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 Bss*
* to X when ss is a locally
finite group, and other components of the mapping space when ss is elementa*
*ry abelian. Results about
the null component are used to give a general criterion for the existence o*
*f 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 ssis a locally
finite group and Xis a simply connected finite dimensional CW-complex then the *
*space of pointed
maps from the classifying space Bssto X is weakly contractible, ie. Map (Bss; X*
*) ' .This result
had immediate applications.Alex Zabrodsky [Z] used it to study maps between cla*
*ssifying 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 dimensiona*
*l CW-complex with
"H (X;Fp)6= 0 thenthere are infinitely many dimensions in which ss (X) has p-to*
*rsion.
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 wit*
*h one component
of the function complex Map (Bss;X) at a time.
Fix a prime numberp.
Theorem1.1. Let ss be a locally finite group and Xa simply connected p-complete*
* space.Assume
that H (X;Fp) is finitely generated as an algebra. Then the component of Map (B*
*ss;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 i*
*s p-complete.
Theorem 1.1 is actually a specialcase of a more general assertion. Recalltha*
*t an unstable
module M over the mod p Steenrod AlgebraAp is said to be locally finite [LS] if*
* each element
x 2 M is contained in a finite Ap submodule. If R is a connectedunstable algebr*
*a over Ap then
the augmentation ideal I(R) is by definitionthe ideal of positive-dimensional e*
*lements and the
module of indecomposables Q(R) is the unstable Apmodule I(R)=I(R)2. An unstable*
* algebra R
overAp is of finite type if each Rk is finite-dimensional as an Fp vector spac*
*e.
Theorem!1.2.!Let ssbe a locally finite group and X a simply connected p-complet*
*espace 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 overAp . Then the component of Map (Bss;X) which co*
*ntains 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 generalizationit is necessary to assume that ss is an*
* elementaryab elian
p-group.!!
!!Given 1.2,the arguments of [MN] goover more or less directly and lead to the *
*following result.
A!CW-complex is of finite type if it hasa finite number of cells in each dimens*
*ion.
!
Both authors were supported in part by the National Science Foundation. The fir*
*stauthor would like to thank the
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. Then *
*there exist
infinitely many ksuch that ssk(X) has p-torsion.
Remark: The example of CP1 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 su*
*itable compact
Lie group then the conclusion of 1.3 canb e obtained by applying [MN] to the lo*
*op space on X.
However, Theorem 1.3 appliesin many previously inaccessible cases; for example,*
* it applies if X
is the Borel construction EGG Y of the action of a compactLie group G on a fini*
*te complex Y
or if X is a quotient space obtained from such a Borel construction by collapsi*
*ng out a skeleton.
We first noticedTheorem 1.1 as part of our work [DW] on calculating fragments*
* of TV 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 TV *
*. 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 fixedfor the rest of the paper; all un*
*specified
cohomology is taken with Fpcoefficients. Thesymbol U (resp. K) willdenote the*
* category of
unstable modules (resp. algebras) [L]overAp . If R2 Kthen U (R) (resp. K(R)) wi*
*ll 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)fdenote the compo*
*nent of
the pointed mapping space Map (K;X) containing f. The component of the unpointe*
*d mapping
space containing f is Map(K;X)f.
x2 The functor TV
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 TV :*
* U ! Uwhich
is left adjoint to the functor given by tensor product (over Fp) with HV and ha*
*s shown that TV
lifts to a functor K! Kwhich 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 TV is 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)= TV (M) TV(R)TV (N)
Now suppose that fl: R ! HV is a K-map.The adjoint of fl is a map TV(R) ! Fp *
*or in other
words a ring homomorphism^fl: TV(R)0 ! Fp. For M2 U(R), let TVfl(M ) be the ten*
*sor pro duct
TV (M) TV(R)0Fp, where the action of TV (R)0 onFp is given by^fl. Note that TVf*
*l(R) 2K.
Proposition 2.2 [DW, 2.1]. For any K-map fl : R! H V thefunctor TfVl() induces *
*functors
U(R) ! U(TfVl(R)) and K(R) ! K(TVfl(R)). The functor TVflis exact, and preserv*
*es tensor
products in the sense that if M and N are objects ofU (R) there is anatural iso*
*morphism
TVfl(MR N) = TVfl(M)TVfl(R)TVfl(N) :