under construction
symmetric monoidal (∞,1)-category of spectra
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
so in higher algebra for $E$ an E-∞ ring there is a natural homomorphism
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 $E$:
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
On cohomology elements this map
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.
By the Bousfield-Kuhn construction there is an equivalence of spectra
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
(see Rezk 06, section 3).
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
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
(This is for instance the case for $L_{K(1)}$tmf).
Then on a finite CW complex $X$ the logarithmic cohomology operation from above
is given by the series
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
The above expression in terms of power operations may be used to establish the string orientation of tmf (Ando-Hopkins-Rezk 10).
…
The logarithmic operation for $p$-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