Elliptic-fiber brotherhood and a rational-homotopy obstruction

Definitions and mathematical statement

The broken source diagram is interpreted by the aid as a finite tower of bundles \[ F_i\longrightarrow M_{i-1}\longrightarrow M_i\qquad(1\leq i\leq k), \] with the fibers $F_1,\ldots,F_k$ and final base $M_k$ fixed topological manifolds. The question asks for a weakened "brotherhood" notion for the total manifolds $M_0$, together with a property of its members not already known for all manifolds satisfying $\sec\geq K$, $\operatorname{diam}\leq1$. The source does not define the weakening, the property, dimensions, compactness, or the quantifiers on $K$.

The aid proposes one specific rational-homotopy formulation. Work with simply connected finite CW complexes. Call $X$ rationally elliptic if \[ \sum_{i\geq2}\dim_{\mathbb{Q}}(\pi_i(X)\otimes{\mathbb{Q}})<\infty; \] finite-dimensional rational cohomology is automatic in this category. Fix a library $\mathcal F$ of simply connected rationally elliptic finite CW complexes. Generate an equivalence relation $\sim_{\mathcal F}$ by finite undirected zigzags of Serre fibrations $E\to B$ whose fiber is homotopy equivalent to some $F\in\mathcal F$, with every vertex still simply connected and of finite CW type. An arrow may be traversed in either direction in the zigzag; it remains an actual fibration in its own direction.

The supplied claim is \[ X\sim_{\mathcal F}Y\quad\Longrightarrow\quad (X\text{ rationally elliptic}\Longleftrightarrow Y\text{ rationally elliptic}). \] On elliptic spaces define the rational homotopy Euler characteristic \[ \chi_\pi(X)=\sum_{i\geq2}(-1)^i\dim_{\mathbb{Q}}(\pi_i(X)\otimes{\mathbb{Q}}), \qquad L_{\mathcal F}=\langle\chi_\pi(F):F\in\mathcal F\rangle_{\mathbb{Z}}\subseteq{\mathbb{Z}}. \] The reported congruence invariant is \[ X\sim_{\mathcal F}Y,\quad X,Y\text{ elliptic} \quad\Longrightarrow\quad \chi_\pi(X)-\chi_\pi(Y)\in L_{\mathcal F}. \] Thus $[\chi_\pi(X)]\in{\mathbb{Z}}/L_{\mathcal F}$ is constant on an elliptic brotherhood class. The finite sum defining $\chi_\pi$ is not assigned to hyperbolic spaces by an unproved alternating-series convention, and the congruence is a necessary obstruction, not a complete classification.

For a directed tower in this category with elliptic fibers and elliptic final base, the aid gives \[ \chi_\pi(M_0)=\chi_\pi(M_k)+\sum_{i=1}^k\chi_\pi(F_i), \] \[ \chi(M_0)=\chi(M_k)\prod_{i=1}^k\chi(F_i), \] where $\chi$ is the ordinary Euler characteristic. The latter multiplicative formula is distinct from the additive rational-homotopy formula.

As a boundary example, the aid reports that the simply connected closed manifold $\#_2(S^2\times S^2)$ is rationally hyperbolic, so it cannot belong to an elliptic $\mathcal F$-brotherhood class. This shows that the proposed relation does not place every simply connected finite CW manifold into one elliptic class. It is not a claim that the example admits or fails to admit a metric in the source's curvature class. The fixed-library zigzag relation is the aid's particular proposal, not the uniquely intended meaning of "brotherhood," and no theorem asserting rational ellipticity for all $\sec\geq K$, $\operatorname{diam}\leq1$ manifolds is supplied.

Short English statement

The supplied proposal replaces fixed bundle towers with zigzags of fibrations using a chosen elliptic fiber library. Rational ellipticity is constant on each class, and a homotopy Euler characteristic gives a congruence obstruction. This is one restricted rational-homotopy model, not a curvature theorem or a complete answer to the requested brotherhood notion.

Sources

• Local UnsolvedMath registry, permanent ID $20003136$, exact source code catalogue entry; canonical aim-topology-notes.json, notes[223], source tag problem.
• AIM workshop list "Open Problems in Non-negative Sectional Curvature," workshop "Manifolds with nonnegative sectional curvature," Problem 11: https://aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf. The supplied verification reports that the broken diagram is a bundle tower; no PDF was fetched here.
• The original literature field is empty. No complete authoritative bibliographic citation for the proposed zigzag relation, rational-homotopy claims, or hyperbolicity of the connected-sum example is supplied.
• Research attempt 1: result reduction, problem status at run open, rigor label proved, novelty confidence low; classification PARTIAL-PROGRESS, registry status partially_solved, suggested difficulty L4. Created and updated 14 August 2026. These are supplied assessments requiring review.
• Attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. The machine-generated research aid is not peer reviewed and requires independent verification before citation.