Resolve the remaining extension problems in the torsion groups

Resolve the unsolved extension problems remaining in the computation of the torsion subgroups of the higher smooth surgery structure sets of complex projective spaces, arising from the long exact surgery sequence for complex projective spaces.

Background

Theorem 1 gives the torsion subgroups T_{n,k} for 1≤n,k≤6 only up to certain extension problems, with the unresolved cases highlighted in the table. The paper distinguishes these from extension problems occurring in the calculation of normal invariants, which were solved using spectral-sequence and stable-homotopy methods.

The remaining problems arise in reconstructing the higher smooth surgery structure sets from the long exact surgery sequence for complex projective spaces. The authors state that resolving them will probably require technology different from that developed in the paper.

References

The table in Theorem~(i) still contains unsolved extension problems. However, these are of a different nature than those mentioned in Remark~\ref{rem:methods-and-organization}. Namely, they arise from the long exact surgery sequence for $\cpn$; see Eq.(3.2). Their solution will probably require different technology from that used in the present paper.

— Higher smooth surgery structure sets of complex projective spaces, part II  (2609.01505 - Kalužný et al., 1 Sep 2026) in Remark immediately following Remark 3.1 (unnumbered remark; following Remark "Methods and Organization") and Theorem 1, Table of T_{n,k}

We could extend the results from Lemma~\ref{lema:coker-J-CP-values} and combine them with Table 5 to obtain

\MCG{\DIFF}(\cp7)\cong(\Z_8{\Sigma{15}}\textcolor{red}{\oplus}\Z_2{\eta\kappa})\rtimes\Z_2{c},

where $\eta\kappa$ comes from $\coker[\Sigma\cp7,\Omega J]\cong\sN_{\partial}{\DIFF}(\cp7\times\text{D}1)$, $\Sigma{15}$ is the connected sum with the generator of $bP_{16}\subseteq\Theta_{15}$. The red color highlights an unsolved extension problem.

— Higher smooth surgery structure sets of complex projective spaces, part II  (2609.01505 - Kalužný et al., 1 Sep 2026) in Remark "Smooth Mapping Class Groups", near the end of Section 1