Foliations and Diffeomorphism Groups — Question 24

Definitions and mathematical statement

Let $S^3$ be the oriented $3$-sphere, and let $H=\mathrm{Homeo}_0(S^3)$ denote the identity component of its homeomorphism group, with the compact-open topology. For a connected based $4$-manifold $M$ and a representation $\rho:\pi_1(M)\to H$, the associated flat oriented sphere bundle is \[ E_\rho=(\widetilde M\times S^3)/\pi_1(M)\longrightarrow M, \qquad \gamma\cdot(x,s)=(\gamma x,\rho(\gamma)(s)). \] Its Euler class is $e(E_\rho)\in H^4(M;{\mathbb{Z}})$, the primary obstruction to a section. If $M$ is closed and oriented, its Euler number is $\langle e(E_\rho),[M]\rangle$; these additional hypotheses are not specified in the supplied question.

The supplied Question 24 asks whether there is a geometric proof of the preceding theorem, specifically through a family of representations \[ \rho_i:\pi_1(M_i)\longrightarrow \mathrm{Homeo}_0(S^3),\qquad i\in I, \] with $M_i$ $4$-manifolds, exhibiting the unboundedness of the Euler class.

Unresolved qualifications. The preceding theorem is absent from the supplied record. Consequently, the precise Euler class whose unboundedness is asserted, the meaning of unboundedness, and any required normalization or norm cannot be recovered. In particular, divergence of Euler numbers alone must not be substituted for an unspecified bounded-cohomological assertion. The record also does not specify the permitted category, compactness, orientation, or boundary conditions of the manifolds, or whether their fundamental group is fixed across the family. The bundle construction above supplies notation but does not resolve these missing hypotheses. Thus the geometric-proof request is preserved, while a complete quantified unboundedness statement remains undetermined.

Short English statement

Find a geometric family of representations of fundamental groups of $4$-manifolds into $\mathrm{Homeo}_0(S^3)$ that demonstrates the Euler-class unboundedness asserted by the theorem preceding Question 24. That theorem and its precise hypotheses are missing, so the intended unboundedness criterion remains unresolved.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $800083$, source problem code catalogue entry, title Foliations and Diffeomorphism Groups --- Question 24. The registry attributes the problem to Nariman and Yazdi, with proposed year $2025$, and labels it open; this is a supplied status label, not an independently verified assessment.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 24. Supplied source URL: https://arxiv.org/abs/2503.18274. The provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not a claim of independent source access or verification.
• Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82. The extracted question refers to an earlier theorem whose text is not included in the supplied record.