CHARACTERIZING CONNECTED

K-THEORY BY HOMOTOPY GROUPS

John Rognes

In this note we provide proofs for two facts related to connected topological K-theory, expected by Ib Madsen.

Let bu denote the connected K-theory spectrum, and let Im J denote the connected image of J spectrum. For an odd prime p we prove that bu_{p}^{∧} is characterized by its homotopy groups and v_{1}-periodic structure. Thereafter we prove that spectrum maps from bu_{p}^{∧} to B Im J_{p}^{∧} are detected on homotopy groups.

The purpose for proving these Lemmas characterizing properties of bu by homotopy group data, is to complete the calculation of M. Bökstedt, W. C. Hsiang and I. Madsen’s cyclotomic trace invariant T C(ℤ_{p}^{∧}, p) receiving a map from K(ℤ_{p}^{∧}), for odd p. See Corollary 3, whose proof is due to Bökstedt and Madsen. Their proof of its hypothesis will appear in [2].

Lemma 1. Suppose X is a p-complete spectrum, with π_{∗}X ≅ π_{∗}bu_{p}^{∧} abstractly, and

π_{∗}(X; 𝔽_{p}) ≅ π_{∗}(bu; 𝔽_{p})

as 𝔽_{p}[v_{1}]-modules. Then X ≃ bu_{p}^{∧}.

Let S denote the sphere spectrum, and H𝔽_{p} the mod p Eilenberg–MacLane spectrum. For any spectrum Y, let Y/p denote the cofiber of the “multiplication by p” map. In all proofs, let all spectra be implicitly completed at p.

Proof. We first prove that X/p ≃ bu/p. The map Σ^{2p−2}S/p ^{v_{1}} S/p smashed with X induces the v_{1}-module structure on mod p homotopy, and appears in the bottom line of :

[figure: Σ^{2p−2}bu/p; \overline{v_{1}}; bu/p; ∨_{i=1}^{p−1}Σ^{2i}H𝔽_{p}]

∨f  ≃

[figure: Σ^{2p−2}X/p; \overline{v_{1}}; X/p; C(v_{1}); Σ^{2p−1}X/p]

where the horizontal lines are (co–)fibration sequences (cf. [1] Corollary 4.8). By the hypothesis about the 𝔽_{p}[v_{1}]-module structure on π_{∗}(X/p), the mapping cofiber C(v_{1}) has the same homotopy groups as ∨_{i=1}^{p−1}Σ^{2i}H𝔽_{p}. The k-invariants of its Postnikov tower thus sit in degrees 3 through 2p−3 in the mod p Steenrod algebra, where all the groups vanish. Hence C(v_{1}) is a wedge of Eilenberg–MacLane spectra, and we can choose an equivalence as in the diagram above.

Typeset by AMS-TEX

Consider the composite bu/p → ∨_{i}Σ^{2i}H𝔽_{p} → C(v_{1}) → Σ^{2p−1}X/p. By the usual Atiyah–Hirzebruch spectral sequence argument (all groups sit in odd degrees), this map must induce a null homotopic map of underlying spaces.

For the corresponding spectrum level conclusion, recall that the spectrum bu is the connected cover of the mapping telescope for the canonical degree two map (multiply by the Bott class) on the suspension spectrum of its zero’th space. Also note that there is an equivalence between maps from a periodic spectrum or its connected cover to a connected periodic spectrum (cf. [4] Proposition II.3.8).

Similarly, the spectrum bu/p can be recovered as the connected cover of the mapping telescope for the v_{1}-map on the suspension spectrum of its zeroth space. Hence by a Milnor exact sequence argument there are no essential spectrum maps from bu/p to Σ^{2p−1}X/p. (The lim^{1}-term vanishes by completeness.)

Thus there exists a lifting f making the diagram above commute. On homotopy, f induces a homomorphism of isomorphic 𝔽_{p}[v_{1}]-modules which is an isomorphism modulo the ideal generated by v_{1}. Hence π_{∗}(f) is an isomorphism and f is an equivalence.

Next consider the diagram :

[figure: bu; \overline{p·}; bu; \overline{q}; bu/p; su]

∨h  f

[figure: X; \overline{p·}; X; X/p; ΣX]

By an Atiyah–Hirzebruch spectral sequence argument as above, there are no essential maps bu → ΣX. Hence the composite f ∘ q lifts to a map h: bu → X. This is a map between p-complete spectra with abstractly isomorphic homotopy groups, which induces an equivalence with mod p-coefficients, and is hence an equivalence. □

I thank Marcel Bökstedt for noting the need for the spectrum level extensions in the above argument.

Let q be a topological generator for the p-adic units ℤ_{p}^{×} ⊂ ℤ_{p}^{∧}. Extend the Adams operation ψ^{q} to a spectrum map bu_{p}^{∧} → bu_{p}^{∧} mapping as q^{−n}ψ^{q} on the 2nth space, and let Im J_{p}^{∧} be the fiber of ψ^{q} − 1: bu_{p}^{∧} → bu_{p}^{∧}.

Lemma 2. The spectrum maps from bu_{p}^{∧} to B Im J_{p}^{∧} are precisely the p-adic multiples of the cofiber map corresponding to ψ^{q} −1. They are all detected on homotopy groups.

Proof. Consider the diagram :

bu

[figure: ϕ∨]

[figure: bu; \overline{ψ^{q} − 1}; bu; B Im J; su]

Again, there are no essential maps from bu to su, so any spectrum map bu → B Im J lifts to a spectrum map ϕ: bu → bu. This is a reduced K-theory operation, and by [3] Formula 2.3 and Remark 2.10, ϕ is expressible as a series :

ϕ = ∑_{r≥0} ∑_{i≥0} a_{r,i}(ψ^{p^{r}+i} − ψ^{i})

with a_{r,i} ∈ ℤ_{p}^{∧}, where a_{r,i} = 0 if i ≥ p^{r+1} − p^{r} or p|i, and the sum may be infinite in r. Note that ψ^{p^{r}+i} − ψ^{i} may be expressed as (ψ^{p^{r}+i} − 1) − (ψ^{i} − 1) with both p^{r} + i and i prime to p in the cases where a_{r,i} ≠ 0, except when r = i = 0.

We claim that any operation ψ^{n} − 1 with n a p-adic unit factors through ψ^{q} − 1. This follows for n = q^{i} with i a natural number from the factorization ψ^{q^{i}} − 1 = (ψ^{q} − 1)(ψ^{q^{i−1}} + ⋯ + ψ^{q} + 1), and for negative i using the operation ψ^{−q}. As q is a topological generator, the general case follows by completion.

On the other hand, any operation ϕ factoring through ψ^{q} −1 has a_{0,0} = 0. For if it were nonzero, let j = v_{p}(a_{0,0}) < ∞ be the p-adic valuation, let i = (p − 1)p^{j}, and consider homotopy groups in degree 2i. For any p-adic unit n, v_{p}(n^{i} − 1) ≥ j + 1, so

v_{p}(π_{2i}(ϕ)) = v_{p}(a_{0,0}) = j

which precludes ϕ from mapping into the image of ψ^{q} − 1, as v_{p}(π_{2i}(ψ^{q} − 1)) = v_{p}(q^{i} − 1) ≥ j + 1 by the above (q is a p-adic unit).

Hence the subspace of operations ϕ which factor through ψ^{q} − 1 is contained in the closed ideal where a_{0,0} = 0 in the compact Hausdorff ring of all stable operations, and contains the dense subset of such operations which are finite sums of Adams operations. As the former subspace is a continuous image of the whole compact ring, it is itself compact, thus closed, and is therefore the whole ideal.

In conclusion, the coefficient a_{0,0} precisely classifies the spectrum maps bu → B Im J, as we wanted to prove. □

Let j denote the connective image of J spectrum, i.e. the connective cover of the fiber of ψ^{q} − 1: K → K, or equivalently the connective cover of the K-localization of S, where K is the periodic K-theory spectrum.

Corollary 3. (Bökstedt–Madsen) Suppose there is a map of p-complete spectra j_{p}^{∧} ∨ Bj_{p}^{∧} → T C(ℤ_{p}^{∧}, p) with cofiber X, injective on homotopy groups, such that π_{∗}X ≅ π_{∗}su_{p}^{∧} abstractly, and π_{∗}(X; 𝔽_{p}) ≅ π_{∗}(su; 𝔽_{p}) as 𝔽_{p}[v_{1}]-modules. Then

T C(ℤ_{p}^{∧}, p) ≃ j_{p}^{∧} ∨ Bj_{p}^{∧} ∨ su_{p}^{∧} .

Proof. By suspending Lemma 1 once, we find X ≃ su. Thus there is a fibration sequence of spectra

[figure: j ∨ Bj; T C(ℤ_{p}^{∧}, p); su; \overline{Σe}; Bj ∨ BBj]

Desuspending the map Σe once, we are led to consider e: bu → j ∨ Bj, which is assumed zero on homotopy groups. There are no essential maps bu → j which are zero on homotopy groups, due to the corresponding fact for maps bu → bu. Any map bu → Bj which is zero on homotopy groups factors through the 1-connected cover B Im J, so by Lemma 2 the map e (and thus Σe) is null homotopic, from which the splitting follows (lift the identity map of su). □

References

[1] J. F. Adams, Lectures on generalized cohomology, Category Theory, Homology Theory and their Applications III, Lecture Notes in Math., vol. 99, Springer, 1969, pp. 1–138.

[2] M. Bökstedt and I. Madsen, The cyclotomic trace of ℤ_{p}^{∧}, in preparation.

[3] I. Madsen, V. Snaith and J. Tornehave, Infinite loop maps in geometric topology, Math. Proc. Camb. Phil. Soc. 81 (1977), 399–429.

[4] J. P. May et al., E_{∞} ring spaces and E_{∞} ring spectra, Lecture Notes in Math., vol. 577, Springer, 1977.

Matematisk Institutt, Universitetet i Oslo