On the lambda algebra and Singer's cohomological transfer

Abstract

Writing $\mathbb{A}$ for the 2-primary Steenrod algebra, which is the algebra of stable natural endomorphisms of the mod 2 cohomology functor on topological spaces. Working at the prime 2, computing the cohomology of $\mathbb{A}$ is an important problem of Algebraic topology, because it is the initial page of the Adams spectral sequence converging to stable homotopy groups of the spheres. A relatively efficient tool to describe this cohomology is the Singer algebraic transfer of rank $n$ in [Math. Z. 202 (1989), 493-523], which passes from a certain subquotient of a divided power algebra to the cohomology of $\mathbb{A}.$ Singer predicted that this transfer is a monomorphism, but this remains open for $n\ge 4.$ This short note is to verify the conjecture in the ranks 4 and 5 and some generic degrees.

arXiv comment5 pages. This paper is an announcement whose details will appear elsewhere. Comments are welcome! arXiv admin note: text overlap with arXiv:2106.14605, arXiv:2106.14606. substantial text overlap with arXiv:1412.1709 by other author

References in corpus (4)

• On the Peterson hit problem
• The "hit" problem of five variables in the generic degree and its application
• On a construction for the generators of the polynomial algebra as a module over the Steenrod algebra
• On the dimension of $H^{\ast }(({\mathbb{Z}}_2{)}^{\times t},{\mathbb{Z}}_2)$ as a module over Steenrod ring

Cited by in corpus (4)

• The "hit" problem of five variables in the generic degree and its application
• On the dimension of $H^{\ast }(({\mathbb{Z}}_2{)}^{\times t},{\mathbb{Z}}_2)$ as a module over Steenrod ring
• A note on the modular representation on the $\mathbb{Z}/2$-homology groups of the fourth power of real projective space and its application
• Structure of the space of $G{L}_4({\mathbb{Z}}_2)$-coinvariants ${\mathbb{Z}}_2{\otimes }_{G{L}_4({\mathbb{Z}}_2)}P{H}_{\ast }({\mathbb{Z}}_2^4,{\mathbb{Z}}_2)$ in some generic degrees and its application