A fundamental problem in topology dating from its conception in the -works
of Hausdorff [3h] and Frechet [27], among others, is the description of the
local and global structure of various function spaces. The appropriate topol-
ogy for such spaces has, itself, been the subject of much work, e.g. [8,15,28,
k6,48,50,61,62]. These problems still directly provide the impetus for cur-
rent research. Indeed particular recent interest centers upon various spaces
of equivalences, in some category. This paper concerns the structure of the
(simplicial) space of piece-wise linear homeomorphisms of a piece-wise linear
Until recently very little was known (even locally) of the structure of
many of the spaces of homeomorphisms. Low dimensional information is now
available through the result of many efforts, e.g. [2,17,25,27,33,50,67,68,
69]. Local information has also become available through the work of others,
[18,23,2^,^9,61]. The situation with regard to global information is similar.
With the exception of the classical groups and the researches of Mason [51]
and Geoghegan [29], results concern a study of the arc components, e.g., [7,
13,14,20,26], or stable information, e.g. [11,19,30,32,42,^,49,59,64,70,72].
As these results are often achieved only through extraordinary effort,
significantly new techniques, or by making substantial restrictions and since,
for many purposes, the resulting information has had such profound conse-
quences, e.g., [^2,*J6], or alternative approaches have been developed, e.g.,
[4l,57,64], one might regard the questions concerning the "unstable" range as
either insurmountable or uninteresting were it not for the fact that not all
manifolds are spheres, i.e., as highly connected, and not all global structure
is reflected in the arc components of an individual space. Thus the research
begun here is only one step in the attempt to provide the means required to
determine the homotopy groups of the (simplicial) piecewise linear homeomor-
phism group of a piecewise linear manifold although the applications indicated
in this paper concern only spheres or Euclidean spaces.
C. Morlet [59] has also begun such a program employing a rather different
approach via his "Lemme de disjonction". Also Akiba [2] and Scott [68] have
considered the low dimensional problems. The latter approach has its roots in
the work of Kuiper and Lashof [44] which also provides the foundation for the
point of view employed here.
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