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\centerline{\bf ISOVARIANT HOMOTOPY THEORY AND}
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\centerline{\bf DIFFERENTIABLE GROUP ACTIONS}
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\centerline{\smc Reinhard Schultz}

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\+& \hfill Department of Mathematics \hfill&\cr
\+& \hfill Purdue University \hfill&\cr
\+& \hfill West Lafayette, Indiana  47907 \quad USA\hfill&\cr
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\centerline{\it Introduction}
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This is an expanded version of notes for lectures I had planned to give
at the Korea Advanced Institute of Science and Technology during the 
Seventh KAIST Mathematics Workshop in Algebra and Topology 
at the Korea Advanced Institute of Science and Technology in Taejon, Korea,
from August 11 to August 14, 1992.  

There are three parts, with each corresponding to 
one of the planned lectures.  The first two discuss joint work with 
G. Dula on the foundations of isovariant homotopy theory, the applications
of this work to classification problems for smooth manifolds with (smooth)
group actions and its relation to work on equivariant surgery over the
past two decades; some of the results in the second part have been obtained
independently by M. Dawson.  The third part discusses joint work 
with S. Kwasik
on a somewhat different but related topic; namely, differentiable actions
of finite groups on homology 3-spheres.  One common theme relating the
second and third parts is the problem of adapting equivariant surgery
to cases where a standard technical condition (the {\sl Gap Hypothesis})
does not hold.   A second relationship is that the 3-dimensional 
questions exhibit some basic features of higher-dimensional problems
with certain technical simplifications.  Finally, advances in the 
geometrization theory of 3-manifolds over the past two decades have
suggested that suitably pseudo-geometric manifolds in higher dimensions
form an especially promising subject for future research in geometric
topology and transformation groups; perhaps ideas resembling those of
Part III will lead to progress in this direction.
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{\smc Acknowledgments. } 
Much of my research on the material in this article was either joint
work or directly related to previous joint work with Giora Dula, 
S{\l}awomir Kwasik, and other collaborators including J. C. Becker, 
K. H. Dovermann, and M. Masuda; needless to say I am grateful to all of 
these collaborators for their input.  I am also grateful to numerous
other topologists for helpful coments in connection with this work,
including Bill Browder, Mike Davis, Mike Dawson, Emmanuel Dror Farjoun, 
Shmuel Weinberger, and Min Yan (others also probably deserve to be 
mentioned, so I apologize in advance for any inadvertent omissions).
At various times my work on these subjects has been
partially supported the U. S. National Science Foundation 
(MPS74-03609, MPS76-08794, MCS78-02913, MCS81-04852, MCS83-00669, DMS86-02543,
DMS89-02622, DMS91-02711),
Sonder\-for\-schungs\-be\-reich 170 (\anfu Geometrie und Analysis``)
at the Mathematical Institute in G\"ottingen, the Max-Planck-Institut f\"ur
Mathematik in Bonn, and the Mathematical Sciences Research Institute in
Berkeley.  I am also grateful to
the Northwestern University Mathematics Department for access to its
computer and library
facilities during the preparation of this paper and related work.
Finally, I would like to thank the KAIST, both for providing the opportunity
to participate in their 1992 workshop and for their kind understanding 
when personal circumstances kept me from participating.
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The manuscript for this paper was prepared using \AmSTeX\ Version 2.0.
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\centerline{\smc Outline of Contents}
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\settabs\+\qquad&IIII.\quad
&1. \quad xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx&\cr
\+&&& \cr
\+&I.&Introduction to isovariant homotopy theory& \cr
\+&&1. \quad Equivariant differential topology &  \cr
\+&&2. \quad Foundations of equivariant homotopy theory &  \cr
\+&&3. \quad Isovariant homotopy theory &  \cr
\+&&4. \quad Isovariance versus equivariance &   \cr
\+&&& \cr
\+&II.&Isovariant homotopy, classification problems, and 
general position\cr
\+&&& \cr
\+&&1. \quad Isovariant homotopy structures &  \cr
\+&&2. \quad Isovariance and the Gap Hypothesis &  \cr
\+&&3. \quad Semilinear differentiable actions on spheres &  \cr
\+&&4. \quad Borderline of the Gap Hypothesis range &  \cr
\+&&5. \quad Extensions to nonsmoothable actions &  \cr
\+&&& \cr
\+&III.&Finite group actions on homology 3-spheres& \cr
\+&&& \cr
\+&&1. \quad Survey of some known results &  \cr
\+&&2. \quad Equivariant surgery in three dimensions &  \cr
\+&&3. \quad Construction of exotic examples &  \cr
\+&&4. \quad Actions on hyperbolic homology spheres &  \cr
\+&&& \cr
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\hfill {\tt Version 1.0 \qquad October, 1992}
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