PRO-ISOMORPHISMS OF HOMOLOGY TOWERS

BROOKE E. SHIPLEY^{∗}

1. Introduction

A. K. Bousfield, in [B], shows that convergence of the generalized Eilenberg-Moore spectral sequence is equivalent to having a pro-isomorphism between the homology groups of two towers of spaces. This motivates our search for criteria which ensure that the homology towers of two towers of spaces are pro-isomorphic. Throughout this paper R is 𝔽_{p}, the field with p elements, and homology is with 𝔽_{p} coefficients unless stated otherwise. Our most general result is the following theorem.

Theorem 1.1. Let f : {X_{s}} → {Y_{s}} be a map of towers of spaces. Assume {H_{∗}X_{s}} is pro-constant and {H_{∗}Y_{s}} is pro-finite type. Assume either

a) {H_{∗}X_{s}} is pro-finite type, or

b) {H_{∗}Y_{s}} is pro-constant. Then f induces a pro-homology isomorphism if and only if it induces a homotopy equivalence \underleftarrow{holim} R_{s}X_{s} → \underleftarrow{holim} R_{s}Y_{s}.

This theorem will be proved in sections 3 and 4.

For applications to the convergence of the generalized Eilenberg-Moore spectral sequence, instead of comparing two general towers of spaces we consider one general tower and its homotopy inverse limit considered as a constant tower. In this case we deduce the following corollary of Theorem 1.1 in section 5.

Corollary 1.2. Let X be the homotopy inverse limit of a tower {X_{s}}. Assume {X_{s}} is pro-p-nilpotent and {H_{∗}X_{s}} is pro-finite type. Assume either

a) H_{∗}X is finite type, or

b) \underleftarrow{lim}H_{∗}X_{s} is finite type.

Then the following are equivalent.

1) X is p-good.

2) {H_{∗}X} → {H_{∗}X_{s}} is a pro-isomorphism.

3) H_{∗}X → \underleftarrow{lim}H_{∗}X_{s} is an isomorphism.

∗The author was supported by an ONR/NDSEG Fellowship.

A tower of p-nilpotent spaces is a pro-p-nilpotent tower. The more general definition of pro-p-nilpotent towers will be discussed in section 2.

This corollary can be used to prove the following theorem. By adding finite type conditions this theorem generalizes Theorem 3.6 in [B] to give convergence conditions for the generalized Eilenberg-Moore spectral sequence for a non-simply connected total space. We use the notation which appears in [B]. {E^{r}(X·; R)} is the generalized Eilenberg-Moore spectral sequence for the cosimplicial space X·. T_{s}(R ˜⊗X·) for s ≥ 0 are the filtration quotients of the normalized double complex of R ˜⊗X·.

Theorem 1.3. Let X· be a pointed fibrant cosimplicial space such that each codegree, X^{m}, is p-nilpotent and {H_{∗}Tot_{s}X·} is pro-finite type. Assume {H_{n}T_{s}(R ˜⊗X·)} is pro-trivial for n ≤ 0 and either H_{∗}Tot X· is finite type or \underleftarrow{lim}H_{∗}Tot_{s}X· is finite type. Then {E^{r}(X·; R)} strongly converges to H_{∗}(Tot X·) if and only if Tot X· is p-good.

This theorem is proved by replacing Lemma 9.3 [B] in Bousfield’s proof by Corollary 1.2.

This project was suggested to me by Bill Dwyer. His advice also helped me form my approach to this work. I would like to thank Jim Turner for many helpful conversations and Pete Bousfield for a useful letter. Finally, I am grateful to Haynes Miller for many enlightening conversations and insightful questions, not to mention his thorough reading of earlier versions of this paper.

We assume the reader is familiar with the material in [B-K, I] on p-completions and the construction of the tower of spaces {R_{s}X}. We will also use nilpotent actions as defined in [B-K, II 4]. The action of a group π on a group G is said to be nilpotent if there is a finite filtration of G such that the sequence of filtrations is a normal series, the filtrations are π-invariant, and π acts trivially on the quotients. A p-nilpotent group is a nilpotent group in which the order of each element divides p^{k} for some fixed k < ∞ [B-K, III 5]. A nilpotent space is one in which π_{1} acts nilpotently on π_{n} for all n ≥ 1. A p-nilpotent space is a nilpotent space for which each π_{n} is a p-nilpotent group.

In the next section we will develop the terminology and notation which will be needed to work with towers of groups and spaces. Section 2 also includes several lemmas which will be necessary for the proof of Theorem 1.1.

2. Towers of Groups and Spaces

This paper will rely heavily on the use of towers of groups and spaces. A good reference here is [B-K, III]. In this section the definitions will be for pointed sets whenever possible. These definitions extend to groups by neglecting their extra structure. A tower of pointed sets, {C_{s}}, is a collection of sets indexed by the nonnegative integers together with maps σ_{s} : C_{s} → C_{s−1}. A morphism of towers is a collection of maps {f_{s}} : {B_{s}} → {C_{s}} such that σ_{s} ∘ f_{s} = f_{s−1} ∘ σ_{s}. We could similarly define a tower of groups or spaces.

A map {f_{s}} : {B_{s}} → {C_{s}} is a pro-isomorphism if for every D it induces an isomorphism

\underrightarrow{lim} Hom_{∗}(C_{s}, D) ≅ \underrightarrow{lim} Hom_{∗}(B_{s}, D).

This is equivalent to the condition that for each s there is a t such that a map from C_{s+t} to B_{s} exists which makes the following diagram commute

[figure: B_{s+t}; C_{s+t}; B_{s}; C_{s}.]

These definitions also extend to towers of groups.

A tower is pro-trivial if the tower is pro-isomorphic to the trivial tower. It is easy to see that a map of towers of groups {f_{s}} : {G_{s}} → {H_{s}} is a pro-isomorphism if and only if the pointed towers of sets {kerf_{s}} and {cokerf_{s}} are pro-trivial [B-K, III 2.2]. By analogy, define a pro-monomorphism (pro-epimorphism) to be a map such that {ker_{s}} ({cokerf_{s}}) is pro-trivial. A map of towers of pointed sets {f_{s}} : {B_{s}} → {C_{s}} will be called a pro-epimorphism if for every s there exists a t such that the image of σ : C_{s+t} → C_{s} is contained in the image of f_{s}. For towers of groups, this definition is equivalent to requiring that {cokerf_{s}} is pro-trivial. A pro-constant tower is pro-isomorphic to a constant tower. Similarly, a pro-finite tower is pro-isomorphic to a tower of finite sets.

One property of towers which we will use often is the Mittag-Leffler condition. For a tower of sets, {C_{s}}, let C_{s}^{(t)} be the image of C_{s+t} in C_{s}. We say {C_{s}} is Mittag-Leffler if for each s there exists an N < ∞ such that C_{s}^{(N)} = lim_{∞←t} C_{s}^{(t)}. In other words, the images eventually stabilize. The following lemma lists some of the useful properties of a Mittag-Leffler tower.

Lemma 2.1. Let {C_{s}} be a tower of groups or a tower of sets.

1) If each C_{s} is finite then {C_{s}} is Mittag-Leffler.

2) {C_{s}} is Mittag-Leffler if and only if {C_{s}} is pro-isomorphic to a tower of epimorphisms.

3) \{C_{s}\} is Mittag-Leffler if and only if the map \{\underleftarrow{lim}C_{s}\} → \{C_{s}\} is a proepimorphism.

4) Given a pro-epimorphism of towers of sets or groups, {B_{s}} → {C_{s}}, if {B_{s}} is Mittag-Leffler then so is {C_{s}}.

5) Let \{G_{s}\} be a tower of groups. If \{G_{s}\} is Mittag-Leffler then \underleftarrow{lim}^{1}G_{s} is trivial.

6) Given an exact sequence of towers of abelian groups, {G_{s}} → {H_{s}} → {K_{s}} → {L_{s}}, if {L_{s}} is pro-trivial and {G_{s}} and {K_{s}} are Mittag-Leffler then {H_{s}} is also Mittag-Leffler. This statement also holds for an extension of groups {1} → {G_{s}} → {H_{s}} → {K_{s}} → {1}.

These statements are easy extensions of the above definitions.

As a generalization of Bousfield and Kan’s definition of a nilpotent action [B-K, II 4], we define a pro-nilpotent action of towers of groups. Let {G_{s}} and {π_{s}} be towers of groups. A tower {π_{s}} acts on {G_{s}} if π_{s} acts on G_{s} for each s and the actions are compatible. We call the action pro-nilpotent if there are pro-isomorphisms of towers {π_{s}} → {π′_{s}} and {f_{s}} : {G_{s}} → {G′_{s}} such that f_{s} is π_{s}-equivariant and π′_{s} acts nilpotently on G′_{s} for each s.

As preparation for the next lemma, let [G, π] be the commutator group of the action. In other words [G, π] is the subgroup generated by {g^{x}g^{−1}|g ∈ G, x ∈ π}. Set G[0] = G. Let G[n + 1] = [G[n], π]. Note that G[n + 1] is a normal subgroup in G[n].

Lemma 2.2. {π_{s}} acts pro-nilpotently on {G_{s}} if and only if {G_{s}[s]} is pro-trivial.

Proof. If {G_{s}[s]} is pro-trivial then {G_{s}} is pro-isomorphic to {G_{s}/G_{s}[s]}. But π_{s} acts nilpotently on G_{s}/G_{s}[s]. Thus {G_{s}} is pro-isomorphic to a tower with level-wise nilpotent actions. By definition, this means {π_{s}} acts pro-nilpotently on {G_{s}}.

For the converse, given a pro-isomorphism to a tower of level-wise nilpotent actions, for each s there will exist a t such that the map G_{s+t} → G_{s} factors through a group with a nilpotent action. Hence, for some N, G_{s+t}[N] → G_{s}[N] will be trivial. This shows that {G_{s}[n]}_{s,n} is pro-trivial, which implies that the diagonal {G_{s}[s]} is also pro-trivial.

Similarly, we define a pro-p-nilpotent tower of groups to be a tower which is proisomorphic to a tower of p-nilpotent groups. Let G[n]_{p} denote the nth filtration in the p-lower central series of G [R]. Then it is clear that a tower {G_{s}} is pro-p-nilpotent if and only if {G_{s}[s]_{p}} is pro-trivial. The proof of this statement follows the proof of the above lemma.

Lemma 2.3. Let {π_{s}} act pro-nilpotently on {G_{s}}.

1) Assume {π_{s}/[π_{s}, π_{s}]} is Mittag-Leffler. If {G_{s}/[G_{s}, π_{s}]} is Mittag-Leffler then so is {G_{s}}.

2) If {G_{s}} is pro-p-nilpotent and {(G_{s}/[G_{s}, G_{s}]) ⊗ ℤ/p} is Mittag-Leffler then {G_{s}/[G_{s}, G_{s}]} and {G_{s}} are Mittag-Leffler.

Proof. Consider the map {G_{s}} → {G_{s}/G_{s}[s]}. It is a pro-epimorphism, with kernel {G_{s}[s]}. By the assumption of a pro-nilpotent action, Lemma 2.2 shows this kernel is pro-trivial. Thus this map is a pro-isomorphism. So, to show {G_{s}} is Mittag-Leffler, we only need to show that {G_{s}/G_{s}[s]} is Mittag-Leffler. We will use the following lemma, which is stated without proof.

Lemma 2.4. Let {G_{s,t}} be a bi-tower. Assume G_{s,t+1} → G_{s,t} is epic for all s and t and {G_{s,t}}_{s} is Mittag-Leffler for all t. Then {G_{s,s}} is Mittag-Leffler.

Since G_{s}/G_{s}[t+1] → G_{s}/G_{s}[t] is always surjective, it will be enough to show that {G_{s}/G_{s}[t]} is Mittag-Leffler for each t. Consider the following tower of extensions

1 → {G_{s}[1]/G_{s}[2]} → {G_{s}/G_{s}[2]} → {G_{s}/G_{s}[1]} → 1.

This induces the following exact sequence in homology

… {H_{1}(π_{s}; G_{s}/G_{s}[1])} → {H_{0}(π_{s}; G_{s}[1]/G_{s}[2])} →
{H_{0}(π_{s}; G_{s}/G_{s}[2])} → {H_{0}(π_{s}; G_{s}/G_{s}[1])} → 0.

Because of the way the filtration was chosen these homology groups are as follows

… {(π_{s}/[π_{s}, π_{s}]) ⊗ (G_{s}/G_{s}[1])} → {G_{s}[1]/G_{s}[2]} →
{G_{s}/G_{s}[1]} → {G_{s}/G_{s}[1]} → 0.

Using Lemma 2.1, Lemma 2.4, and the fact that the tensor product preserves surjections, we can prove the tensor product of two Mittag-Leffler towers is Mittag-Leffler. Because the last map above is an isomorphism we conclude that {G_{s}[1]/G_{s}[2]} is Mittag-Leffler by Lemma 2.1 part four. Now, using Lemma 2.1 part six applied to the coefficient sequence above, we see that {G_{s}/G_{s}[2]} is Mittag-Leffler. We proceed by induction to see that {G_{s}/G_{s}[t]} is Mittag-Leffler for each t. This is what was needed to finish the proof of part 1).

Now we prove part 2). Because {G_{s}} is pro-p-nilpotent, there exists a proisomorphic tower {H_{s}} where each H_{s} is p-nilpotent. This pro-isomorphism induces a pro-isomorphism {(G_{s}/[G_{s}, G_{s}])⊗ℤ/p} → {(H_{s}/[H_{s}, H_{s}])⊗ℤ/p}. Thus the latter is also Mittag-Leffler. Let {H_{s}/[H_{s}, H_{s}]} = {A_{s}}. Each A_{s} is p-nilpotent by [B-K, III 5.8]. We have an exact sequence as follows

{A_{s} ⊗ ℤ/p} → {A_{s} ⊗ ℤ/p^{2}} → {A_{s} ⊗ ℤ/p} → 0.

By Lemma 2.1 part six we see that {A_{s} ⊗ ℤ/p^{2}} is Mittag-Leffler. Similarly, we deduce that {A_{s} ⊗ ℤ/p^{n}} is Mittag-Leffler for all n. Consider the bi-tower {A_{s} ⊗ ℤ/p^{n}}_{s,n}. For fixed s the maps are all surjections. For fixed n we have seen that {A_{s} ⊗ ℤ/p^{n}}_{s} is Mittag-Leffler. Thus by Lemma 2.4 {A_{s} ⊗ ℤ/p^{s}} is Mittag-Leffler. In the bi-tower there will be only finitely many groups for each fixed s which are not equal to A_{s} because each A_{s} is p-nilpotent. Hence the bi-tower is pro-isomorphic to {A_{s}}_{s,n}. So the diagonals are pro-isomorphic, and thus {A_{s}} = {H_{s}/[H_{s}, H_{s}]} is Mittag-Leffler. We conclude that {G_{s}/[G_{s}, G_{s}]} is Mittag-Leffler. Thus by part 1) {G_{s}} is Mittag-Leffler.

Now consider towers of spaces. A tower map {X_{s}} → {Y_{s}} is a weak pro-homotopy equivalence if it induces a pro-isomorphism {π_{0}X_{s}} → {π_{0}Y_{s}} and for each i and s there exists a t such that for each vertex v ∈ X_{s+t} there exists a map π_{i}(Y_{s+t}, v) → π_{i}(X_{s}, v) making the following diagram commute

[figure: π_{i}(X_{s+t}, v); π_{i}(Y_{s+t}, v); π_{i}(X_{s}, v); π_{i}(Y_{s}, v).]

For pointed towers of connected spaces this condition is equivalent to having a proisomorphism for each tower of homotopy groups. A weak pro-homotopy equivalence induces a pro-homology isomorphism, i.e. {H_{∗}X_{s}} → {H_{∗}Y_{s}} is a pro-isomorphism [B, 8.5].

We define a space to be p-nilpotent if each of its path components is p-nilpotent as defined in [B-K, III 5]. Here we will always have conditions which require that in some sense there are only finitely many components. In general one may want to include a finiteness requirement in the definition. A tower of spaces, {X_{s}}, will be called pro-p-nilpotent if the map {X_{s}} → {R_{s}X_{s}} is a weak pro-homotopy equivalence. The following lemma gives equivalent conditions for a pointed tower of connected spaces to be pro-p-nilpotent.

Lemma 2.5. Let {X_{s}} be a pointed tower of connected spaces. Then the following are equivalent.

1) {X_{s}} is pro-p-nilpotent.

2) {X_{s}} is weak pro-homotopy equivalent to a tower of p-nilpotent spaces.

3) {π_{i}X_{s}} is pro-p-nilpotent and {π_{1}X_{s}} acts pro-nilpotently on {π_{i}X_{s}} for all i.

Proof. The only difficult step in proving this lemma is 3) implies 1). Because the {π_{i}X_{s}} are pro-p-nilpotent for all i we see that {π_{i}X_{s}[s]_{p}} is pro-trivial. Hence {π_{i}X_{s}} is pro-isomorphic to {π_{i}X_{s}/π_{i}X_{s}[s]_{p}}, which is a tower of p-nilpotent groups. Thus {K(π_{i}X_{s}/π_{i}X_{s}[s]_{p}, i)} is a tower of p-nilpotent spaces. Therefore it is weakly pro-homotopy equivalent to {R_{s}K(π_{i}X_{s}/π_{i}X_{s}[s]_{p}, i)} [B-K, III 5.4]. Consider the following diagram

{K(π_{i}X_{s}, i)}  ⟶  {R_{s}K(π_{i}X_{s}, i)}
↓  ↓
{K(π_{i}X_{s}/π_{i}X_{s}[s]_{p}, i)} ⟶ {R_{s}K(π_{i}X_{s}/π_{i}X_{s}[s]_{p}, i)}.

The left hand map is a weak pro-homotopy equivalence and thus a pro-homology isomorphism [B-K, III 3.4]. By the following lemma, we conclude that the right hand map is a weak pro-homotopy equivalence. Thus the top map is also a weak pro-homotopy equivalence. So {K(π_{i}X_{s}, i)} is pro-p-nilpotent for all i. We have also shown that the property of being pro-p-nilpotent is preserved under weak prohomotopy equivalence.

Lemma 2.6. Given a map of towers of fibrations {X_{s}} → {Y_{s}}, {H_{∗}X_{s}} → {H_{∗}Y_{s}} is a pro-isomorphism if and only if {R_{s}X_{s}} → {R_{s}Y_{s}} is a weak pro-homotopy equivalence.

This lemma is stated in [B-K, III 6.6] for towers of connected spaces. It is easy to generalize using our definition of a weak pro-homotopy equivalence for towers of non-connected spaces.

We know that {π_{1}X_{s}} acts pro-nilpotently on {π_{i}X_{s}} for all i. Thus {π_{i}X_{s}[s]} is pro-trivial. So {π_{i}X_{s}} → {π_{i}X_{s}/π_{i}X_{s}[s]} is a pro-isomorphism. We proved above that {K(π_{1}X_{s}, 1)} was pro-p-nilpotent. Now, by induction, we assume that {P_{n−1}X_{s}} is pro-p-nilpotent and prove that {P_{n}X_{s}} is pro-p-nilpotent. Consider the following map of towers of level-wise fibrations

{K(π_{n}X_{s}, n)} ⟶ {(K(π_{n}X_{s}/π_{n}X_{s}[s], n)}
↓  ↓
\{P_{n}X_{s}\} ⟶  \{\overline{P}_{n}X_{s}\}
↓  ↓
{P_{n−1}X_{s}} ⟶  {P_{n−1}X_{s}}.

Here \overline{P}_{n}X_{s} is the space created by killing off the subgroup π_{n}X_{s}[s] of π_{n}X_{s} and all of the higher homotopy groups. The maps for the fibers and bases are both weak pro-homotopy equivalences. Thus, by the long exact sequence in homotopy and the five-lemma for towers [B-K, III 2.7], we see that the map of total spaces is also a weak pro-homotopy equivalence.

For each level of the tower of fibrations on the right the fundamental group of the base acts nilpotently on the homology of the fiber. Thus the fiber of the map of towers \{R_{s}\overline{P}_{n}X_{s}\} → \{R_{s}P_{n−1}X_{s}\} is weakly pro-homotopy equivalent to {R_{s}K(π_{n}X_{s}/π_{n}X_{s}[s], n)} by [B-K, III 7.1] and a diagonal argument. We proved above that {K(π_{n}X_{s}, n)} is pro-p-nilpotent. Hence {K(π_{n}X_{s}/π_{n}X_{s}[s], n} is pro-p-nilpotent because these two towers are weakly pro-homotopy equivalent. Therefore we have a map of towers of the homotopy long exact sequences of the right fibration to the sequence of their respective R_{s} towers. This map is a pro-isomorphism on the homotopy towers of the fibers as just noted and on the bases due to the assumption that {P_{n−1}X_{s}} is pro-p-nilpotent. Hence by the five-lemma for towers, [B-K, III 2.7], the map is a weak pro-homotopy equivalence on the total spaces. Thus \{\overline{P}_{n}X_{s}\} is pro-p-nilpotent. Because \{P_{n}X_{s}\} is weakly pro-homotopy equivalent to \{\overline{P}_{n}X_{s}\}, {P_{n}X_{s}} is also pro-p-nilpotent Thus {P_{n}X_{s}} is pro-p-nilpotent for all n. It is easy to deduce from this that {X_{s}} is pro-p-nilpotent.

We will also need a Hurewicz Theorem for pro-connected tower of spaces. M. Artin and B. Mazur wrote down a Hurewicz Theorem for towers of connected spaces in [A-M, 4.5].

Lemma 2.7. [Hurewicz Theorem for towers] Consider a tower of spaces {X_{s}} such that \{π_{0}X_{s}\} is pro-trivial. If \{π_{i}X_{s}\} is pro-trivial for all i < n then \{\overline{H}_{i}(X_{s}; ℤ)\} is pro-trivial for all i < n. Also, for n > 1, {π_{n}X_{s}} is pro-isomorphic to {H_{n}(X_{s}; ℤ)}. For n = 1 {π_{1}X_{s}/[π_{1}X_{s}, π_{1}X_{s}]} is pro-isomorphic to {H_{1}(X_{s}; ℤ)}.

Proof. Note that if \{π_{0}X_{s}\} is pro-trivial then \{\overline{H}_{0}(X_{s}; ℤ)\} is also pro-trivial. Given {π_{0}X_{s}} pro-trivial, we can consider the base point components and assume we have a tower of connected spaces. Then for each s we know π_{1}X_{s}/[π_{1}X_{s}, π_{1}X_{s}] ≅ H_{1}(X_{s}; ℤ). Therefore {π_{1}X_{s}/[π_{1}X_{s}, π_{1}X_{s}]} is pro-isomorphic to {H_{1}(X_{s}; ℤ)}.

For n > 1 we can consider the tower of (n−1)-connected covers {X_{s}<n−1>}. {X_{s}<n−1>} will be weakly pro-homotopy equivalent to {X_{s}} because {π_{i}X_{s}} is pro-trivial for i < n. They will thus be pro-homology equivalent [B-K, III 3.4]. So we can assume we have a tower of (n−1)-connected spaces. Then, applying the standard Hurewicz Theorem on each level, we conclude that \{\overline{H}_{i}(X_{s}; ℤ)\} is pro-trivial for all i < n and {π_{n}X_{s}} is pro-isomorphic to {H_{n}(X_{s}; ℤ)}.

3. Proof of Theorem 1.1

In this section we will prove Theorem 1.1.

By the usual process of replacing a map by a homotopy equivalent fibration, we can replace both {X_{s}} and {Y_{s}} by towers of fibrations. The homology towers will remain unchanged. This replacement process also replaces {R_{s}X_{s}} and {R_{s}Y_{s}} by homotopy equivalent towers of fibrations because the functor R_{s} preserves both weak equivalences and fibrations. Thus we can assume that both {X_{s}} and {Y_{s}} are towers of fibrations. The homotopy inverse limits in the statement become inverse limits.

First we prove part a). We assume that both {H_{∗}X_{s}} and {H_{∗}Y_{s}} are pro-finite type. From this we see that the homotopy groups of {R_{s}X_{s}} and {R_{s}Y_{s}} are all pro-isomorphic to towers of finite p-groups. We will call such towers of groups prop-groups. A map of pro-p-groups {G_{s}} → {H_{s}} is a pro-isomorphism if and only if it induces an isomorphism of their inverse limits. A tower of spaces whose homotopy groups are pro-p-groups will be called a pro-p-space. By the Milnor sequence, a map of pro-p-spaces is a weak pro-homotopy equivalence if and only if the map of inverse limits is a homotopy equivalence. Thus {R_{s}X_{s}} and {R_{s}Y_{s}} are weakly pro-homotopy equivalent if and only if \underleftarrow{lim}R_{s}X_{s} is homotopy equivalent to \underleftarrow{lim}R_{s}Y_{s}. Lemma 2.6 completes the proof of part a).

In order to prove part b) we show that it is enough to prove that {H_{∗}R_{s}X_{s}} → {H_{∗}R_{s}Y_{s}} is a pro-isomorphism. Then we see that it is actually enough to prove that we have a pro-homology isomorphism on certain towers of components of {R_{s}X_{s}} and {R_{s}Y_{s}}. To deduce this we assume that we have proved the theorem in the case of towers of connected spaces. The proof of the connected case is completed in the next section.

First note that Lemma 2.6 proves the forward implication of the theorem in case b). So we are left with the converse. Consider the following diagram

{X_{s}} ⟶ {Y_{s}}
↓  ↓
{R_{s}X_{s}} ⟶ {R_{s}Y_{s}}.

The left and right maps are pro-homology isomorphisms [B-K, III 6.5]. So to prove that the top map is a pro-homology isomorphism it is enough to prove that the bottom map is a pro-homology isomorphism. This also implies that {H_{∗}R_{s}X_{s}} is pro-constant and {H_{∗}R_{s}Y_{s}} is pro-constant and pro-finite type. Since {H_{∗}Y_{s}} is pro-finite type, \{R_{s}Y_{s}\} is a pro-p-space. Thus \underleftarrow{lim}^{1}π_{1}R_{s}Y_{s} is trivial. So, by the Milnor sequence [B-K, IX 3.1], we see that π_{0} \underleftarrow{lim}R_{s}Y_{s} ≅ \underleftarrow{lim}π_{0}R_{s}Y_{s}. We know \{π_{0}R_{s}Y_{s}\} is pro-constant and pro-finite so \underleftarrow{lim}π_{0}R_{s}Y_{s} is finite. Since \underleftarrow{lim}R_{s}X_{s} ≃ \underleftarrow{lim}R_{s}Y_{s}, the action of \underleftarrow{lim}^{1}π_{1}R_{s}X_{s} on π_{0}(\underleftarrow{lim}R_{s}X_{s}) factors through \underleftarrow{lim}^{1}π_{1}R_{s}Y_{s}, which is trivial. Hence π_{0} \underleftarrow{lim}R_{s}X_{s} ≅ \underleftarrow{lim}π_{0}R_{s}X_{s} and is finite.

Given ϕ ∈ π_{0} \underleftarrow{lim}R_{s}X_{s}, let (\underleftarrow{lim}R_{s}X_{s})_{ϕ} be the component determined by ϕ. Using the map \underleftarrow{lim}R_{s}X_{s} → R_{s}X_{s}, let (R_{s}X_{s})_{ϕ} be the component of R_{s}X_{s} which is hit by (\underleftarrow{lim}R_{s}X_{s})_{ϕ}. \{π_{0}R_{s}X_{s}\} is pro-constant because \{H_{∗}R_{s}X_{s}\} is pro-constant. Thus {π_{0}_{∐ϕ}(R_{s}X_{s})_{ϕ}} → {π_{0}R_{s}X_{s}} is a pro-isomorphism. Then, from the definition, it follows that {_{∐ϕ}(R_{s}X_{s})_{ϕ}} → {R_{s}X_{s}} is a weak pro-homotopy equivalence. Because {H_{∗}R_{s}Y_{s}} is also pro-constant, {_{∐ϕ}(R_{s}Y_{s})_{ϕ}} → {R_{s}Y_{s}} is also a weak pro-homotopy equivalence. Consider the following diagram

{_{⨁ϕ}H_{∗}(R_{s}X_{s})_{ϕ}} ⟶ {_{⨁ϕ}H_{∗}(R_{s}Y_{s})_{ϕ}}
↓  ↓
{H_{∗}R_{s}X_{s}} ⟶ {H_{∗}R_{s}Y_{s}}.

Both the left and right maps are pro-isomorphisms. Because {H_{∗}R_{s}X_{s}} is proconstant, {H_{∗}(R_{s}X_{s})_{ϕ}} is pro-constant for all ϕ. Similarly, {H_{∗}R_{s}Y_{s}} is proconstant and pro-finite type so {H_{∗}(R_{s}Y_{s})_{ϕ}} is pro-constant and pro-finite type for all ϕ. Because \underleftarrow{lim}R_{s}X_{s} ≃ \underleftarrow{lim}R_{s}Y_{s}, \underleftarrow{lim}(R_{s}X_{s})_{ϕ} ≃ \underleftarrow{lim}(R_{s}Y_{s})_{ϕ} for all ϕ. Thus, for each ϕ, we have exactly the requirements for applying the theorem. We also note that in this case we are dealing with towers of connected spaces. We state this case in the following proposition.

Proposition 3.1. Let f : {X_{s}} → {Y_{s}} be a map of towers of fibrations of connected spaces. Assume {H_{∗}X_{s}} is pro-constant and {H_{∗}Y_{s}} is pro-constant and pro-finite type. Then f induces a pro-homology isomorphism if and only if it induces a homotopy equivalence \underleftarrow{lim}R_{s}X_{s} → \underleftarrow{lim}R_{s}Y_{s}.

This proposition is proved in the next section. Assuming the proposition, we conclude that {H_{∗}(R_{s}X_{s})_{ϕ}} → {H_{∗}(R_{s}Y_{s})_{ϕ}} is a pro-isomorphism for all ϕ. Thus the top map in the diagram above is a pro-isomorphism. Hence {H_{∗}R_{s}X_{s}} → {H_{∗}R_{s}Y_{s}} is a pro-isomorphism. But this is enough to show that {H_{∗}X_{s}} → {H_{∗}Y_{s}} is a pro-isomorphism, which is what we wanted.

4. Proof of Proposition 3.1

Lemma 2.6 proves that if \{H_{∗}X_{s}\} → \{H_{∗}Y_{s}\} is a pro-isomorphism then \underleftarrow{lim}R_{s}X_{s} is homotopy equivalent to \underleftarrow{lim}R_{s}Y_{s}. The proof of the converse is more involved.

Consider the diagram of towers

{X_{s}} ⟶ {Y_{s}}
↓  ↓
{R_{s}X_{s}} ⟶ {R_{s}Y_{s}}.

The left and right maps are pro-homology isomorphisms [B-K, III 6.5]. So to prove the top map is a pro-homology isomorphism it is enough to see that the bottom map is a pro-homology isomorphism.

Let F_{s} be the homotopy fiber of the map R_{s}X_{s} → R_{s}Y_{s}. This creates a tower of level-wise fibrations

{F_{s} → R_{s}X_{s} → R_{s}Y_{s}}.

To prove {H_{∗}R_{s}X_{s}} is pro-isomorphic to {H_{∗}R_{s}Y_{s}} we show that {H_{∗}F_{s}} is protrivial. Then we apply a Serre spectral sequence argument. We first must prove that both \underleftarrow{lim}π_{i}F_{s} and \underleftarrow{lim}^{1}π_{i}F_{s} vanish for all i. i To do this we need the following lemma.

Lemma 4.1. Consider a commutative diagram of abelian groups

A_{1} ⟶ A_{2} ⟶ A_{3} ⟶ A_{4} ⟶ A_{5}
f_{1}↓  ^{f_{2}}↓  ^{f_{3}}↓  ^{f_{4}}↓  ^{f_{5}}↓
B_{1} ⟶ B_{2} ⟶ B_{3} ⟶ B_{4} ⟶ B_{5}

where the horizontal sequences are exact. If the induced map kerf_{1} → kerf_{2} is epic and kerf_{4} → kerf_{5} and cokerf_{1} → cokerf_{2} are monic then f_{3} is monic. If the induced maps kerf_{4} → kerf_{5} and cokerf_{1} → cokerf_{2} are epic and cokerf_{4} → cokerf_{5} is monic then f_{3} is epic.

The proof is a diagram chase.

We apply this lemma to sections of the long exact sequence in homotopy for the tower of level-wise fibrations {F_{s}} → {R_{s}X_{s}} → {R_{s}Y_{s}}. Choose one of the five towers centered on {π_{i}F_{s}} for i ≥ 2, then substitute the direct product of all of the groups in the tower for the tower itself. Let this sequence be both of the horizontal sequences for the application of the lemma. Let f_{i} be 1−σ where σ is the shift map induced from the maps in each respective tower.

Because {H_{∗}Y_{s}} is pro-finite type, {π_{i}R_{s}Y_{s}} are pro-p-groups for all i. Thus \underleftarrow{lim}^{1}π_{i}R_{s}Y_{s} vanishes for all i. Since we have assumed \underleftarrow{lim}R_{s}X_{s} ≃ \underleftarrow{lim}R_{s}Y_{s}, the Milnor sequence shows that \underleftarrow{lim}π_{i}R_{s}X_{s} ≅ \underleftarrow{lim}π_{i}R_{s}Y_{s} and that \underleftarrow{lim}^{1}π_{i}R_{s}X_{s} vanishes for all i [B-K, IX 3.1]. Thus the induced maps of the kernels and cokernels which are mentioned in the lemma are all isomorphisms. We conclude that 1 − σ : ∏_{s} π_{i}F_{s} → ∏_{s} π_{i}F_{s} is an isomorphism. So, for i ≥ 2, \underleftarrow{lim}π_{i}F_{s} and \underleftarrow{lim}^{1}π_{i}F_{s} are trivial. Using the “exactness” of the homotopy sequence for i = 1 and i = 0, we can adapt the above argument to see that \underleftarrow{lim}π_{1}F_{s}, \underleftarrow{lim}π_{0}F_{s}, and \underleftarrow{lim}^{1}π_{1}F_{s} are trivial.

Now we show that {H_{∗}F_{s}} is pro-trivial. We use an inductive Serre spectral sequence argument which at the nth stage proves that \{\overline{H}_{i}F_{s}\} is pro-trivial for i ≤ n, and thus {H_{n}R_{s}X_{s}} → {H_{n}R_{s}Y_{s}} is a pro-isomorphism.

Consider the long exact sequence in homotopy for the tower of fibrations, {F_{s} → R_{s}X_{s} → R_{s}Y_{s}}. Because each X_{s} and Y_{s} is connected, each R_{s}X_{s} and R_{s}Y_{s} is also connected [B-K, I 6.2]. So {π_{0}R_{s}X_{s}} and {π_{0}R_{s}Y_{s}} are pro-trivial. Thus {π_{1}R_{s}Y_{s}} → {π_{0}F_{s}} is a pro-epimorphism of towers. Since {π_{1}R_{s}Y_{s}} is a pro-p-group, it is Mittag-Leffler by Lemma 2.1 part 1. We see that {π_{0}F_{s}} is Mittag-Leffler by applying the fourth part of Lemma 2.1. But \underleftarrow{lim}π_{0}F_{s} vanishes. So, by the third part of Lemma 2.1, {π_{0}F_{s}} must be pro-trivial. From this it follows that \{\overline{H}_{0}F_{s}\} is pro-trivial.

We proceed by induction. For each n, using a Serre spectral sequence argument and Lemma 2.3, we prove {H_{n}F_{s}} is Mittag-Leffler. Lemma 2.3 is used again to see that this implies \{π_{n}F_{s}\} is Mittag-Leffler. But, since \underleftarrow{lim}π_{n}F_{s} vanishes, this shows that {π_{n}F_{s}} is pro-trivial. By the Hurewicz Theorem for towers, Lemma 2.7, we then conclude that {H_{n}F_{s}} is pro-trivial. We first consider n = 1.

Before turning to the analysis of the Serre spectral sequence we need to see that {H_{i}(R_{s}Y_{s}; H_{0}F_{s})} is pro-constant for all i. Consider the following short exact sequence

0 → \{\overline{H}_{0}F_{s}\} → \{H_{0}F_{s}\} → \{𝔽_{p}\} → 0.

Since \{\overline{H}_{0}F_{s}\} is pro-trivial, the long exact sequence induced by this coefficient sequence shows that {H_{i}(R_{s}Y_{s}; H_{0}F_{s})} is pro-isomorphic to {H_{i}(R_{s}Y_{s}; 𝔽_{p})}.

We have assumed that {H_{∗}Y_{s}} is pro-constant. As noted before, {Y_{s}} → {R_{s}Y_{s}} is a pro-homology equivalence [B-K, III 6.5]. Thus {H_{∗}R_{s}Y_{s}} is pro-constant. By the pro-isomorphism established above we conclude that {H_{i}(R_{s}Y_{s}; H_{0}F_{s})} is pro-constant. Similarly, we assumed {H_{∗}X_{s}} is pro-constant so {H_{∗}R_{s}X_{s}} is proconstant.

Consider the tower of Serre long exact sequences for the tower of level-wise fibrations {F_{s} → R_{s}X_{s} → R_{s}Y_{s}}

{H_{2}R_{s}X_{s}} → {H_{2}(R_{s}Y_{s}; H_{0}F_{s})} → {H_{0}(R_{s}Y_{s}; H_{1}F_{s})}

→ {H_{1}R_{s}X_{s}} → {H_{1}(R_{s}Y_{s}; H_{0}F_{s})} → 0.

Every term except {H_{0}(R_{s}Y_{s}; H_{1}F_{s})} is pro-constant. Thus {H_{0}(R_{s}Y_{s}; H_{1}F_{s})} is pro-constant and hence Mittag-Leffler. Let I_{s} be the kernel of the augmentation of the group ring, R[π_{1}R_{s}Y_{s}]. Then {H_{0}(R_{s}Y_{s}; H_{1}F_{s})} = {H_{1}F_{s}/I_{s}H_{1}F_{s}}. So {H_{1}F_{s}/I_{s}H_{1}F_{s}} is pro-constant. It is also the first filtration quotient in the lower central series for the action. Since a finite p-group acts nilpotently on any 𝔽_{p}-vector space, a pro-p-group acts pro-nilpotently on any tower of 𝔽_{p}-vector spaces. Thus {π_{1}R_{s}Y_{s}} acts pro-nilpotently on {H_{1}F_{s}}. Since {π_{1}R_{s}Y_{s}} is Mittag-Leffler, we can apply Lemma 2.3 and conclude that {H_{1}F_{s}} is Mittag-Leffler.

Note that F_{s} is the fiber of a map between p-nilpotent spaces, and thus is itself p-nilpotent. So {π_{1}F_{s}} will be pro-p-nilpotent. The first filtration quotient is {(π_{1}F_{s}/[π_{1}F_{s}, π_{1}F_{s}]) ⊗ ℤ/p}. But this is exactly {H_{1}F_{s}}, which we just proved is Mittag-Leffler. Thus, by Lemma 2.3, we conclude that {π_{1}F_{s}} is Mittag-Leffler. Because \underleftarrow{lim}π_{1}F_{s} vanishes, the fact that \{π_{1}F_{s}\} is Mittag-Leffler implies it is protrivial. Using the Hurewicz Theorem for towers, Lemma 2.7, we finally conclude that {H_{1}F_{s}} is pro-trivial.

To continue the induction, at the nth stage we have proved that \{\overline{H}_{i}F_{s}\} is protrivial for all i < n. So in the Serre spectral sequence rows 1 through n − 1 are protrivial. Thus the Serre long exact sequence extends to include {H_{0}(R_{s}Y_{s}; H_{n}F_{s})}. Then, following the same steps as the proof above for n = 1, we deduce that {H_{n}F_{s}} is pro-trivial.

The conclusion of this induction is that the tower of Serre spectral sequences is pro-trivial except for the very first row. We have seen that this first row is proisomorphic to {H_{∗}R_{s}Y_{s}}. Because the spectral sequence is converging to {H_{∗}R_{s}X_{s}}, we conclude that {H_{∗}R_{s}X_{s}} is pro-isomorphic to {H_{∗}R_{s}Y_{s}}. This is what was necessary to finish the proof of the proposition.

5. Applications

The first application of Theorem 1.1 is for a pro-p-complete tower. A pro-p-complete tower is a tower of spaces which is weakly pro-homotopy equivalent to a tower of p-complete spaces.

Corollary 5.1. Let X be the homotopy inverse limit of a tower {X_{s}}. Assume {X_{s}} is pro-p-complete and {H_{∗}X_{s}} is pro-finite type. Assume either

a) H_{∗}X is finite type, or

b) \underleftarrow{lim}H_{∗}X_{s} is finite type.

Then the following are equivalent.

1) X is p-complete.

2) {H_{∗}X} → {H_{∗}X_{s}} is a pro-isomorphism.

3) H_{∗}X → \underleftarrow{lim}H_{∗}X_{s} is an isomorphism.

Proof. As in the proof of Theorem 1.1, we can assume that {X_{s}} is actually a tower of fibrations.

The following lemma shows that case b) here is equivalent to case b) in Theorem 1.1.

Lemma 5.2. Let {C_{s}} be a pro-finite tower of sets. Then {C_{s}} is pro-constant if and only if \underleftarrow{lim}C_{s} is finite.

Proof. To prove this lemma we can easily reduce to the case when {C_{s}} is actually a tower of finite sets. Assume {C_{s}} is pro-constant. This means it is pro-isomorphic to the constant tower \{\underleftarrow{lim}C_{s}\}. Thus for some s \underleftarrow{lim}C_{s} → C_{s} is one-to-one. So \underleftarrow{lim}C_{s} is finite.

For the converse, consider two distinct elements of \underleftarrow{lim}C_{s}. Because they are distinct in the inverse limit, their images under the map \underleftarrow{lim}C_{s} → C_{s} must be distinct for some s. Since \underleftarrow{lim}C_{s} is finite, we can find an N such that for every two distinct elements of the inverse limit their images in C_{N} are distinct. This means that the map \underleftarrow{lim}C_{s} → \{C_{s}\} is a pro-monomorphism. Now, note that \{C_{s}\} is Mittag-Leffler by Lemma 2.1 part one. Hence by Lemma 2.1 part three \{\underleftarrow{lim}C_{s}\} → \{C_{s}\} is a pro-epimorphism. Thus {C_{s}} is pro-constant.

Consider the pro-p-complete tower {X_{s}}. By definition there is a tower of pcomplete spaces {Y_{s}} and a map {X_{s}} → {Y_{s}} which is a weak pro-homotopy equivalence. This implies that {R_{s}X_{s}} → {R_{s}Y_{s}} is also a weak pro-homotopy equivalence. The inverse limit of \{R_{s}Y_{s}\} is homotopic to \underleftarrow{lim}R_{∞}Y_{s}, which in turn is homotopic to \underleftarrow{lim}Y_{s} because each Y_{s} is p-complete. Thus \underleftarrow{lim}R_{s}X_{s} is also homotopic to \underleftarrow{lim}X_{s} ≃ X. So we can conclude that X is p-complete if and only if R_{∞}X ≃ \underleftarrow{lim}R_{s}X_{s}. Thus the equivalence of 1) and 2) is just a restatement of Theorem 1.1.

Next we show that 2) and 3) are equivalent. In case a) we know that both {H_{∗}X} and {H_{∗}X_{s}} are pro-finite type. Thus they are pro-isomorphic if and only if their inverse limits are isomorphic. In case b), by Lemma 5.2, {H_{∗}X_{s}} is pro-constant. In other words, \{\underleftarrow{lim}H_{∗}X_{s}\} → \{H_{∗}X_{s}\} is a pro-isomorphism. So \{H_{∗}X\} is proisomorphic to \{H_{∗}X_{s}\} if and only if it is pro-isomorphic to \{\underleftarrow{lim}H_{∗}X_{s}\}. Thus we see that 2) is equivalent to 3) in both cases a) and b).

The second application of Theorem 1.1 is Corollary 1.2 which concerns pro-p-nilpotent towers. This corollary can be deduced from Corollary 5.1 by using the following lemma and noting that a pro-p-nilpotent tower is pro-p-complete since {R_{s}X_{s}} is always a tower of p-complete spaces.

Lemma 5.3. Assume \{X_{s}\} is a pro-p-nilpotent tower of fibrations. Let X = \underleftarrow{lim}X_{s}. Then X is p-good if and only if X is p-complete.

Proof. Consider the following diagram

X ⟶ \underleftarrow{lim}X_{s}
↓  ↓
R_{∞}X ⟶ \underleftarrow{lim}R_{s}X_{s}.

The top map is a homotopy equivalence by definition. The right map is a homotopy equivalences because {X_{s}} is pro-p-nilpotent. Thus the bottom map is surjective in homotopy. Under the assumption that X is p-good, the left map is a homology isomorphism. The bottom map is then also a homology isomorphism. Consider R_{∞} of this map, it will be a weak homotopy equivalence [B-K, I 5.5]

R_{∞}X ⟶ \underleftarrow{lim}R_{s}X_{s}
↓  ↓
R_{∞}R_{∞}X ⟶ R_{∞} \underleftarrow{lim}R_{s}X_{s}.

Because X is p-good the left map is also a weak homotopy equivalence [B-K, I 5.5]. Thus the top map must be injective in homotopy. So R_{∞}X → \underleftarrow{lim}R_{s}X_{s} is in fact a weak homotopy equivalence. We conclude that X → R_{∞}X is a weak homotopy equivalence; in other words X is p-complete. So if X is p-good then it is p-complete. The converse is always true.

Corollary 1.2 applies to towers of spaces which are not necessarily connected. The following lemma provides conditions for checking whether such a tower is prop-nilpotent. Lemma 2.5 is useful for providing further equivalent conditions. We use the same notation for components as developed in section 3.

Lemma 5.4. Assume \{H_{∗}X_{s}\} is pro-finite type. Also assume that π_{0} \underleftarrow{lim}X_{s} → \underleftarrow{lim}π_{0}X_{s} is an isomorphism of finite sets. Then \{X_{s}\} is pro-p-nilpotent if and only if \{X_{s,ϕ}\} → \{R_{s}(X_{s,ϕ})\} is a weak pro-homotopy equivalence for all ϕ ∈ π_{0} \underleftarrow{lim}X_{s}.

Proof. We can assume that {X_{s}} is a tower of fibrations. Since {H_{∗}X_{s}} is profinite type, {π_{0}X_{s}} is pro-finite. Lemma 5.2 shows that {π_{0}X_{s}} is pro-constant because we have assumed \underleftarrow{lim}π_{0}X_{s} is finite. As in section 3, for ϕ ∈ π_{0} \underleftarrow{lim}X_{s} let X_{s,ϕ} be the image of the component (\underleftarrow{lim}X_{s})_{ϕ} under the map \underleftarrow{lim}X_{s} → X_{s}. Thus we see that {π_{0}(_{∐ϕ}X_{s,ϕ})} → {π_{0}X_{s}} is a pro-isomorphism. It then follows that {_{∐ϕ}X_{s,ϕ}} → {X_{s}} is in fact a weak pro-homotopy equivalence.

Consider the following diagram

{_{∐ϕ}X_{s,ϕ}} ⟶ {X_{s}}
↓  ↓
{R_{s}(_{∐ϕ}X_{s,ϕ})} ⟶ {R_{s}X_{s}}.

Because the top map is a weak pro-homotopy equivalence, the bottom map is also by Lemma 2.6. Thus the left map is a weak pro-homotopy equivalence if and only if {X_{s}} is pro-p-nilpotent.

The tower {H_{∗}X_{s}} is pro-finite type. This implies that {R_{s}X_{s}} is a pro-p-space. Thus {R_{s}(_{∐ϕ}X_{s,ϕ})} is also a pro-p-space. By the above remarks, {_{⨁ϕ}H_{∗}(X_{s,ϕ})} → {H_{∗}X_{s}} is a pro-isomorphism. So for each ϕ, {H_{∗}(X_{s,ϕ})} is pro-finite type. Hence \{R_{s}(X_{s,ϕ})\} is a pro-p-space. Since π_{0} \underleftarrow{lim}X_{s} is finite, \{_{∐ϕ}R_{s}(X_{s,ϕ})\} is a pro-p-space. Bousfield and Kan show that _{∐ϕ}R_{∞}X_{ϕ} → R_{∞}(_{∐ϕ}X_{ϕ})} is a weak homotopy equivalence [B-K, I 7.1]. Thus \underleftarrow{lim}(_{∐ϕ}R_{s}X_{s,ϕ}) ≃ \underleftarrow{lim}R_{s}(_{∐ϕ}X_{s,ϕ}). Hence {_{∐ϕ}R_{s}(X_{s,ϕ})} → {R_{s}(_{∐ϕ}X_{s,ϕ})} is a weak pro-homotopy equivalence since both tower are pro-p-spaces. Therefore {_{∐ϕ}X_{s,ϕ}} → {_{∐ϕ}R_{s}(X_{s,ϕ})} is a weak prohomotopy equivalence if and only if {_{∐ϕ}X_{s,ϕ}} → {R_{s}_{∐ϕ}X_{s,ϕ}} is a weak prohomotopy equivalence. In turn, as we saw above, this is equivalent to having {X_{s}} → {R_{s}X_{s}} be a weak pro-homotopy equivalence. This is equivalent to the statement in the lemma.

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Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA