Foliations and Diffeomorphism Groups — Question 71

Definitions and mathematical statement

Let $X=B\Gamma_H$ denote the space appearing in the supplied source. The record does not define $\Gamma_H$ or specify the construction of $B\Gamma_H$; consequently, identifying the intended space remains an unresolved prerequisite for a standalone formulation. Write $H_2(X;{\mathbb{Z}})$ for its second singular homology group with integer coefficients. For a closed oriented surface $\Sigma$ and a continuous map $f\colon\Sigma\to X$, the associated surface cycle is $f_*[\Sigma]\in H_2(X;{\mathbb{Z}})$, where $[\Sigma]$ is the fundamental class.

The source supplies the background assertions \[ \pi_2(X)=0, \qquad f_*[T^2]=0\quad\text{for every continuous }f\colon T^2\to X, \] where $T^2=S^1\times S^1$. It also states that there are no nonzero spherical cycles, meaning that $g_*[S^2]=0$ for every continuous map $g\colon S^2\to X$. These assertions are retained as supplied background, without independent verification.

The question, attributed in the source to Tsuboi, is: \[ \text{Is }H_2(B\Gamma_H;{\mathbb{Z}})=0\text{?} \] The intended meaning of $B\Gamma_H$ must be recovered from the cited source before this question can be regarded as fully specified.

Short English statement

Determine whether the second integral homology of the source's space $B\Gamma_H$ vanishes. The source states that spherical and torus cycles vanish, but the supplied record does not define the space itself.

Sources

• Local UnsolvedMath registry: permanent ID $800130$, exact source problem code catalogue entry, title Foliations and Diffeomorphism Groups --- Question 71. The supplied registry labels the problem open; this label is not an independently verified mathematical or literature assessment.
• Nariman, Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 71. Supplied source URL: https://arxiv.org/abs/2503.18274. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent access.
• The question is attributed within the supplied statement to Tsuboi, with reference identifier MR3726712. No full bibliographic entry for that reference is supplied.