Spaces of Null Homotopic Maps
William G. Dwyer and Clarence W. Wilkerson
University of Notre Dame
Wayne State University
Abstract. We study the null component of the space of pointed maps from Bs*
*s to X when ss is a locally
finite group, and other components of the mapping space when ss is element*
*ary abelian. Results about
the null component are used to give a general criterion for the existence *
*of torsion in arbitrarily high
dimensions in the homotopy of X.
x1. Introduction
In 1983 Haynes Miller [M] proved a conjecture of Sullivan and used it to show*
* that if ss is a locally
finite group and X is a simply connected finite dimensional CW-complex then the*
* space of pointed
maps from the classifying space Bss to X is weakly contractible, ie. Map *(Bss;*
* X) ' * . This result
had immediate applications. Alex Zabrodsky [Z] used it to study maps between cl*
*assifying spaces
of compact Lie groups. McGibbon and Neisendorfer [MN] applied Miller's theorem *
*to answer a
question of Serre; they proved that if X is a simply connected finite dimension*
*al CW-complex with
H"*(X; Fp)6= 0 then there are infinitely many dimensions in which ss*(X) has p-*
*torsion.
The goal of this note is to use the functor T Vof [L] to generalize Miller's *
*theorem and some of
its corollaries to a large class of infinite dimensional spaces (see [LS2] for *
*closely related earlier
work in this direction). This generalization comes at the expense of working wi*
*th one component
of the function complex Map *(Bss; X) at a time.
Fix a prime number p.
Theorem 1.1. Let ss be a locally finite group and X a simply connected p-comple*
*te space. Assume
that H*(X; Fp) is finitely generated as an algebra. Then the component of Map **
*(Bss; X) which
contains the constant map is weakly contractible.
Remark: There is a standard way [M, 1.5] to relax the assumption in 1.1 that X *
*is p-complete.
Theorem 1.1 is actually a special case of a more general assertion. Recall t*
*hat an unstable
module M over the mod p Steenrod Algebra Ap is said to be locally finite [LS] i*
*f each element
x 2 M is contained in a finite Ap submodule. If R is a connected unstable algeb*
*ra over Ap then
the augmentation ideal I(R) is by definition the ideal of positive-dimensional *
*elements and the
module of indecomposables Q(R) is the unstable Ap module I(R)=I(R)2. An unstabl*
*e algebra R
over Ap is of finite type if each Rk is finite-dimensional as an Fp vector spac*
*e.
Theorem 1.2. Let ss be a locally finite group and X a simply connected p-comple*
*te space such
that H*(X; Fp) is of finite type. Assume that the module of indecomposables Q(*
*H*(X; Fp)) is
locally finite as a module over Ap . Then the component of Map *(Bss; X) which*
* contains the
constant map is weakly contractible.
Remark: Theorem 1.1 does in fact follow from Theorem 1.2, since if H*(X; Fp) is*
* finitely gener-
ated as an algebra then Q(H*(X; Fp)) is a finite Ap module.
Remark: Theorem 1.2 has a converse, at least if p = 2 (see Theorem 3.2). There*
* is also a
generalization of 1.2 that deals with other components of the mapping space Map*
* *(Bss; X) (see
Theorem 4.1) but for this generalization it is necessary to assume that ss is a*
*n elementary abelian
p-group.
Given 1.2, the arguments of [MN] go over more or less directly and lead to th*
*e following result.
A CW-complex is of finite type if it has a finite number of cells in each dimen*
*sion.
____________________________
Both authors were supported in part by the National Science Foundation. The fir*
*st author would like to thank the
University of Chicago Mathematics Department for its hospitality during the cou*
*rse of this work.
2 W. Dwyer and C. Wilkerson
Theorem 1.3. Suppose that X is a two-connected CW-complex of finite type. Assu*
*me that
H"*(X; Fp)6= 0 and that Q(H*(X; Fp)) is locally finite as a module over Ap . T*
*hen there exist
infinitely many k such that ssk(X) has p-torsion.
Remark: The example of CP 1 shows that it would not be enough in Theorem 1.3 to*
* assume
that X is 1-connected.
Some instances of 1.3 were previously known; for instance, if X = BG for G a *
*suitable compact
Lie group then the conclusion of 1.3 can be obtained by applying [MN] to the lo*
*op space on X.
However, Theorem 1.3 applies in many previously inaccessible cases; for example*
*, it applies if X
is the Borel construction EG xG Y of the action of a compact Lie group G on a f*
*inite complex Y
or if X is a quotient space obtained from such a Borel construction by collapsi*
*ng out a skeleton.
We first noticed Theorem 1.1 as part of our work [DW] on calculating fragment*
*s of T V with
Smith theory techniques. The proof of 1.1 given here does not use the localizat*
*ion approach of
[DW]; it is partly for this reason that the proof generalizes to give 1.2.
Organization of the paper. Section 2 recalls some properties of the functor T V*
*. In x3 there is
a proof of 1.2 and in x4 a generalization of 1.2 to other components of the map*
*ping space. Section
5 uses the ideas of [MN] to deduce 1.3 from 1.2.
Notation and terminology. The prime p is fixed for the rest of the paper; all u*
*nspecified
cohomology is taken with Fp coefficients. The symbol U (resp. K) will denote *
*the category of
unstable modules (resp. algebras) [L] over Ap. If R 2 K then U(R) (resp. K(R)) *
*will stand for the
category of objects of U (resp. K) which are also R-modules (resp. R-algebras) *
*in a compatible
way [DW].
For a pointed map f : K ! X of spaces we will let Map *(K; X)f denote the com*
*ponent of
the pointed mapping space Map *(K; X) containing f. The component of the unpoin*
*ted mapping
space containing f is Map (K; X)f.
x2 The functor T V
Let V be an elementary abelian p-group, ie., a finite-dimensional vector spac*
*e over Fp, and HV
the classifying space cohomology H*BV . Lannes [L] has constructed a functor T *
*V: U ! U which
is left adjoint to the functor given by tensor product (over Fp) with HV and ha*
*s shown that T V
lifts to a functor K ! K which is similarly left adjoint to tensoring with HV .
Proposition 2.1 [L]. For any object R of K the functor T Vinduces functors U(R)*
* ! U(T V(R))
and K(R) ! K(T V(R)). The functor T Vis exact, and preserves tensor products in*
* the sense that
if M and N are objects of U(R) there is a natural isomorphism
T V(M R N) ~=T V(M) TV (R)T V(N)
Now suppose that fl : R ! HV is a K-map. The adjoint of fl is a map T V(R) ! *
*Fp or in other
words a ring homomorphism ^fl: T V(R)0 ! Fp. For M 2 U(R), let TfVl(M) be the t*
*ensor product
T V(M) TV (R)0Fp, where the action of T V(R)0 on Fp is given by ^fl. Note that *
*TfVl(R) 2 K.
Proposition 2.2 [DW, 2.1]. For any K-map fl : R ! HV the functor TfVl(-) induce*
*s functors
U(R) ! U(TfVl(R)) and K(R) ! K(TfVl(R)). The functor TfVlis exact, and preserv*
*es tensor
products in the sense that if M and N are objects of U(R) there is a natural is*
*omorphism
TfVl(M R N) ~=TfVl(M) TVfl(R)TfVl(N) :
The following proposition is a straightforward consequence of the above two.
Null Homotopic Maps *
* 3
Lemma 2.3. Suppose that ff : R1 ! R2 and fi : R2 ! HV are morphisms of K, and l*
*et fl : R1 !
HV denote the composite fi . ff.
(1)If ff is a surjection and M 2 U(R2) is treated via ff as an object of U(R*
*1), then the natural
map TfVl(M) ! TfVi(M) is an isomorphism.
(2)If M 2 U(R1) then the natural map TfVi(R2) TVfl(R1)TfVl(M) ! TfVi(R2 R1 M*
*) is an
isomorphism.
There is a natural map X : T V(H*X) ! H* Map(BV; X) for any space X. If g : B*
*V ! X is a
map which induces the cohomology homomorphism fl : H*X ! HV then X passes to a *
*quotient
map
X;g: TfVl(H*X) ! H* Map(BV; X)g:
A lot of the geometric usefulness of T Vis explained by the following theorem.
Theorem 2.4 [L2]. Let X be a 1-connected space, g : BV ! X a map, and fl : H*X *
*! HV the
induced cohomology homomorphism. Assume that H*X is of finite type, that TfVlH**
*X is of finite
type, and that TfVlH*X vanishes in dimension 1. Then X;gis an isomorphism.
For any object M of U the adjunction map M ! HV Fp T V(M) can be combined wit*
*h the
unique algebra map HV ! Fpto give a map M ! T V(M); call this map ffl. (If M = *
*H*X for some
space X, then ffl fits into a commutative diagram involving X and the cohomolog*
*y homomorphism
induced by the basepoint evaluation map Map (BV; X) ! X.)
Theorem 2.5 [LS, 6.3.2]. The map ffl : M ! T V(M) is an isomorphism iff M is lo*
*cally finite as
a module over Ap.
If R 2 K, M 2 U(R) and fl : R ! HV is a K-map, we will denote the composite M*
* !fflT V(M) !
TfVl(M) by fflfl. Theorem 2.5 leads to the following result, which we will need*
* in x4.
Proposition 2.6. Let M be an object of U(HV ) and : HV ! HV the identity map. *
* Then
ffl : M ! T V(M) is an isomorphism iff M splits as a tensor product HV Fp N for*
* some N 2 U
which is locally finite as a module over Ap.
Proof: The fact that ffl is an isomorphism if M has the stated tensor product *
*decompositon
follows directly from 2.3(2), 2.5 and [L, 4.2]. Conversely, under the assumpti*
*on that ffl is an
isomorphism Proposition 2.4 of [DW] guarantees that M splits as a tensor produc*
*t HV FpN for
some N 2 U; the fact that N is locally finite is again a consequence of 2.3(2) *
*and 2.5.
x3 The null component
In this section we will prove Theorem 1.2. Before doing this we will recast t*
*he conclusion of the
theorem in a slightly different form.
Lemma 3.1. Let K be a finite pointed CW-complex, X a 1-connected space, and f :*
* K ! X a
pointed map. Then Map*(K; X)f is weakly contractible if and only if the inclusi*
*on of the basepoint
in K induces a weak equivalence Map (K; X)f ! X.
Proof: As in [M, 9.1] the inclusion * ! K gives rise to a fibration sequence Ma*
*p *(K; X)f !
Map (K; X)f ! X.
The arguments of [M, x9] now show that Theorem 1.2 follows directly from the *
*following result.
Theorem 3.2. Let V be an elementary abelian p-group and X a 1-connected p-compl*
*ete space
such that H*X is of finite type. Let f : BV ! X be a constant map and OE : H*X *
*! HV the
induced cohomology homomorphism. Consider the following three conditions:
(1)QH*X is locally finite as an Ap module
(2)the map fflOE: H*X ! TOVEH*X is an isomorphism
(3)the inclusion of the basepoint * ! BV induces a weak equivalence Map (BV;*
* X)f ! X.
4 W. Dwyer and C. Wilkerson
Then (1) =) (2) =) (3). Moreover, if p = 2 then (3) =) (1).
Remark 3.3: It is likely that the three conditions of Theorem 1.2 are equivalen*
*t for any prime p;
the proof would depend on the odd primary version of the results in [S].
Proof of 3.2: First consider the implication (1) =) (2). Let R = H*X and let I*
* R be
the augmentation ideal. Pick s 0. The fact that the action of R on Is=Is+1 fac*
*tors through
the augmentation R ! Fp implies that the action of T V(R) on T V(Is=Is+1) facto*
*rs through
the map T V(R) ! T V(Fp) ~= Fp induced by augmentation; since this last map is *
*adjoint to
OE : R ! H*(BV ) it follows from 2.3(1) that the quotient map T V(Is=Is+1) ! TO*
*VE(Is=Is+1) is an
isomorphism. Moreover, Is=Is+1, as a quotient of (I=I2)s , is the union of its *
*finite Ap submodules
so by 2.5 the map ffl : Is=Is+1 ! T V(Is=Is+1) is an isomorphism. Putting these*
* two facts together
shows that fflOE: Is=Is+1 ! TOVE(Is=Is+1) is an isomorphism. By induction and e*
*xactness, then, the
map fflOE: R=Is+1 ! TOVE(R=Is+1) is an isomorphism. The map TOVE(R) ! TOVE(Fp) *
*~=Fp induced by
augmentation is an epimorphism, so by exactness TOVE(I) vanishes in dimension 0*
*. By Lemma 2.2
and exactness, TOVE(Is+1) vanishes up to and including dimension s, and hence a*
*gain by exactness
the map TOVE(R) ! TOVE(R=Is+1) induced by the quotient projection R ! R=Is+1 is*
* an isomorphism
up through dimension s. It follows immediately that fflOE: R ! TOVE(R) is an is*
*omorphism.
The implication (2) =) (3) is an easy consequence of Theorem 2.4.
For (3) =) (1), assume p = 2. According to [S, proof of 3.1] condition (3) im*
*plies that the loop
space cohomology H*(X) is locally finite as an Ap module, ie., in the terminolo*
*gy of [S], that
H*(X) 2 N ilkfor all k. According to [S, 2.1(iii)], this implies that -1QH*X 2 *
*N ilkfor all k.
This amounts to the assertion that -1QH*X (or equivalently QH*X) is locally fin*
*ite [S, proof
of 3.1].
x4 Other mapping space components
In this section we will give a generalization of Theorem 1.2 to mapping space*
* components other
than the component containing the constant map; this generalization is limited,*
* however, in that
it deals with elementary abelian p-groups rather than with arbitrary locally fi*
*nite groups.
Given an elementary abelian p-group V , call an object M of U(HV ) f-split if*
* M is isomorphic to
HV Fp N for some N 2 U which is locally finite as a module over Ap. Suppose tha*
*t fl : R ! HV
is a map in K with image S HV and kernel I R. Say that fl is almost f-split if
(i)S is a Hopf subalgebra of HV , and
(ii)for each s 0 the tensor product HV S (Is=Is+1) is f-split as an object o*
*f U(HV ).
Recall from 3.1 that Map *(K; X)f is weakly contractible iff evaluation at th*
*e basepoint gives an
equivalence Map (K; X)f ~=X.
Theorem 4.1. Let V be an elementary abelian p-group and X a 1-connected p-compl*
*ete space
such that H*X is of finite type. Let g : BV ! X be a map and fl : H*X ! HV the *
*induced
cohomology homomorphism. Consider the following three conditions:
(1)fl is almost f-split
(2)the map fflfl: H*X ! TfVlH*X is an isomorphism
(3)the inclusion of the basepoint * ! BV induces a weak equivalence Map (BV;*
* X)g ! X.
Then (1) =) (2) =) (3). Morever, if p = 2 then (3) =) (2) =) (1).
Remark 4.2: As in the case of Theorem 3.2, it is likely that the three conditio*
*ns of Theorem 4.1
are equivalent for any prime p.
Lemma 4.3. Let K be a pointed CW-complex, X a pointed 0-connected space, g : K *
*! X a
map, and f : K ! X a constant map. Assume that there exists a map m : K x X ! X*
* which
is 1X on the axis * x X and g : K ! X on the axis K x *. Then the basepoint e*
*valuation
Null Homotopic Maps *
* 5
map ef : Map (K; X)f ! X is a weak equivalence if and only if the corresponding*
* map eg :
Map (K; X)g ! X is a weak equivalence.
Proof: Construct a commutative diagram
=
K ----! K
? ?
a?y ?yb
K x X -----! K x X
(pr1; m)
in which a(k) = (k; *), b(k) = (k; g(k)) and pr1 is projection on the first fac*
*tor. Since the
lower horizontal map is a weak equivalence, it follows that the induced map c :*
* Map (K; K x
X)a ! Map (K; K x X)b is a weak equivalence. It is clear that c commutes with *
*the natural
projections from its domain and range to Map (K; K)i, where i is the identity m*
*ap of K. The
lemma follows from the fact that the domain of c is Map (K; K)ix Map(K; X)f whi*
*le the range is
Map (K; K)ix Map(K; X)g.
Lemma 4.4. Let K be a pointed CW-complex, X a pointed 0-connected space, g : K *
*! X a map,
and f : K ! X a constant map. Assume that the basepoint evaluation map eg : Map*
*(K; X)g ! X
is a weak equivalence. Then the basepoint evaluation map ef : Map (K; X)f ! X i*
*s also a weak
equivalence.
Proof: The map m required in 4.3 is provided up to weak equivalence by the eval*
*uation map
K x Map(K; X)g ! X.
Lemma 4.5. Let V be an elementary abelian p-group, R a connected object of K, f*
*l : R ! HV
a map, and OE : R ! HV the trivial map (ie. the map which factors through the a*
*ugmentation
R ! Fp). Assume there exists a map : R ! HV Fp R which gives 1R when combined *
*with
the augmentation map of HV and fl : R ! HV when combined with the augmentation *
*map of R.
Then fflOE: R ! TOVE(R) is an isomorphism if and only if fflfl: R ! TfVl(R) is *
*an isomorphism.
Proof: This is essentially the proof of 4.3 with the arrows reversed. Construct*
* a commutative
diagram =
HV ---- HV
x x
ff?? fi??
in1.
HV Fp R ---- HV Fp R
in which ff is the product of 1HV with the augmentation of R, fi is (1HV ) . fl*
*, and in1 is the map
from HV to the tensor product obtained using the unit of R. Since the lower hor*
*izontal map is
an isomorphism, it follows that the induced map O : TfVi(HV Fp R) ! TfVf(HV Fp *
*R) is an
isomorphism. It is clear that O respects the natural structures of its domain a*
*nd range as modules
over T V(HV ), where the identity map of HV . The lemma follows from the fact *
*[DW, 2.2] that
the domain of O is T V(HV ) Fp TfVl(R) while the range is T V(HV ) Fp TOVE(R).
Lemma 4.6. Let V be an elementary abelian p-group, R a connected object of K, f*
*l : R ! HV a
map and OE : R ! HV the trivial map. Assume that fflfl: R ! TfVl(R) is an isomo*
*rphism. Then
fflOE: R ! TOVE(R) is also an isomorphism.
Proof: The map required in 4.5 is provided by the map R ! HV Fp TfVl(R) which *
*is adjoint
to the identity map of TfVl(R).
Remark 4.7: It follows from 4.5, 4.6 and 3.2 that at least if p = 2 the three c*
*onditions of 4.1 are
equivalent to a fourth, namely, that QH*X is locally finite as an Ap module and*
* there exists a K
map H*X ! HV Fp H*X which satisfies the conditions of 4.5.
6 W. Dwyer and C. Wilkerson
Lemma 4.8. Let V be an elementary abelian p-group and : S ! HV the inclusion o*
*f a subalgebra
over Ap. Then ffl : S ! T V(S) is an isomorphism if and only if includes S as*
* a Hopf subalgebra
of HV .
Proof: Suppose that ffl is an isomorphism. In this case the adjunction homomo*
*rphism S !
HV Fp T V(S) provides a map S : S ! HV Fp S which fits into a commutative diagr*
*am
S
S ----! HV Fp S
? ?
?y ?y
HV ----! HV Fp HV
HV
where is the identity map of HV and we have used the fact [L, 4.2] that ffl :*
* HV ! HV is an
isomorphism. It is easy to see that HV is the Hopf algebra comultiplication map*
* on HV . It now
follows from the fact that the comultiplication on HV is cocommutative that S(S*
*) S Fp S
and thus that S is a Hopf subalgebra of HV .
Suppose conversely that S is a Hopf subalgebra of HV , and let OE : S ! HV be*
* the trivial map
which factors through the augmentation S ! Fp. The Hopf algebra HV is primitive*
*ly generated,
and the associated restricted Lie algebra of primitives [MM, 6.7] is a free abe*
*lian restricted Lie
algebra on a finite collection of generators (in dimensions 1 and 2). It follow*
*s from [MM, 6.13-
6.16] that S is primitively generated and is isomorphic as an algebra to a fini*
*te tensor product
of exterior and polynomial algebras; in particular, Q(S) is a finite unstable A*
*p module. By the
proof of (1) =) (2) in Theorem 3.2 the map fflOE: S ! TOVE(S) is an isomorphism*
*. Since the
comultiplication of S produces the map required for Lemma 4.5, an application *
*of this lemma
finishes the proof.
Proof of 4.1: Let R denote H*X, I the kernel of fl : R ! HV , S the image of fl*
* and : S ! HV
the inclusion map. We will use f to stand for a constant map BV ! X and OE for *
*the cohomology
homomorphism induced by f.
(1) =) (2). The assumption that S is a Hopf subalgeba of HV implies by 4.8 that*
* ffl : S ! T V(S)
and hence (2.3(1)) fflfl: S ! TfVl(S) are isomorphisms. Pick s 1 and let M = I*
*s=Is+1. If we can
show that fflfl: M ~=TfVl(M) we will be able to finish up by imitating the proo*
*f of (1) =) (2) in
Theorem 3.2. By 2.3(1) it is enough to show that ffl : M ~=T V(M). Proposition*
* 2.6 ensures that
ffl : HV S M ! T V(HV S M) is an isomorphism, where is the identity map of HV *
*. By 2.3(2)
and [L, 4.2] , however, the map ffl is S ffl , so the desired result follows f*
*rom the fact that HV is
free [MM, 4.4] and therefore faithfully flat as a module over S.
(2) =) (3). This is an immediate consequence of 2.4.
(3) =) (2). By Lemma 4.4 and Theorem 3.2 the map fflOE: R ! TOVE(R) is an isomo*
*rphism. The
evaluation map m : BV x Map(BV; X)g ! X induces a cohomology homomorphism : R !
HV Fp R which satisfies the conditions of 4.5, so the implication follows from *
*the conclusion of
4.5.
(2) =) (1). This implication does not in fact require the assumption that p = 2*
*. The map
TfVl(R) ! TfVl(S) is surjective and it follows immediately from naturality that*
* fflfl: S ! TfVl(S)
is surjective. The map TfVl(S) ! TfVl(HV ) is injective, and it follows from n*
*aturality and the
fact that HV ! TfVl(HV ) is injective [L, 4.2] that S ! TfVl(S) is injective. *
*By 2.3(1) the map
ffl : S ! T V(S) is an isomorphism and hence (4.8) S is a Hopf subalgebra of H*
*V .
By exactness the map Is ! TfVl(Is) is seen to be an isomorphism if s = 1 and *
*a monomorphism
if s > 1; this first fact, though, combines with the tensor product formula (2.*
*2) and exactness to
Null Homotopic Maps *
* 7
show that Is ! TfVl(Is) is an epimorphism for s 1. Thus by exactness and 2.3(*
*1) the maps
ffl : Is=Is+1 ! T V(Is=Is+1) are isomorphisms. The proof is finished by runnin*
*g in reverse the
argument used above at the end of the proof of (1) =) (2).
x5 Torsion in homotopy groups
In this section we will use a slight variation on the ideas of [MN] to prove *
*Theorem 1.3.
Let Z denote the ring of integers, Z^pthe additive group of p-adic integers, *
*and Z=pn the cyclic
group of order pn. The group Z=p1 is by definition the locally finite group obt*
*ained by taking the
direct limit of the groups Z=pn under the standard inclusion maps.
Lemma 5.1. For any finitely-generated abelian group A the cohomology group Hk(B*
*Z=p1 ; A) is
isomorphic to Z^p A if k > 0 is even and is zero if k is odd. The natural map A*
* ! Z^p A induces
isomorphisms Hk(BZ=p1 ; A) ~=Hk(BZ=p1 ; Z^p A) for all k > 0.
Sketch of proof: One way to do this is to calculate the homology H*(BZ=p1 ; Z) *
*as a direct
limit lim!nH*(BZ=pn ; Z) and then pass to cohomology by using the universal coe*
*fficient theorem.
The key algebraic ingredient is the fact that
ExtZ(Z=p1 ; Z) ~=ExtZ(Z=p1 ; Z^p) ~=Z^p:
Let PnX stand for the n'th Postnikov stage of the space X and kn+1(X) for the*
* Postnikov
invariant of X which lies in Hn+1(Pn-1X; ssnX)(see [W, IX]).
Lemma 5.2. If Y is a loop space X and Y has finitely-generated homotopy groups,*
* then the
Postnikov invariants of Y are torsion cohomology classes.
Proof: This follow from [MM, p. 263]. In effect, Milnor and Moore show that the*
* rationalized
Postnikov invariants
kn+1(Y ) Q 2 Hn+1(Pn-1Y; ssn(Y ) Q)
are zero. Under the stated finite generation assumption this implies that the P*
*ostnikov invariants
themselves are torsion.
Proof of 1.3: Let S1 be the set of all k such that ssk(X) Z^p6= 0 and S2 the s*
*et of all
k such that sskX contains p-torsion. The set S1 is non-empty (because H*(X; Fp*
*) 6= 0) and
clearly contains S2. Suppose that S2 is finite. In that case we can find an int*
*eger k in S1 such
that no integer j greater than k belongs to S2. Let Y = k-2X. (Note that bec*
*ause X is
2-connected the integer k is greater than 2 and Y is a loop space.) By Lemma *
*5.1 the space
Map *(BZ=p1 ; P1Y ) is contractible and hence Map *(BZ=p1 ; P2Y ) ~=Map *(BZ=p1*
* ; K(ss2Y; 2)).
Because of the way in which k was chosen we can thus, by Lemma 5.1 again, find *
*an essential
map f : BZ=p1 ! P2Y which remains essential in the p-completion (P2Y )^p. The *
*obstructions to
lifting f to a map g : BZ=p1 ! Y are the pullbacks to BZ=p1 of the Postnikov *
*invariants of Y
[W, p. 450]; by Lemma 5.2 these obstructions are torsion, but by Lemma 5.1 and *
*the choice of k
they lie in torsion-free abelian groups. Therefore the obstructions vanish, and*
* the lift g exists. The
composite h of g with the completion map Y ! Y ^pis non-trivial because the com*
*posite of h with
the projection map Y ^p! P2(Y ^p) ~=(P2Y )^pis essential. The adjoint of h is t*
*hen non-zero element
of ssk-2Map *(BZ=p1 ; X), an element which by Theorem 1.2 cannot exist. This c*
*ontradiction
shows that S2 is infinite and proves the theorem.
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Dedicated to the memory of Alex Zabrodsky
University of Notre Dame, Notre Dame, Indiana 46556
Wayne State University, Detroit, Michigan 48202