This book contains the results of work done during the years 1967-1970 on fixed-point-free involutions on manifolds, and is an enlarged version of the author's doctoral dissertation [54J written under the direction of Professor William Browder. The subject of fixed-paint-free involutions, as part of the subject of group actions on manifolds, has been an important source of problems, examples and ideas in topology for the last four decades, and receives renewed attention every time a new technical development suggests new questions and methods ([62, 8, 24, 63J). Here we consider mainly those properties of fixed-point-free involutions that can be best studied using the techniques of surgery on manifolds. This approach to the subject was initiated by Browder and Livesay. Special attention is given here to involutions of homotopy spheres, but even for this particular case, a more general theory is very useful. Two important related topics that we do not touch here are those of involutions with fixed points, and the relationship between fixed-point-free involutions and free Sl-actions. For these topics, the reader is referred to [23J, and to [33J, [61J, [82J, respectively. The two main problems we attack are those of classification of involutions, and the existence and uniqueness of invariant submanifolds with certain properties. As will be seen, these problems are closely related. If (T, l'n) is a fixed-point-free involution of a homotopy sphere l'n, the quotient l'n/Tis called a homotopy projective space.
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ISBN-13
978-3-540-05092-6 (9783540050926)
DOI
10.1007/978-3-642-65012-3
Schweitzer Klassifikation
0 Notation, Conventions, Preliminaries.- I The Browder-Livesay Invariants.- I.1 Involutions of Spheres.- I.1.1 Desuspensions and Characteristic Submanifolds.- I.1.2 Equivariant Surgery.- I.1.3 The Desuspension Theorem.- I.1.4 Concordance of Desuspensions.- I.2 Involutions of Simply Connected Manifolds.- I.2.1 Characteristic Submanifolds and h-regularity.- I.2.2 The Browder-Livesay Invariants.- II Realization of the Browder-Livesay Invariants.- II.1 The Realization Theorem.- II.2 Constructions with Involutions.- II.2.1 M?TM*.- II.2.2 (T, Mn) # N.- II.3 Proof of Theorem II.1-D.- II.4 Proof of Theorem II.1-A.- II.5 Homology 3-Spheres.- II.6 Manifolds with the Same Regular Homotopy Type as Line Bundles over Projective Spaces.- II.7 Invariant Codimension 2 Spheres.- III Relations with Non-simply-connected Surgery Obstructions.- III.1 Normal Invariants.- III.1.1 Normal Maps, Cobordisms and Invariants.- III.1.2 G/PL, G/0.- III.1.3 Homotopy Triangulations and Smoothings.- III.2 Non-simply-connected Surgery Obstructions.- III.2.1 The Groups Ln(?, ?).- III.2.2 The Obstruction Groups for ?=?2.- III.3 Relations with the Browder-Livesay Invariants.- III.3.1 ln: BLn(?) ? Ln?1(?2, ? ?).- III.3.2 Ambient vs. Abstract Surgery Obstructions.- III.3.3 Proof of Lemma 2, II.4.- III.3.4 Proof of Theorem II.1-C.- IV Combinatorial Classification of Involutions.- IV.1 Some Maps and Exact Sequences.- IV.2 Computation of [Pn, G/PL].- IV.3 The Classification Theorem.- IV.3.1 Homotopy Projective Spaces.- IV.3.2 Suspension.- IV.3.3 Two Useful Theorems.- IV.3.4 The Classification Theorem.- IV.3.5 Proof of Theorem II.1-B (p.l. case).- IV.4 Another Relation with Non-simply-connected Surgery Obstructions.- IV.4.1 ?= ± ?.- IV.4.2 On Theorem II.1-C.- IV.5 On the Topological Classification of Involutions.- V Smooth Involutions.- V.1 General Remarks on Smooth Involutions of Spheres.- V.2 Normal Invariants and Browder-Livesay Invariants.- V.2.1 Involutions of Spheres.- V.2.2 Involutions of Simply-connected Manifolds.- V.2.3 ?= ± ? again.- V.3 Differentiable Structure of Spheres and Other Double Coverings.- V.4 Proof of Theorem II.1-B, Smooth Case.- V.5 Involutions and the Generalized Kervaire Invariant.- V.5.1 The Generalized Kervaire Invariant.- V.5.2 Odd Multiples of an Involution.- V.6 Smooth Involutions of Spheres of Low Dimension.- V.6.1 Involutions of 7-Spheres.- V.6.2 Spheres that Admit Involutions.- V.7 Action of ?n(??) (After Browder).- V.8 How to Prove ? = ?.- VI Codimension 2 Invariant Spheres.- VI.1 Invariant vs. Characteristic Spheres.- VI.2 Applications.- VI.2.1 A Non-embedding Result.- VI.2.2 Some Brieskorn-Hirzebruch Involutions.- VI.3 Knotted and Unknotted Codimension 2 Invariant Spheres.- VI.4 Cobordism Classes of Invariant Knots.- Some Unsolved Problems.- References.- List of Symbols.
