Orientation reversal of manifolds - UniversitÃ¤t Bonn

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12 2 Known examples and obstructions If f ∶ M → N is a continuous map, we have for all a, b ∈ H k (N). Q( f ∗ a, f ∗ b) = deg( f ) ⋅ Q(a, b) Proof. See e. g. [HatcherAT, Section 3.3, p. 249 ff.]. The naturality statement follows from the naturality properties of the cup product and the Kronecker pairing. Theorem 2 implies that a manifold is homotopically chiral if its intersection form Q is not isomorphic to its negative −Q. If k is odd, the intersection form on A ∶= H k (M) free is an antisymmetric bilinear form on a finitely generated free abelian group. By [MH, Cor. 3.5], A has even rank and there is a basis (e 1 , . . . , e 2m ) of A such that Q has the form ( 0 I m −I m 0 ) , where I m denotes the identity matrix of size m. The base change e i ↦ −e i for all i ≤ m changes this matrix to its negative. Thus, if the dimension of a manifold is congruent 2 mod 4, the intersection form is always isomorphic to its negative and cannot provide an obstruction to amphicheirality. If k is even, the intersection form is symmetric. Unlike the antisymmetric case, symmetric unimodular bilinear forms over Z have not been classified. In particular, the number of positive (or equivalently, negative) definite forms grows rapidly with the rank [MH, p. 28]. Fortunately, this does not cause complications for the question of chirality because a definite form is never isomorphic to its negative. Indefinite symmetric unimodular form over Z are distinguished by their rank, signature and type [MH, Thm. 5.3]. Since the rank and the type of a form and its negative are the same, the signature is the only invariant which can distinguish a form Q from −Q. Summarising, we have Proposition 3: [MH] Let Q ∶ A × A → Z be a symmetric unimodular bilinear form on a finitely generated free abelian group A. Then Q is isomorphic to −Q if and only if its signature is zero. The signature of an oriented manifold can be defined as the signature of its intersection form if the dimension is a multiple of 4. Otherwise, the signature is set to zero. Since a self-map of degree ±1 induces an isomorphism in cohomology (see the proof of Lemma 1 and convert it from homology to cohomology), we have the following statement: Corollary 4 A manifold with nonzero signature admits no self-map of degree −1.
2.3 The linking form 13 Corollary 5 A 4k-dimensional manifold with odd (2k)-th Betti number b 2k admits no self-map of degree −1. Proof. Since the rank of the middle homology group is odd and the intersection form is symmetric in the present case, the signature must be nonzero. The observations in the two preceding lemmas and similar statements for the linking form were already made in 1938 by Rueff [Rueff]. We point out that by Proposition 3 the signature is the only algebraic obstruction to orientation reversal which can be obtained from the intersection form. This is not a mathematically rigid statement since the term “obstruction to amphicheirality” has not been given a mathematically well-defined meaning. Nevertheless, it should be clear what is meant by this statement: There might be chiral manifolds with signature 0 (in fact, there are), but there must be characteristics of these manifolds other than the intersection form that cause chirality. In conclusion, we want to record the obstructions to orientation reversal from this section. Most generally, the cup product structure can be made responsible in the case of CP 2n since an even power of a cohomology element t that generates a cohomology group of rank 1 evaluates nontrivially on the fundamental class. More specifically, the signature of manifold is a homotopyinvariant obstruction. The point of view of the signature as a characteristic number will be taken up in Section 2.5. 2.3 The linking form For odd-dimensional manifolds, the linking form is the analogue to the intersection form. Theorem 6 Let M be a closed, oriented topological manifold of odd dimension 2k − 1. Then there is a nondegenerate, (−1) k -symmetric bilinear form L ∶ Tor H k (X) × Tor H k (X) → Q/Z, which is called the linking form. Furthermore, if f ∶ N → M is a continuous map then L( f ∗ a, f ∗ b) = deg( f ) ⋅ L(a, b). Although this theorem is well-known, a proof of all properties in one piece is given in Appendix A.1. The cohomological version of the linking form is preferred because naturality can be handled more easily in this setting. A definition of the homological version can be found in [Ranicki, Ex. 12.44 (i)].
Page 1 and 2: Orientation reversal of manifolds D
Page 3 and 4: Preface There was a topologist, who
Page 5: Contents 1 Introduction . . . . . .
Page 8 and 9: 2 1 Introduction the total space is
Page 10 and 11: 4 1 Introduction homology and const
Page 12 and 13: 6 1 Introduction depending on the c
Page 14 and 15: 8 1 Introduction Since deg f = ±1,
Page 16 and 17: 10 2 Known examples and obstruction
Page 35 and 36: 3 Examples in every dimension ≥ 3
Page 37 and 38: 3.1 Examples in every odd dimension
Page 43 and 44: 3.2 Products of chiral manifolds 37
Page 49 and 50: 4 Simply-connected chiral manifolds
Page 51 and 52: 4.1 Results in low dimensions 45 al
Page 53 and 54: 4.1 Results in low dimensions 47 Le
Page 55 and 56: 4.1 Results in low dimensions 49 Th
Page 57 and 58: 4.2 Dimensions 10 and 17 51 Accordi
Page 59 and 60: 4.2 Dimensions 10 and 17 53 Lemma 4
Page 61 and 62: 4.2 Dimensions 10 and 17 55 structu
Page 63 and 64: 4.2 Dimensions 10 and 17 57 For the
Page 65 and 66: 4.3 Dimensions 9 and 13 59 Given a
Page 67 and 68: 4.3 Dimensions 9 and 13 61 homomorp
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4.3 Dimensions 9 and 13 63 z ⊗ g
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4.3 Dimensions 9 and 13 65 We fix a
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4.3 Dimensions 9 and 13 67 the imag
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4.3 Dimensions 9 and 13 69 Lemma 62
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4.3 Dimensions 9 and 13 71 Thom the
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4.3 Dimensions 9 and 13 73 and its
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4.3 Dimensions 9 and 13 75 sional m
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5 Bordism questions So far, example
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5.2 Even dimensions ≥ 6 79 To sho
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5.2 Even dimensions ≥ 6 81 Recall
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5.3 Dimension 4 and signature 0 83
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92 6 Products of Lens spaces Theore
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94 6 Products of Lens spaces Since
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96 6 Products of Lens spaces Thus,
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98 6 Products of Lens spaces since
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100 6 Products of Lens spaces There
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102 6 Products of Lens spaces A fin
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7 Orientation-reversing diffeomorph
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A Appendix A.1 The linking form In
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A.1 The linking form 109 [α] ⊗ [
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A.2 A.2 Homotopy equivalences betwe
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A.2 Homotopy equivalences between p
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A.3 Diffeomorphisms between product
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A.3 Diffeomorphisms between product
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120 References [EMcL] Samuel Eilenb
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122 References [McCleary] John McCl
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Summary We study the phenomenon of
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Orientation reversal of manifolds - UniversitÃ¤t Bonn

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