The transfer in homological algebra
Let A be the Steenrod algebra and M a graded right A-module. Then in homological degree s we write \[ H^ A_ s(M)=\oplus_{t\geq 0}Tor^ A_{s,t}(M,F_ 2) \] for the graded s-dimensional homology of A with coefficients in M. Let \(P_ s=F_ 2[t_ 1,...,t_ s]\) be the graded polynomial algebra on s generators \(t_ i\) each of dimension 1. Both A and \(GL_ s=GL_ s(F_ 2)\) act on \(P_ s\) and the actions commute. This paper defines a ``transfer'' map of graded vector spaces \[ \phi_ s: H^ A_ s(M)\to (M\otimes_ AP_ s)^{GL_ s}. \] The principal results of the paper deal with the case where M is \(F_ 2\). The author shows that \[ \phi: H^ A_*(F_ 2)\to P_*=\oplus_{s\geq 0}(F_ 2\otimes_ AP_ s)^{GL_ s} \] is a morphism of bigraded co- algebras. In particular it is shown that \(\phi_ 2\) is an isomorphism of graded vector spaces, \(\phi_ 3\) and \(\phi_ 4\) are isomorphisms in a range of internal degrees, and \(\phi_ 5\) is not an isomorphism. The proofs are technical, relying on explicit calculation and invariant theory. The author hints at but does not elaborate upon a geometrical context for the work.
scientific article; zbMATH DE number 4126314
scientific article