ABSTRACT
The problem that we are concerned with is the existence and construction
of embeddings of a given G-CW complex (G-manifold) in another G-CW com-
plex (G-manifold) having a prescribed homotopy type and a prescribed family
of isotropy subgroups.
It is shown that for the case of compact complexes, one can specify
obstructions lying in the quotient of Grothendieck groups of certain cate-
gories of equivariant complexes. In the non-compact case, the embedding is
possible under favorable hypotheses. Techniques similar to general position
surgery are used to construct an equivariant thickening functor from the
equivariant homotopy category to the category of smooth G-manifolds. Appli-
cations are made to the problem of existence of actions on the euclidean
spaces and other simply-connected open manifolds with a prescribed family of
isotropy subgroups. In particular, it is shown that a finite group of order
not a prime power can act on a euclidean space—of reasonable dimension,
close to the best possible one—without stationary points and with a pre-
scribed family of isotropy subgroups.
Topological and PL actions of finite groups on spheres leaving any
finite simplicial complex stationary are demonstrated. Classification of
concordance classes of actions and construction of equivariant embeddings are
discussed. A final chapter is devoted to answering some problems in trans-
formation groups by Bredon, Raymond, and P. A. Smith.
1980 Mathematics Subject Classification:
57S15, 57S17, 57S25, 57R65.
IV
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