3
       The  Homotopic  Uniqueness of  B S



                          William G. Dwyer

                          Haynes R. Miller

                       Clarence W. Wilkerson



1     Introduction


Let!p!be a fixed prime number, Fp the field with p elements, and S3 the unit

sphere!in!R4 considered as the multiplicative Lie group of norm 1 quaternions.

The!purpose!of this paper is to prove the following theorem.
!!
1.1!Theorem!.!If!X is any space with H (X;Fp) isomorphic to H (BS3;Fp)

as!an!algebra!over!the  mod p Steenrod algebra, then the p-completion of X

is!homotopy!equivalent!to!the p-completion of  BS3:
!!!
1.2!Remark!.!It is easy to see that 1.1 implies that thereis up to homotopy only
!
one!space B whose loop space is homotopy equivalent to the p-completion of S3:
!
To!get such a strong uniqueness result it is definitely necessary to work one p*
 *rime

at!a!time:  Rector [R] has produced an uncountable number of homotopically

distinct!spaces!Y with loop space homotopy equivalent to S3itself.

!!  Rector's deloopings fY g have the property that Yp '(B S3)^pfor all primes

p:!Theorem!1.1 implies that this condition is forced. Thus Rector's classificat*
 *ion
!
    The research of the authors was partially supported by the NSF, and that of*
 * the third

author by sabbatical funds from Wayne State University.





of!the!genus of BS3 actually classifies all deloopings of S3: McGibb on[Mc 1]

proved!in!1978 that any delooping of S3 is stably equivalent at each odd prime

to!the!standard BS3: Rector [R] for odd primes and McGibbon [Mc 2] for p = 2

showed!that!the!existence of a maximal torus inthe sense of Rector distinguishes

BS3!from!other!members of its genus, and hence by1.1, from other deloopings
!
of!S3:!!
!!!
1.3!Remark!.!!Theorem1.1!is!in some sense a delooping of the results of [M]

and![D-M],!although!our!techniques are somewhat different, esp ecially in the

complicated!case!p=2.!!

!!
1.4!Organization!of!this!paper. Section 2 contains an account of the main back-

ground!material!we will need from [La] and [D-Z]. Section3 treats the odd

primary!case!of 1.1, and section 5 the case p=2. The intervening section  4

describes!a!new way of homotopically constructing BS3at the prime 2. Section

6!is!essentially an appendix which contains the proof of an auxiliary result ne*
 *eded

in!section!5.

!
1.5!Notation!and terminology. Some of the methods in this paper are based

on simplicial techniques, so we will occasionally use "space" to mean "simplici*
 *al

set" and tacitly assume that any topological space involved in theargument has

been replaced by its singular complex [Ma]. In particular, X^p(or X^; if p is

understood) will denote the simplicial p-completion of the space X in the sense

of [B-K,VII,5.1]. The space X  is p-complete if the natural map X ! X^pis a

weak equivalence. A p-completion Xp^is itself p-complete iff the map X !Xp^

induces an isomorphism on mod p homology;this map does give an isomorphism,

for instance, if H1(X ;Fp) = 0 or if X is connected and ss1(X) is a finite group

[B-K]. Theorem 1.1 is equivalent to the claim that any p-complete space with

the stated cohomology is homotopy equivalent to the p-completion of BS 3:

    If X  and Y  are spaces, then Hom(X;Y ) denotes the full function complex

of maps X ! Y ;the subscripted variant Hom(X; Y )fstands for the component

of Hom(X; Y) containing a particular map f: As usual,[X; Y] denotes the set

of components of Hom(X; Y); i.e., the set of homotopy classes of maps from

X  toY :

    If G  is a (simplicial) group, then EG ! BG  is the functorial universal

simplicial principal Gbundle [Ma,p83].  If G  is abelian,  then BG  is also

an abelian simplicial group and the classifying process can be iteratedto form