7 problems

The conjecture. Under these hypotheses, the map

Let $X$ be a pointed simplicial set and let $p$ be a prime. Consider the algebraic Goodwillie spectral sequence … together with the algebraic Adams spectral sequence, the Rector sp…

Let $X$ be a simply-connected space of finite type, and let $\operatorname{rank}_p(\pi_n(X))$ and $\operatorname{rank}_{\mathbb{F}_p}(\operatorname{H}_n(\Omega X;\mathbb{F}_p))$ de…

Charming-map conjecture. The following statements are true:

Localize all spaces and maps at an odd prime $p$. Let $W_n$ be the homotopy fibre of the double suspension … and let $T^{2np+1}(p)$ denote Anick's space. Let $BW_n$ be a classifyin…

Goodwillie–Whitehead conjecture. The GSS for $S^1$ collapses at $E_2$. This is a calculus analogue of the $2$-primary Whitehead conjecture and would explain the convergence of the…

The preceding constructions involve spectra $mathbf L(i)\wedge \mathbf T(2^{n-i}-1)$, together with the spectra appearing in the complex of Theorem; the final cofibration has sourc…