On stable homotopy equivalences
by
R. R. Bruner, F. R. Cohen1, and C. A. McGibbon
A fundamentalconstruction in the study of stable homotopy is the free infinite
loop space generated by a space X. This is the colimit QX = lim!nnX.
The ith homotopy group of QX is canonically isomorphic to theith stable
homotopy group of X. Thus, one may obtain stable information about X by
obtaining topological results about QX. One such result is the Kahn-Priddy
theorem [7]. In another direction,Kuhn conjectured in [8] that the homotopy
type of QX determines the stable homotopy type of X. In this note we prove
his conjecture for a finite CW-complex X; that is, we prove the following.
Theorem If X and Y are finite CW-complexes, then QX and QY are ho-
motopy equivalent if and only if nXand nY are homotopy equivalent for
some sufficiently large integer n. 2
Of course, one direction of our theorem is obvious. A homotopy equiv-
alence nX ! nY clearly induces a homotopy equivalence QX ! QY .
The proof in the other direction has three steps. Here is a brief outline of
!
it.!!We begin with the case where X and Y are connected and we prove a
p-local!version!of our result for each prime p. In doing so,we use results of
!
Wilkerson,![11],!to first express the stable p-local homotopy types of X and
!
Y!!as bouquets of prime retracts. The samesort of analysis is then carried
out,!p-locally,!on QX and QY, through an appropriate range ofdimensions.
!
The!assumption!that QX and QY are homotopy equivalent then forces the
!
stable!prime!retracts of X(p)to match upwith those of Y(p).
!
!
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1Partially supported by NSF