For p an odd prime, let En = (Z/p)n act on itself by translations to define an embedding as a subgroup of the alternating group Apn on pn letters. We compute the mod p homology of the image of the restriction Res(Apn, En) : H & lowast;(Apn; Fp) -> H & lowast;(En; Fp) with differential given by the Milnor operation Qj, for every n and j. In doing so, we disprove an analog of Pengelley-Sinha's conjecture for symmetric groups by constructing critical elements in this Margolis homology. The Margolis homology of ImRes(Apn,En) is non-zero in both even and odd degrees, unlike the case of similar restriction maps for symmetric groups, which are concentrated in even degrees. Our calculation is a necessary step in calculating Margolis homology of alternating groups, which in turn plays a key role in our broader program to understand Morava K-theory of alternating groups.
In the 1980's Harer explicitly constructed a spine for the decorated Teichmu & uml;ller space of orientable surfaces with at least one puncture and negative Euler characteristic. In this paper we point out some instances where his computation of the dimension of this spine is off by 1 and give the correct dimension.
We completely compute the mod 2 Margolis homology of the M & ugrave;i algebra M-n = F-2[x(1), ... , x(n)](GLn,2), i.e. the homology of Mn with the differential to be the Milnor operation Q(j), for every n and j, where GL(n,2) denotes the Sylow 2-subgroup, which is also the Borel subgroup of GL(n, F2). The motivation for this problem is that the Margolis homology of the M & ugrave;i algebra plays a key role in study of the Morava K-theory K(j)& lowast;(S-m,S-2) of the Sylow 2-subgroup of the symmetric group Sm on m letters. We prove that for n = 1 or 2, the j-th Margolis homology of the n-th M & ugrave;i algebra is represented by elements in M-n(2) only. For n >= 3, this phenomenon is no longer true, because of the occurrence of some "critical elements" h(m,s1,...,sk), which we create, in the homology for 0 < m + 1 < s1 < & centerdot;& centerdot; & centerdot;< sk <= n. The mod 2 Margolis homology of the M & ugrave;i algebra is concentrated in even degrees. This is analogous to the mod 2 Margolis homology of the Dickson algebra and the mod p Margolis homology of the Dickson-M & ugrave;i algebra, for p an odd prime.
We point out a relationship between the norm residue isomorphism theorem of Suslin-Voevodsky-Rost and the theory of birational motives, as well as its generalisation to "higher jets".
