nLab
logarithmic cohomology operation

Contents

under construction

Context

Higher algebra

Stable Homotopy theory

Contents

Idea

In analogy to how in ordinary algebra the natural logarithm of positive rational numbers is a group homomorphism from the group of units to the completion of the rationals by the (additive) real numbers

log: >0 × log \;\colon\; \mathbb{Q}^\times_{\gt 0}\longrightarrow \mathbb{R}

so in higher algebra for EE an E-∞ ring there is a natural homomorphism

n,p:gl 1(E)L K(n)E \ell_{n,p} \;\colon\; gl_1(E) \longrightarrow L_{K(n)} E

from the ∞-group of units of EE to the K(n)-local spectrum obtained from EE (see Rezk 06, section 1.7).

On the cohomology theory represented by EE this induces a cohomology operation called, therefore, the “logarithmic cohomology operation”.

More in detail, for XX the homotopy type of a topological space, then the cohomology represented by gl 1(E)gl_1(E) in degree 0 is the ordinary group of units in the cohomology ring of EE:

H 0(X,gl 1(E))(E 0(X)) ×. H^0(X, gl_1(E)) \simeq (E^0(X))^\times \,.

In positive degree the canonical map of pointed homotopy types GL 1(E)=Ω gl 1(E)Ω EGL_1(E) = \Omega^\infty gl_1(E) \to \Omega^\infty E is in fact an isomorphism on all homotopy groups

π 1GL 1(E)π 1Ω E. \pi_{\bullet \geq 1} GL_1(E) \simeq \pi_{\bullet \geq 1} \Omega^\infty E \,.

On cohomology elements this map

π q(gl 1(E))H˜ 0(S q,gl 1(E))(1+R˜ 0(S q)) ×(R 0(S q)) × \pi_q(gl_1(E)) \simeq \tilde H^0(S^q, gl_1(E)) \simeq (1+ \tilde R^0(S^q))^\times \subset (R^0(S^q))^\times

is logarithm-like, in that it sends 1+xx1 + x \mapsto x.

But there is not a homomorphism of spectra of this form. This only exists after K(n)-localization, and that is the logarithmic cohomology operation.

Definition

By the Bousfield-Kuhn construction there is an equivalence of spectra

L K(n)gl 1(E)L K(n)E L_{K(n)} gl_1(E) \simeq L_{K(n)}E

between the K(n)-local spectrum induced by the abelian ∞-group of units of EE (regarded as a connective spectrum) with that induced by EE itself. The logarithm on EE is the composite of that with the localization map

n,p:gl 1(E)L K(n)gl 1(E)L K(n)E. \ell_{n,p} \;\colon\; gl_1(E) \stackrel{}{\longrightarrow} L_{K(n)}gl_1(E) \stackrel{\simeq}{\to} L_{K(n)} E \,.

(see Rezk 06, section 3).

Properties

Action on cohomology groups

For every E-∞ ring EE and spaces XX, prime number pp and natural number nn, the logarith induces a homomorphism of cohomology groups of the form

n,p:(E 0(X)) ×(L K(n)E) 0(X). \ell_{n,p} \;\colon\; (E^0(X))^\times \longrightarrow (L_{K(n)}E)^0(X) \,.

Explicit formula in terms of power operations

Under some conditions there is an explicit formula of the logarithmic cohomology operation by a series of power operations.

Let EE be a K(1)-local E-∞ ring such that

  • the kernel of π 0L K(1)𝕊π 0E\pi_0 L_{K(1)}\mathbb{S} \longrightarrow \pi_0 E contains the torsion subgroup of π 0L K(1)𝕊\pi_0 L_{K(1)}\mathbb{S}.

(This is for instance the case for L K(1)L_{K(1)}tmf).

Then on a finite CW complex XX the logarithmic cohomology operation from above

1,p:(E 0(X)) ×E 0(X) \ell_{1,p}\;\colon\; (E^0(X))^\times \longrightarrow E^0(X)

is given by the series

1,p:x (11pψ)log(x) =1plogx pψ(x) = k=1 (1) kp k1k(θ(x)x p) k \begin{aligned} \ell_{1,p} \colon x & \mapsto \left( 1 - \frac{1}{p}\psi \right) log(x) \\ & = \frac{1}{p} log \frac{x^p}{\psi(x)} \\ & = \sum_{k=1}^\infty (-1)^k \frac{p^{k-1}}{k}\left( \frac{\theta(x)}{x^p}\right)^k \\ \end{aligned}

which converges p-adically.

(Rezk 06, theorem 1.9, see also Ando-Hopkins-Rezk 10, prop. 4.5)

Here θ\theta ….

In the special case that x=1+ϵx = 1 + \epsilon with ϵ 2=0\epsilon^2 = 0 then this reduces to

1,p(1+ϵ)=ϵ1pψ(ϵ). \ell_{1,p}(1+ \epsilon)= \epsilon - \frac{1}{p}\psi(\epsilon) \,.

Relation to the string-orientation of tmftmf

The above expression in terms of power operations may be used to establish the string orientation of tmf (Ando-Hopkins-Rezk 10).

References

The logarithmic operation for pp-complete K-theory was first described in

  • Tammo tom Dieck, The Artin-Hasse logarithm for λ-rings, Algebraic topology (Arcata, CA, 1986), 409–415, Lecture Notes in Math., 1370, Springer, Berlin, 1989.

The formulation in terms of the Bousfield-Kuhn functor and the expression in terms of power operations is due to

The application of this to the string orientation of tmf is due to

Last revised on March 26, 2014 at 13:26:33. See the history of this page for a list of all contributions to it.