\(E_{n}\) genera (Q906181)

There is a whole hierarchy of notions of ring spectra. At the lower end, there are spectra whose multiplication is only unital, associative or commutative up to homotopy. If we replace unitality and associativity by their most homotopy coherent versions, we get the notion of an \(A_\infty\)- or \(E_1\)-ring spectrum. Commutativity has again a hierarchy of homotopy coherent versions, ranging from \(E_2\) to \(E_\infty\). One important aspect is that the smash product of two \(R\)-modules of an \(E_2\)-ring spectrum \(R\) has again the structure of an \(R\)-module. An important class of (homotopy commutative) ring spectra are the complex-oriented ones, i.e. those who receive a (homotopy) ring map from \(MU\), the complex bordism spectrum. Some of these have actually a more coherent multiplication. For example, the Lubin--Tate spectra and the spectra \(TMF_1(n)\) (complex-oriented for \(n\geq 2\)) have an \(E_\infty\)-multiplication and \(BP\) has an \(E_4\)-multiplication. The article under review asks whether one can refine the homotopy ring maps from \(MU\) to \(E_2\) or even better ring maps. The first no-go theorem for this was proven by Johnson--Noel: If the map \(MU \to R\) factors through \(BP\), then it is in many cases not possible to refine it to an \(E_\infty\)-map (or even to an \(H_\infty\)-map, which is a slightly weaker notion). The authors add a new no-go result: There are explicit (homotopy) ring self-maps of \(MU\) that cannot be lifted to an \(E_4\)-map. On the positive side, there are older results by Ando--Hopkins--Rezk and Ando giving an \(E_\infty\)-orientation \(MString \to TMF\) and giving \(H_\infty\)-orientations from \(MU\) to Lubin--Tate spectra. The paper under review has the first general result in this direction. It shows that \textit{every} (homotopy) ring map \[ MU \to R \] into an \(E_2\)-ring spectrum with homotopy concentrated in even degrees can be refined to an \(E_2\)-map. In particular, this is also true for the Quillen idempotent on \(MU\) defining \(BP\). This gives a new proof that \(BP\) has an \(E_2\)-ring structure. Note that by Johnson--Noel this idempotent cannot be lifted to an \(E_\infty\)-map in general. The authors actually give methods to attack to whole space of \(E_n\)-maps out of \(MU\) into an \(E_n\) or better \(E_{n+1}\)-ring spectrum \(R\) (if it is non-empty). In the latter case they identify the space of \(E_n\)-maps from \(MU\) to \(R\) with the based mapping space \[ \mathrm{Map}_*(B^nBU, B^nSL_1R). \] (The corresponding statement is also true for an arbitrary \(E_n\)-Thom spectrum instead of \(MU\).) By Bott periodicity, the spaces \(B^nBU\) are for even \(n\) just the stages of the Whitehead tower of \(BU\). The authors restrict themselves to the cases \(n=2,\) for concrete calculations because the cohomology rings of the \(B^nBU\) turn out to be just polynomial rings concentrated in even degrees for \(n=2,4\).