Miller Spaces and Spherical Resolvability of Finite Complexes Jeffrey Strom Abstract We show that if K is a nilpotent finite complex, then K can be built from spheres using fibrations and homotopy (inverse) limits. This is ap- plied to show that if map *(X, Sn) is weakly contractible for all n, then map *( X, K) is weakly contractible for any nilpotent finite complex K . AMS Classification numbers Primary: Secondary: Keywords: Miller Spaces, Spherical Resolvability, Resolving Class, Closed Class, Homotopy Limit Discussion of Results A Miller space is a CW complex X with the property that the space of pointed maps from X to K , written map *(X, K), is weakly contractible for every nilpotent finite complex K . They are named for Haynes Miller, who proved in [12 ] that the spaces B Z =p are all Miller spaces; in fact, he proved that map *(B Z =p, K) is weakly contractible for every finite dimensional CW complex K . In the stable category, one can define a Miller spectrum by requiring that the mapping spectrum F (X, K) ' * for every finite spectum K . Since cofibrations and fibrations are the same in the stable category, a finite spectrum K with m cells is the fiber in a fibration K -! L -! Sn in which L has only m - 1 cells; in the terminology of [5, 11], this means that K is spherically resolvable with weight m. An easy induction shows that X is a Miller spectrum if and only if F (X, Sn) ' * for every n. 1 Our goal is to prove the following unstable analog of this observation: if map *(X, Sn) is weakly contractible for all n, then X is a Miller space. The proof of the stable version is not available to us because cofibrations are not fibrations, unstably. To prove our result, it is necessary to deter- mine the extent to which a finite complex can be constructed from spheres in a more general way, i.e., by arbitrary homotopy (inverse) limits [3] and by extensions by fibrations. To be more precise, we require some new terminology. We call a nonempty class R of spaces a resolving class if it is closed under weak equivalences and pointed homotopy (inverse) limits (all spaces and homotopy limits will be pointed). It is a strong resolving class if it is further closed under extensions by fibrations, i.e., if whenever F -! E -! B is a fibration with F, B 2 R, then E 2 R. Resolving classes are dual to closed classes as defined in [4] and [6, p. 45]. Notice that every resolving class R contains * (cf. [6, p. 47]). From this, it follows that if F -! E -! B is a fibration with E, B 2 R, then F 2 R. Similarly, if Aff2 R for each ff then the categorical product ffAff2 R also. The weak product effAffis the homotopy colimit of the finite subproducts; if for each i only finitely many of the groups ßi(Aff) are nonzero, then the weak product has the same weak homotopy type as the categorical product. Let_S be the smallest resolving class that contains Sn for each n, and let S be the smallest strong resolving class that contains_Sn_for each n. We say that a space K is spherically resolvable if kK 2 S for some k. This concept is related to, but not the same as, the notion of spherical resolvability described in [5, 11]. We list some other important examples of (strong) resolving classes below. Examples (a) If f : A -! B is any map then the class of all f -local spaces is a resolving class [6, p. 5]. This includes, for example, the class of all spaces with ßi(X) = 0 for i > n, or all h*-local spaces, where h* is a homology theory. (b) If P is a set of primes, then the class of all P -local spaces is a strong resolving class. 2 (c) If f : W -! *, then the class of all f -local spaces is a strong resolv- ing class [6, p. 5]. This includes, for example the class {K+ }, where K+ denotes the Quillen plus construction on K [6, p. 27]. (d) More generally, if F is a covariant functor that commutes with ho- motopy limits (and fibrations) and R is a (strong) resolving class, then the class {K | F (K) 2 R} is also a (strong) resolving class. This applies, for example to the functor F (K) = map *(X, K). (e) The class {K | K is weakly contractible} is a strong resolving class. Our proofs will proceed by induction on a certain kind of cone length [1]. Let F denote the collection of all finite type wedges of spheres. The F - cone length clF(K) of a space K is the least integer n for which there are cofibrations Si- ! Ki- ! Ki+1, 0 i < n, with K0 ' *, Kn ' K and each Si 2 F . If no such n exists, then clF(K) = 1. Clearly every finite complex K has clF(K) < 1. With these preliminaries in place, we can state our main result. Theorem 1 If K is a nilpotent space with clF(K) = n < 1, then __ (a) K 2 S , and (b) nK 2 S . In particular, every nilpotent finite complex K is spherically resolvable in our sense. Our application to Miller spaces follows from the following more general consequence of Theorem 1. Theorem 2 Let R be a strong resolving class, and assume that X has the property that map *(X, Sn) 2 R for each n. Then map *( X, K) 2 R for each nilpotent space K with clF(K) < 1. This result has many corollaries; we list a few. Corollary 3 Let X be a space and P a set of primes. 3 (a) If map *(X, Sn) is weakly contractible for all n, then map *( X, K) is weakly contractible for all nilpotent K with clF(K) < 1. In other words, X is a Miller space. (b) If map *(X, Sn) is P -local for all n, then map *( X, K) is P -local for all nilpotent K with clF(K) < 1. We end by making the surprising observation that a (non-nilpotent, of course) finite complex can be a Miller space! Example Let A be a connected 2-dimensional acyclic finite complex. (The classifying space of the Graham-Higman group [7] is such a space; so is the space obtained by removing a point from a non-simply connected 3-dimensional Poincar'e sphere). Since ß1(A) is equal to its commuta- tor subgroup, there are no nontrivial homomorphisms from ß1(A) to any nilpotent group. It follows that if f : A -! K with K a nilpotentWfinite complex, then ß1(f) = 0 and so f factors through q : A -! A=A1 ' S2. Since [A, S2] ~=H2(A) = 0, we conclude f ' *. Thus A is a Miller space. One of the key points in the proof of Theorem 1 is an explicit description of the homotopy fiber of a map B -! B [CA, which we state as Proposition 5. In an appendix, we use this result to give simple proofs of the Blakers- Massey excision theorem [2] and the James splitting of X [10 ]. I would like to thank Robert Bruner and Charles McGibbon for suggesting I think about Miller spaces. This work owes much to McGibbon in partic- ular - Theorem 3(a) was conjectured in joint work with him. Thanks to Bill Dwyer for directing me to the result of [9], which is the key to Propo- sition 4; thanks are also due to Daniel Tanr'e for bringing Proposition 5 to my attention. 1 Proof of Theorem 1 We begin with two supporting results. Proposition 4 Let K be a connected nilpotent space, let R be a resolv- ing class and let F be a functor that preserves fibrationsWand homotopy (inverse) limits over diagrams indexed on n and L. If F ( mi=1 K) 2 R for each m, then F (K) 2 R. 4 Proof This follows from a result of Hopkins [9, p. 222], which says that K is homotopy equivalent to the homotopy (inverse) limit of a tower A0- A1- . .-. An- . . . of spaces, each of whichWis a homotopy (inverse) limit over n of a diagram of spaces of the form mi=1 K . 2 W m It follows that to show that kK 2 R itWsuffices to show that k( i=1 K) 2 R for all m; more generally, if F ( k( mi=1 K)) 2 R for all m then F ( kK) 2 R. Proposition 5 Let A -! B -! C be a cofibration, and let F be the ho- motopy fiber of B -! C . Then F ' A _ ( A ^ C). Proof Convert each of the maps A -*! C , B -! C and C -=! C to a fibrations. The total spaces and fibers form the commutative diagram A x |CJ ______________//FA | JJJ | AAA | JJJ | AAA | JJ%% | A__ | C ______|______//* | | | | | | | | | | | fflffl| || fflffl| | A _________________//JB | JJJ | @@@ | JJJ | @@ | JJJ | @@ | JJ$fflffl|$ _fflffl|_ * ______________//C, in which the bottom square is a homotopy pushout. A result of V. Puppe [13 ] shows that the top square is also a homotopy pushout. Hence, the cofiber F of the map F -! * has the same homotopy type as the cofiber of A x C -! C , namely A _ ( A ^ C), as can be seen from the diagram A x C _______// C _____________//_ F | | || | pushout | || fflffl|* fflffl| || A ________//A * C_____// A _ ( A ^ C). 2 5 Proof of Theorem 1 Notice that the assumption on X implies that X is connected; we may therefore assume that K is also connected. We prove both assertions in parallel, by induction on clF(K). If clF(K) = 1 then K is homotopyWequivalent to a connected finite type wedge of spheres. Then each mi=1 K is a simply-connected finite typeWwedge of spheres, and the Hilton-Milnor theorem [8, 14] shows that ( mi=1 K) is homotopy equivalent to the weak product e Snff2 S . Since the wedge is of finite type and simply connected, all but finitely many factors are not i-connected for each i, so the weak product has the same homotopy type as the categorical product. Proposition 4 proves both assertions in the initial case. Now assume that both statements are known for all nilpotent spaces with F -cone length less than n, and that K is nilpotentWwith clF(K) = n. ByWProposition 4,_ it is enough to show that n( mi=1 2K) 2 S and ( mi=1 2K) 2 S for each m. Wm 2 W m Write V = i=1 K . Notice that clF( i=1K)W clF(K), and the double suspension of an F -cone decomposition of mi=1K is an F -cone decom- position of V . Thus we may assume that V has an F -cone decomposi- tion Si- ! Vi- ! Vi+1, 0 i < n with each Si and Vi simply-connected. Therefore, we have a cofibration L -! V - ! W with L simply-connected, clF(L) < n and W a simply-connected finite type wedge of spheres. Let F denote the homotopy fiber of V - ! W . Consider the fibration sequences F -! V - ! W and nV - ! nW -! n-1F. __ __ Since kW 2 S S for all k > 0, it suffices to show that F 2 S and that n-1F 2 S . Now we use Proposition 5 to determine the homotopy type of F : `` ' F ' L _ (L ^ W ) ' L ^ Snff ff which is a finite type wedge of suspensionsWof L. If we smash an F -cone length decomposition of L with the space ffSnff we obtain an F -cone length decompositionWfor F - in other words, clF( F ) < n and, more importantly, clF( li=1 F ) < n for each l. 6 W l __ n-1 W l By the inductive hypothesis, ( i=1 F ) 2 S and ( i=1 F ) 2 S for each l. Since L, V and W are_each simply-connected, so is F , and Proposition 4 implies that F 2 S and that n-1F 2 S , as desired. 2 2 Proof of Theorem 2 and Corollary 3 Proof of Theorem 2 Let M be the class of all spaces Y such that map *(X, Y ) 2 R; we have already seen that M is a strong resolving_ class. Since Sn 2 M for each n by assumption, it follows that S M. By Theorem 1, M contains K for every nilpotent space K with clF(K) < 1. 2 Proof of Corollary 3 Only statement 3 needs proof. Let M be the class of all spaces K for which map *(X, cK) is weakly contractible. By assumption Sn 2 M for each n. Since the Sullivan completion functor commutes with fibrations and homotopy (inverse) limits indexed on n and L, 3 Appendix: Applying Proposition 5 This section is essentially an advertisement for Proposition 5. Note that the proof of Proposition 5 uses only elementary results on homotopy pushouts and pullbacks. Nevertheless, it has many very important conse- quences, among which are elementary proofs of the Blakers-Massey exci- sion theorem [2] and the James splitting of X [10 ]. Theorem(Blakers-Massey) Let A -! B -! C be a cofibration of simply- connected spaces with A (a - 1)-connected and C (c - 1)-connected. If F is the homotopy fiber of B -! C then the natural map A -! F is an (a + c - 2)-equivalence. Proof The assumptions clearly imply that F is simply-connected. The result follows from the fact that the map A -! F ' A _ ( A ^ C), 7 is an (a + c - 1)-equivalence. 2 We use the notation X(n) for the n-fold smash product of X with itself. W (n) Theorem(James) If X is a connected space, then X ' n X . Proof We apply Proposition 5 to the cofibration X -! * - ! X . The homotopy fiber of * -! X is X , so we conclude that X ' X _ (X ^ ( X)) ' X _ ( X ^ X) _ (X ^ X ^ ( X))) ' . ... 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