarrow \left\lbrack {X, Y \times Z}\right\rbrack \]\n\n\[ \left( {\left\lbrack {f : X \rightarrow Y}\right\rbrack ,\left\lbrack {g : X \rightarrow Z}\right\rb... | Proof. Nothing can possibly go wrong in any attempt to prove this lemma. | No |
Lemma 121.3. The maps \[ X \mapsto \text{free loop space}\Omega \left( X\right) \mathrel{\text{:=}} \left\{ {\varphi \in {X}^{\left\lbrack 0,1\right\rbrack } \mid \varphi \left( 0\right) = \varphi \left( 1\right) }\right\} \] \[ \left\lbrack {f : X \rightarrow Y}\right\rbrack \mapsto \left\lbrack \begin{matrix} \Omega ... | Proof. There are only two statements one needs to think about: given a map \( f : X \rightarrow \) \( Y \) one needs to show that \( \Omega \left( f\right) \) and \( \sum \left( f\right) \) are continuous, and given homotopic maps \( f, g : X \rightarrow Y \) one needs to show that the corresponding maps on the right h... | Yes |
Lemma 121.4. Let \( X \) be a topological space and let \( G \) be an (abelian) topological group, e.g. we could take \( G = {S}^{1} \) . The set \( \left\lbrack {X, G}\right\rbrack \) together with the multiplication map\n\n\[ \nu : \left\lbrack {X, G}\right\rbrack \times \left\lbrack {X, G}\right\rbrack \rightarrow \... | Proof. In fact the only thing one needs to verify is that the map \( \nu \) is actually well-defined. But this follows easily from the fact that the group multiplication \( G \times G \rightarrow G \) is continuous. We leave it to the reader to fill in the details. | No |
Lemma 121.5. Let \( \left( {X,{x}_{0}}\right) ,\left( {Y,{y}_{0}}\right) \) and \( \left( {Z,{z}_{0}}\right) \) be pointed topological spaces. The map\n\n\[ \left\langle {\left( {X,{x}_{0}}\right) ,\left( {Y,{y}_{0}}\right) }\right\rangle \times \left\langle {\left( {Y,{y}_{0}}\right) ,\left( {Z,{z}_{0}}\right) }\right... | Proof. The proof of this lemma is almost identical to the proof of Lemma 18.7 and it is left to the insomniac reader. | No |
Lemma 121.6. Let \( \left( {X,{x}_{0}}\right) \) and \( \left( {Y,{y}_{0}}\right) \) be pointed topological spaces.\n\n(1) Let \( f : \left( {X,{x}_{0}}\right) \rightarrow \left( {Y,{y}_{0}}\right) \) be a map. The corresponding morphism\n\n\[ \left\lbrack f\right\rbrack \in {\operatorname{Mor}}_{\mathcal{P}\mathcal{{H... | Proof. The first statement is, as in the case of Lemma 121.1, an immediate consequence of the definitions. The second statement is a reformulation of the first statement. | No |
Lemma 121.7. Let \( \left( {X,{x}_{0}}\right) ,\left( {Y,{y}_{0}}\right) \) and \( \left( {Z,{z}_{0}}\right) \) be pointed topological spaces. The map\n\n\[ \chi : \left\langle {\left( {X,{x}_{0}}\right) ,\left( {Y,{y}_{0}}\right) }\right\rangle \times \left\langle {\left( {X,{x}_{0}}\right) ,\left( {Z,{z}_{0}}\right) ... | Proof. Conveniently enough the proof is identical to the proof of Lemma 121.2 | No |
Lemma 121.8. Let \( \\left( {X,{x}_{0}}\\right) ,\\left( {Y,{y}_{0}}\\right) \) and \( \\left( {Z,{z}_{0}}\\right) \) be pointed topological spaces. The map\n\n\[ \n\\nu : \\left\\langle {\\left( {X,{x}_{0}}\\right) ,\\left( {Z,{z}_{0}}\\right) }\\right\\rangle \\times \\left\\langle {\\left( {Y,{y}_{0}}\\right) ,\\lef... | Proof. The statement of the lemma is, in contrast to Lemma 121.5 and 121.7, nontrivial. It is a reformulation of Lemma 18.27 which traces its ancestry to the formidable Theorem 5.16. | No |
Lemma 121.9. The maps \[ \left( {X,{x}_{0}}\right) \mapsto \left( {\Omega \left( {X,{x}_{0}}\right) ,{c}_{{x}_{0}}}\right) \] \[ \left\lbrack {f : \left( {X,{x}_{0}}\right) \rightarrow \left( {Y,{y}_{0}}\right) }\right\rbrack \mapsto \left\lbrack \begin{matrix} \Omega \left( {X,{x}_{0}}\right) & \rightarrow & \Omega \l... | Proof. The lemma follows easily basically from Lemma 5.5 (1) and (3), one just needs to make a few minute modifications to factor in the base points. We leave this elementary task to the inexhaustible reader. | No |
Lemma 121.10. Let \( \\left( {X,{x}_{0}}\\right) \) be a pointed topological space. The multiplication map\n\n\[ \n\\mu : \\Omega \\left( {X,{x}_{0}}\\right) \\times \\Omega \\left( {X,{x}_{0}}\\right) \\rightarrow \\Omega \\left( {X,{x}_{0}}\\right) \n\]\n\nhas the following properties.\n\n(1) The map \( \\mu : \\Omeg... | Proof \( \\left( *\\right) \) . Let \( \\left( {X,{x}_{0}}\\right) \) be a pointed topological space.\n\n(1) The statement that \( \\mu \) is continuous is precisely the content of Lemma 114.22 (1). The statements that \( \\mu \) is base-point preserving and natural follows basically immediately from the definitions. | Yes |
Proposition 121.11. Let \( \left( {X,{x}_{0}}\right) \) be a pointed topological space and let \( k \in \mathbb{N} \) . The map \[ \overset{\text{subset of the topological space }{X}^{{\left\lbrack 0,1\right\rbrack }^{k}}}{\overbrace{\left\{ F \in {X}^{{\left\lbrack 0,1\right\rbrack }^{k}} \mid F\left( \partial {\left\... | Proof \( \left( *\right) \) . To simplify the notation we only discuss the case \( k = 2 \) . The general case is treated the same way. We write \( I = J = \left\lbrack {0,1}\right\rbrack \) . We start out with the following subtle observation: (*) The inclusion \( \Omega \left( {X,{x}_{0}}\right) \rightarrow {X}^{I} \... | Yes |
(1) The maps\n\n\[ \left( {X,{x}_{0}}\right) \mapsto \left( {S\left( {X,{x}_{0}}\right) ,{s}_{{x}_{0}}}\right) \]\n\n\[ \left( {f : \left( {X,{x}_{0}}\right) \rightarrow \left( {Y,{y}_{0}}\right) }\right) \mapsto \left( \begin{aligned} {S\left( f\right) : S\left( {X,{x}_{0}}\right) \rightarrow S\left( {Y,{y}_{0}}\right... | (1) The first statement is proved basically the same way as we proved Lemma 24.4. Note that the proof of the statement that if \( f, g : \left( {X,{x}_{0}}\right) \rightarrow \left( {Y,{y}_{0}}\right) \) are homotopic implies that \( S\left( f\right), S\left( g\right) \) are homotopic is somewhat delicate, see the proo... | No |
Proposition 121.13. Let \( \left( {X,{x}_{0}}\right) \) and \( \left( {Y,{y}_{0}}\right) \) be pointed topological spaces. The map\n\n\[ \Upsilon : \left\langle {\left( {S\left( {X,{x}_{0}}\right) ,{s}_{{x}_{0}}}\right) ,\left( {Y,{y}_{0}}\right) }\right\rangle \rightarrow \left\langle {\left( {X,{x}_{0}}\right) ,\left... | Proof. Let \( \left( {X,{x}_{0}}\right) \) and \( \left( {Y,{y}_{0}}\right) \) be pointed topological spaces. First we consider the map\n\n\[ \overset{ = {Y}^{X\times \lbrack - 1,1\rbrack }}{\overbrace{\{ \text{set of maps }X \times \left\lbrack {-1,1}\right\rbrack \rightarrow Y\} }} \rightarrow \overset{ = {\left( {Y}... | No |
Theorem 121.14. If \( \\left( {X,{x}_{0}}\\right) \\in \\mathcal{W} \), then for any \( k \\in \\mathbb{N} \) the iterated loop space \( {\\Omega }^{k}\\left( {X,{x}_{0}}\\right) \) also lies in \( \\mathcal{W} \) . | Proof. If \( \\left( {X,{x}_{0}}\\right) \) is a pointed CW-complex, then [Miln59, Corollary 3] implies that the pointed topological space \( \\left( {\\Omega \\left( {X,{x}_{0}}\\right) ,{c}_{{x}_{0}}}\\right) \) lies in \( \\mathcal{W} \) . (Alternatively we refer to FrPi90a, Corollary 5.3.7] for a proof of a very si... | Yes |
Lemma 121.15. Let \( \left( {K,{k}_{0},\varphi }\right) \) and \( \left( {L,{l}_{0},\psi }\right) \) be two polarized Eilenberg-Maclane spaces of the same type. There exists a unique class \( {}^{1715}\Xi = {\Xi }_{\left( {K,{k}_{0}}\right) ,\left( {L,{l}_{0}}\right) } \in {\left\langle \left( K,{k}_{0}\right) ,\left( ... | Proof. Let \( n \in \mathbb{N} \), let \( G \) be an (abelian) group and let \( \left( {K,{k}_{0},\varphi }\right) \) and \( \left( {L,{l}_{0},\psi }\right) \) be two polarized Eilenberg-Maclane spaces of type \( \mathrm{K}\left( {G, n}\right) \) . Since \( \left( {K,{k}_{0}}\right) \) and \( \left( {L,{l}_{0}}\right) ... | Yes |
Lemma 121.16. We equip \( {S}^{1} \) with the base point \( * = \left( {1,0}\right) \) and we equip \( {\mathbb{{CP}}}^{\infty } \) with the base point \( \left\lbrack {1 : 0 : 0\ldots }\right\rbrack \) . As on page 2859 we consider the polarized Eilenberg-Maclane spaces \( \left( {{S}^{1},*,\varphi }\right) \) of type... | Proof (*). Let \( i : {\left\lbrack 0,1\right\rbrack }^{2}/\partial \left( {\left\lbrack 0,1\right\rbrack }^{2}\right) \rightarrow {\mathbb{{CP}}}^{\infty } \) be the above inclusion map. It follows immediately from the definition of \( \Xi : {S}^{1} \rightarrow \Omega \left( {{\mathbb{{CP}}}^{\infty }, * }\right) \) t... | Yes |
Proposition 121.17. Let \( n \in \mathbb{N} \), let \( G \) be an abelian group and let \( \left( {X,{x}_{0}}\right) \) be a pointed topological space. The above multiplication turns the set \( \left\langle {\left( {X,{x}_{0}}\right) ,\mathrm{K}\left( {G, n}\right) }\right\rangle \) into a group. Furthermore the neutra... | Proof. Let \( n \in \mathbb{N} \), let \( G \) be an abelian group and let \( \left( {X,{x}_{0}}\right) \) be a pointed topological space. Recall that we have the identification\n\n\[ \n{\Xi }_{ * } : \left\langle {\left( {X,{x}_{0}}\right) ,\mathrm{K}\left( {G, n}\right) }\right\rangle = \left\langle {\left( {X,{x}_{0... | No |
Lemma 121.18. Let \( \left( {K,{k}_{0}}\right) \) be a pointed topological space. The map\n\n\[ \left\langle {\left( {X,{x}_{0}}\right) ,\Omega \left( {K,{k}_{0}}\right) }\right\rangle \times \left\langle {\left( {X,{x}_{0}}\right) ,\Omega \left( {K,{k}_{0}}\right) }\right\rangle \]\n\n\[ \text{map}\chi \text{from Lemm... | Proof. As we will see shortly, this statement follows fairly easily from Lemma 121.10. In particular we will see that the statements have preciously little to do with \( \left( {X,{x}_{0}}\right) \) . In an effort to lighten the notation we henceforth write \( X \) instead of the more accurate \( \left( {X,{x}_{0}}\rig... | Yes |
Lemma 121.20. Let \( \Xi : {S}^{1} \rightarrow \Omega \left( {{\mathbb{{CP}}}^{\infty }, * }\right) \) be the homotopy equivalence defined in Lemma 121.16. Given any topological space \( X \) we have the following commutative diagram where the horizontal maps are bijections: \n\nWe outsource this verification to Exercise 121.1. | No |
Proposition 121.22. Let \( \\left( {X,{x}_{0}}\\right) \) and \( \\left( {K,{k}_{0}}\\right) \) be pointed topological spaces. The following diagram commutes:\n\n\[ \n\\left\\langle {\\left( {X,{x}_{0}}\\right) ,\\Omega \\left( {K,{k}_{0}}\\right) }\\right\\rangle \\times \\left\\langle {\\left( {X,{x}_{0}}\\right) ,\\... | Proof. Let \( f, g : \\left( {X,{x}_{0}}\\right) \\rightarrow \\Omega \\left( {K,{k}_{0}}\\right) \) be two maps. One can easily verify that\n\n\[ \n\\left( {\\Upsilon \\circ {\\mu }_{ * } \\circ \\chi }\\right) \\left( {\\left\\lbrack f\\right\\rbrack ,\\left\\lbrack g\\right\\rbrack }\\right) = \\left\\lbrack \\begin... | Yes |
Lemma 121.23. Let \( n \in \mathbb{N} \) . The maps \[ G \mapsto \mathrm{K}\left( {G, n}\right) \] \[ \left( {\varphi : G \rightarrow H}\right) \mapsto \left\lbrack \varphi \right\rbrack \in \langle \mathrm{K}\left( {G, n}\right) ,\mathrm{K}\left( {H, n}\right) \rangle \] define a covariant functor from the category of... | Proof. The fact that the maps define a covariant functor is an immediate consequence of the uniqueness statement in Proposition 120.15. | No |
Proposition 121.24. Let \( n \in \mathbb{N} \) . (1) Let \( G \) be an abelian group. The maps \[ \left( {X,{x}_{0}}\right) \mapsto \left\langle {\left( {X,{x}_{0}}\right) ,\mathrm{K}\left( {G, n}\right) }\right\rangle \] \[ \left( {f : \left( {X,{x}_{0}}\right) \rightarrow \left( {Y,{y}_{0}}\right) }\right) \mapsto \l... | Proof. Let \( n \in \mathbb{N} \) . (1) We fix an abelian group \( G \) . First note that it follows from Lemma 15.4 that the assignment \( \left( {X,{x}_{0}}\right) \rightarrow \left\langle {\left( {X,{x}_{0}}\right) ,\mathrm{K}\left( {G, n}\right) }\right\rangle \) is a contravariant functor from the category of poin... | No |
Proposition 121.25. Let \( \left( {X,{x}_{0}}\right) \) and \( \left( {Y,{y}_{0}}\right) \) be two pointed CW-complexes. If \( Y \) is path-connected, if \( {\pi }_{1}\left( {Y,{y}_{0}}\right) \) is abelian and if \( {\pi }_{i}\left( {Y,{y}_{0}}\right) = 0 \) for \( i \geq 2{}^{1723} \) then the natural map \( \left\la... | Proof. Let \( \left( {Y,{y}_{0}}\right) \) be a pointed path-connected CW-complex such that the fundamental group \( {\pi }_{1}\left( {Y,{y}_{0}}\right) \) is abelian and such that \( {\pi }_{i}\left( {Y,{y}_{0}}\right) = 0 \) for \( i \geq 2 \) . By Proposition 38.7 (1) it remains to show that the map \( \left\langle ... | Yes |
Corollary 121.26. Let \( n \in \mathbb{N} \), let \( G \) be an abelian group and let \( X \) be a connected CW-complex. The definition of the group structure on \( \left\lbrack {X,\mathrm{\;K}\left( {G, n}\right) }\right\rbrack \) does not depend on the choice of \( {x}_{0} \in {X}^{0} \) . | Proof (*). Let \( n \in \mathbb{N} \), let \( G \) be an abelian group and let \( X \) be a connected CW-complex. Now let \( {x}_{0},{x}_{1} \in X \) be two points in the 0 -skeleton of \( X \) . Since \( X \) is connected we know by Proposition 38.10 that there exists a map \( f : X \rightarrow X \) with the following... | Yes |
Lemma 121.28. Given \( m, n \in \mathbb{N} \) there exists a canonical homeomorphism\n\n\[ \n{S}^{m} \land {S}^{n}\overset{ \cong }{ \rightarrow }{S}^{m + n} \n\] | Proof. We write \( I = \left\lbrack {0,1}\right\rbrack \) . Given any \( k \in {\mathbb{N}}_{0} \) we have the following identifications of\n\npointed topological spaces:\n\n\[ \n\left( {{I}^{k}/\partial {I}^{k},\left\lbrack {\partial {I}^{k}}\right\rbrack }\right) \underset{ \uparrow }{\overset{ \cong }{ \rightarrow }... | Yes |
Theorem 121.30. Let \( X = \left( {X,{x}_{0}}\right) \) be a pointed CW-complex and let \( m, n \in \mathbb{N} \) . The cup product\n\n\[ \cup : \langle X,\mathrm{\;K}\left( {\mathbb{Z}, m}\right) \rangle \times \langle X,\mathrm{\;K}\left( {\mathbb{Z}, n}\right) \rangle \rightarrow \langle X,\mathrm{\;K}\left( {\mathb... | Proof. We will not make use of this theorem, thus we will also not provide a proof. Instead  does not explicitly say that \( X \) needs to be a CW-complex, but implicitly the discussion on Stro11, Chapter 21] seems... | No |
Let \( \mathcal{H} \) be a Hilbert space. The set \( \mathcal{L}\left( \mathcal{H}\right) \) of all bounded linear operators on \( \mathcal{H} \) is a unital Banach algebra, with the operator norm, and the map \( T \mapsto {T}^{ * }\left( {T}^{ * }\right. \) being the adjoint of \( T \) ) is an involution that makes \(... | Here is the verification of (1.2): On the one hand, we have \( \begin{Vmatrix}{{T}^{ * }T}\end{Vmatrix} \leq \begin{Vmatrix}{T}^{ * }\end{Vmatrix}\parallel T\parallel = \parallel T{\parallel }^{2} \) . On the other, for any unit vector \( u \in \mathcal{H},\begin{Vmatrix}{{T}^{ * }T}\end{Vmatrix} \geq \left\langle {{T}... | Yes |
Let \( {l}^{1} = {l}^{1}\left( \mathbb{Z}\right) \) be the space of all sequences \( a = {\left( {a}_{n}\right) }_{-\infty }^{\infty } \) such that \( \parallel a\parallel = \mathop{\sum }\limits_{{-\infty }}^{\infty }\left| {a}_{n}\right| < \infty .{l}^{1} \) is a unital Banach algebra if we define multiplication to b... | For \( k \in \mathbb{Z} \), let \( {\delta }^{k} \in {l}^{1} \) be defined by \( {\left( {\delta }^{k}\right) }_{n} = 1 \) if \( n = k \) , \( {\left( {\delta }^{k}\right) }_{n} = 0 \) otherwise. (In particular, \( {\delta }^{0} = \delta \) .) It is easily verified that \( {\delta }^{j} * {\delta }^{k} = {\delta }^{j +... | No |
1.3 Lemma. If \( \parallel x\parallel < 1 \) then \( e - x \) is invertible, and\n\n\[{\left( e - x\right) }^{-1} = \mathop{\sum }\limits_{0}^{\infty }{x}^{n}\] | Proof. The usual proof that the geometric series \( \mathop{\sum }\limits_{0}^{\infty }{t}^{n} \) converges to \( 1/\left( {1 - t}\right) \) for \( \left| t\right| < 1 \) works equally well in any unital Banach algebra. | Yes |
Proposition 1.10(c) says that \( \sigma \left( \mathcal{A}\right) \) is a subset of the closed unit ball \( B \) of \( {\mathcal{A}}^{ * } \) . We make \( \sigma \left( \mathcal{A}\right) \) into a topological space by imposing its weak* topology as a subset of \( {\mathcal{A}}^{ * } \), that is, the topology of pointw... | The conditions \( h\left( e\right) = 1 \) and \( h\left( {xy}\right) = h\left( x\right) h\left( y\right) \) are clearly preserved under pointwise limits, so \( \sigma \left( \mathcal{A}\right) \) is a closed subset of \( B \) in the weak* topology. By Alaoglu’s theorem, then, \( \sigma \left( \mathcal{A}\right) \) is a... | Yes |
1.27 Proposition. If \( \mathcal{A} \) is a nonunital \( {C}^{ * } \) algebra, there is a unique norm on \( \widetilde{\mathcal{A}} \) that makes \( \widetilde{\mathcal{A}} \) into a \( {C}^{ * } \) algebra with involution (1.26). This norm agrees with the original norm on \( \mathcal{A} \) . | Proof. Since \( \mathcal{A} \) is an ideal in \( \widetilde{\mathcal{A}} \), each \( \left( {x, a}\right) \in \widetilde{\mathcal{A}} \) acts on \( \mathcal{A} \) by left multiplication: \( \left( {x, a}\right) \left( {y,0}\right) = \left( {{xy} + {ay},0}\right) \) . We define \( \parallel \left( {x, a}\right) \paralle... | Yes |
The Fourier transform satisfies \( {\left( f * g\right) }^{ \frown } = \widehat{f}\widehat{g} \) | It is not hard to show that these are all the multiplicative functionals on \( {L}^{1}\left( \mathbb{R}\right) \) ; we shall give the proof in \( §{4.1} \) (see Theorems 4.3 and 4.6(a)). | No |
1.31 Theorem. If \( \mathcal{A} \) is a nonunital commutative \( {C}^{ * } \) algebra, \( {\Gamma }_{\mathcal{A}} \) is an isometric \( * \) -isomorphism from \( \mathcal{A} \) to \( {C}_{0}\left( {\sigma \left( \mathcal{A}\right) }\right) \) . | Proof. This is simply a matter of combining our previous results. We make \( \widetilde{\mathcal{A}} \) into a \( {\mathrm{C}}^{ * } \) algebra according to Proposition 1.27. Then \( {\Gamma }_{\widetilde{\mathcal{A}}} \) is an isometric \( * \) -isomorphism from \( \widetilde{\mathcal{A}} \) to \( C\left( {\sigma \lef... | Yes |
2.69 Theorem. On any locally compact group \( G \) we have \( {L}^{1}\left( G\right) * \) \( {L}^{p}\left( G\right) = {L}^{p}\left( G\right) \) for \( 1 \leq p < \infty \) . Moreover, \( {L}^{1}\left( G\right) * {L}^{\infty }\left( G\right) = {L}^{1}\left( G\right) * \) \( {C}_{lu}\left( G\right) = {C}_{lu}\left( G\rig... | This theorem has a rather complicated history. The fact that \( {L}^{1}\left( G\right) * \) \( {L}^{1}\left( G\right) = {L}^{1}\left( G\right) \) was first proved by Salem for \( G = \mathbb{T} \) and then by Rudin for \( G = {\mathbb{R}}^{n} \times {\mathbb{T}}^{m}\left( {m, n \geq 0}\right) \) ; for general \( G \) i... | "No" |
If we identify \( \widehat{\mathbb{R}} \) with \( \mathbb{R} \) by the pairing \( \langle x,\xi \rangle = {e}^{2\pi i\xi x} \) , then Lebesgue measure is self-dual. | This can be seen by considering \( g\left( x\right) = {e}^{-\pi {x}^{2}} \) . We have \( \widehat{g}\left( \xi \right) = \int {e}^{-{2\pi i\xi x} - \pi {x}^{2}}{dx} \) ; differentiation under the integral followed by integration by parts shows that \( {\left( \widehat{g}\right) }^{\prime }\left( \xi \right) = - {2\pi \... | Yes |
If we identify \( {\widehat{\mathbb{Q}}}_{p} \) with \( {\mathbb{Q}}_{p} \) as in Theorem 4.13, the Haar measure on \( {\mathbb{Q}}_{p} \) such that \( \left| {\mathbb{Z}}_{p}\right| = 1 \) is self-dual. | To see this, let \( f \) be the characteristic function of \( {\mathbb{Z}}_{p} \). The restriction of any character \( {\xi }_{y} \) on \( {\mathbb{Q}}_{p} \) to the compact group \( {\mathbb{Z}}_{p} \) is a character on \( {\mathbb{Z}}_{p} \). Hence, if \( \left| {\mathbb{Z}}_{p}\right| = 1 \) and we identify \( {\xi ... | Yes |
The groups \( \mathbb{T} \) and \( \mathbb{Z} \) are dual to each other; the natural dual measures on them are normalized Lebesgue measure \( {d\theta }/{2\pi } \) on \( \mathbb{T} \) and counting measure on \( \mathbb{Z} \) . The Fourier inversion theorem for functions on \( \mathbb{T} \) reads: | \[ \widehat{f}\left( n\right) = {\int }_{0}^{2\pi }f\left( \theta \right) {e}^{-{in\theta }}\frac{d\theta }{2\pi },\;f\left( \theta \right) = \mathop{\sum }\limits_{{-\infty }}^{\infty }\widehat{f}\left( n\right) {e}^{in\theta }. \] | Yes |
4.26 Theorem (The Plancherel Theorem). The Fourier transform on \( {L}^{1}\left( G\right) \cap {L}^{2}\left( G\right) \) extends uniquely to a unitary isomorphism from \( {L}^{2}\left( G\right) \) to \( {L}^{2}\left( \widehat{G}\right) \) . | Proof. If \( f \in {L}^{1} \cap {L}^{2} \) then \( f * {f}^{ * } \in {L}^{1} \cap \mathcal{P} \) by Corollary 3.16, and \( {\left( f * {f}^{ * }\right) }^{ \frown } = {\left| \widehat{f}\right| }^{2} \), so by Theorem 4.22,\n\n\[ \n\int {\left| f\left( x\right) \right| }^{2}{dx} = f * {f}^{ * }\left( 1\right) = \int {\... | Yes |
If \( \sigma \) is the trivial representation of the trivial subgroup \( H = \{ 1\} \), then \( {\operatorname{ind}}_{H}^{G}\left( \sigma \right) \) is the ordinary left regular representation of \( G \). | If \( \left\lbrack \pi \right\rbrack \in \widehat{G} \), mult \( \left( {\sigma ,\pi \mid H}\right) \) clearly equals \( {d}_{\pi } \), so Frobenius reciprocity recaptures part of the Peter-Weyl theorem: each \( \left\lbrack \pi \right\rbrack \in \widehat{G} \) occurs in the regular representation with multiplicity equ... | No |
Take \( G \) to be \( {SO}\left( 3\right), H \) the subgroup that leaves the point \( \left( {1,0,0}\right) \) fixed, and \( \sigma \) the trivial representation of \( H \) on \( \mathbb{C} \) . Then \( G/H \) can be identified with the unit sphere \( {S}^{2} \subset {\mathbb{R}}^{3} \), and \( {\operatorname{ind}}_{H}... | To analyze this situation, let us replace \( {SO}\left( 3\right) \) by its double cover \( {SU}\left( 2\right) \) . The calculations leading to Theorem 5.44, together with (5.46), show that the subgroup \( \widetilde{H} \) of \( {SU}\left( 2\right) \) corresponding to \( H \) is the group\n\n\( \{ F\left( \theta \right... | Yes |
Let \( G \) and \( H \) be as in Example 2. \( H \) acts on the tangent plane to \( {S}^{2} \) at the point \( \left( {1,0,0}\right) \) (essentially the \( {yz} \) -plane in \( {xyz} \) - space) by rotations,\n\n(6.11)\n\n\[ \left( {y, z}\right) \mapsto \left( {y\cos \theta - z\sin \theta, y\sin \theta + z\cos \theta }... | We can identify the irreducible subspaces as follows. First, the Euclidean metric allows us to identify vector fields on \( {S}^{2} \) with differential 1-forms. The exterior derivative \( d \) maps functions to 1-forms and commutes with the action of \( {SO}\left( 3\right) \), so by Schur’s lemma it is either zero or ... | Yes |
Every Abelian group is type I. | Indeed, if \( \pi \) is a representation of an Abelian group \( G,\mathcal{A}\left( \pi \right) \) is commutative. Hence, if \( \pi \) is primary we must have \( \pi \left( x\right) = \langle x,\xi \rangle I \) for some \( \xi \in \widehat{G} \), and a choice of orthonormal basis for \( {\mathcal{H}}_{\pi } \) then exh... | No |
Example 3. The Heisenberg groups \( {H}_{n} \) discussed in \( §{6.7} \) are type I. | Indeed, the center \( Z \) of \( {H}_{n} \) is the set of elements of the form \( \left( {0,0, t}\right) \), so if \( \pi \) is a primary representation of \( {H}_{n} \), the operators \( \pi \left( {0,0, t}\right) \) must be scalar multiples of the identity, so that \( \pi \left( {0,0, t}\right) = {e}^{2\pi iht}I \) f... | Yes |
Let us consider the discrete Heisenberg group \( \mathrm{H} \) and its central quotients \( {\mathrm{H}}_{q}\left( {q \in {\mathbb{Z}}^{ + }}\right) \) discussed in \( §{6.8} \). For each \( q \) , \( \{ \left( {j, k, l}\right) : q \mid j\} \) is an Abelian normal subgroup of finite index of \( {\mathrm{H}}_{q} \) , so... | We have \( \widehat{\mathrm{H}} = \mathop{\bigcup }\limits_{{\omega \in \mathbb{R}/\mathbb{Z}}}{\widehat{\mathrm{H}}}^{\omega } \), where \( {\widehat{\mathrm{H}}}^{\omega } \) is the set of equivalence classes of irreducible representations with central character \( {\chi }_{\omega } \), and we know from the results i... | Yes |
7.19 Proposition. Let \( \left\{ {\mathcal{H}}_{\alpha }\right\} ,\left\{ {e}_{j}\right\} \) be a measurable field of Hilbert spaces over \( A \), with \( \dim {\mathcal{H}}_{\alpha } = d\left( \alpha \right) \in \left\lbrack {1,\infty }\right\rbrack \) . Then \( \{ \alpha \in A : d\left( \alpha \right) = m\} \) is mea... | Proof. First, define a sequence \( \left\{ {f}_{j}\right\} \) of vector fields inductively as follows: \( {f}_{1}\left( \alpha \right) \) is the first of the vectors \( {e}_{1}\left( \alpha \right) ,{e}_{2}\left( \alpha \right) ,\ldots \) that is nonzero; for \( j > 1 \) , \( {f}_{j}\left( \alpha \right) \) is the firs... | Yes |
For \( x \in \left\lbrack {0,1}\right\rbrack \), let\n\n\[ f\left( x\right) = \left\{ \begin{array}{ll} \frac{1}{x} & \text{ if }x \neq 0 \\ 0 & \text{ if }x = 0 \end{array}\right. \]\n\nThis time \( f \) is defined for \( x = 0 \), but \( f \) is still not in \( B\left\lbrack {0,1}\right\rbrack \) . | Although one can see this by thinking of the graph of \( f \), we give a more formal argument here. For every \( M > 1,\frac{1}{M + 1} \in \left\lbrack {0,1}\right\rbrack \) and \( f\left( \frac{1}{M + 1}\right) = \) \( M + 1 > M \) . Thus \( f \) is not in \( B\left\lbrack {0,1}\right\rbrack \) because no constant \( ... | Yes |
Example 0.1.3. Let\n\n\[ f\left( x\right) = \left\{ \begin{array}{ll} 1 & \text{ if }x \in \mathbb{Q} \\ 0 & \text{ if }x \notin \mathbb{Q} \end{array}\right. \]\n\nThis function is known as the Dirichlet function and is often denoted by \( {\mathcal{X}}_{\mathbb{Q}}\left( x\right) \) . Because \( \left| {{\mathcal{X}}... | The Dirichlet function shows that not all functions in \( B\left\lbrack {a, b}\right\rbrack \) are continuous. On the other hand, if \( f \) is continuous on \( \left\lbrack {a, b}\right\rbrack \), then \( f \in B\left\lbrack {a, b}\right\rbrack \) by the Extreme Value Theorem, a result found in most texts used for a f... | Yes |
Proposition 0.1.6. Let \( f \in B\left\lbrack {a, b}\right\rbrack \) . For any partition \( P \) of \( \left\lbrack {a, b}\right\rbrack \) ,\n\n\[ m\left( {b - a}\right) \leq L\left( {f, P}\right) \leq U\left( {f, P}\right) \leq M\left( {b - a}\right) ,\]\n\nwhere\n\n\[ m = \mathop{\inf }\limits_{{x \in \left\lbrack {a... | The proof of this proposition is straightforward and is left to the reader as an exercise (See Exercise 1). | No |
Consider the Dirichlet function \( {\mathcal{X}}_{\mathbb{Q}}\left( x\right) \) as described in Example 0.1.3 For any partition \( P \) of \( \left\lbrack {0,1}\right\rbrack \) , | \[ L\left( {{\mathcal{X}}_{\mathbb{Q}}, P}\right) = 0\;\text{ and }\;U\left( {{\mathcal{X}}_{\mathbb{Q}}, P}\right) = 1. \] Therefore, \[ {\int }_{0}^{1}{\mathcal{X}}_{\mathbb{Q}}\left( x\right) {dx} = 0\;\text{ and }\;\overline{{\int }_{0}^{1}}{\mathcal{X}}_{\mathbb{Q}}\left( x\right) {dx} = 1. \] | Yes |
Lemma 0.2.2. Let \( f \in B\left\lbrack {a, b}\right\rbrack \) .\ni) If \( {P}^{ * } \) is a refinement of the partition \( P \) of \( \left\lbrack {a, b}\right\rbrack \), then\n\n\[ L\left( {f, P}\right) \leq L\left( {f,{P}^{ * }}\right) \leq U\left( {f,{P}^{ * }}\right) \leq U\left( {f, P}\right) .\n\]\n\nii) If \( {... | Proof. Let \( f \in B\left\lbrack {a, b}\right\rbrack \) . We will only give a sketch of the argument and leave some of the details to the reader.\n\ni) The main step for this part is to show the result is true if \( {P}^{ * } \) is just \( P \) with one additional point. To this end, let \( P \) be the partition\n\n\[... | No |
Corollary 0.2.3. Let \( f \in B\left\lbrack {a, b}\right\rbrack \) . Then\n\n\[{\int }_{a}^{b}f\left( x\right) {dx} \leq \overline{{\int }_{a}^{b}}f\left( x\right) {dx}.\] | Proof. By Lemma 0.2.2, given any two partitions \( {P}_{1} \) and \( {P}_{2} \) of \( \left\lbrack {a, b}\right\rbrack \) ,\n\n\[L\left( {f,{P}_{1}}\right) \leq U\left( {f,{P}_{2}}\right)\]\n\nHence, \( U\left( {f,{P}_{2}}\right) \) is an upper bound for \( \{ L\left( {f, P}\right) \mid P \) is a partition of \( \left\... | Yes |
Let\n\n\[ f\\left( x\\right) = \\left\\{ \\begin{array}{ll} 2 & \\text{ if }x \\neq \\frac{1}{2} \\\\ 5 & \\text{ if }x = \\frac{1}{2} \\end{array}\\right.\n\]\n\nWe will first show \( f \) is Riemann integrable. | One option is to compute the lower integral and upper integral by comparing all lower sums and all upper sums. Instead, we will use Theorem 0.2.4 Let \( \\epsilon > 0 \) be small. Let \( {P}_{\\epsilon } \) be the partition\n\n\[ {P}_{\\epsilon } = \\left\\{ {0,\\frac{1}{2} - \\frac{\\epsilon }{7},\\frac{1}{2} + \\frac... | Yes |
Theorem 0.2.6. Let \( f \) be continuous on \( \left\lbrack {a, b}\right\rbrack \) . Then \( f \in R\left\lbrack {a, b}\right\rbrack \) . | Proof. We will use Theorem 0.2.4 Let \( \epsilon > 0 \) be given. Since \( f \) is continuous on the closed and bounded interval \( \left\lbrack {a, b}\right\rbrack, f \) must be uniformly continuous on \( \left\lbrack {a, b}\right\rbrack \) . Thus, there is a \( \delta > 0 \) so that\n\n\[ \left| {f\left( x\right) - f... | No |
We will compute the Lebesgue outer measure of \( A = \{ 3\} \) . | Let \( \epsilon > 0 \) . Set \( S = \{ \left\lbrack {3 - \epsilon ,3 + \epsilon }\right\rbrack \} \) . Thus,\n\n\[ 0 \leq {m}^{ * }\left( A\right) \leq \sigma \left( S\right) = {2\epsilon }.\]\n\nSince \( \epsilon \) was arbitrary, it follows that \( {m}^{ * }\left( A\right) = 0 \) . | Yes |
The Lebesgue outer measure of \( \varnothing \) is 0. | To see this, let \( \epsilon > 0 \) be given. Then \( S = \{ \left\lbrack {-\epsilon ,\epsilon }\right\rbrack \} \) is a covering of \( \varnothing \) by closed intervals. Therefore,\n\n\[ \n{m}^{ * }\left( \varnothing \right) \leq \sigma \left( S\right) = {2\epsilon } \n\]\n\nSince \( \epsilon \) was arbitrary, it fol... | Yes |
Let \( A = \left\lbrack {0,1}\right\rbrack \) . The Lebesgue outer measure of \( A \) is 1. | This should come as no surprise. After all, the length of this interval is 1 . In fact, \( S = \{ \left\lbrack {0,1}\right\rbrack \} \) is a covering of \( A \) by a single closed interval. Therefore,\n\n\[ \n{m}^{ * }\left( A\right) \leq \sigma \left( S\right) = 1 \n\]\n\nHowever, it is not an easy matter to prove tha... | No |
Let \( E = \left\{ {\left( {x,0}\right) \in {\mathbb{R}}^{2} \mid 0 \leq x \leq 1}\right\} \) . Let \( \epsilon > 0 \) be given. Set\n\n\[ \n{I}_{\epsilon } = \left\{ {\left( {x, y}\right) \in {\mathbb{R}}^{2} \mid 0 \leq x \leq 1, - \epsilon \leq y \leq \epsilon }\right\} .\n\]\n\nThen \( S = \left\{ {I}_{\epsilon }\r... | Since \( \epsilon \) was arbitrary, \( {m}^{ * }\left( E\right) = 0 \) . | Yes |
Proposition 1.1.8. If \( A \subseteq B \subseteq {\mathbb{R}}^{n} \), then \( {m}^{ * }\left( A\right) \leq {m}^{ * }\left( B\right) \) . | Proof. Let \( S \) be a covering of \( B \) by closed intervals. It follows that \( S \) is also a covering of \( A \) by closed intervals. Thus,\n\n\[ \n{m}^{ * }\left( A\right) \leq \sigma \left( S\right) \n\]\n\nwhere \( S \) is any covering of \( B \) by closed intervals. Hence,\n\n\( {m}^{ * }\left( A\right) \leq ... | Yes |
Proposition 1.1.9. The following additivity properties hold for Lebesgue outer measure:\n\n(i) For any two sets \( A \) and \( B \) ,\n\n\[ \n{m}^{ * }\left( {A \cup B}\right) \leq {m}^{ * }\left( A\right) + {m}^{ * }\left( B\right) .\n\] | Proof. (i) If either \( {m}^{ * }\left( A\right) \) or \( {m}^{ * }\left( B\right) \) is infinite, then \( {m}^{ * }\left( {A \cup B}\right) \) is also infinite by Proposition 1.1.8 Thus, in this case the result is true by the convention that \( + \infty \leq + \infty \) . So, assume both \( {m}^{ * }\left( A\right) \)... | Yes |
Corollary 1.1.10. If \( A \subseteq B \subseteq {\mathbb{R}}^{n} \) and \( {m}^{ * }\left( B\right) \) is finite, then\n\n\[ {m}^{ * }\left( B\right) - {m}^{ * }\left( A\right) \leq {m}^{ * }\left( {B \smallsetminus A}\right) . \] | ## Proof. This is Exercise 7. | No |
Proposition 1.1.11. For any closed interval \( I \subseteq {\mathbb{R}}^{n},{m}^{ * }\left( I\right) = v\left( I\right) \) . | Proof. We need to show that \( v\left( I\right) \leq {m}^{ * }\left( I\right) \) . Because of the above discussion, we only need to prove that if \( S = \left\{ {I}_{k}\right\} \) is a countably infinite covering of \( I \) by closed intervals, then \( v\left( I\right) \leq \sigma \left( S\right) \) . Given\n\n![8c56b0... | No |
We can now compute the Lebesgue outer measure of \( B = \left\lbrack {-1,2}\right\rbrack \cup \{ 3\} \) . | By Proposition 1.1.9,\n\n\[ \n{m}^{ * }\left( B\right) \leq {m}^{ * }\left( \left\lbrack {-1,2}\right\rbrack \right) + {m}^{ * }\left( {\{ 3\} }\right) .\n\]\n\nBy Proposition 1.1.11 and Example 1.1.3,\n\n\[ \n{m}^{ * }\left( \left\lbrack {-1,2}\right\rbrack \right) = 2 - \left( {-1}\right) = 3\;\text{ and }\;{m}^{ * }... | Yes |
Theorem 1.1.13. Let \( A \subseteq {\mathbb{R}}^{n} \) be a set with finite outer measure. For every \( \epsilon > 0 \) there is an open set \( G \) such that \( A \subseteq G \) and\n\n\[ \n{m}^{ * }\left( G\right) < {m}^{ * }\left( A\right) + \epsilon \n\] | Proof. Given \( \epsilon > 0 \) there is a covering of \( A \) by closed intervals \( S = \left\{ {I}_{k}\right\} \) such that\n\n\[ \n\sigma \left( S\right) = \sum v\left( {I}_{k}\right) < {m}^{ * }\left( A\right) + \frac{\epsilon }{2}. \n\]\n\n(Here we are using the assumption that \( {m}^{ * }\left( A\right) \) is f... | Yes |
Corollary 1.1.14. Let \( A \subseteq {\mathbb{R}}^{n} \) . For every \( \epsilon > 0 \) there is an open set \( G \) such that \( A \subseteq G \) and \[ {m}^{ * }\left( G\right) \leq {m}^{ * }\left( A\right) + \epsilon \] | Proof. If \( {m}^{ * }\left( A\right) \) is finite, we can use the open set \( G \) from the previous theorem. In the case that \( {m}^{ * }\left( A\right) \) is infinite, use \( G = {\mathbb{R}}^{n} \) . | Yes |
We will show that \( E = \{ 3\} \) is Lebesgue measurable. | Given \( \epsilon > 0 \) let \( G = \left( {3 - \frac{\epsilon }{3},3 + \frac{\epsilon }{3}}\right) \) . By Propostion 1.1.8 and by Proposition 1.1.11,\n\n\[ \begin{matrix} {m}^{ * }\left( {G \backslash E}\right) & = & {m}^{ * }\left( {\left( {3 - \frac{\epsilon }{3},3}\right) \cup \left( {3,3 + \frac{\epsilon }{3}}\ri... | Yes |
Every set with Lebesgue outer measure 0 is measurable. | To verify this, suppose \( E \subseteq {\mathbb{R}}^{n} \) is a set with \( {m}^{ * }\left( E\right) = 0 \) . Given \( \epsilon > 0 \), by Theorem 1.1.13, there is an open set \( G \) containing \( E \) with\n\n\[ \n{m}^{ * }\left( G\right) < {m}^{ * }\left( E\right) + \epsilon = \epsilon .\n\]\n\nBy Proposition 1.1.8,... | Yes |
Theorem 1.2.5. Let \( \\left\\{ {E}_{k}\\right\\} \) be a countable collection of Lebesgue measurable sets. Then\n\n\[ E = \\bigcup {E}_{k} \]\n\nis Lebesgue measurable and\n\n\[ m\\left( E\\right) \\leq \\sum m\\left( {E}_{k}\\right) \] | Proof. Let \( \\epsilon > 0 \) be given. We must show there exists an open set \( G \) containing \( E = \\bigcup {E}_{k} \) such that \( {m}^{ * }\\left( {G \\smallsetminus E}\\right) < \\epsilon \) .\n\nFor each \( k \) there exists an open set \( {G}_{k} \) containing \( {E}_{k} \) such that\n\n\[ {m}^{ * }\\left( {... | Yes |
Let \( I \subseteq {\mathbb{R}}^{n} \) be a closed interval in \( {\mathbb{R}}^{n} \). Then \( I \) is the union of its interior, which is an open set, and its sides. | The open interior is measurable by Example 1.2.3. The sides are subsets of hyperplanes, which have Lebesgue outer measure 0. Thus, the sides have Lebesgue outer measure 0 and are Lebesgue measurable by Example 1.2.4 Consequently, \( I \) is the countable union of measurable sets. By Theorem 1.2.5, \( I \) is Lebesgue m... | Yes |
Lemma 1.2.9. Let \( {\left\{ {I}_{n}\right\} }_{n = 1}^{M} \) be a finite collection of pairwise nonoverlapping closed intervals. Then\n\n\[ m\left( {\mathop{\bigcup }\limits_{{n = 1}}^{M}{I}_{n}}\right) = \mathop{\sum }\limits_{{n = 1}}^{M}v\left( {I}_{n}\right) \] | Proof. It follows from Example 1.2.6 and Theorem 1.2.5 that \( \mathop{\bigcup }\limits_{{n = 1}}^{M}{I}_{n} \) is measurable and\n\n\[ m\left( {\mathop{\bigcup }\limits_{{n = 1}}^{M}{I}_{n}}\right) \leq \mathop{\sum }\limits_{{n = 1}}^{M}v\left( {I}_{n}\right) \]\n\nWe need only establish the reverse inequality.\n\nAs... | No |
Lemma 1.2.10. Every nonempty open set \( G \subseteq {\mathbb{R}}^{n} \) can be written as the countable union of pairwise nonoverlapping closed intervals. | Proof. Let \( G \subseteq {\mathbb{R}}^{n} \) be an open set. Divide \( {\mathbb{R}}^{n} \) into nonoverlapping intervals along the hyperplanes \( {x}_{i} = k \), where \( k \in \mathbb{Z} \), thus creating a countable collection of closed intervals. Set aside those closed intervals which are completely contained in \(... | Yes |
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