Orientation reversal of manifolds - UniversitÃ¤t Bonn

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2 1 Introduction the total space is easily controllable. This reduces the problem to an algebraic problem on the non-existence of solutions of certain matrix equations (Lemma 29) depending on the monodromy. A family of appropriate matrices for the monodromy is presented, and we prove the non-existence of solutions mostly by linear algebra and comparing eigenvalues but also by appealing in one step to the fundamental theorem of Galois theory. Examples of chiral manifolds in even dimensions are then obtained by cartesian products of odd-dimensional chiral manifolds. Apart from the cup product structure in cohomology, we use a theorem by Hopf on the Umkehr homomorphism and Betti numbers to exclude degree −1 for all self-maps. The odd-dimensional examples in Theorem A are Eilenberg-MacLane spaces, and the proof of chirality uses as a substantial ingredient that the effect of a self-map on homology is completely determined by the induced map on the fundamental group. Therefore, we next ask for obstructions other than the fundamental group und restrict the analysis to simply-connected manifolds. In Chapter 4, it is shown that in dimensions 3, 5 and 6, a nontrivial fundamental group is a necessary characteristic of chiral manifolds. From dimension 7 on, we prove that there exist simply-connected chiral manifolds in every dimension: Theorem B In dimensions 3, 5 and 6, every simply-connected, closed smooth (or PL or topological) manifold is amphicheiral in the respective category. A closed, simply-connected, topological 4-manifold admits an orientation-reversing homeomorphism if its signature is zero. If the signature is nonzero, the orientation cannot even be reversed by a continuous map of degree −1. In every dimension ≥ 7 there is a closed, simply-connected, chiral smooth manifold. The results in dimension 3 to 5 are obtained almost immediately from the powerful classification theorems for simply-connected manifolds in these dimension. The classification of simply-connected 6-manifolds needs more complicated invariants. We provide the necessary details on the invariants from the proof of the classification [Zhubr] and complete the argument by analysing the homology of the first Postnikov stage of the manifolds in question and the effect of automorphisms of the first Postnikov approximation. For the evidence of simply-connected chiral manifolds in dimensions ≥ 7, examples in all dimensions except 9, 10, 13 and 17 can be constructed with methods used already in the previous chapters. Since simply-connected chiral manifolds in the remaining dimensions are more difficult to obtain, we split the proof of existence into two parts: (1) find a mechanism or an obstruction to orientation reversal in the partial homotopy type and (2) realise the obstruction by a simply-connected manifold.
3 The first step is done with the help of the Postnikov tower: In every instance, we construct an appropriate finite tower of principal K(π, n)-fibrations (or simply a single stage) and fix an element in the integral homology of one of the stages that is to be the image of the fundamental class of the manifold. Then we prove that (by the mechanism that lies in the particular construction) this homology class can never be mapped to its negative under any self-map of a single Postnikov stage or of the partial Postnikov tower. In the second step, the obstruction is realised by proving that there is indeed a manifold with the correct partial homotopy type and the correct image of the fundamental class in the Postnikov approximation. This step involves bordism computations and surgery techniques. For simply-connected chiral manifolds in dimensions 10 and 17, is is sufficient to construct a single Postnikov stage. The obstruction is manifest in the mod 3 Steenrod algebra in the cohomology of Eilenberg-MacLane spaces. The bordism computation in the second step is done in this and all further proofs with the help of the Atiyah-Hirzebruch spectral sequence. For the surgery step, we use the surgery theory of Kreck [Kreck99]. The examples in dimensions 9 and 13 require a more complicated setup of the Postnikov tower. Here, we construct a three-stage Postnikov tower by appropriate k-invariants. Together with the construction, we analyse the possible automorphisms of this Postnikov tower in each step. The analysis is made possible by rational homotopy theory. However, the information which is obtained from the rational homotopy type is not enough in our case, and we also include information about the automorphisms of the integral Postnikov tower. Again, the Atiyah-Hirzebruch spectral sequence and Kreck’s surgery theory are applied for the realisation part of the proof. Here, we extend a proposition in [Kreck99] in order to prove that surgery in rational homology in the middle dimension is possible in our setting. Next, in order to further characterise the properties of manifolds which allow or prevent orientation reversal, we consider the question whether every manifold is bordant to a chiral one. This allows also an approximation to the (not mathematically precise) question “how many” manifolds are chiral or if “the majority” of manifolds is chiral or amphicheiral. The following statement is proved in Chapter 5. Theorem C In every dimension ≥ 3, every closed, smooth, oriented manifold is oriented bordant to a manifold of this type which is connected and chiral. Summarising, we prove this by showing that the existing obstructions in our examples can be kept when we change the bordism class via connected sums of manifolds. A special case, for which an entirely new example is necessary, are nullbordant manifolds in dimension 4. We translate this problem into group
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 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
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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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