nLab
logarithmic cohomology operation

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

(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

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