The A-decomposability of the Singer construction

Let RsM denote the Singer construction on an unstable module M over the Steenrod algebra A at the prime two; RsM is canonically a subobject of Ps⊗M, where Ps=F2[x1,…,xs] with generators of degree one and F2 is the field with two elements. Passage to A-indecomposables gives the natural transformation...

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Bibliographic Details
Published in:Journal of algebra 2019-01, Vol.517, p.186-206
Main Authors: Hưng, Nguyễn H.V., Powell, Geoffrey
Format: Article
Language:English
Subjects:
Algebraic Topology
Destabilization
Indecomposables
Mathematics
Singer functor
Steenrod algebra
Unstable module
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Summary:Let RsM denote the Singer construction on an unstable module M over the Steenrod algebra A at the prime two; RsM is canonically a subobject of Ps⊗M, where Ps=F2[x1,…,xs] with generators of degree one and F2 is the field with two elements. Passage to A-indecomposables gives the natural transformation RsM→F2⊗A(Ps⊗M), which identifies with the dual of the composition of the Singer transfer and the Lannes–Zarati homomorphism. The main result of the paper proves the weak generalized algebraic spherical class conjecture, which was proposed by the first author. Namely, this morphism is trivial on elements of positive degree when s>2. The condition s>2 is necessary, as exhibited by the spherical classes of Hopf invariant one and those of Kervaire invariant one.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2018.09.030