Title: On behavior of the algebraic transfer
Authors: Robert R. Bruner, Le Minh Ha, and Nguyen H. V. Hung
MSC-class: 55P47, 55Q45, 55S10, 55T15
Paper: math.AT/0205170
Addresses:
Robert R. Bruner
Department of Mathematics
Wayne State University
Detroit, MI 48202
USA
rrb@math.wayne.edu
Le Minh Ha
IHES,
F-91440, Bures-sur-Yvette
France
lha@ihes.fr
Nguyen H. V. Hung
Department of Mathematics
Wayne State University
Detroit, MI 48202
USA
nhvhung@math.wayne.edu
Abstract: Let V be a mod 2 vector space of rank k. W. Singer
defined a transfer homomorphism from the GL(k,2) coinvariants of
the primitives in the homology of BV to the cohomology of the
Steenrod algebra, as an algebraic version of the geometric
transfer from the stable homotopy of BV to the stable homotopy
of spheres. It has been shown that the algebraic transfer is
highly nontrivial and, more precisely, that it is an isomorphism
for k=1, 2, or 3. However, Singer showed that it is not an
epimorphism for k=5. In this paper, we prove that it also fails
to be an epimorphism when k=4. Precisely, it does not detect
the non zero elements in the g family, in stems 20, 44, 92, and
in general, 12*2^s - 4, for each s > 0. The transfer still
fails to be an epimorphism even after inverting Sq^0, thereby
giving a negative answer to a prediction by Minami.