Spherical classes in \(H_*(\Omega ^lS^{l+n};{\mathbb {Z}}/2)\) for \(4\pmb {\leqslant } l \pmb {\leqslant } 8\)

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Abstract

We show that for \(l\in \{4,5,6,7,8\}\), the only spherical classes in \(H_*\Omega ^lS^{n+l}\) arise from the inclusion of the bottom cell, or the Hopf invariant one elements. Together with our previous work in Zare (Topol Appl 224:1–18, 2017), this verifies Eccles conjecture when we restrict to finite loop spaces \(\Omega ^lS^{n+l}\) with \(l<9\) and \(n>0\). Moreover, we prove a generalised Browder theorem for spherical classes in \(H_*\Omega ^lS^{n+l}\) showing that for \(1\leqslant l<+\infty \), if \(\xi ^2\in H_*\Omega ^lS^{n+l}\) is given with \(\dim \xi +1\ne 2^t\) and \(\dim \xi +1\equiv 2\text { mod }4\) and \(n>l\), then \(\xi ^2\) is not spherical. We call this as a generalised Browder theorem. We also record an observation, probably well known, that for \(n>0\) and \(1\leqslant l\leqslant +\infty \), if \(\xi \in H_*\Omega ^lS^{n+l}\) is spherical with \(\sigma _*^l\xi =0\) where \(\sigma _*^i\) is the iterated homology suspension for \(i\leqslant l\), then for some \(j<+\infty \) we have \(\sigma _*^j\xi =\zeta ^2\) for some A-annihilated primitive class where \(\zeta \) is odd dimensional. This reduces study of spherical classes to those which are square. We call this the reduction theorem as it reduces the problem to the study of spherical classes that are square. As an application, we show that if \(f\in {_2\pi _{2d}}QS^n\) is given with \(h(f)\ne 0\), \(\sigma _*h(f)=0\), and \(d+1\equiv 2{\mathrm {mod}}4\) then the dimension of the sphere of origin of f is bounded below in the sense that if f pulls back to an element of \({_2\pi _{2d}}\Omega ^lS^{n+l}\) then \(l>n\). This is the first type of this result that we know of in the existing literature providing a lower bounded on the dimension of sphere of origin of an element of \({_2\pi _*^s}\).

Keywords

Loop space James–Hopf map Dyer–Lashof algebra Steenrod algebra 

Mathematics Subject Classification

55P35 55S12 55S10 

Notes

Acknowledgements

I am grateful to Drew Heard for the communication on MathOverFlow regarding \({_2\pi _{64}^s}\) and the current state of knowledge about stable stems in these dimensions who drew my attention to [15]. I am very grateful to an anonymous referee for his/her comments and suggestion on the earlier version of this paper which resulted in an improved version of the earlier submission.

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Copyright information

© Sociedad Matemática Mexicana 2018

Authors and Affiliations

  1. 1.School of Mathematics, Statistics, and Computer Science, College of ScienceUniversity of TehranTehranIran

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