Morava \(E\)-theory of symmetric groups.

*(English)*Zbl 0912.55012
Topology 37, No. 4, 757-779 (1998); correction ibid. 38, No. 4, 931 (1999).

In his ‘abstract’ the author summarizes the contents of this paper as ‘the computation of the completed \(E(n)\) cohomology of the classifying spaces of the symmetric groups, and the relation of the answer to the theory of finite subgroups of formal groups’. Here, and very roughly speaking, the coefficients of the theory \(E(n)\) are obtained from those of Brown-Peterson theory by inverting the generator \(v_n\) and setting all generators of higher degree equal to zero. Thus, at least locally, \(E(2)\) has the flavour of elliptic cohomology. Formally the main theorem is stated in terms of \(E^0 (DS^0)\), where \(DS^0\) is equal to the wedge of augmented classifying spaces \(B\Sigma_{k+}\) \((k\) greater than or equal to 0). This turns out to be a formal power series ring under a product induced by a stable transfer map from degree \((k+l)\) to bidegree \((k,l)\), the indecomposable elements are carefully described, together with an \(E^0\)-basis of monomials in the Chern classes of the natural permutation representations of the groups \(\Sigma_{p^k}\). The reviewer wishes to emphasize the care shown in this paper – given the fact that wreath products are ‘good’ in the cohomology groups concerned, and that the importance of exterior powers of the permutation representation have been known for a long time, something like the main theorem was to be expected. But the devil was in the details, here set out very clearly.

Reviewer: C.B.Thomas (Cambridge)

##### MSC:

55R35 | Classifying spaces of groups and \(H\)-spaces in algebraic topology |

55N20 | Generalized (extraordinary) homology and cohomology theories in algebraic topology |