Problem 8.7 — (V.

Definitions and mathematical statement

For an abelian group $H$, an Eilenberg--Mac Lane space $K(H,2)$ has $\pi_2\cong H$ and all other positive homotopy groups zero. Maps from suitable manifold/CW inputs to this target are classified by degree-two cohomology classes with coefficients in $H$.

An HQFT is a homotopy quantum field theory: in a standard functorial formulation it assigns state spaces to structured manifolds equipped with maps to a target and linear maps to structured cobordisms carrying compatible target maps, with gluing, tensor and homotopy-invariance axioms. For spin or $\operatorname{spin}^c$ HQFTs the cobordisms and boundary objects have the corresponding tangential structures.

Study these theories with target $K(H,2)$ in all three spacetime dimensions \[ 1+1,\qquad2+1,\qquad3+1, \] for the group rendered as "$\mathbb Z N$" in the extraction. If this means $H=\mathbb Z/N\mathbb Z$, it is the cyclic-group version; $\mathbb Z^N$ would instead be a free-abelian-group target. The missing subscript/superscript typography must be confirmed, and neither reading is silently chosen.

The request "study" does not prescribe a construction, classification equivalence, output invariants, coefficient category, unitarity or extendedness convention. It retains both spin and $\operatorname{spin}^c$ variants and each dimension; no known lower-dimensional or aspherical-target theory is substituted for the $K(H,2)$ problem. The summary provides only an open/triage label and no precise theorem, so current status requires primary-literature review.

Short English statement

Develop spin and $\operatorname{spin}^c$ HQFTs with target $K(H,2)$ in dimensions two, three and four. The group typography and the intended construction/classification scope are missing, and all three dimensions remain separate targets.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $10400153$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 8.7, attributed to V. Turaev, PDF page 124: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Supplied PDF-font-structure extraction accessed 2026-07-29; provenance distinguishes 231 numbered items.
• No exact group-typography recovery, detailed HQFT convention or separate current theorem reference is supplied.