2

KENNETH C. MILLETT

The basic technique is to relate (or reduce) questions concerning the

homotopy groups of homeomorphism spaces to questions concerning homotopy

groups of various embedding spaces and, subsequently, to questions concerning

the arc components of related embedding spaces. The appropriate connection

is hinted at in the work of J.F.P. Hudson, "Concordance, Isotopy and

Diffeotopy" [39],"which provides one model for the methods developed here.

The result is that this paper is primarily concerned "with the connections be-

tween the structure of concordance spaces and isotopy spaces. From another

point of view, the concern is the relationship between the "simplicial" em-

bedding space and the "block" embedding space of Rourke and Sanderson [6k],

In Section 1 the basic definitions and statements of the main results

are given. The proofs of these results are given in Section 2, assuming

Theorem 1.8. The techniques required in proof of Theorem 1.8, given in

Section 6, are developed in Sections 3? k and 5.

Questions of substantial difficulty remain before the central problem

can be effectively answered in a completely general setting. These include

the ubiquitous low codimension problems as well as problems concerning the

arc components of various concordance (or embedding) spaces and the homotopy

groups beyond the range of the main theorem. Many of these are indicated in

this paper while others will be discussed in a sequel.

I wish to thank the University of Warwick and Princeton University for

the atmosphere and support provided during this research and to R. Edwards,

R. Lashof and C. Rourke for their stimulating encouragement and suggestions.