Orientation reversal of manifolds - UniversitÃ¤t Bonn

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4 1 Introduction homology and construct a series of finite groups G such that H 4 (G) contains an element of order > 2 which is invariant under all automorphisms of G. The proof is again completed by bordism and surgery arguments. The majority of the theorems so far aimed at proving that certain manifolds or families of manifolds are chiral. The opposite problem, however, namely proving amphicheirality in nontrivial circumstances, is also an interesting question. In general, this is even more challenging since not only one obstruction to orientation reversal must be identified and realised but for the opposite direction every possible obstruction must vanish. Surgery theory is a framework for comparing diffeomorphism classes of manifolds, and smooth amphicheirality can be considered a showcase of surgery theory: Given the manifolds M and −M it must be decided if M and −M are oriented diffeomorphic. Surgery provides powerful theorems and some recipes for classification problems but not a generally applicable algorithm, so that a particular problem must still be solved individually. We carry out the surgery programme of [Kreck99] for some products of 3-dimensional lens spaces. We prove the following theorem. Theorem D Let r 1 and r 2 be coprime odd integers and let L 1 and L 2 be (any) 3-dimensional lens spaces with fundamental groups Z/r 1 resp. Z/r 2 . Then the product L 1 × L 2 admits an orientation-reversing self-diffeomorphism. The question why these products constitute a relevant problem is discussed in the introduction of Chapter 6. The proof is facilitated by the fact that the products are known to be homotopically amphicheiral. This is not a necessary input to Kreck’s surgery programme but we use it here since it simplifies the first part of the proof. We then carry out the bordism computation in the Atiyah-Hirzebruch spectral sequence. This uses the fact that we are dealing with a product manifold to a great extent, and we employ the module structure of the spectral sequence heavily. In the final surgery step, it is not necessary to analyse individual surgery obstructions, but we show that the obstruction group vanishes, using results by [Bak] and from the book [Oliver]. In Chapter 7, we add a new facet to the results of the previous chapters by showing that the order of an orientation-reversing map can be relevant. From the literature, we present examples of manifolds which admit an orientationreversing diffeomorphism but none of finite order. We complement this with manifolds where the minimal order of an orientation-reversing map is finite: Theorem E For every positive integer k, there are infinitely many lens spaces which admit an orientation-reversing diffeomorphism of order 2 k but no orientationreversing self-map of smaller order. The nonexistence of orientation-reversing maps of smaller order is shown by a well-known formula for the degree of maps between lens spaces. Lens
1.1 Conventions and notation 5 spaces with an orientation-reversing map of the desired order in infinitely many dimensions are given explicitly, and the map itself can be written by a simple formula in complex coordinates in the universal covering. 1.1 Conventions and notation Throughout this work, all manifolds are compact and orientable. Also, boundaryless manifolds are understood without further notice (except for bounding manifolds in a bordism, but this will be clear from the context). With the exception of links (in the context of knot theory) and manifolds in the oriented bordism groups, every manifold is connected. Unless indicated otherwise, we consider smooth (i. e. differentiable of class C ∞ ) manifolds. Finally, all manifolds are required to be second-countable Hausdorff spaces. The following list clarifies some notations, which otherwise follow common practice. • If M is an oriented manifold, the same manifold with the opposite orientation is denoted −M (as in bordism theory, not ̅M as in algebraic geometry). Often, the initial orientation does not matter, and we speak of M and −M for orientable manifolds, meaning that an arbitrary orientation for M is fixed. • Care has been taken in using the equal sign. Often in mathematics, when this detail is not important, not only equal objects are related by = but also isomorphic objects. Since naturality is a crucial detail in some proofs, the equal sign is reserved in this text to correspondences which are canonical (i. e. do not depend on choices) or natural (i. e. functorial). Otherwise, isomorphisms are denoted as usual by ≅. • An arrow with a tilde ∼ → also denotes an isomorphism. This is a little less standard notation than ↪ for injective and ↠ for surjective maps but there is no danger of confusion since weak equivalences in model categories, for which the symbol ∼ → is also used, do not occur in this text. • Coefficients in homology and cohomology are always the integers Z if not stated otherwise. In order to distinguish relative (co-)homology from the notation with coefficient groups, the coefficient are separated by a semicolon (compare H ∗ (A, B) and H ∗ (A; Q)). In the (co-)homology of Eilenberg-MacLane spaces, the “K” is often omitted. Thus, H ∗ (Z/3, 3; Z/3) denotes the integral homology of an Eilenberg-MacLane space K(Z/3, 3) with Z/3-coefficients. • As a general rule, abelian groups are written additively and arbitrary groups multiplicatively. Thus, the trivial group is either denoted 0 or 1,
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 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
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4.2 Dimensions 10 and 17 55 structu
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4.2 Dimensions 10 and 17 57 For the
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4.3 Dimensions 9 and 13 59 Given a
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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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