Contemporary Mathematics

E_{∗}-homology spheres for a connective spectrum E

Yousuke Kamiya and Katsumi Shimomura

Abstract. For a given ring spectrum E, L_{E} denote the category of E-local category. We call a spectrum X ∈ L_{E} an E_{∗}-homology sphere if E_{∗}(X) = E_{∗} as an E_{∗}-module. A spherical spectrum X is an E_{∗}-homology sphere with an additional condition so that the E_{1}-terms of the E-based Adams spectral sequences abutting to π_{∗}(X) and π_{∗}(S^{0}) agree. We show here that if the cofiber of the unit map of E is 0-connected, then a spherical spectrum is nothing but the E-local sphere spectrum. We also show that a collection of homotopy classes of E-spherical spectra is a set and it is described by using the E_{r}-terms of the E-based Adams spectral sequence abutting to π_{∗}(S^{0}). As an application, we have a proof of Pic(L_{HZ/p}) = Z.

Introduction

Let S_{p} denote the stable homotopy category of p-local spectra for a fixed prime number p. For a spectrum E ∈ S_{p}, we call a spectrum X E-local if homotopy groups [C, X]_{∗} = 0 for any C with E ∧ C = pt. We denote the full subcategory of E-local spectra by L_{E}, and we work in this category. Then there is the Bousfield localization functor L_{E}: S_{p} → L_{E}. Now we assume that E is a ring spectrum. We call an E-local spectrum X an E_{∗}-homology sphere if E_{∗}(X) is isomorphic to E_{∗} = E_{∗}(S^{0}) as an E_{∗}-module. The sphere spectrum L_{E}S^{0} plays an important role in the L_{E} and so a E_{∗}-homology sphere X is also important. We study E_{∗}- homology sphere by using the Adams tower. To do so, we need additional structure on it. We call a spectrum X ∈ L_{E} is E-spherical if there is a homotopy equivalence u_{X}: E → X ∧ E such that (u_{X} ∧ 1)(i ∧ 1) = (1 ∧ i ∧ 1)u_{X}, where i: S^{0} → E denotes the unit map of the ring spectrum E. Note that an E-spherical spectrum is an E_{∗}-homology sphere. Let E_{r}^{∗,∗}(X) denote the E_{r}-term of the E-based Adams spectral sequence abutting to π_{∗}(X). We call a ring spectrum E semi-connective (resp. good) if the E_{r}-term E_{r}^{r,r−1}(S^{0}) is zero (resp. a torsion group) for each r > 1.

Theorem A. Let E be a semi-connective ring spectrum. Then an E-spherical spectrum is the sphere spectrum L_{E}S^{0}.

\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

2000 Mathematics Subject Classification. Primary 55P43; Secondary 55P60, 55P65.

Key words and phrases. Bousfield localization, E-based Adams spectral sequence, connective spectrum.

We call a ring spectrum E is strongly connective if the cofiber of the unit map i: L_{E}S^{0} → E is 0-connected. The Eilenberg-MacLane spectra HZ and HZ/p, and the Brown-Peterson spectrum BP are examples of strongly connective spectra. Since a strongly connective spectrum is semi-connective, we have

Corollary. Let E be a strongly connective ring spectrum. Then an E-spherical spectrum is the sphere spectrum L_{E}S^{0}.

Note that L_{HZ/p}S^{0} is the p-completed sphere spectrum S_{p}^{0}, and so an HZ/pspherical spectrum is S_{p}^{0}. We call a spectrum X is invertible if there is a spectrum Y such that L_{HZ/p}(X ∧Y) = L_{HZ/p}S^{0}. We then see that an invertible spectrum X is an HZ/p-spherical spectrum up to suspension. Therefore, we have the following corollary that is shown in [5] in a different way, where Pic(L_{HZ/p}) denotes the collection of homotopy equivalence classes of invertible spectra.

Corollary. Pic(L_{HZ/p}) = Z.

Let S_{E} denote the collection of homotopy equivalence classes of E-spherical spectra. In general, we have no idea whether or not S_{E} is a set. The following theorem shows that S_{E} is a set if E is good. Using the differential of the E-based Adams spectral sequence, we define a filtration {F_{s}} such that F_{1} = S_{E}, F_{r} ⊃ F_{r+1} and ⋂_{s} F_{s} = 0 (see (3.1)). We show that F_{r}/F_{r+1} is a group and write GS_{E} as the direct sum of F_{r}/F_{r+1} for r > 0.

Theorem B. Let E be a good ring spectrum. Then the associated graded group GS_{E} is contained in the direct sum of the E_{s}-terms E_{s}^{s,s−1}(S^{0}) for s > 1 of the E-based Adams spectral sequence abutting to π_{∗}(S^{0}).

Note that there is an E-spherical spectrum other than L_{E}S^{0} if E is the Johnson-Wilson spectrum E(1) at the prime 2 [3], or E(2) at the prime 3 [4]. The Johnson-Wilson spectrum E(n) for each n ≥ 0 is seen to be good. We have no idea whether or not there is a ring spectrum that is not good.

In the next section we recall the definition of the E-based Adams spectral sequence. Then we study the E-spherical spectrum and a condition under which E_{∗}-homology sphere is E-spherical in §2. In §3, we prove Theorems A and B. In §4, we study the relation between invertible spectra and E-spherical spectra.

1. E-based Adams spectral sequences

In this paper, E denotes a ring spectrum and we work in the category L_{E} of the E-local spectra, where a spectrum X is E-local if the homotopy groups [C, X]_{∗} are zero for any C such that E ∧ C = pt.

Let \overline{E} be a spectrum defined by the cofiber sequence S^{0} ⟶^{i} E ⟶^{j} \overline{E} ⟶^{k} ΣS^{0}, where i: S^{0} → E is the unit map. Then we obtain a cofiber sequence \overline{E}^{s} ⟶^{i∧1_{s}} I_{s} ⟶^{j∧1_{s}} \overline{E}^{s+1} ⟶^{k∧1_{s}} Σ\overline{E}^{s} for each integer s ≥ 0. Here \overline{E}^{s} denotes the s-fold smash product of \overline{E},

I_{s} = E ∧ \overline{E}^{s},

and 1_{s}: \overline{E}^{s} → \overline{E}^{s} denotes the identity map.

E_{∗}-HOMOLOGY SPHERES FOR A CONNECTIVE SPECTRUM E 3

Define the spectrum \overline{E}_{s} by the cofiber sequence

Σ^{−s}\overline{E}^{s} ⟶^{k^{s}} S^{0} ⟶ \overline{E}_{s},

where k^{s} denotes the s-fold smash product of k: Σ^{−1}\overline{E} → S^{0}. Since we have the commutative diagram

Σ^{−s−1}\overline{E}^{s+1}  S_{0}  _{\overline{E}_{s+1}} Σ^{−s}\overline{E}^{s}  S^{0}  _{\overline{E}_{s}} Σ^{−s}E ∧ \overline{E}^{s} = Σ^{−s}I_{s}  Σ^{−s+1}I_{s}^{k^{s+1}} w_{_{k∧1_{s}} u}  w _{u_{k_{s}}} w_{k^{s}}_{_{i∧1_{s}} u}  w _{u_{i_{s}}}

we obtain a cofiber sequence \overline{E}_{s+1} ⟶^{k_{s}} \overline{E}_{s} ⟶^{i_{s}} Σ^{−s+1}I_{s} ⟶^{j_{s}} Σ\overline{E}_{s+1} and the Ebased Adams tower

pt  E = \overline{E}_{1}  ⋯ Σ^{s−1}\overline{E}_{s}  Σ^{s}\overline{E}_{s+1}  ⋯ I_{0}  I_{1}  ⋯  I_{s}  · · · ._{’ ’’)}  _{’ ’)}_{d_{0}=i_{1}} u  u^{k_{1}}  _{’ ’)}_{i_{s}} u^{k_{s−1}}  u ^{k_{s}}  u_{^{[ [} ^{[ [}} _{^{d_{0}}  w} _{^{d_{1}} w} _{^{d_{s−1}} w} _{^{[ []}^{j_{s}}}_{^{d_{s}} w}

Let X be a spectrum. Then smashing X with this tower defines the E-based Adams spectral sequence abutting to π_{∗}(X) with E_{1}-term E_{1}^{s,∗} = π_{∗}(I_{s}). We write

E_{r}^{s,t}(X)

as the E_{r}-term of the spectral sequence. We call a sequence {X ∧ I_{s}, 1 ∧ d_{s}} an E-resolution of X. E-resolutions {X ∧ I_{s}, 1 ∧ d_{s}} and {Y ∧ I_{s}, 1 ∧ d_{s}} of X and Y are isomorphic if there are homotopy equivalences u_{s}: X ∧ I_{s} → Y ∧ I_{s} such that each diagram

X ∧ I_{s}  Y ∧ I_{s} X ∧ I_{s+1}  Y ∧ I_{s+1}^{u_{s}} w_{_{1∧d_{s}} u}  _{_{1∧d_{s}} u}_{^{u_{s+1}} w}

commutes. The spectrum E X̂ = lim_{s}(X ∧ \overline{E}_{s}) is called an E-nilpotent completion of X and the E-based Adams spectral sequence converges to π_{∗}(E X̂) if there is a finite r such that E_{r}^{s,t}(X) = E_{∞}^{s,t}(X) for each s, t [2]. An E-nilpotent completion E X̂ of X is E-local, and E X̂ is homotopy equivalent to L_{E}X in some cases [2].

2. E-spherical spectra

We call an E-local spectrum X an E_{∗}-homology sphere if E_{∗}(X) = E_{∗}(S^{0})(= E_{∗}) as an E_{∗}-module.

Lemma 2.1. If X is an E_{∗}-homology sphere, then we have an homotopy equivalence u_{X}: E → X ∧ E.

Proof. Let u be a generator of E_{∗}(X) as an E_{∗}-module. Then the composite u_{X} = (1 ∧ μ)(u ∧ 1): S^{0} ∧ E → X ∧ E ∧ E → X ∧ E induces an isomorphism E_{∗} → E_{∗}(X) sending 1 = {i} ∈ E_{∗} to u ∈ E_{∗}(X). □

This means that an E_{∗}-homology sphere X is quasi E_{∗}-equivalent to the sphere spectrum S^{0}. For E = KO, Yosimura has some results on quasi E_{∗}-equivalence.

We call an E-local spectrum X E-spherical if there is an homotopy equivalence u_{X}: E → X ∧ E such that the diagram

E  E ∧ E X ∧ E  X ∧ E ∧ E^{i∧1} w_{_{u_{X}} u}  _{_{u_{X}∧1} u}_{^{1∧i∧1} w}

commutes.

Proposition 2.2. Let E_{∗}(E) be flat as E_{∗}-module. Then if X is an E-spherical spectrum, then E_{∗}(X) is isomorphic to E_{∗} as an E_{∗}(E)-comodule.

Lemma 2.3. The E-resolution {X ∧ I_{s}} of an E-spherical spectrum X is isomorphic to the E-resolution {I_{s}} of the sphere spectrum S^{0}.

Proof. By definition, d_{s} = d_{0} ∧ 1_{s}: I_{s} = E ∧ \overline{E}^{s} → I_{s+1} = I_{1} ∧ \overline{E}^{s}. Since d_{0} = (1 ∧ j)(i ∧ 1), we have the commutativity (u_{X} ∧ 1)d_{0} = (1 ∧ d_{0})u_{X}: E → X ∧ I_{1}. Therefore, (u_{X} ∧ 1_{s+1})d_{s} = (1 ∧ d_{s})(u_{X} ∧ 1_{s}), which gives an isomorphism {u_{X} ∧ 1_{s}}: {I_{s}} → {X ∧ I_{s}} of E-resolutions. □

Corollary 2.4. If X is E-spherical, then E_{2}^{∗}(X) = E_{2}^{∗}(S^{0}).

Lemma 2.5. Let X be an E_{∗}-homology sphere for a ring spectrum E, and u ∈ E_{∗}(X) a generator as an E_{∗}-module. If the composite E ⟶^{u∧1} X ∧ E ∧ E ⟶^{1∧j∧1} X ∧ \overline{E} ∧ E is zero, then X is E-spherical.

Proof. Define the homotopy equivalence u_{X}: E → X ∧ E as the composite (1 ∧ μ)(u ∧ 1), where μ: E ∧ E → E is the multiplication of the ring spectrum E. The cofibration S^{0} →^{i} E →^{j} \overline{E} induces the split exact sequence

[E, X ∧ E]_{0}  [E, X ∧ E ∧ E]_{0}  [E, X ∧ \overline{E} ∧ E]_{0}. v  w \overline{}_{(1 ∧ μ)_{∗}}^{(1∧i∧1)_{∗}}  ww^{(1∧j∧1)_{∗}}

We see that u∧1 = (1∧i∧1)(1∧μ)(u∧1), since both of u∧1 and (1∧i∧1)(1∧μ)(u∧1) in [E, X ∧ E ∧ E]_{0} are sent to 0 ∈ [E, X ∧ \overline{E} ∧ E]_{0} and u_{X} ∈ [E, X ∧ E]_{0} under the maps (1 ∧ j ∧ 1)_{∗} and (1 ∧ μ)_{∗}, respectively. Consider the commutative diagram

S_{0} ∧ E  X ∧ E ∧ E  X ∧ E E ∧ E  X ∧ E ∧ E ∧ E  X ∧ E ∧ E.^{u∧1} w_{_{i∧1} u}  ^{1∧μ} w_{_{1∧1∧i∧1} u} _{‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚‚ ‚} _{u_{1∧i∧1}}_{^{u∧1∧1} w}  _{^{1∧μ∧1}w}

Then (1∧i∧1)u_{X} = (1∧i∧1)(1∧μ)(u∧1) = u∧1 = (u_{X} ∧1)(i∧1) as desired. □

3. E-spherical spectra for a good ring spectrum E

Let S_{E} denote a collection of homotopy equivalence classes of E-spherical spectra. Then L_{E}S^{0} ∈ S_{E} by setting u_{L_{E}S^{0}} = 1_{E}. Hereafter, we abuse notation by writing X ∈ S_{E} when X is E-spherical. If X and Y are E-spherical, then L_{E}(X∧Y) is E-spherical by putting u_{L_{E}(X∧Y)} = (1_{X} ∧ u_{Y})u_{X}. If S_{E} is a set, then it is a semigroup. Indeed, the product X · Y for X, Y ∈ S_{E} is given by L_{E}(X ∧ Y) and the unit is L_{E}S^{0}. For X ∈ S_{E}, 1 ∈ E_{1}^{0,0}(X) = E_{1}^{0,0}(S^{0}) denotes the class of the unit map i: S^{0} → E of E. Consider a filtration {F_{r}} of S_{E} defined by

(3.1)  F_{r} = {X ∈ S_{E} | d_{s}(1) = 0 ∈ E_{s}^{s,s−1}(X) for s < r}.

Here E_{s}^{s,s−1}(X) denotes the E_{s}-term of the E-based Adams spectral sequence abutting to π_{∗}(X). Then F_{1} = S_{E} and F_{r} ⊃ F_{r+1}.

Lemma 3.2. If X ∈ ⋂_{r} F_{r}, then X = L_{E}S^{0}.

Proof. Since d_{r}(1) = 0 for any r, 1 is a permanent cycle and gives an element u of π_{∗}(X) such that u_{∗}: E_{∗}(S^{0}) → E_{∗}(X) is an isomorphism. Therefore, u induces an equivalence L_{E}S^{0} = L_{E}X. □

We call a ring spectrum E good if E_{s}^{s,s−1}(S^{0}) for each s > 1 is a torsion group. Lemma 3.3. Let E be a good ring spectrum. Then we have the following state-

ments.

(1) For any X, Y ∈ F_{r}, then L_{E}(X ∧ Y) ∈ F_{r}.

(2) If X ∈ F_{r}, then there exists an element X′ such that L_{E}(X ∧ X′) ∈ F_{r+m}

for any m.

(3) For Y ∈ S_{E}, if L_{E}(X ∧ Y), X ∈ F_{r}, then Y ∈ F_{r}.

Proof. We have a homotopy equivalence (1 ∧ u_{Y})u_{X}: E → X ∧ Y ∧ E, which makes L_{E}(X ∧Y) E-spherical. Then the naturality of the Adams differential shows the first statement.

Suppose that d_{r}(1) = y ≠ 0 ∈ E_{r}^{r,r−1}(X). Put y_{0} = y and Y_{0} = S^{0}, and assume that we have Y_{k} ∈ S_{E} such that L_{E}(X ∧ Y_{k}) ∈ F_{r+k} and d_{r+k}(1) = y_{k} ∈ E_{r+k}^{r+k,r+k−1}(X ∧ Y_{k}). Let m be the order of y_{k} and consider the (m − 1)- fold smash product Z = L_{E}((X ∧ Y_{k})^{(m−1)}). Then we see that Z ∈ S_{E}, and L_{E}(X ∧ Y_{k} ∧ Z) ∈ S_{E}. Therefore, we obtain L_{E}(X ∧ Y_{k} ∧ Z) ∈ F_{r+k} from the naturality of the Adams differentials. Put now Y_{k+1} = L_{E}(Y_{k} ∧Z) and d_{r+k+1}(1) = y_{k+1} ∈ E_{r+k+1}^{r+k+1,r+k}(X ∧ Y_{k+1}). Inductively we obtain the second.

The third statement follows immediately from the naturality of the Adams differentials. □

We define an equivalence relation ∼ on F_{r} by X ∼ Y if X ∧ Y′ ∈ F_{r+1} for Y′ given in Lemma 3.3(2), and write F_{r}/F_{r+1} = F_{r}/ ∼.

Lemma 3.4. Let E be a good ring spectrum. Then F_{r}/F_{r+1} is isomorphic to a subgroup of E_{r}^{r,r−1}(S^{0}).

Proof. Let φ: F_{r}/F_{r+1} → E_{r}^{r,r−1}(S^{0}) be a correspondence defined by φ([X]) = d_{r}(1) in E_{r}^{r,r−1}(X) = E_{r}^{r,r−1}(S^{0}), where [X] denotes the equivalence class of X. By definition, this is well defined. If φ(X) = φ(Y) = y ≠ 0, then φ(Y′) = −y for Y′ given in Lemma 3.3(2), since 0 = d_{r}(1_{Y ∧Y′}) = y + φ(Y′) by the naturality of the differential d_{r}. Then, X ∧ Y′ ∈ F_{r+1}, and X ∼ Y. Therefore, φ is monomorphic and F_{r}/F_{r+1} is a set. By the standard argument of semigroups, we see that F_{r}/F_{r+1} is a group with multiplication given by the smash product. Indeed, F_{r+1} is a unitary cofinal sub-semigroup of F_{r} by Lemma 3.3. By the naturality of the Adams differential, this is a homomorphism of groups. □

Let GS_{E} denote the graded group associated to the filtration {F_{r}}. This lemma together with Lemma 3.2 show Theorems A and B in the introduction.

Corollary 3.5. If E is a good ring spectrum, then S_{E} is a semigroup and F_{r} is a unitary cofinal sub-semigroup of S_{E}, that is, for any x ∈ S_{E}, there is an element y such that x · y ∈ F_{r}.

Proof. The first statement follows from Lemma 3.4. Indeed, S_{E} is a set. The other follows from Lemma 3.3. □

Lemma 3.3(2) also implies the following:

Corollary 3.6. If E is a good ring spectrum, then S_{E}/F_{r} is a group.

4. Invertible spectra

An E-local spectrum X is called invertible if there exists a spectrum Y such that L_{E}(X ∧ Y) = L_{E}S^{0}. If there is an integer s_{0} such that E_{s}^{s,s−1}(S^{0}) = 0 for s > s_{0}, then F_{s_{0}+1} = {L_{E}S^{0}} by Lemma 3.2. By Lemma 3.3(2), there is a spectrum X′ such that L_{E}(X ∧ X′) ∈ F_{s_{0}+1}. Thus we obtain the following:

Proposition 4.1. Let E be a good ring spectrum, and suppose that there exists an integer s_{0} such that E_{s}^{s,s−1}(S^{0}) = 0 for s > s_{0}. Then an E-spherical spectrum is invertible.

Lemma 4.2. For E = HZ/p, an invertible spectrum X is spherical up to suspension.

Proof. Let X be an invertible spectrum. Then there is a spectrum Y such that L_{HZ/p}(X∧Y) = L_{HZ/p}S^{0} = S_{p}^{0}. Here S_{p}^{0} denotes the p-completed sphere spectrum. Therefore, HZ/p_{∗}(X ∧Y) = Z/p and we have HZ/p_{∗}(X) = Z/p. Then there is an integer k such that (HZ/p)_{k}(X) = Z/p and let u be a generator of π_{k}(X∧HZ/p) = (HZ/p)_{k}(X). Since the E_{1}-term E_{1}^{r,r−1}(Σ^{−k}X) of the classical Adams spectral sequence abutting to π_{∗}(Σ^{−k}X) is zero, u ∈ E_{1}^{0,0}(Σ^{−k}X) = (HZ/p)_{k}(X) is a permanent cycle and u is pulled back to an element of π_{k}(X) under the map (1∧i). Therefore, (1 ∧ j)u = 0, and we see that the composite HZ/p ⟶^{u∧1} Σ^{−k}X ∧ HZ/p ∧ HZ/p ⟶^{1∧j∧1} Σ^{−k}X ∧ \overline{HZ/p} ∧ HZ/p is zero. Therefore, Σ^{−k}X is spherical by Lemma 2.5. □

Since HZ/p is strongly connective, we see that S_{HZ/p} = {S_{p}^{0}} by Theorem A. Let Pic(L_{HZ/p}) denote the collection of the homotopy equivalence classes of invertible spectra. Then the lemma implies immediately the following corollary since the collection of the homotopy equivalence classes of the p-completed sphere spectra S_{p}^{n} for n ∈ Z form a group Z.

Corollary 4.3. Pic(L_{HZ/p}) = Z.

References

[1] J. F. Adams, Stable homotopy and generalised homology, University of Chicago Press, Chicago, 1974.

[2] A. K. Bousfield, The localization of spectra with respect to homology, Topology, 18 (1979), 257–281.

[3] M. J. Hopkins, Minimal atlases of real projective spaces, Lecture Note in Math. 1370, Springer-Verlag, 1986, 171–192.

[4] Y. Kamiya and K. Shimomura, The Picard group of E(2)-localized category at the prime 3, preprint.

[5] Y. Kamiya and K. Shimomura, The Picard groups of some local categories, preprint.

[6] D. C. Ravenel, Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies, 128, Princeton University Press, 1992

Department of Mathematics, Faculty of Science, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan

    E-mail address: Yousuke.Kamiya@ma7.seikyou.ne.jp

Department of Mathematics, Faculty of Science, Kochi University, Kochi,780-8520, Japan

    E-mail address: katsumi@math.kochi-u.ac.jp