Series Logo Volume 00, Number 00, Xxxx 19xx INFINITE LOOP SPACES WITH ODD TORSION FREE HOMOLOGY MICHAEL SLACK Abstract.It is shown that an infinite loop space with no odd torsion in * *its integral homology also has no odd torsion in its homotopy. Combined with known results of Steve Wilson, this gives a complete classification; all* * such spaces are products of the Wilson spaces, which are the building blocks * *of the spaces in the omega spectrum for BP. 1.Introduction Infinite loop spaces are fundamental objects of study in homotopy theory, due to the fact that they are in one-to-one correspondence with connective cohomol- ogy theories. They are especially useful in understanding the relationship betw* *een stable and unstable information. The Main Theorem of this paper translates un- stable information (lack of torsion in the homology of an infinite loop space) * *into a stable result (lack of torsion in the stable homotopy of the corresponding om* *ega spectrum). Theorem 1.1. Suppose X is an infinite loop space and p is an odd prime. If the integral homology of X is p-torsion free, then the homotopy groups of X are also p-torsion free. A related result due to Steve Wilson [7] completely classifies H-spaces whose homology and homotopy are both p-torsion free. Theorem 1.1 can be used together with Wilson's results to completely classify all infinite loop spaces with p-to* *rsion free homology. Wilson's result can be summarized as follows. For each prime p and each non- negative integer n there is a unique atomic infinite loop space, denoted B(n; p* *), whose homology and homotopy are free of p-torsion. We call B(n; p) a Wilson space of type (n; p). Wilson proved that every finite type CW H-space with p-to* *rsion free homology and homotopy must be (p-locally) a product of these spaces. The following Corollary of Theorem 1.1 is now immediate, given Wilson's classificat* *ion. Corollary 1.2.Suppose X is an infinite loop space and p is an odd prime. If the integral homology of X is p-torsion free, then X is (p-locally) homotopy equiva* *lent to a product of Wilson spaces. It is interesting to note that Theorem 1.1 is false if the hypothesis that th* *e prime is odd is dropped. Simple counterexamples exist, most notably certain spaces in ____________ Received by the editors November 20, 1995. 1991 Mathematics Subject Classification. Primary 55xxx; Secondary 55xxx. Oc0000 American Mathematical Society 0000-0000/00 $1.00 + $.25 per pa* *ge 1 2 MICHAEL SLACK the omega spectrum of bo, the real connective K-theory spectrum. For example, the infinite loop space BSp has 2-torsion free homology, but has torsion of ord* *er 2 in its homotopy. It is an interesting problem to consider which infinite loop s* *paces have 2-torsion free homology, and what limits that assumption puts on the torsi* *on in homotopy. The author plans to address these questions in future work. The breakdown of the papers is as follows. In Section 2, some necessary techn* *ical background material from the author's previous papers [2] and [3] is summarized. Section 3 is then comprised of the proof of Theorem 1.1. Throughout the paper we shall assume all of our spaces to be based and compactly generated and such that (X; *) is an NDR pair. We shall also assume that H*(X; Z(p)) has finite type. T* *he letter p will be reserved to denote an arbitrary odd prime number. All homology and cohomology is taken to be reduced. 2. Background Material One of the basic techniques in the proof of Theorem 1.1 is the use of the hig* *her projective planes defined in [3]. The main idea behind these constructions is t* *hat they geometrically encode the primary delooping information for X, that is, the part of the infinite loop space structure of X which determines the Dyer-Lashof operations on H*(X; Z=p). The Main Theorem of [2] is then used to give an ex- plicit relationship between the Steenrod operations in the mod p cohomology of * *the spectrum PpX of projective planes, and the Dyer-Lashof operations in the mod p homology of X. More explicitly, let C1;pX denote the pth filtration of the May-Milgram [4] construction C1;1 X which approximates 1 1 X. The subscripts of C1;1 X correspond to filtrations by the dimension, and number of "little cubes", respe* *c- tively. Recall that the first filtration C1;1X of C1;1 X is homotopy equivalent to X. The quotient C1;pX=C1;1X is denoted by eC1;pX. An infinite loop space comes equipped with a structure map C1;pX ! X which extends the homotopy equivalence C1;1X -'!X. Conversely, if X has such a structure map (just enough to define mod p Dyer-Lashof operations), then X is called an H1p-space (a weaker criterion than being an infinite loop space). In [3], the structure map is use* *d to construct a related map h : 1 eC1;pX ! 1 X, which can be defined only stably in general. This is what we call the higher Hopf construction. The spectrum PpX* * is then defined to be the cofiber of h. One of the main results of [3] is that Pp * *may be made into a functor from an appropriate category of H1p-spaces to the homotopy category of spectra, and that Pp is left adjoint to the zero space functor 1 in* * this setting. PpX should be thought of as a partial infinite delooping of X. Hence t* *he cohomology of PpX will be a reflection of the cohomology of the deloopings of X. The spectrum PpX comes equipped with a very useful filtration in the following manner. The usual filtration of 1 1 X (1) X ! X ! 22X ! . .!.1 1 X is modeled by a corresponding filtration of C1;1 X (2) X ! C1;1X ! C2;1X ! . .!.C1;1 X: Restricting this to the pthfiltrations of these models and then passing to the * *quo- tient of each by the first filtration gives (3) * ! eC1;pX ! eC2;pX ! . .!.eC1;pX: INFINITE LOOP SPACES 3 This in turn gives rise to a filtration of the Hopf construction, and hence of * *PpX (4) 1 X ! -11 Pp1X ! -21 Pp2X ! . .!.PpX: We then filter H*(PpX; Z=p) by the formula (5) F k+1= ker{H*(PpX; Z=p) ! H*(-k1 PpkX; Z=p)}: The cohomological results needed from [2] are best stated in terms of the bas* *ic cofibration sequence (6) 1 eC1;pX -h!1 X -i!PpX -j!1 eC1;pX: The mod p cohomology of eC1;pX is computed in [1], and summarized in [2]. It is an algebraic functor of the mod p cohomology of X. Specifically, given an e* *l- ement x 2 H*(X; Z=p), there are functorially defined elements eQ(n-1)(p-1)-fflx* * 2 H*(Ce1;pX; Z=p) (ffl 2 {0; 1}) which are related to the Dyer-Lashof operations * *of the same lower indices. These elements are also related to the Steenrod operati* *ons in H*(PpX; Z=p) by the following Theorem, which is one of the main results of [* *2]. Theorem 2.1. Let n = 2s + 1 - ffi for some ffi 2 {0; 1} and s ffi. If x 2 H2q+ffi(PpX; Z=p), and s > 0, then (7) P q+sx= u1 . j*(oe1 eQ(n-1)(p-1)-1x) mod F n+1; fiP q+sx= u2 . j*(oe1 eQ(n-1)(p-1)x) mod F n+1; where i*x = oe1 x defines x 2 H2q+ffi(X; Z=p), and u1; u2 are units in Z=p. The utility of Theorem 2.1 is in the relationship between the cohomology ele- ments eQ(n-1)(p-1)-fflx (external operations) and the Dyer-Lashof (internal) op* *er- ations on H*(X; Z=p). What we will need can be summarized by saying that the triviality of P q+sx(and hence of fiP q+sx) modulo F n+1will imply the nontrivi- ality of the Dyer-Lashof operations Q(n-1)(p-1)-1and Q(n-1)(p-1)on H*(X; Z=p), since the map h can be used to define homology operations. Roughly speaking, an element dual to x will have a nontrivial action of each of these operations * *on it. The fact that fiQ(n-1)(p-1)= Q(n-1)(p-1)-1(here fi is the homology Bock- stein operation) will then imply that X has p-torsion in its homology if Pq+sx * *= 0 mod F n+1. If x is defined as in Theorem 2.1 by the formula i*x = oe1 x, then by exactne* *ss, h*(oe1 x) = 0. In this case we call the element x 2 H*(X; Z=p) H1p-primitive. It is an elementary fact that the subalgebra of H*(X; Z=p) generated by the H1p- primitive elements is also a sub-Hopf algebra, since it is primitively generate* *d. At this point, a few results about the Steenrod algebra action on H*(Ce1;pX; * *Z=p) are needed. The main tool here is the Nishida relations. Let ffi 2 {0; 1} and d* *efine a function by (2j + ffi) = ffi. 4 MICHAEL SLACK Proposition 2.2.The following formulas hold for every element x 2 Hq(X; Z=p) and every r > 0: [k=p]X[r=2] + (p - 1)(q=2 - i) (i) PkQerx = u1 . Qer+2(p-1)(k-pi)Pix i=0 k - pi [k=p]X[r=2] + (p - 1)(q=2 - i) - 1 + u2 . (r - 1) eQr+2(p-1)(k-pi)-pfiP ix; i=0 k - pi - 1 (ii) fiQe2r-1x = u3 . eQ2rx; where u1; u2; u3 2 Z=p are units. Let M* denote the subspace of H*(Ce1;pX; Z=p) generated by the dual external Pontryagin product elements, let M+ and M- denote the vector spaces generated by elements of the form eQ(n-1)(p-1)z and eQ(n-1)(p-1)-1z, respectively, where * *n > 1 and z 2 H*(X; Z=p). For any space X the vector space M* is closed under the action of the Steenrod algebra. Though the vector spaces M+ and M- are not closed under the action of the full Steenrod algebra, the condition that H*(X; * *Z(p)) is torsion free can be combined with the Nishida relations to show that M+ and M- are closed under the action of the subalgebra generated by the reduced power operations. In other words, there can be no crossing from M+ to M- , or vice ve* *rsa, without the presence of Bockstein operations. 3. Proof of the Main Theorem The proof is similar in principle to the proof of the Main Theorem in [6]. It will follow from a series of three Propositions. The first two of these, Propos* *itions 3.1 and 3.2, have as their goal proving that the mod p cohomology of X is a free commutative algebra. Once this is established, then a simple argument yiel* *ds Proposition 3.3, which states essentially that all of the loop spaces of X also* * must have free commutative mod p cohomology. The Main Theorem then follows easily. The bulk of the work occurs in the proof of Proposition 3.1, which shows that a certain sub-Hopf algebra of the mod p cohomology of X is a free commutative algebra. Recall that an element x 2 H*(X; Z=p) is called H1p-primitive if h*(oe1 x) = * *0, and that the algebra generated by H1p-primitive elements is a sub-Hopf algebra * *of H*(X; Z=p). The strategy used in the proof of Proposition 3.1 is to show that t* *he vanishing of certain pth powers in H*(X; Z=p) implies the vanishing of some par- ticular high excess Steenrod operations in H*(PpX; Z=p), which in turn imply the nontriviality of certain Dyer-Lashof operations in H*(X; Z=p). The nonvanishing Dyer-Lashof operations that are found will detect torsion in the integral homol* *ogy of X, which is assumed not to be possible. Therefore the assumption that these pthpowers were trivial must have been erroneous. Proposition 3.1.Suppose X is an infinite loop space whose integral homology is p-torsion free. Then the sub-Hopf algebra of H*(X; Z=p) generated by the H1p- primitive elements is a free commutative algebra. INFINITE LOOP SPACES 5 Proof.In order to prove the Proposition, it suffices to show that every H1p-pri* *mitive element of Heven(X; Z=p) has infinite height. Since the pth power of an H1p- primitive element is H1p-primitive, it suffices to show that the pthpower of ev* *ery even degree H1p-primitive element is nonzero. This will be done using an induct* *ive argument. Let r and s be non-negative integers. Define a function OE(r; s) by the recur* *sive formula (8) OE(0; s)= s OE(r; s)= p . OE(r - 1; s) - 1: Then it is straightforward to see that any positive integer can be expressed un* *iquely as OE(r; s) if we restrict s to positive integers not equivalent to -1 mod p. * *There is also a useful alternative definition which is easy to verify r - 1 (9) OE(r; 0)= - p_____p - 1 OE(r; s)= OE(r; s - 1) + pr: From this it is evident that a nonrecursive definition of OE(r; s) is r - 1 (10) OE(r; s) = spr - p_____p:- 1 Assuming s is not equivalent to -1 mod p, it will be shown by upward inducti* *on on r that the pth power map is monic on the H1p-primitives of H2OE(r;s)(X; Z=p). Let x 2 H2OE(r;s)(X; Z=p) be an H1p-primitive element, and assume that xp = 0. This will be shown to lead to a contradiction to the hypothesis that H*(X; Z(p)) is free of p-torsion. Assume by induction that the pthpower map is monic on the H1p-primitives of H2OE(t;s)(X; Z=p) for all 0 t < r and all s. Using the basic cofibration sequence (6) from the previous section, choose an element x 2 H2OE(r;s)(PpX; Z=p) for which i*(x) = oe1 x. The assumption that x is H1p-primitive is precisely the condition we need in order to find such an * *el- ement x. Thenrthe Theorem 2.1 implies the following formulas, which hold in H2pOE(r;s)+2p (p-1)(PpX; Z=p) r+2 (11) P OE(r;s+1)x= u1 . j*(oe1 eQpr(p-1)-1x) mod F p ; r+2 fiP OE(r;s+1)x= u2 . j*(oe1 eQpr(p-1)x) mod F p : A contradiction will be reached if it is shown that r+2 (12) POE(r;s+1)x= 0 mod (M* + M+ ) [ F p ; r+2 fiP OE(r;s+1)x= 0 mod M* [ F p : This also follows by the results of [2] (see the comments following Theorem 2.1* *), since the elements oe1 eQpr(p-1)-1x and oe1 eQpr(p-1)x, connected by the cohomo* *l- ogy Bockstein, will then be in the image of h* (modulo elements which can be ignored). This in turn implies the existence of two nontrivial Dyer-Lashof oper* *a- tions in H*(X; Z=p) connected by a homology Bockstein, which can't happen due to the hypotheses on X. 6 MICHAEL SLACK The Adem relations will be used to prove P OE(r;s+1)x= 0. Specifically, the relation of interest is r-1 r OE(r;s)pX OE(r;s+1)-ii (13) Pp P = aiP P ; i=0 where the constants ai2 Z=p are certain binomial coefficients. The only one whi* *ch will be explicitly needed is (14) a0= (p - 1)OE(r;ps)r- 1 = OE(r + 1;ps)r- OE(r; s) r+1 r = OE(r + 1; 0) +pspr - OE(r; 0) - sp r+1 r = sp - (sp+r1)p = -(s + 1) mod p: Because s is assumed to be not equal to -1 modulo p, a0 is nonzero. The next step is to show that Pix = 0 in H*(X; Z=p) for 0 < i pr-1. This will be where the inductive hypothesis is used. Note that when r = 0 the statement Pix = 0 for 0 < i pr-1 is empty, hencemno verification is needed to begin the induction. It suffices to show P p x = 0 for 0 m r - 1,msince these are the corresponding indecomposable operations. The element P p x is H1p-primitive, and its degree is 2OE(r; s) + 2pm (p - 1) = 2OE(m; OE(rm- m; s) + p - 1).mThere* *fore the inductive hypothesis implies thatmthe pthmpower+of1Pp x is nonzero if Pp x 6=* * 0. But it is well known that (P p x)pm= P p (xp), which is zero, by assumption. Hence it must be the case that Pp x is also zero. Since 0 = oe1 Pix = i*(P ix), it must be the case that P ixis in the image of j*. Suppose P ix= j*(y), for some y 2 H*(1 eC1;pX; Z=p). The degree of y is 2OE(r; s) + 2i(p - 1). Since y is an element of the cohomology of the suspen* *sion spectrum of the suspension of a space, the unstable condition implies Pny is ze* *ro if n is greater than or equal to OE(r; s) + i(p - 1). Compare this to OE(r; s +* * 1) - i; since (15) OE(r; s + 1)=-OiE(r; s) + pr - i = OE(r; s) + i(p - 1) + p(pr-1- i); we may conclude that POE(r;s+1)-iy = 0 if 0 < i pr-1. Now calculate (16) POE(r;s+1)-iPix= POE(r;s+1)-i(j*y) = j*P OE(r;s+1)-iy = 0 when 0 < i pr-1: Using the Adem relation (13) together with (16), there is significant simplif* *ica- tion, and what remains is r OE(r;s) (17) -(s + 1) POE(r;s+1)x= Pp P x: INFINITE LOOP SPACES 7 In order to complete the proof of the Proposition, by (12) and (17) it remains * *to show that r OE(r;s) pr+2 (18) Pp P x = 0 mod (M* + M+ ) [ F : Recall the original assumption that xp = 0. This implies (19) i*(P OE(r;s)x)= POE(r;s)(i*x) = oe1 POE(r;s)x = oe1 (xp) = 0: Therefore POE(r;s)xis in the image of j*. There must be elements zn 2 H*(X; Z=p) for which 1X (20) P OE(r;s)x= j*(oe1 eQn(p-1)-1zn) mod M* + M+ : n=1 By induction it will be shown that enough of the elements zn are zero so that in fact 1X (21) POE(r;s)x= j*(oe1 eQpr+1k(p-1)-1zpr+1k) mod M* + M+ k=1 r+2 = 0 mod (M* + M+ ) [ F p : To ground the induction, a routine calculation shows that zn can have integral degree only if n = pk for some k, in which case (22) |zn| = 2OE(r; s) - k(p - 1): By induction, assume 1X (23) P OE(r;s)x= j*(oe1 eQptk(p-1)-1zptk) mod M* + M+ k=1 for some 1 t r. In this case the degree of zptkis given by (24) |zptk| = 2OE(r; s) - pt-1k(p - 1): t-1 The strategy here will be to apply the operation Pp to both sides of equation (23). The resulting left hand side will be shown to be zero, and via a partial computation of the right hand side this will force zptkto be zero unless p | k. For the left hand side the following Adem relation will be needed t-2 t-1 OE(r;s)pX OE(t-1;pOE(r-t;s))-ii (25) Pp P = biP P : i=0 8 MICHAEL SLACK The first coefficient b0 may be computed explicitly, (26) b0= (p - 1)OE(r;ps)t--11 = OE(r +p1;ts)--1OE(r; s) r+1 r = sp -p(st+-1)p1 = 0 mod p; because t - 1 < r. Therefore one obtains the slightly simpler looking relation t-2 t-1 OE(r;s)pX OE(t-1;pOE(r-t;s))-ii (27) Pp P = biP P : i=1 It has already been shown that P ixis in thetimage-of1j* for each i above, since 1 i pt-2 pr-2. Thus in order to prove Pp POE(r;s)xis zero, it suffices (as before) to check that OE(t-1; pOE(r -t; s))-i is less than or equal to OE(r; s)* *+i(p-1) (28) OE(t - 1; pOE(r - t; s))=-OiE(r; s) + pt-1- i = OE(r; s) + i(p - 1) + p(pt-2- i): t-1 OE(r;s) The criterion is met, and this together with (25) implies Pp P x = 0. Combining this with equation (23) gives 1X t-1 (29) j*(oe1 Pp eQptk(p-1)-1zptk) = 0 mod M* + M+ : k=1 The left hand side can be computed using the Nishida relations (Proposition 2.2* *); the specific relation which is used is t-2 t-1 pX j (30) Pp eQptk(p-1)-1zptk= u1 . cjeQ(ptk+2pt-1-2pj)(p-1)-1Pzptk; j=0 where the cj are certain binomial coefficients. Only the coefficient c0 is expl* *icitly needed; it is ptk(p-1)-2_ pt-1k(p-1)_ (31) c0= 2 + (p - 1)(OE(r; s) - 2 ) pt-1 pt-1k(p-1)_ = OE(r + 1; s) - OE(r; s) + 2 pt-1 r+1 r 1_ t t-1 = sp - (s + 1)p + 2k(p - p ) pt-1 = -1_2k mod p: The last line follows since t r. The coefficient c0 is therefore nonzero whene* *ver k is not divisible by p. The terms Pjzptkin the expansion above are in differing INFINITE LOOP SPACES 9 degrees for distinct pairs (j; k), as we may see by explicitly computing (32) |zptk|= 2OE(r; s) - pt-1k(p - 1); |P jzptk|= 2OE(r; s) - pt-1k(p - 1) + 2j(p - 1); using the fact that j pt-2. Because c0 is nonzero whenever k is not divisible by p, equations (29) and (3* *0) imply that t-1 (33) oe1 Pp eQptk(p-1)-1zptk will be a nonzero element of kerj* = im h* modulo M* + M+ if zptkis nonzero for some k not divisible by p. Combining this with the main results of [2], we * *can then find nontrivial Dyer-Lashof operations in H*(X; Z=p) which are connected by the Bockstein operation. This would contradict the hypotheses on X. Hence all such zptkmust be zero, and the induction is complete. We may now conclude that POE(r;s)x= 0 mod (M*+M+ ) [F pr+2. Since the vector space (M*+M+ ) [F pr+2 is closed under the action of the reduced power operations, equation (18) is al* *so_ satisfied, and hence the proof of the Proposition is complete. * *|__| Proposition 3.2.Suppose X is an infinite loop space whose integral homology is p-torsion free. Then the mod p cohomology of X is a free commutative algebra. Proof.First it will be shown that the even degree Hopf algebra primitives of H*(X; Z=p) all have infinite height. Suppose x 2 P H2k(X; Z=p). Then X (34) h*(oe1 x) = oe1 ( anQen(p-1)-1yn + bnQen(p-1)zn); n1 where yn 2 Hodd(X; Z=p), zn 2 Heven(X; Z=p) and an; bn are undetermined nonzero coefficients. If the right hand side above is zero, then x is an H1p-pr* *imitive, and hence has infinite height. So assume that the right hand side is nonzero, 0 = xp = Pkx, and that the degree of x is the smallest in which the pthpower of* * a Hopf algebra primitive element is zero. Then the formula above together with the Nishida relations yield X (35) 0 = h*(oe1 Pkx)= oe1 ( anP keQn(p-1)-1yn + bnP keQn(p-1)zn) n1 X = oe1 ( bnQen(p-1)zpn): n1 The only way the right hand side of the this equation can be zero is if all of * *the zn are zero by the assumption of the choice of x. Therefore at least one of the* * yn must be nonzero. In that case, the torsion free assumption of the homology of X implies X (36) 0 = h*(oe1 fix)= oe1 ( anfiQen(p-1)-1yn) n1 X = oe1 ( anQen(p-1)yn): n1 This is a contradiction, since the an are all nonzero and at least one of the yn is nonzero. Hence it may be concluded that the sub-Hopf algebra of H*(X; Z=p) 10 MICHAEL SLACK generated by the even degree Hopf algebra primitive elements is a free commutat* *ive algebra. Proposition 7:21 of Milnor and Moore's paper [5] implies that the Hopf algebra H*(X; Z=p) splits as a tensor product of Hopf algebras (37) H*(X; Z=p) ~=E H*(X; Z=p)==E; where E is an exterior algebra on odd degree generators, and H*(X; Z=p)==E is e* *ven dimensional. Furthermore, by the results above, the primitives of H*(X; Z=p)==E must have infinite height. We will show that every element of H*(X; Z=p)==E has infinite height. Suppose that there is an element w 2 H*(X; Z=p)==E for which wp = 0. Then it must be that w is not primitive, so that X (38) (w) = 1 w + w 1 + w0i w00i; i where w0iand w00iare nonzero for at least one i. Choose w so that it is in the * *lowest degree in which the pthpower map is trivial. Then X (39) 0 = (wp) = (1 w + w 1 + w0i w00i)p X i = (w0i)p (w00i)p: i This is impossible since the elements w0iand w00ihave nonzero pthpowers. Theref* *ore_ H*(X; Z=p)==E is a polynomial algebra, and the Proposition is proved. |_* *_| For the next Proposition, we need to consider connective covers of X. Recall that the k-connected cover, X, of X is defined to be the space obtained from* * X by killing the homotopy groups in degrees k and below. Proposition 3.3.Suppose X is an infinite loop space whose integral homology is p-torsion free. Then the integral homology of nX is also p-torsion free. Proof.The proof proceeds by upward induction on n. The case n = 0 is just the assumption that X has p-torsion free homology. Assume by induction that the integral homology of n-1X is p-torsion free. Then the 0-component n-1X must also have p-torsion free homology. By Proposition 3.2, the mod p cohomology of n-1X is a free commutative algebra. It is a routine verification using the Eilenberg-Moore spectral sequence that a connected H-spa* *ce with p-torsion free homology whose mod p cohomology is a free commutative algeb* *ra has its loop space also free of p-torsion in homology (the only way new torsion* * could arise would be from truncations of even degree elements). Thus the Proposition_* *is proved. |__| It is now routine to verify the Main Theorem. Since ssn(X) = ss0(nX), it follows from Proposition 3.3 and the Hurewicz Theorem. References [1]F. Cohen, T. Lada and J. P. May, The Homology of Iterated Loop Spaces, Lect* *ure Notes in Mathematics 533 (Springer, Berlin-New York, 1976). [2]M. Foskey and M. Slack, On the odd primary cohomology of higher projective * *planes, Pac. J. Math., to appear. INFINITE LOOP SPACES 11 [3]N. Kuhn, M. Slack and F. Williams, Hopf constructions and higher projective* * planes for iterated loop spaces, Trans. Amer. Math. Soc. 347 (1995), 1201-1238. [4]J. P. May, The Geometry of Iterated Loop Spaces, Lecture Notes in Mathemati* *cs 271 (Springer, Berlin-New York, 1972). [5]J. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. Math. 81* * (1965), 211-264. [6]M. Slack, Infinite loop spaces with trivial Dyer-Lashof operations, Math. P* *roc. Camb. Phil. Soc. 113 (1993), 311-328. [7]W. S. Wilson, The omega spectrum for BP homology part II, Amer. J. Math. 97* * (1975), 101-123. Department of Mathematics and Statistics, Western Michigan University, Kalama- zoo, MI 49008-5152 E-mail address: slack@wmich.edu