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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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