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Lemma 74.6 (3) to \( p \), finally apply Proposition 82.4 (1) to \( p \) | \[ \begin{array}{l} = \left\langle {\overline{{\mathrm{{PD}}}_{Z}}\left( {{i}_{ * }\left( a\right) }\right) \cup {\mathrm{{PD}}}_{Z}\left( {{i}_{ * }\left( b\right) }\right) ,\left\lbrack Z\right\rbrack }\right\rangle \underset{ \uparrow }{ = }\overline{{Q}_{Z}}\left( {{i}_{ * }\left( a\right) ,{i}_{ * }\left( b\right)... | No |
Proposition 100.15. Let \( M \) be a compact oriented \( n \) -dimensional topological manifold. Furthermore 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 = \part... | (1) We consider the map\n\n\[ \n\Psi : {\mathrm{{FH}}}_{k}\left( {M, A;\mathbb{Z}}\right) \overset{{\mathrm{{PD}}}_{M}}{ \rightarrow }{\mathrm{{FH}}}^{n - k}\left( {M, B;\mathbb{Z}}\right) \overset{\mathrm{{ev}}}{ \rightarrow }\operatorname{Hom}\left( {{\mathrm{{FH}}}_{n - k}\left( {M, B;\mathbb{Z}}\right) ,\mathbb{Z}}... | Yes |
Lemma 100.16. Let \( M \) be a compact oriented connected \( n \) -dimensional topological manifold and suppose that we are given two compact \( \left( {n - 1}\right) \) -dimensional submanifolds \( A \) and \( B \) of \( \partial M \) . We suppose that one of the following two conditions is satisfied:\n\n(i) We have \... | Proof \( \left( *\right) \) . We write \( X = \partial M \smallsetminus \overset{ \circ }{A} \) and \( Y = \partial M \smallsetminus \overset{ \circ }{B} \). \n\n(1) Given \( v \in {\mathrm{H}}_{k}\left( {M, A;\mathbb{Z}}\right) \) and \( w \in {\mathrm{H}}_{n - k}\left( {M, B;\mathbb{Z}}\right) \) we have the followin... | Yes |
Proposition 100.19. Let \( M \) be a compact oriented \( m \) -dimensional topological manifold and let \( W \subset M \) be a compact codimension-zero submanifold with corner. \( {}^{1413} \) Let \( A \) and \( B \) be disjoint unions of components of \( \partial M \) with \( {\partial }_{1}W \subset A \) . We denote ... | Proof \( \\left( *\\right) \) . We perform the following calculation:\n\n\[ \n{Q}_{W,{\partial }_{1}W,{\partial }_{0}W}^{\mathrm{{as}}}\left( {\alpha ,{\left( {w}_{ * }\right) }^{-1}\left( {{p}_{ * }\left( \beta \right) }\right) = \\left\\langle {{\\mathrm{{PD}}}_{W}\left( \alpha \right) \\cup {\\mathrm{{PD}}}_{W}\left... | Yes |
(1) Let \( \alpha \in {\mathrm{{FH}}}_{n}\left( {M;\mathbb{Z}}\right) \) . The following two statements are equivalent:\n\n(a) \( \alpha \in \operatorname{im}\left( {{i}_{ * } : {\mathrm{{FH}}}_{n}\left( {\partial M;\mathbb{Z}}\right) \rightarrow {\mathrm{{FH}}}_{n}\left( {M;\mathbb{Z}}\right) }\right) \) .\n\n(b) For ... | Proof. We denote by \( i : \partial M \rightarrow M \) and \( j : \left( {M,\varnothing }\right) \rightarrow \left( {M,\partial M}\right) \) the obvious maps. Before we do anything else we recall that we have a long exact sequence\n\n\[ \ldots \rightarrow {\mathrm{H}}_{n}\left( {\partial M;\mathbb{Z}}\right) \overset{{... | No |
Lemma 101.1. Let \( R \) be a commutative ring.\n\n(1) Let \( A \) be a (anti-) symmetric \( {}^{1417}n \times n \) -matrix over \( R \) . We consider the map\n\n\[ \varphi \left( A\right) : {R}^{n} \times {R}^{n} \rightarrow R \]\n\n\[ \left( {v, w}\right) \mapsto {v}^{T}{Aw} \]\n\nThe following statements hold:\n\n(a... | Proof (*).\n\n(1) (a) This statement follows almost immediately from the definitions and the observation that for any \( v, w \in {R}^{n} \) we have\n\n\[ \varphi \left( A\right) \left( {w, v}\right) = {w}^{T}{Av}\underset{ \uparrow }{ = }{\left( {w}^{T}{A}^{T}v\right) }^{T} = {v}^{T}{A}^{T}w = \varphi \left( {A}^{T}\r... | No |
Proposition 101.2. If \( \left( {V,\varphi }\right) \) is a non-singular anti-symmetric form over \( \mathbb{R} \) or over \( \mathbb{Z} \) , then there exists a \( k \in {\mathbb{N}}_{0} \) such that \( {}^{1418}\left( {V,\varphi }\right) \cong k \cdot \varphi \left( S\right) \), i.e. \( \varphi \) is represented by t... | SKETCH OF PROOF. First we remark that throughout the proof on many occasions we use Lemma 101.1 without saying so explicitly. After this preamble, let \( \left( {V,\varphi }\right) \) be a non-singular anti-symmetric form over \( \mathbb{R} \) of rank \( n \) . We pick an \( \left( {n \times n}\right) \)-matrix \( A \)... | No |
Proposition 85.34 (4) by Theorem 88.6 we have \( {\dim }_{\mathbb{R}}\left( {{\mathrm{H}}_{i}\left( {M;\mathbb{R}}\right) }\right) = {\dim }_{\mathbb{R}}\left( {{\mathrm{H}}_{{4k} + 2 - i}\left( {M;\mathbb{R}}\right) }\right) \) | \n\n\( \equiv 0\;{\;\operatorname{mod}\;2} \)\n\n\( \uparrow \)\n\nby Proposition 100.4 and the facts that \( M \) is closed and that \( {2k} + 1 \) is odd we know\n\nthat the intersection form on \( {\mathrm{H}}_{{2k} + 1}\left( {M;\mathbb{R}}\right) \) is anti-symmetric and non-singular,\n\nby Proposition 101.2 we th... | Yes |
Proposition 101.6. If \( \left( {V,\varphi }\right) \) is a non-singular symmetric form over \( \mathbb{R} \) of rank \( n \), then there exists a \( k \in {\mathbb{N}}_{0} \) such that \( \left( {V,\varphi }\right) \cong k \cdot \left( {+1}\right) \oplus \left( {n - k}\right) \cdot \left( {-1}\right) \) . | Sketch of PROOF. Let \( \left( {V,\varphi }\right) \) be a non-singular symmetric form over \( \mathbb{R} \) of rank \( n \) . In the subsequent arguments below we use the conventions and notions we introduced in Proposition 101.2 In particular we will also use the operations of type (i), (ii) and (iii) that we introdu... | No |
Proposition 101.8. If \( D \) is a diagonal \( \left( {n \times n}\right) \) -matrix where the diagonal entries \( {d}_{1},\ldots ,{d}_{n} \) are non-zero real numbers, then\n\n\( {b}^{ + }\left( {\varphi \left( D\right) }\right) = \) number of positive \( {d}_{i} \) ’s \( \; \) and \( \;{b}^{ - }\left( {\varphi \left(... | Proof. After reordering the diagonal entries we can assume that the first \( k \) entries of \( D \) are positive and that the remaining \( \left( {n - k}\right) \) entries are negative. We write\n\n\[ {V}^{ + } \mathrel{\text{:=}} \left\{ {\left( {{x}_{1},\ldots ,{x}_{k},0,\ldots ,0}\right) \in {\mathbb{R}}^{n} \mid {... | Yes |
Corollary 101.9.\n\n(1) Let \( \left( {V,\varphi }\right) \) be a non-singular symmetric form over \( \mathbb{R} \) . We write \( n = {\dim }_{\mathbb{R}}\left( V\right) \) . The\n\nfollowing statements hold:\n\n(a) We have \( {b}^{ \pm }\left( {-\varphi }\right) = {b}^{ \mp }\left( \varphi \right) \), in particular we... | Proof.\n\n(1) (a) This statement follows immediately from the definitions.\n\n(b) By Proposition 101.6 and Lemma 101.7 it suffices to prove the statement for non-singular symmetric forms that are represented by diagonal matrices. But for those forms the statement follows immediately from Proposition 101.8.\n\n(c) This ... | Yes |
Theorem 101.10. (Sylvester's Theorem a. k. a. Sylvester's law of inertia) Let \( \\left( {V,\\varphi }\\right) \) and \( \\left( {W,\\psi }\\right) \) be two non-singular symmetric forms over \( \\mathbb{R} \) . Then\n\n\( \\left( {V,\\varphi }\\right) \) and \( \\left( {W,\\psi }\\right) \) are isometric \( \\; \\Left... | Proof. The \ | No |
Lemma 101.11. Let \( k \in \mathbb{N} \) .\n\n(1) Let \( M \) be a compact oriented \( {4k} \) -dimensional topological manifold. The form \( {Q}_{M} \otimes \mathbb{R} \) is isometric to the form\n\n\[ \n{\mathrm{H}}^{2k}\left( {M,\partial M;\mathbb{R}}\right) \times {\mathrm{H}}^{2k}\left( {M,\partial M;\mathbb{R}}\r... | Proof.\n\n(1) This statement follows easily from the various definitions and Corollary 57.21. | No |
If \( {\varphi }_{1} \) and \( {\varphi }_{2} \) are two non-singular indefinite symmetric forms over the integers, then the following holds:\n\n\( {\varphi }_{1} \) and \( {\varphi }_{2} \) are isometric \( \Leftrightarrow {\varphi }_{1} \) and \( {\varphi }_{2} \) have the same rank, signature and parity. | The proof of the theorem is provided in [MH73, Theorem II.5.3] (see also [GoS99, Theorem 1.2.14 for an outline of the proof). The proof relies crucially on a significant input from algebraic number theory, namely the Hasse-Minkowski theorem. This theorem says that given \( {a}_{1},\ldots ,{a}_{n} \in \mathbb{Q} \smalls... | Yes |
Corollary 101.17. If \( \varphi \) is a non-singular indefinite odd symmetric form over the integers, then\n\n\[ \varphi \cong {b}^{ + }\left( \varphi \right) \cdot \left( 1\right) \oplus {b}^{ - }\left( \varphi \right) \cdot \left( {-1}\right) . \] | Proof. Let \( \varphi \) be a non-singular indefinite odd symmetric form. It follows from Proposition 101.8 that the rank, signature and parity of the two non-singular indefinite odd forms in the statement of the corollary agree. So these two forms are isometric by Theorem [101.16] | No |
(1) The \( {\mathbb{F}}_{2} \) -valued intersection form \( {Q}_{M}^{{\mathbb{F}}_{2}} \) on a closed \( {2n} \) -dimensional topological manifold \( M \) is symmetric and non-singular.\n\n(2) The \( {\mathbb{F}}_{2} \) -valued intersection forms of homeomorphic closed topological manifolds are isometric. | Proof. The proof is a very mild variation on the proofs of Proposition 100.4 and Lemma 100.6 | No |
Proposition 101.22. Two non-singular symmetric forms \( \left( {V,\kappa }\right) \) and \( \left( {W,\lambda }\right) \) over \( {\mathbb{F}}_{2} \) are isometric if and only if \( \dim \left( V\right) = \dim \left( W\right) \) and if \( \kappa \) and \( \lambda \) have the same parity. | Proof. We will prove Proposition 101.22 in Exercise 101.13. For peace of mind we point out that the proposition is proved in Hopk15, Theorem 2.7. It can also be deduced from Szy97, Corollary 6.3.1]. | No |
Proposition 102.2. The K3-surface is a closed simply connected complex manifold of complex dimension 2 or equivalently of real dimension 4. Furthermore we have intersection form of the K3-surface \( \cong 2 \cdot {E}_{8} \oplus 3 \cdot H \) . | Proof. All the statements are proved in GoS99, Theorem 1.3.8 or alternatively [McS16, p. 176]. The intersection form is also calculated in [Huy16, Proposition 3.5]. The calculation of the intersection form, perhaps somewhat disappointingly, makes use of Corollary 101.19 | Yes |
Proposition 102.4. Let \( X \) be a closed oriented 4-dimensional smooth manifold.\n\n(1) If \( X \) is spin, then the intersection form \( {Q}_{X} \) is even.\n\n(2) If the intersection form \( {Q}_{X} \) is even and if \( {\mathrm{H}}_{1}\left( {X;\mathbb{Z}}\right) \) has no 2-torsion, then the smooth manifold \( X ... | Proof. The proposition is the content of GoS99, Corollary 5.7.6], which in turn is proved using the \ | No |
Theorem 102.5. (Rokhlin 1952) Let \( X \) be a closed oriented 4-dimensional smooth manifold. If \( X \) is spin, then \( \operatorname{sign}\left( X\right) \) is divisible by 16. | Proof. This theorem was proved by Vladimir Rokhlin [Rok52] [Rok86a, p. 21] in 1952. More modern proofs of Rokhlin's Theorem are given in [FK78], LaM89, Theorem II.2.13 and Corollary IV.1.2] and [Sav02, Theorem 2.1]. | No |
Theorem 102.7. (Milnor 1958) Let \( X \) and \( Y \) be two closed oriented simply connected 4-dimensional topological manifolds. Then \( X \) and \( Y \) are homotopy equivalent \( \Leftrightarrow {Q}_{X} \) is isometric to \( \pm {Q}_{Y} \) . | Proof. This theorem was proved for smooth manifolds by John Milnor [Miln58b, Theorem 3] in 1958, building on work of John Whitehead [WhdJ49a]. Furthermore [GoS99], Theorem 1.2.25 and Kir89, Theorem 2.1] sketch a proof that also works for topological manifolds. | No |
Theorem 102.9. There exist closed orientable 4-dimensional topological manifolds that do not admit a smooth structure. | Proof. At this point we can combine the above results to give two types of examples of such smooth manifolds:\n\n(1) It follows from Theorem 102.8 (2) and the above Remark (3) that given any nonsingular odd symmetric form \( Q \) there exists a closed oriented 4-dimensional topological manifold \( X \) with \( {Q}_{X} ... | Yes |
Proposition 103.3. Let \( M \) be a compact \( n \) -dimensional smooth manifold. Suppose that for \( i = 1,\ldots, m \) we are given smooth embeddings\n\n\[ \n{\varphi }_{i} : {\bar{B}}^{n - {k}_{i}} \times {S}^{{k}_{i} - 1} \rightarrow \partial \left( {M{ \cup }_{{\varphi }_{1}}{h}^{{k}_{1}}\cdots { \cup }_{{\varphi ... | Proof. To simplify the notation we prove the proposition for the special case \( m = 1 \) . The proof of the general case is not much harder and we leave it to the reader to fill in the details.\n\nThus suppose that we are given a smooth embedding \( \varphi : {\bar{B}}^{n - k} \times {S}^{k - 1} \rightarrow \partial M... | No |
Corollary 103.4. Let \( M \) be a compact \( n \) -dimensional smooth manifold. If \( M \) admits a handle decomposition, then it also admits a handle decomposition with the same number of handles of each index, but which is now standard. | Proof. This follows immediately from iteratively applying Proposition 103.3 to reorder the handles. | No |
Lemma 103.6. Let \( X \) be an \( n \) -dimensional smooth manifold and let \( \varphi : {\bar{B}}^{n - k} \times {S}^{k - 1} \rightarrow \partial X \) be a smooth embedding. Let \( X{ \cup }_{\varphi }{h}^{k} \mathrel{\text{:=}} X{ \cup }_{\partial X\overset{\varphi }{ \leftarrow }{\bar{B}}^{n - k} \times {S}^{k - 1}}... | Sketch of PROOF OF LEMMA 103.6. Once one has digested the statement of the lemma, one realizes, that the lemma comes close to being a tautology. One can easily verify that the \ | No |
Corollary 103.7. Let \( M \) be a closed connected non-empty \( n \) -dimensional smooth manifold. For every handle decomposition of \( M \) we have the inequality:\n\n\[ 1 + \# \left( {n - 1}\right) \text{-handles - #n-handles} \geq \text{minimal size of a generating set of}{\pi }_{1}\left( M\right) \text{.} \] | Proof. Let \( M \) be a closed connected non-empty \( n \) -dimensional smooth manifold that is equipped with a handle decomposition of \( M \) . We have the following (in-) equalities:  | No |
Lemma 103.8. Let \( M \) be a compact \( n \) -dimensional smooth manifold. Suppose that we are given a thickened \( \left( {k - 1}\right) \) -sphere in the boundary \( \partial M \), i.e. suppose we are given a smooth embedding \( \varphi : {\bar{B}}^{n - k} \times {S}^{k - 1} \rightarrow \partial M \). The following ... | Proof. The statements for \( k > 1 \) and \( k = 1 \) follow easily from Lemma 2.62. The statement for \( k = 0 \) follows from the fact that adding a 0 -handle is just taking the disjoint union with the 0-handle. | Yes |
Lemma 103.10. Let \( M \) be a compact oriented \( n \) -dimensional smooth manifold, let \( F \) be a component of \( \partial M \) and let \( \varphi ,\psi : {\bar{B}}^{n - 1} \times {S}^{0} \rightarrow F \) be two thickened 0 -spheres. If \( \varphi \) and \( \psi \) are both orientable or if they are both non-orien... | Proof. Let \( \varphi ,\psi : {\bar{B}}^{n - 1} \times {S}^{0} \rightarrow F \) be two thickened 0 -spheres. First assume that \( n \geq 3 \) . As in Proposition 103.9 (1) let \( \tau : {S}^{0} \rightarrow {S}^{0} \) be the map that swaps the two points \( \pm 1 \) . By Proposition 103.9 we can, if convenient, replace ... | No |
Proposition 103.11. Let \( M \) be a compact \( n \) -dimensional smooth manifold, let \( k \in \mathbb{N} \) and let \( \varphi : {\bar{B}}^{n - k} \times {S}^{k - 1} \rightarrow \partial M \) be a thickened \( \left( {k - 1}\right) \) -sphere. We pick \( \left( {{x}_{0},{y}_{0}}\right) \in {S}^{n - k - 1} \times {S}^... | Proof. The statements follow fairly easily from a slight generalization of the Seifert-van Kampen Theorem 22.2 and the HNN-Seifert-van Kampen Theorem 26.3 (b). Overall the proof is very similar to the proof of Proposition 37.11 and we leave it to the reader to make the obvious alterations to obtain a proof for the pres... | No |
Corollary 103.12. Let \( M \) be a compact connected non-empty smooth manifold. For every handle decomposition of \( M \) we have the inequality:\n\n\( 1 + \# 1 \) -handles - #0-handles \( \geq \) minimal size of a generating set of \( {\pi }_{1}\left( M\right) \) . | Proof. Let \( M \) be a compact connected non-empty smooth manifold that is equipped with a handle decomposition. We denote by \( r \) the number of 0 -handles and we denote by \( s \) the number of 1-handles. First note, that by Corollary 103.4 we can assume that the handle decomposition is standard. We pick a base po... | Yes |
Lemma 103.13. Let \( M \) be a compact \( n \) -dimensional smooth manifold, let \( k \in {\mathbb{N}}_{0} \), let \( \varphi : {\bar{B}}^{n - k} \times {S}^{k - 1} \rightarrow \partial M \) be a thickened \( \left( {k - 1}\right) \) -sphere and let \( x \in {\bar{B}}^{n - k} \) . We denote by \( i : M \rightarrow M{ \... | Proof. We consider the following diagram: \n\nHere the upper sequence is the long exact sequence coming from a slight generalization of the Mayer-Vietoris Theorem 46.10 for Smooth Manifolds. The maps emanating from... | No |
Lemma 103.14. Let \( M \) be a connected \( n \) -dimensional smooth manifold such that \( \partial M \) is also connected. Let \( \varphi : {\bar{B}}^{n - k} \times {S}^{k - 1} \rightarrow \partial M \) be a thickened \( \left( {k - 1}\right) \) -sphere. The following statements hold:\n\n(1) If \( 1 \leq k < n - 1 \),... | Proof. First we prove Statement (1). By Proposition 103.1 we have\n\n\[ \partial \left( {M{ \cup }_{\varphi }{h}^{k}}\right) = \underset{\text{since }\partial M\text{ is connected }}{\underbrace{\left( {\partial M \smallsetminus \varphi \left( {{B}^{n - k} \times {S}^{k - 1}}\right) }\right) { \cup }_{\varphi }{ \mid }... | Yes |
Corollary 103.15. Let \( n \in \mathbb{N} \) with \( n \geq 4 \) . Let \( M \) be a compact connected non-empty \( n \) -dimensional smooth manifold that is equipped with a handle decomposition.\n\n(1) If all handles have index \( \leq n - 3 \), then the boundary \( \partial M \) is connected and the inclusion induced ... | Proof. Let \( M \) be a compact connected non-empty \( n \) -dimensional smooth manifold that is equipped with a handle decomposition. If \( M \) has a single 0 -handle, then the statement follows immediately from Lemma 103.14.\n\nNow suppose that \( M \) has in fact more than one 0-handle. In Corollary ?? we will see ... | No |
Lemma 103.16. Let \( M \) be an \( n \) -dimensional smooth manifold. Suppose we are given a thickened \( \left( {k - 1}\right) \) -sphere \( \varphi : {\bar{B}}^{n - k} \times {S}^{k - 1} \rightarrow \partial M \). Let \( j \in {\mathbb{N}}_{0} \). The following statements hold:\n\n(1) If \( j \leq n - k - 2 \) and if... | Proof. The lemma follows from applying the Mayer-Vietoris Theorem 46.10 to the decomposition \( \partial M = \left( {\partial M \smallsetminus \varphi \left( {{B}^{n - k} \times {S}^{k - 1}}\right) }\right) \cup \left( {{\bar{B}}^{n - k} \times {S}^{k - 1}}\right) \), to \( M{ \cup }_{\varphi }{h}^{k} \) and finally to... | No |
Lemma 104.1. Let \( M \) be an n-dimensional smooth manifold and let \( f : M \rightarrow \mathbb{R} \) be a smooth function. Let \( P \in M \smallsetminus \partial M \) be a critical point. We pick a chart \( \Phi : U \rightarrow V \) for \( P \) . We write \( Q \mathrel{\text{:=}} \Phi \left( P\right) \) . The congru... | Proof. There are two approaches to proving the lemma:\n\n(1) Let \( \Psi : X \rightarrow Y \) be some other chart around \( P \) . We set \( R \mathrel{\text{:=}} \Psi \left( Q\right) \) and we set \( Z \mathrel{\text{:=}} \mathrm{D}{\left( \Psi \circ {\Phi }^{-1}\right) }_{P} \) . A slightly heroic calculation, see Ga... | Yes |
Corollary 104.3. Let \( U \subset {\mathbb{R}}^{n} \) be an open convex subset that contains the origin and let \( f : U \rightarrow \mathbb{R} \) be a smooth function with \( f\left( 0\right) = 0 \) . If \( 0 \) is a critical point of \( f \), then there exist smooth functions \( {h}_{ij} : U \rightarrow \mathbb{R}, i... | Proof of Corollary 104.3. First note that by Proposition 6.16 there exist smooth maps \( {a}_{1},\ldots ,{a}_{n} : U \rightarrow \mathbb{R} \) such that for \( i = 1,\ldots, n \) we have \( {a}_{i}\left( 0\right) = \frac{\partial f}{\partial {x}_{i}}\left( 0\right) = 0 \) and such that\n\n\[ f\left( x\right) = \mathop{... | Yes |
Corollary 104.4. Let \( M \) be a smooth manifold and let \( f : M \rightarrow \mathbb{R} \) be a smooth function such that no critical point is contained in \( \partial M \) .\n\n(1) The set of non-degenerate critical points of \( f \) is a discrete subset of \( M \) .\n\n(2) If \( M \) is compact and if all critical ... | Proof.\n\n(1) It follows immediately from the Morse Lemma 104.2 that every non-degenerate critical point admits an open neighborhood which contains no other critical point. But this means, by definition, that the non-degenerate critical points of \( f \) form a discrete subset of \( M \) .\n\n(2) In Exercise 6.29 we sh... | No |
Proposition 104.5. Let \( M \) be a closed smooth submanifold of some \( {\mathbb{R}}^{n} \). (1) The set of \( Q \in {\mathbb{R}}^{n} \) for which the map \( M \rightarrow \mathbb{R} \) given by \( x \mapsto \parallel x - Q{\parallel }^{2} \) is a Morse function is a set of full measure. | Sketch of Proof. Let \( M \) be a closed smooth submanifold of some \( {\mathbb{R}}^{n} \). (1) Given \( Q \in {\mathbb{R}}^{n} \) we consider the map \[ {f}_{Q} : M \rightarrow \mathbb{R} \] \[ x \mapsto \parallel x - Q{\parallel }^{2}. \] Furthermore we consider \[ N \mathrel{\text{:=}} \left\{ {\left( {p, w}\right) ... | No |
Theorem 104.8. (Smooth Manifold Product Theorem) Let \( M \) be a compact smooth manifold, let \( f : M \rightarrow \mathbb{R} \) be a smooth function and let \( a < b \in \mathbb{R} \) . We suppose that for every \( P \in {f}^{-1}\left( \left\lbrack {a, b}\right\rbrack \right) \) the differential \( \mathrm{D}{f}_{P} ... | Sketch of a proof of Theorem 104.8. Note that using Proposition 9.1 and Proposition 8.1 we can assume that \( M \) is a submanifold of some \( {\mathbb{R}}^{n} \) . Given \( P \in M \) we denote by \( {}^{1457} \) grad \( {f}_{P} \in {\mathrm{V}}_{P}M \) the gradient of \( f \) at \( P{}^{1458} \) Note that our hypothe... | Yes |
Theorem 104.9. (Handle Addition Theorem) Let \( M \) be a compact smooth manifold and let \( f : M \rightarrow \mathbb{R} \) be a smooth function. Furthermore let \( a < b \in \mathbb{R} \) such that \( a \) and \( b \) are regular values and such that \( {f}^{-1}\left( \left\lbrack {a, b}\right\rbrack \right) \cap \pa... | Proof. Let \( M \) be a compact smooth manifold, let \( f : M \rightarrow \mathbb{R} \) be a smooth function and let \( a < b \in \mathbb{R} \) be regular values such that \( {f}^{-1}\left( \left\lbrack {a, b}\right\rbrack \right) \cap \partial M = \varnothing \) . In the following we will only prove Statement (1). Sta... | No |
Proposition 104.10. Let \( M \) be a closed smooth manifold. If \( f : M \rightarrow \mathbb{R} \) is a Morse function, then there exists a standard handle decomposition of \( M \) such that for each \( k \in {\mathbb{N}}_{0} \) the number of \( k \) -handles equals the number of critical points of \( f \) of index \( ... | Proof. Let \( M \) be a closed smooth manifold and furthermore let \( f : M \rightarrow \mathbb{R} \) be a Morse function. Recall that by Corollary 104.4 (2) we know that \( f \) has only finitely many critical points. We order the corresponding critical values by \( {\lambda }_{1} < \cdots < {\lambda }_{s} \) . Next w... | Yes |
Theorem 104.11. (Handle Decomposition Theorem) Every closed smooth manifold admits a standard handle decomposition. | Proof. Let \( M \) be a closed smooth manifold. By Proposition 104.6 we know that there exists a Morse function \( f : M \rightarrow \mathbb{R} \) . It follows from Proposition 104.10 that \( M \) admits a standard handle decomposition. | No |
Proposition 64.6. Let \( M \) be a closed \( n \) -dimensional smooth manifold.\n\n(1) For every \( k > n \) we have \( {\mathrm{H}}_{k}\left( {M;\mathbb{Z}}\right) = 0 \).\n\n(3) Given any \( P \in M \) the fundamental group \( {\pi }_{1}\left( {M, P}\right) \) is finitely presented.\n\n(4) All homology groups of \( M... | Sketch of Proof. Let \( M \) be a closed \( n \) -dimensional smooth manifold. By the Handle Decomposition Theorem 104.11 we know that \( M \) admits a handle decomposition. It follows easily from a straightforward induction argument using Proposition 103.11 that for any \( P \in M \) the fundamental group \( {\pi }_{1... | Yes |
Proposition 104.12. Let \( M \) be a closed smooth manifold.\n\n(1) If \( f \) is a Morse function for \( M \), then \( M \) is homotopy equivalent to a CW-complex where the number of \( k \) -cells equals precisely the number of critical points of index \( k \) . | Proof.\n\n(1) This statement follows immediately from Corollary 104.4 (2), Theorem 104.9 and Proposition 103.18. | Yes |
Proposition 104.14. Let \( n \in \mathbb{N} \) .\n\n(1) If \( \varphi : {S}^{n - 1} \rightarrow {S}^{n - 1} \) is a homeomorphism, then \( {\bar{B}}^{n}{ \cup }_{\varphi }{\bar{B}}^{n} \) is homeomorphic to \( {S}^{n} \) . | Proof.\n\n(1) Let \( \varphi : {S}^{n - 1} \rightarrow {S}^{n - 1} \) be a homeomorphism. We apply the Alexander trick to get an extension of \( \varphi : {S}^{n - 1} \rightarrow {S}^{n - 1} \) to the ball \( {\bar{B}}^{n} \) . More precisely we consider the following map:\n\n\[ \Phi : {\bar{B}}^{n} \rightarrow {\bar{B... | Yes |
Proposition 105.1. Let \( M \) be a compact \( n \) -dimensional smooth manifold. Furthermore let \( \partial M = A \sqcup B \) be a decomposition of \( \partial M \) into two disjoint unions of components \( A \) and \( B \). Finally let \[ M = \left( {\left\lbrack {0,1}\right\rbrack \times A}\right) { \cup }_{{\varph... | There exists canonical handle decomposition \[ {M}^{\prime } = \left( {\left\lbrack {0,1}\right\rbrack \times {B}^{\prime }}\right) { \cup }_{{\psi }_{m}}\underset{0\text{-th dual handle }}{\underbrace{{\widetilde{h}}^{n - {k}_{m}} \cup \cdots { \cup }_{{\psi }_{0}}\underset{m\text{-th dual handle }}{\underbrace{{\wide... | Yes |
Lemma 105.2. Let \( M \) be a compact smooth manifold and let \( C \) be a union of components of \( \partial M \) . Let \( \varphi : C \rightarrow C \) be a diffeomorphism. Given any neighborhood \( U \) of \( C \subset M \) there exists a diffeomorphism \[ M \rightarrow \underset{\text{smooth manifold by Proposition ... | Proof of Lemma 105.2. As is explained in Proposition 8.15 (1), to say that the topological space \( \left( {M \sqcup \left( {\left\lbrack {0,1}\right\rbrack \times C}\right) }\right) / \sim \) is a smooth manifold we actually need to pick a collar \( \left\lbrack {-1,0}\right\rbrack \times C \) with \( C = \{ 0\} \time... | Yes |
Proposition 105.3. Let \( M \) be a compact \( n \) -dimensional smooth manifold and let \( N \) be a compact \( n \) -dimensional smooth manifold. Furthermore let \( A \) be a union of boundary components of \( M \) and let \( B \) be a union of boundary components of \( N \) . Next let\n\n\[ M = \left( {\left\lbrack ... | Proof. We have the following diffeomorphisms:\n\n\[ M{ \cup }_{\Theta }N = M{ \cup }_{\Theta }((\lbrack 0,1\rbrack \times B){ \cup }_{{\psi }_{0}}{h}^{{l}_{0}} \cup \cdots { \cup }_{{\psi }_{s}}{h}^{{l}_{s}}) = (M{ \cup }_{\Theta }(\lbrack 0,1\rbrack \times B)){ \cup }_{{\psi }_{0}}{h}^{{l}_{0}} \cup \cdots { \cup }_{{... | Yes |
Lemma 105.5. Let \( N \) be a closed smooth manifold and let \( f : \left\lbrack {0,1}\right\rbrack \times N \rightarrow \mathbb{R} \) be a smooth function that is pointing inwards on \( \{ 0\} \times N \) and that has no critical points. There exists a smooth function \( g : \left\lbrack {0,1}\right\rbrack \times N \r... | Sketch of A PROOF OF LEMMA 105.5. We pick some \( s \in \mathbb{R} \) that is less than the minimum of \( f \) on \( \left\lbrack {0,1}\right\rbrack \times N \) . Using Lemma 6.13 it is pretty straightforward to see that there exist smooth maps \( \alpha ,\beta : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{R} \... | No |
Proposition 105.9. Let \( M \) be a compact smooth manifold and suppose we are given a decomposition \( \partial M = A \sqcup B \) where \( A \) and \( B \) are unions of components of \( \partial M \) .\n\n(1) Let \( \varphi : {\bar{B}}^{n - k} \times {S}^{k - 1} \rightarrow B \) be a thickened \( \left( {k - 1}\right... | (1) This statement is proved by running the proof of Theorem 104.9 \ | No |
Proposition 105.10. Let \( M \) be a compact \( n \) -dimensional smooth manifold and suppose we are given a decomposition \( \partial M = A \sqcup B \) where \( A \) and \( B \) are unions of components of \( \partial M \) . There exists a self-indexing Morse function \( f : M \rightarrow \mathbb{R} \) with respect to... | Sketch of PROOF. Let \( M \) be a compact \( n \) -dimensional smooth manifold and suppose we are given a decomposition \( \partial M = A \sqcup B \) where \( A \) and \( B \) are unions of components of \( \partial M \) . By the Handle Decomposition Theorem 104.11 there exists a standard handle decomposition\n\n\[ M =... | No |
Lemma 106.1. Let \( M \) be a compact \( n \) -dimensional smooth manifold together with a standard handle decomposition \[ M \cong \varnothing \underset{0\text{-handles }}{\underbrace{{ \cup }_{{\varphi }_{0,1}}{h}^{0}\cdots { \cup }_{{\varphi }_{0,{r}_{0}}}{h}^{0}}} \cup \ldots \underset{n\text{-handles }}{\underbrac... | Proof. Not surprisingly the proof of the lemma is very similar to the proof of Lemma 48.1, But completeness sake let us go through the motions. (1) We consider the following diagram  The left vertical map is induce... | Yes |
Lemma 106.2. Let \( M \) be compact smooth manifold together with a standard handle decomposition. Given \( k \in {\mathbb{N}}_{0} \) we denote by \( d = {d}_{k} \) the map \[ {\mathrm{H}}_{k}\left( {{M}^{k},{M}^{k - 1}}\right) \xrightarrow[]{{\partial }_{k}}\underset{ = : d = {d}_{k}}{\underbrace{{\mathrm{H}}_{k - 1}\... | Proof. This proof is easy. We just need to copy-paste the proof of Lemma 48.2. Thus we consider the following commutative diagram of maps  The map \( {\partial }_{k} \circ {j}_{k} : {\mathrm{H}}_{k}\left( {M}^{k}\r... | Yes |
Proposition 106.3. Let \( M \) be compact smooth manifold together with a standard handle decomposition. Given any \( k \in {\mathbb{N}}_{0} \) there exists a uniquely determined natural \( {}^{1466} \) isomorphism\n\n\[{\Phi }_{M} : {\mathrm{H}}_{k}\left( M\right) \overset{ \cong }{ \rightarrow }{\mathrm{H}}_{k}^{\tex... | Proof of Proposition 106.3. The proof of Proposition 106.3 is almost embarrassingly similar to the proof of Proposition 48.4 (1). For the reader’s convenience we nonetheless provide an abbreviated version of the argument. As in the proof of Proposition 48.4 (1) we add a couple of maps to the commutative diagram that we... | No |
Corollary 106.4. Let \( M \) be a compact smooth manifold. Given any handle decomposition for \( M \) the following equality holds:\n\n\[ \mathop{\sum }\limits_{{k \in {\mathbb{N}}_{0}}}{\left( -1\right) }^{k} \cdot \text{ number of }k\text{-handles } = \chi \left( M\right) . \] | Proof. Let \( M \) be a compact smooth manifold together with a handle decomposition. Given \( k \in {\mathbb{N}}_{0} \) we denote by \( {c}_{k} \) the number of \( k \) -handles. By Corollary 103.4 we can find a standard handle structure on \( M \) such that for every \( k \in {\mathbb{N}}_{0} \) the number of \( k \)... | Yes |
Corollary 106.5. The Euler characteristic of every closed odd-dimensional smooth manifold is zero. | Proof. Let \( M \) be a closed \( \left( {{2n} + 1}\right) \) -dimensional smooth manifold. By the Handle Decomposition Theorem 104.11 we can equip \( M \) with a handle decomposition. By Proposition 103.5 we can also consider the corresponding dual handle decomposition. Now we see that | No |
Corollary 106.4\n\n\( \chi \left( M\right) = \mathop{\sum }\limits_{{k = 0}}^{{ \downarrow {2n} + 1}}{\left( -1\right) }^{k} \cdot \underset{\text{ the handle decomposition }}{\text{ number of }k\text{-handles in }}\overset{ \downarrow }{ = }\mathop{\sum }\limits_{{k = 0}}^{{{2n} + 1}}{\left( -1\right) }^{k} \cdot \und... | \n\[ = - \mathop{\sum }\limits_{{m = 0}}^{{{2n} + 1}}{\left( -1\right) }^{m} \cdot \frac{\text{ number of }m\text{-handles in the }}{\text{ dual handle decomposition }} = - \chi \left( M\right) . \]\nsubstitution \( m = {2n} + 1 - k\; \) Corollary 106.4\n\nThus we see that \( \chi \left( M\right) = 0 \) . | Yes |
Corollary 106.6. Let \( M \) be a closed smooth manifold. For every Morse function \( f : M \rightarrow \) \( \mathbb{R} \) we have\n\n\[ \mathop{\sum }\limits_{{k \in {\mathbb{N}}_{0}}}{\left( -1\right) }^{k} \cdot \text{ #critical points of }f\text{ of index }k = \chi \left( M\right) . \] | Proof. Let \( f : M \rightarrow \mathbb{R} \) be a Morse function on a closed smooth manifold. It follows from Proposition 104.10 that \( M \) admits a handle decomposition such that for each \( k \in {\mathbb{N}}_{0} \) the number of \( k \) -handles equals the number of critical points of index \( k \) . The desired ... | Yes |
Proposition 106.7. Let \( M \) be an oriented \( n \) -dimensional smooth manifold together with a standard handle decomposition\n\n\[ M \cong \varnothing \underset{0\text{-handles }}{\underbrace{{ \cup }_{{\varphi }_{0,1}}{h}^{0}\cdots { \cup }_{{\varphi }_{0,{r}_{0}}}{h}^{0}}} \cup \ldots \underset{n\text{-handles }}... | We assume that the inclusion maps of all the handles are orientation-preserving. Given \( k \in {\mathbb{N}}_{0} \) and given \( i \in \left\{ {1,\ldots ,{r}_{i}}\right\} \) we denote by \( {\Phi }_{k, i} : \{ 0\} \times {\bar{B}}_{i}^{k} \rightarrow {M}^{k} \) the map that is given by the inclusion map of the core of ... | Yes |
Lemma 106.9. If \( F\overset{\varphi }{ \rightarrow }G \rightarrow H \rightarrow 0 \) is an exact sequence where \( F \) and \( G \) are finitely generated free abelian groups, then \( \operatorname{rank}\left( F\right) \geq \) minimal number of generators of the torsion subgroup of \( H \) . | Proof of Lemma 106.9. We write \( m = \operatorname{rank}\left( F\right) \) and \( n = \operatorname{rank}\left( G\right) \) . Furthermore we set \( k \mathrel{\text{:=}} \min \{ m, n\} \) . Since \( \mathbb{Z} \) is a principal ideal domain it follows from the Smith Normal Form Theorem, see Exercise 19.18, that we can... | Yes |
Proposition 106.10. (1) Let \( M \) be a compact smooth manifold that is equipped with a handle decomposition. Given \( k \in {\mathbb{N}}_{0} \) we denote by \( {d}_{k} \) the number of \( k \) -handles in the handle decomposition. (a) Given any \( k \in {\mathbb{N}}_{0} \) we have the inequality \[ \mathop{\sum }\lim... | Proof. (1) Let \( M \) be a compact smooth manifold that is equipped with a handle decomposition. Given \( k \in {\mathbb{N}}_{0} \) we denote by \( {d}_{k} \) the number of \( k \) -handles in the handle decomposition. It follows from Corollary 103.4 that without loss of generality we can assume that the handle decomp... | Yes |
Theorem 106.11. (Morse Inequalities) Let \( M \) be a closed smooth manifold together with a Morse function \( f : M \rightarrow \mathbb{R} \). Given \( k \in {\mathbb{N}}_{0} \) we write\n\n\[ \n{d}_{k} \mathrel{\text{:=}} \text{number of critical points of index}k\text{.\n\]\n\nThe following two inequalities hold:\n\... | Proof of Theorem 106.11. Let \( M \) be a closed smooth manifold together with a Morse function \( f : M \rightarrow \mathbb{R} \). By Proposition 104.10 we know that \( M \) admits a standard handle decomposition of \( M \) such that for each \( \overline{k \in {\mathbb{N}}_{0}} \) the number of \( k \) -handles equal... | Yes |
Lemma 106.14. Let \( M \) be a closed orientable simply connected 4-dimensional smooth manifold. A handle decomposition is minimal if and only if it has no 1-handles and no 3-handles. | Proof. Let \( M \) be a closed orientable simply connected 4-dimensional smooth manifold. Note that in Exercise 88.19 we saw that it follows easily from the Hurewicz Theorem 52.5, \( {}^{1472} \) We will sketch a proof in Exercise ??. Corollary 17.4, the Poincaré Duality Theorem 88.1 and the Universal Coefficient Theor... | No |
Proposition 22.10. Let \( \pi = \left\langle {{g}_{1},\ldots ,{g}_{k} \mid {r}_{1},\ldots ,{r}_{l}}\right\rangle \) be a finite presentation. Given any \( n \geq 4 \) there exists a closed orientable connected non-empty \( n \) -dimensional smooth manifold \( W \) with \( {\pi }_{1}\left( W\right) \cong \pi \) . | Proof. Let \( \pi = \left\langle {{g}_{1},\ldots ,{g}_{k} \mid {r}_{1},\ldots ,{r}_{l}}\right\rangle \) be a finite presentation and let \( n \geq 4 \) . By Proposition 107.1 there exists a compact orientable connected non-empty \( n \) -dimensional smooth manifold with the following properties:\n\n(1) \( {\pi }_{1}\le... | Yes |
Proposition 107.2. Let \( M \) be a compact orientable n-dimensional topological manifold. For every \( j \in {\mathbb{N}}_{0} \) there exists an isomorphism\n\n\[ \n{\mathrm{H}}_{j}\left( {\mathrm{D}M}\right) \cong {\mathrm{H}}_{j}\left( M\right) \oplus \operatorname{Hom}\left( {{\mathrm{H}}_{n - j}\left( M\right) ,\m... | Proof. Let \( i : M \rightarrow \mathrm{D}M \) be the natural inclusion and let \( r : \mathrm{D}M \rightarrow M \) be the folding map that we introduced on page 1164. Note that by definition we have \( r \circ i = {\operatorname{id}}_{M} \) . We consider the long exact sequence in homology of the pair \( \left( {\math... | Yes |
Corollary 107.3. Let \( \pi = \left\langle {{g}_{1},\ldots ,{g}_{k} \mid {r}_{1},\ldots ,{r}_{l}}\right\rangle \) be a finite presentation.\n\n(1) Given any \( n \geq 5 \) there exists a closed orientable connected non-empty \( n \) -dimensional\n\nsmooth manifold \( W \) with \( {\pi }_{1}\left( W\right) \cong \pi \) ... | Proof. As in the proof of Proposition 22.10 that we just gave on page 2550 we use Proposition 107.1 to obtain a compact orientable connected non-empty \( n \) -dimensional smooth manifold with the following properties:\n\n(1) \( {\pi }_{1}\left( M\right) \cong \pi \) .\n\n(2) \( M \) admits a handle decomposition with ... | Yes |
(1) For every closed connected non-empty 3-dimensional smooth manifold \( M \) the fundamental group \( {\pi }_{1}\left( M\right) \) admits a balanced presentation. | (1) We will prove this statement in Exercise 107.2. Alternatively the statement is proved in Theorem ?? | No |
Lemma 107.8. If \( N \geq {2r} + 2 \), then for every smooth embedding \( \varphi : {S}^{r} \rightarrow {\mathbb{R}}^{N} \) there exists an orientation-preserving smooth embedding \( \Phi : {\bar{B}}^{N - r} \times {S}^{r} \rightarrow {\mathbb{R}}^{N} \) such that \( \Phi \left( {0, P}\right) = \) \( \varphi \left( P\r... | Proof. The statement of the lemma is basically the content of the Tubular Neighborhood Theorem 8.24 (5). For the reader's convenience we recall the proof. First we consider the standard smooth embedding \( \theta : {S}^{r} \rightarrow {\mathbb{R}}^{r + 1} \times {\mathbb{R}}^{N - r - 1} = {\mathbb{R}}^{N} \) which is g... | Yes |
Lemma 107.9. Let \( X \) be a finite CW-complex and let \( \alpha : {S}^{k - 1} \rightarrow X \) be a map. If \( X \) is homotopy equivalent to a compact orientable smooth manifold \( M \), then there exists a compact orientable smooth manifold \( N \) and an orientation-preserving thickened \( \left( {k - 1}\right) \)... | Proof of Lemma 107.9. Let \( f : X \rightarrow M \) be a homotopy equivalence between a finite CW-complex \( X \) and a compact orientable \( m \) -dimensional smooth manifold \( M \) . Furthermore let \( \alpha : {S}^{k - 1} \rightarrow X \) be a map.\n\nWe pick some \( r \geq \max \{ k,{2k} - \dim \left( M\right) \} ... | Yes |
Proposition 107.10. Let \( X \) be a finite \( k \) -dimensional CW-complex. There exists an \( N \geq {2k} + 1 \) such that for every \( n \geq N \) there exists a closed orientable \( n \) -dimensional smooth manifold \( M \) which admits a standard handle decomposition and which admits a homotopy equivalence\n\n\[ \... | Proof. Let \( X \) be a finite \( k \) -dimensional CW-complex. By Proposition 107.7 there exists a homotopy equivalence \( \psi : X \rightarrow V \) from \( X \) to a compact smooth manifold \( V \) which has the nice property that it admits a handle decomposition such that all handles are of index \( \leq k \) . By C... | Yes |
Proposition 108.1. Every compact connected non-empty 1-dimensional smooth manifold is diffeomorphic either to \( {S}^{1} \) or to \( \left\lbrack {0,1}\right\rbrack \) . | Proof. Let \( M \) be a compact connected non-empty 1-dimensional smooth manifold. By the Handle Decomposition Theorem 104.11 we can equip \( M \) with a standard handle decomposition.\n\nClaim. There exists a standard handle decomposition for \( M \) with a single 0-handle.\n\nSince \( M \) is non-empty we know that \... | Yes |
Lemma 108.2. Let \( M \) be a 2-dimensional smooth manifold, let \( C \) be a boundary component and let \( \varphi : {S}^{1} \rightarrow C \) be a diffeomorphism. The diffeomorphism type of \( M{ \cup }_{\varphi }{h}^{2} \) does not depend on the choice of \( \varphi \) . | Proof. Let \( M \) be a 2-dimensional smooth manifold, let \( C \) be a boundary component of \( M \) and let \( \varphi ,\psi : {S}^{1} \rightarrow C \) be two diffeomorphisms. We consider the self-diffeomorphism \( f \mathrel{\text{:=}} \varphi \circ {\psi }^{-1} : {S}^{1} \rightarrow {S}^{1} \) . By Proposition 30.1... | Yes |
Lemma 108.4. Every closed connected non-empty 2-dimensional smooth manifold admits a handle decomposition with the following properties:\n\n(1) There exists a single 0-handle.\n\n(2) The attaching maps of all of the 1-handles take values in the boundary of the 0-handle.\n\n(3) There exists a single 2-handle. | Sketch of Proof. By the Handle Decomposition Theorem 104.11 we can equip \( M \) with a standard handle decomposition. Since \( M \) is non-empty we know that there exists at least one 0-handle. Suppose that \( M \) has more than one 0-handle. Since \( M \) is connected it follows from the fact that the handle decompos... | No |
Lemma 108.5. Let \( M \) be a 2-dimensional smooth manifold \( M \) with a completely standard handle decomposition. The following two statements are equivalent:\n\n(1) \( M \) is orientable.\n\n(2) All 1-handles are orientable. | Proof. In Proposition 8.15 (6) we dealt with the orientability of smooth manifolds that are given by gluing oriented smooth manifolds along unions of boundary components. In our setting we only glue along codimension-zero submanifolds of the boundary. But the same statements hold and we see that our lemma is basically ... | No |
Theorem 108.7. Let \( \sum \) be a compact connected non-empty 2-dimensional smooth manifold.\n\n(1) If \( \sum \) is orientable, then \( \sum \) is diffeomorphic to the surface \( {\sum }_{g, n} \) for some \( g, n \in {\mathbb{N}}_{0} \).\n\n(2) If \( \sum \) is non-orientable, then \( \sum \) is diffeomorphic to the... | Proof of Theorem 108.7 for CLOSED MANIFOLDS. By the Handle Decomposition Theorem 104.11 together with Lemma 108.4 it suffices to prove the proposition for a closed 2-dimensional smooth manifold that is equipped with a completely standard handle decomposition, i.e. that is equipped with a handle decomposition with the f... | No |
Theorem 109.4. The first Steenrod operation \( {\mathrm{{Sq}}}^{1} \) is the Bockstein homomorphism corresponding to the short exact sequence \( 0 \rightarrow {\mathbb{Z}}_{2}\xrightarrow[]{\left\lbrack a\right\rbrack \mapsto \left\lbrack {2a}\right\rbrack }{\mathbb{Z}}_{4} \rightarrow {\mathbb{Z}}_{2} \rightarrow 0 \)... | Proof. We will not make use of this theorem, thus we decline to provide a proof. The theorem is proved in [Bre93, p. 418] or alternatively in [MTa68, p. 23] or [Hat02, Theorem 4.L.12]. | No |
Lemma 109.5. For any wedge of finitely many spheres all Steenrod operations in degrees \( \geq 1 \) are zero. | Proof. We will prove the lemma in Exercise 109.2 | No |
Lemma 109.6. Let \( \left( {X, A}\right) \) be a pair of topological spaces. (1) Given any \( \varphi ,\psi \in {\mathrm{H}}^{ * }\left( {X, A;{\mathbb{Z}}_{2}}\right) \) we have \[ \mathrm{{Sq}}\left( {\varphi \cup \psi }\right) = \mathrm{{Sq}}\left( \varphi \right) \cup \mathrm{{Sq}}\left( \psi \right) \] (2) Given a... | Proof. The first statement follows easily from the Cartan formula. Indeed, we see that Cartan formula, i.e. Axiom (4) \[ \mathrm{{Sq}}\left( {\varphi \cup \psi }\right) = \mathop{\sum }\limits_{{k \in {\mathbb{N}}_{0}}}{\mathrm{{Sq}}}^{k}\left( {\varphi \cup \psi }\right) \overset{ \downarrow }{ = }\mathop{\sum }\limit... | Yes |
Corollary 109.7. Let \( X \) be a topological space. For any cohomology class \( \varphi \in {\mathrm{H}}^{1}\left( {X;{\mathbb{Z}}_{2}}\right) \) of degree one, any \( k \in \mathbb{N} \) and any \( i \in \mathbb{N} \) we have \[ {\operatorname{Sq}}^{i}\left( {\varphi }^{k}\right) = \left( \begin{matrix} k \\ i \end{m... | Proof. We calculate that \[ \begin{aligned} \mathrm{{Sq}}\left( {\varphi }^{k}\right) & = \mathrm{{Sq}}{\left( \varphi \right) }^{k} & = {\left( {\mathrm{{Sq}}}^{0}\left( \varphi \right) + {\mathrm{{Sq}}}^{1}\left( \varphi \right) \right) }^{k} & = {\left( \varphi + {\varphi }^{2}\right) }^{k} & = {\varphi }^{k} \cup {... | Yes |
Lemma 109.6 (2) by Axiom (3) Axioms (1) and (2) since the ring \( \left( {{\mathrm{H}}^{ * }\left( {X;{\mathbb{Z}}_{2}}\right) , \cup }\right) \) since \( \varphi \in {\mathrm{H}}^{1}\left( {X;{\mathbb{Z}}_{2}}\right) \) | The desired conclusion follows from comparing the terms in degree \( k + i \) on both sides. | No |
Lemma 109.10. Let \( X \) be a topological space, let \( \left( {Y, B}\right) \) be a pair of topological spaces and let \( R \) be a commutative ring. If \( \varphi \in {\mathrm{H}}^{m}\left( {X;R}\right) \) and \( \psi \in {\mathrm{H}}^{n}\left( {B;R}\right) \), then ![448f61af-e517-4f9c-831f-f6ce5868f6c0_2581_0.jpg]... | Proof of LEMMA 109.10. This equality follows fairly easily from Lemma 81.2, the definition of the cross product via the cup product and the explicit description of the connecting homomorphism on page 1820. We leave it to the reader to fill in the details. | No |
Proposition 109.12. Given any topological space \( X \), any \( n \in \mathbb{N} \) and any \( i \in {\mathbb{N}}_{0} \) the following diagram commutes:  | Proof. We consider the following diagram \n\nThe first square commutes by Proposition 109.8 The two other squares commute by the naturality of the Steenrod operation \( {\mathrm{{Sq}}}^{i} \) . Thus we see that, as... | Yes |
Lemma 109.14. If a map \( \varphi : {S}^{m} \rightarrow {S}^{n} \) is homotopic to a constant map, then the corresponding mapping cone \( \operatorname{Cone}\left( {\varphi : {S}^{m} \rightarrow {S}^{n}}\right) \) is homotopy equivalent to \( {S}^{m + 1} \vee {S}^{n} \) . | Proof. We have\n\n\[ \operatorname{Cone}\left( {\varphi : {S}^{m} \rightarrow {S}^{n}}\right) \underset{ \uparrow }{ \simeq }\operatorname{Cone}\left( {\text{ constant map } : {S}^{m} \rightarrow {S}^{n}}\right) \underset{ \uparrow }{ \cong }{S}^{m + 1} \vee {S}^{n}. \]\n\nby Lemma 24.12 since \( \varphi \) is homotopi... | No |
Proposition 81.6. (1) There exist diagonal approximations. (2) Given any two diagonal approximations \( \Phi \) and \( \Psi \) there exists a natural chain homotopy equivalence between \( \Phi \) and \( \Psi \) . | Proof. We gave an explicit example of a diagonal approximation in Lemma 81.5. In Proposition 80.17 we gave a proof of (2) using the Acyclic Model Theorem 80.16 (2). | Yes |
Theorem 109.17. (Acyclic Model Theorem) Let Top be the category of topological spaces and let \( \mathcal{C} \) be the category of generalized \( {}^{1490} \) chain complexes.\n\n(1) Let \( S : {\mathrm{C}}_{ * }\left( X\right) \rightarrow {\mathrm{C}}_{ * }\left( X\right) \otimes {\mathrm{C}}_{ * }\left( X\right) \) b... | Proof of the Acyclic Model Theorem 109.17\n\n(1) Given any topological space \( X \) we consider \( G\left( X\right) \mathrel{\text{:=}} \left( {{\mathrm{C}}_{ * }\left( X\right) \otimes {\mathrm{C}}_{ * }\left( X\right) }\right) \left\lbrack {-d}\right\rbrack \) . Here we use the notation introduced on page 2039, name... | No |
Lemma 109.18. Let \( {\mathcal{C}}_{ * } \) and \( {\mathcal{D}}_{ * } \) be chain complexes and let \( R \) be a commutative ring. Given \( p, q \in {\mathbb{N}}_{0} \) we consider the map \( {}^{1492} \n\n\[ \n{\Xi }_{p, q} : \operatorname{Hom}\left( {{\mathcal{C}}_{p}, R}\right) \otimes \operatorname{Hom}\left( {{\m... | Proof. The statement follows basically immediately from the definitions. | No |
Lemma 109.19. Let \( X \) be a topological space and let \( R \) be a commutative ring. Furthermore let \( {\Delta }_{0} : {\mathrm{C}}_{ * }\left( X\right) \rightarrow {\mathrm{C}}_{ * }\left( X\right) \otimes {\mathrm{C}}_{ * }\left( X\right) \) be a diagonal approximation. We consider the cochain map \[ \operatornam... | Proof (*). We denote by \( d : X \rightarrow X \times X \) the diagonal map that is given by \( d\left( x\right) = \left( {x, x}\right) \) . Let \( \Theta : {\mathrm{C}}_{ * }\left( {X \times X}\right) \rightarrow {\mathrm{C}}_{ * }\left( X\right) \otimes {\mathrm{C}}_{ * }\left( X\right) \) denote a Eilenberg-Zilber m... | Yes |
Lemma 109.21. Let \( X \) be a topological space. Furthermore let \( R \) be a commutative ring with \( {1}_{R} + {1}_{R} = {0}_{R} \). For any \( \alpha ,\beta \in {\mathrm{C}}^{ * }\left( {X;R}\right) \) we have the following equality:\n\n\[ {h}_{n + 1}\left( {\delta \left( {\alpha \otimes \beta }\right) }\right) + \... | Proof. Recall, see Theorem 109.16, that one of the defining properties of the sequence \( {\Delta }_{n} \) is that for every \( n \in {\mathbb{N}}_{0} \) the following equality holds:\n\n\[ \partial \circ {\Delta }_{n + 1} + {\left( -1\right) }^{n} \cdot {\Delta }_{n + 1} \circ \partial = \left( {T + {\left( -1\right) ... | No |
Lemma 109.22. Let \( X \) be a topological space and let \( R \) be a commutative ring. If \( 1 + 1 = 0 \) in \( R \), then for any \( \alpha \in {\mathrm{C}}^{q}\left( {X;R}\right) \) we have\n\n\[ \n{h}_{n + 1}\left( {{\delta \alpha } \otimes {\delta \alpha }}\right) = \delta {h}_{n + 1}\left( {\alpha \otimes {\delta... | Proof. Since we assume that \( 1 + 1 = 0 \) in \( R \) we can ignore conveniently enough ignore signs throughout the proof. Thus we compute\n\n\[ \n\text{since}\delta \left( {c \otimes d}\right) = {\delta c} \otimes d + c \otimes {\delta d}\;\text{Lemma 109.21} \n\]\n\n\[ \n{h}_{n + 1}\left( {{\delta \alpha } \otimes {... | Yes |
Lemma 109.23. Let \( X \) be a topological space and let \( R \) be a commutative ring with \( 1 + 1 = \) 0 . For any \( \alpha ,\beta \in {\mathrm{C}}^{q}\left( {X;R}\right) \) the following equality holds:\n\n\[ \n{h}_{n}\left( {\left( {\alpha + \beta }\right) \otimes \left( {\alpha + \beta }\right) }\right) = {h}_{n... | Proof of Lemma 109.23. This follows from \ | No |
Proposition 109.24. The definition of the maps \( {\mathrm{{Sq}}}^{i} \) does not depend on the choice of the natural maps \( {\Delta }_{n} \) . | Proof. Suppose we are given another sequence of maps \( {\widetilde{\Delta }}_{n} \) that have the properties described in Theorem 109.16 (1). We define the maps \( {\widetilde{h}}_{i} \) in the obvious way. By Theorem 109.16 (2) there exists a sequence of natural maps\n\n\[ \n{\mathrm{H}}_{n} : {\mathrm{C}}_{ * }\left... | Yes |
Theorem 109.25. The Steenrod operations \( {\mathrm{{Sq}}}^{i} \) have the following properties:\n\n(1) \( {\mathrm{{Sq}}}^{0} = \mathrm{{id}} \) .\n\n(2) For any \( \varphi \in {\mathrm{H}}^{i}\left( {X, A;{\mathbb{Z}}_{2}}\right) \) we have \( {\operatorname{Sq}}^{i}\left( \varphi \right) = {\varphi }^{2} \) .\n\n(3)... | Proof.\n\n(1) The notation might suggest that this statement is surely trivial, but that is not the case. For time, space and mental energy reasons we will not give the proof, which at first glance does not seem terribly enlightening anyway. Instead we refer to Spa95, p. 274] for details.\n\n(2) First we deal with the ... | No |
Theorem 110.1. (Adem Relations) For any \( 0 < a < {2b} \) we have the following equality of cohomology operations.\n\n\[ \n{\mathrm{{Sq}}}^{a} \circ {\mathrm{{Sq}}}^{b} = \mathop{\sum }\limits_{{j = 0}}^{\left\lfloor \frac{a}{2}\right\rfloor }\left( \begin{matrix} b - 1 - j \\ a - {2j} \end{matrix}\right) \cdot {\math... | Proof. Since the proof would consume too much of our valuable time we will not provide it. Instead we refer to the original paper by Adem Ade52, Ade57 or alternatively to [BuM82], [SE62, Theorem VIII.1.5], [MTa68, p. 23], or [Hat02, Theorem 4.L.13] for a proof. | No |
Lemma 110.3. Let \( n, k \in {\mathbb{N}}_{0} \) . If we write \( n = \mathop{\sum }\limits_{{i \in {\mathbb{N}}_{0}}}{n}_{i} \cdot {2}^{i} \) and \( k = \mathop{\sum }\limits_{{i \in {\mathbb{N}}_{0}}}{k}_{i} \cdot {2}^{i} \) with \( {n}_{i},{k}_{i} \in \{ 0,1\} \), then \[ \left( \begin{array}{l} n \\ k \end{array}\r... | Proof. Let \( n, k \in {\mathbb{N}}_{0} \) . We write \( n = \mathop{\sum }\limits_{{i \in {\mathbb{N}}_{0}}}{n}_{i} \cdot {2}^{i} \) and \( k = \mathop{\sum }\limits_{{i \in {\mathbb{N}}_{0}}}{k}_{i} \cdot {2}^{i} \) with \( {n}_{i},{k}_{i} \in \{ 0,1\} \) In the polynomial ring \( {\mathbb{Z}}_{2}\left\lbrack x\right... | Yes |
Theorem 110.4. Let \( i \in {\mathbb{N}}_{0} \) . If \( i \) is not a power of 2, then \( {\mathrm{{Sq}}}^{i} \) is decomposable, in the sense that it can be written as the sum of compositions of Steenrod operations of smaller degree. | Proof. Suppose that we are given \( i \in {\mathbb{N}}_{0} \) that is not a power of two. This means that we can write \( i = a + b \) where \( b = {2}^{k} \) and \( a \in \left\{ {1,\ldots ,{2}^{k} - 1}\right\} \) . Since \( 0 < a < {2b} \) we obtain an Adem relation for \( {\mathrm{{Sq}}}^{a} \circ {\mathrm{{Sq}}}^{b... | Yes |
Corollary 110.5. Let \( X \) be a topological space and let \( n \in {\mathbb{N}}_{0} \) . We consider the Steenrod operation \[ {\mathrm{{Sq}}}^{i} : {\mathrm{H}}^{n}\left( {X;{\mathbb{Z}}_{2}}\right) \rightarrow {\mathrm{H}}^{n + i}\left( {X;{\mathbb{Z}}_{2}}\right) \] If the cohomology groups inbetween vanish, i.e. ... | Proof. Since \( i \) is not a power of 2 we know by Theorem 110.4 that \( {\mathrm{{Sq}}}^{i} \) can be written as the sum of compositions of Steenrod operations of smaller degree. But by our hypothesis for any \( j \in \{ 1,\ldots, i - 1\} \) we have \( {\mathrm{H}}^{n + j}\left( {X;{\mathbb{Z}}_{2}}\right) = 0 \) . B... | Yes |
Lemma 110.7. Let \( m \in \mathbb{N} \) . We denote by \( \varphi : \mathbb{Z} \rightarrow {\mathbb{Z}}_{m} \) the canonical map. Given any topological space \( X \) there exists a long exact sequence\n\n\[ \ldots \rightarrow {\mathrm{H}}^{k}\left( {X;\mathbb{Z}}\right) \overset{m}{ \rightarrow }{\mathrm{H}}^{k}\left( ... | Proof of Lemma 110.7. We denote by \( \psi : \mathbb{Z} \rightarrow \mathbb{Z} \) the map that is given by multiplication by \( m \) . We have the short exact sequence\n\n\[ 0 \rightarrow \mathbb{Z}\overset{\psi }{ \rightarrow }\mathbb{Z}\overset{\varphi }{ \rightarrow }{\mathbb{Z}}_{m} \rightarrow 0. \]\n\nBy the disc... | No |
Lemma 110.8. Let \( n \geq 2 \) . If \( f : {S}^{{2n} - 1} \rightarrow {S}^{n} \) is a map, then there exists an isomorphism\n\n\[ \left( {{\mathrm{H}}^{ * }\left( {\operatorname{Cone}\left( f\right) ;\mathbb{Z}}\right) , \cup }\right) \cong {R}_{n}\left( {\operatorname{Hopf}\left( f\right) }\right) \]\nof superalgebra... | Proof. This statement follows immediately from Lemma 91.5 and the definition of the Hopf invariant, see page 2217. | No |
Proposition 110.9. Let \( n \geq 2 \) .\n\n(1) The Hopf invariant Hopf: \( {\pi }_{{2n} - 1}\left( {S}^{n}\right) \rightarrow \mathbb{Z} \) is a homomorphism. | Proof. The proof follows from Proposition 9.1.7, Corollary 91.8, Lemma 91.11 and Propositions 91.13 and 91.12. | No |
Theorem 110.12. Let \( p \) be an odd prime. There exists a family \( {}^{1500} \n\n\[ \n{P}^{i} : {\mathrm{H}}^{n}\left( {X, A;{\mathbb{Z}}_{p}}\right) \rightarrow {\mathrm{H}}^{n + {2i}\left( {p - 1}\right) }\left( {X, A;{\mathbb{Z}}_{p}}\right) \;\text{ with }i \in \overline{{\mathbb{N}}_{0}}\text{ and }n \in {\math... | Proof. This theorem is proved in [SE62, Chapter VI]. A very different proof is given in [Hat02, Theorem 4.L.16]. | Yes |
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