I’ve just uploaded to arXiv my paper “The homology of ,” which is an outgrowth of a project I was working on last summer. The main result of the paper is a description, well-known in the field but never written down in detail, of the mod cohomology of the spectrum of (connective) topological modular forms, as a module over the Steenrod algebra: one has

where is the Steenrod algebra and is the 64-dimensional subalgebra generated by and . This computation means that the Adams spectral sequence can be used to compute the homotopy groups of ; one has a spectral sequence

Since is finite-dimensional, the *entire* page of the ASS can be computed, although the result is quite complicated. Christian Nassau has developed software to do these calculations, and a picture of the page for is in the notes from André Henriques‘s 2007 talk at the Talbot workshop. (Of course, the determination of the differentials remains.)

The approach to the calculation of in this paper is based on a certain eight-cell (2-local) complex , with the property that

where is a quotient of the classical Brown-Peterson spectrum by a regular sequence. The usefulness of this equivalence, a folk theorem that is proved in the paper, is that the spectrum is a complex-orientable ring spectrum, so that computations with it (instead of ) become much simpler. In particular, one can compute the cohomology of (e.g., from the cohomology of ), and one finds that it is cyclic over the Steenrod algebra. One can then try to “descend” to the cohomology of . This “descent” procedure is made much simpler by a battery of techniques from Hopf algebra theory: the cohomologies in question are graded, *connected* Hopf algebras. (more…)