1 Introduction ................................................ 1
1.1 Conventions and notation ............................... 5
1.2 Why the name "chiral manifold"? ........................ 6
1.3 Chirality in various categories ........................ 7
2 Known examples and obstructions ............................. 9
2.1 Dimensions 0 to 2 ..................................... 10
2.2 The cup product and the intersection form ............. 10
2.3 The linking form ...................................... 13
2.4 Lens spaces ........................................... 15
2.5 Characteristic numbers ................................ 17
2.6 Exotic spheres ........................................ 19
2.7 3-manifolds ........................................... 20
3 Examples in every dimension ≥ 3 ............................ 29
3.1 Examples in every odd dimension ≥ 3 ................... 30
3.2 Products of chiral manifolds .......................... 37
4 Simply-connected chiral manifolds .......................... 43
4.1 Results in low dimensions ............................. 44
4.2 Dimensions 10 and 17 .................................. 51
4.3 Dimensions 9 and 13 ................................... 58
5 Bordism questions .......................................... 77
5.1 Odd dimensions ≥ 3 .................................... 78
5.2 Even dimensions ≥ 6 ................................... 79
5.3 Dimension 4 and signature 0 ........................... 82
6 Products of Lens spaces .................................... 91
7 Orientation-reversing diffeomorphisms of minimal order .... 105
A Appendix .................................................. 107
A.l The linking form ..................................... 107
A.2 Homotopy equivalences between products of lens
spaces ............................................... 1ll
A.3 Diffeomorphisms between products of lens spaces ..... 114
References ................................................... 119
Summary ...................................................... 125
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