nLab logarithmic cohomology operation

under construction

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

$$log \;\colon\; \mathbb{Q}^\times_{\gt 0}\longrightarrow \mathbb{R}$$

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

$$\ell_{n,p} \;\colon\; gl_1(E) \longrightarrow L_{K(n)} E$$

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

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

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

$$H^0(X, gl_1(E)) \simeq (E^0(X))^\times \,.$$

In positive degree the canonical map of pointed homotopy types $GL_1(E) = \Omega^\infty gl_1(E) \to \Omega^\infty E$ is in fact an isomorphism on all homotopy groups

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

On cohomology elements this map

$$\pi_q(gl_1(E)) \simeq \tilde H^0(S^q, gl_1(E)) \simeq (1+ \tilde R^0(S^q))^\times \subset (R^0(S^q))^\times$$

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

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

$$L_{K(n)} gl_1(E) \simeq L_{K(n)}E$$

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

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

(see Rezk 06, section 3).

Properties

Action on cohomology groups

For every E-∞ ring $E$ and spaces $X$, prime number $p$ and natural number $n$, the logarith induces a homomorphism of cohomology groups of the form

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

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

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

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

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

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

is given by the series

$$\begin{aligned} \ell_{1,p} \colon x & \mapsto \left( 1 - \frac{1}{p}\psi \right) log(x) \\ & = \frac{1}{p} log \frac{x^p}{\psi(x)} \\ & = \sum_{k=1}^\infty (-1)^k \frac{p^{k-1}}{k}\left( \frac{\theta(x)}{x^p}\right)^k \\ \end{aligned}$$

which converges p-adically.

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

Here $\theta$ ….

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

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

Relation to the string-orientation of $tmf$

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 $p$-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

• Charles Rezk, The units of a ring spectrum and a logarithmic cohomology operation, J. Amer. Math. Soc. 19 (2006), 969-1014 (arXiv:math/0407022)

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

• Matthew Ando, Mike Hopkins, Charles Rezk, Multiplicative orientations of KO-theory and the spectrum of topological modular forms, 2010 (pdf)