4 problems

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Curtis conjecture on spherical classes in $H_*QS^0$
Curtis conjecture. Only the Hopf invariant one and Kervaire invariant one elements survive under $h$ .
Conjecturesolvedspherical-classes stable-homotopy-theoryarXiv posted 2 days ago
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Eccles's conjecture on spherical classes and detection by homology operations
Eccles conjecture. Under these hypotheses, the stable adjoint of $f$ either is detected by homology or is detected by a primary operation in its mapping cone.
Conjectureopenspherical-classes stable-homotopy-theoryarXiv posted 2 days ago
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Madsen–Curtis spherical class conjecture
Let $X$ be a pointed space and write $QX=\Omega^\infty\Sigma^\infty X$ . Consider the Hurewicz map … For $X=S^0$ , the image in positive degree should consist only of the images of e…
Conjecturesolvedhomology spherical-classes stable-homotopy-theoryarXiv posted 2 days ago
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Conjecture on the suspension-kernel classes generated by $[\eta_i]_6$
Suspension-kernel conjecture. The class $[\eta_i]_6$ dies under $\sigma_*$ . Consequently, the subalgebra of $H_*Q_0S^{-2^i+6}$ generated by the classes $Q^I[\eta_i]_6$ belongs to…
Conjectureopenhomology-of-infinite-loop-spaces spherical-classes stable-homotopy-theoryarXiv posted 2 days ago

Curtis conjecture on spherical classes in H∗QS0H_*QS^0H∗​QS0