AbstractAfter recent work of Hill, Hopkins and Ravenel on the Kervaire invariant one problem [M. A. Hill, M. J. Hopkins and D. C. Ravenel, On the non-existence of elements of Kervaire invariant one, Ann. Math. (2), 184 (2016), 1–262], as well as Adams’ solution of the Hopf invariant one problem [J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. Math. (2), 72 (1960), 20–104], an immediate consequence of Curtis conjecture is that the set of spherical classes in H∗Q0S0 is finite. Similarly, Eccles conjecture, when specialized to X=Sn with n> 0, together with Adams’ Hopf invariant one theorem, implies that the set of spherical classes in H∗QSn is finite. We prove a filtered version of the above finiteness properties. We show that if X is an arbitrary CW-complex of finite type such that for some n, HiX≃0 for any i>n, then the image of the composition π∗ΩlΣl+2X→π∗QΣ2X→H∗QΣ2X is finite; the finiteness remains valid if we formally replace X with S−1. As an application, we provide a lower bound on the dimension of the sphere of origin on the potential classes of π∗QSn which are detected by homology. We derive a filtered finiteness property for the image of certain transfer maps ΣdimgBG+→QS0 in homology. As an application to bordism theory, we show that for any codimension k framed immersion f:M↬ℝn+k which extends to an embedding M→ℝd×ℝn+k, if n is very large with respect to d and k then the manifold M as well as its self-intersection manifolds are boundaries. Some results of this paper extend results of Hadi [Spherical classes in some finite loop spaces of spheres. Topol. Appl., 224 (2017), 1–18] and offer corrections to some minor computational mistakes, hence providing corrected upper bounds on the dimension of spherical classes H∗ΩlSn+l. All of our results are obtained at the prime p = 2.
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