DIAGRAM COHOMOLOGY AND ISOVARIANT HOMOTOPY 3

theories and other standard homotopy-theoretic notions in the almost isovariant

category. As a justification for the machinery introduced in Sections 1-4 we shall

prove an isovariant analog of the Whitehead theorems for recognizing ordinary

and equivariant homotopy equivalences of ordinary and equivariant CW com-

plexes (see Theorem 4.10 and Corollaries 4.11-12). In the final three sections of

the paper we describe some of the applications to isovariant obstruction theories.

Section 5 begins with an obstruction theory for deforming an equivariant map

to an almost isovariant map by an equivariant homotopy, and Section 6 contains

almost isovariant analogs of the Barratt-Federer spectral sequence for homotopy

groups of ordinary and equivariant function spaces in [Bar, Fed] and [M0, Sc2]

respectively. Section 7 calculates the homotopy groups for spaces of (almost)

isovariant self-maps of certain spheres with orthogonal actions. These function

spaces are relevant to the geometric results in [Sc76] and [Sc87], and in Section

8 we outline their application in [Sc91] to construct infinite families of smooth

actions on spheres that are topologically linear but not detectable by previously

defined invariants.

Frequently in this work we have found it helpful in the long run to introduce

new definitions and notation for concepts and objects that are needed. These

are summarized in an index at the end for the convenience of the reader.

At many points in this article it is clear that results could be stated in greater

generality. This was not done because (i) the extra notation, additional concepts,

and longer mathematical arguments would make the main ideas less apparent,

(ii) in some cases it is not clear what the optimal notation or generalization

should be, (in) the present setting is adequate for some applications to trans-

formation groups. Some of these issues will be addressed elsewhere.

ACKNOWLEDGMENTS.

We would like to thank Jim Becker, Bill Browder,

Bill Dwyer, Emmanuel Dror Farjoun, and Soren Illman for helpful discussions

regarding their work. Some suggestions by Mark Mahowald and David Blanc

to the first named author were extremely useful, and the second named author

is indebted to Donald W. Kahn for lending his copy of a difficult to locate pa-

per by D. O. Baladze [BaL2]. Both authors are grateful to the Northwestern

University Mathematics Department for access to its facilities during portions

of the preparation of this paper. Finally, the work of the second named au-

thor on this subject has been partially supported at various times the U. S. Na-

tional Science Foundation (MPS74-03609, MPS76-08794, MCS78-02913, MCS81-

04852, MCS83-00669, DMS86-02543, DMS89-02622, DMS91-02711), Sonderfor-

schungsbereich 170 („Geometrie und Analysis") at the Mathematical Institute

in Gottingen, the Max-Planck-Institut fur Mathematik in Bonn, and the Math-

ematical Sciences Research Institute in Berkeley.

The manuscript for this paper was prepared using

AMS-T$£

Version 2.0.