Problem 3.3 — .

Definitions and mathematical statement

Let $\Gamma_\infty$ be the stable mapping class group under the source's genus-stabilization convention, and $B\Gamma_\infty$ its classifying space. In the cohomological reading of the extracted notation, the problem asks for a direct description of some particular simple torsion classes in \[ H^*(B\Gamma_\infty;{\mathbb{Z}}). \] A torsion class here is a nonzero $x\in H^q(B\Gamma_\infty;{\mathbb{Z}})$ with $mx=0$ for some integer $m>0$. The target is explicit examples with a direct construction and justified torsion/nonvanishing, not merely an abstract existence result or a computation of rational classes. The source does not specify degrees, orders, a simplicity criterion or the allowed meaning of direct.

It additionally asks to study the relation of higher Reidemeister torsion from [10] to the classes $\kappa_i$, noting that both [10] and [14] use parametrized Morse theory, and asks whether "modular higher Reidemeister torsion" could relate to this problem. The usual $\kappa_i$ are Miller--Morita--Mumford surface-bundle classes, expressed by fiber integration of the $(i+1)$-st power of the vertical Euler class, in degree $2i$; the intended relative/boundary version is not supplied. No definition of modular torsion or comparison identity is printed.

The following context concerns a different stable group: for $F_g$ free on $g$ generators, extend each automorphism by fixing the new generator to form $A_\infty=\varinjlim_g\operatorname{Aut}(F_g)$. This is the stable group denoted $\operatorname{Aut}(F_\infty)$ in the source, not automatically the full automorphism group of a countably generated free group. The source reports a perfect index-two subgroup, forms $BA_\infty^+$, and attributes to A. Hatcher [8] a splitting in which the stable sphere infinite-loop space, printed $\Omega^\infty S^\infty$, is a homotopy direct factor of ${\mathbb{Z}}\times BA_\infty^+$. This background is not strengthened to an equivalence or used as a solution to the mapping-class torsion problem.

Short English statement

Construct explicit simple integral torsion classes in stable mapping-class cohomology, and investigate whether a modular form of higher Reidemeister torsion explains them or their relation to the $\kappa_i$ classes.

Sources

• Problems on Mapping Class Groups and Related Topics, edited by Farb, Problem 3.3 and following discussion, supplied PDF page 104. https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Recorded access 2026-07-29; extraction pdf-font-structure.
• Source [10] for higher Reidemeister torsion, [14] for the other parametrized-Morse-theory construction, and A. Hatcher [8] for the stable free-group splitting. Full bibliography entries are absent. The supplied summary attributes torsion existence to Galatius and higher-Reidemeister-torsion expressions to Igusa without complete citations.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $11000126$, exact source code catalogue entry; supplied OPEN-TRIAGE summary reports no located direct elementary construction.