Problem 8.5 — Formulate and find spin c TQFT’s.

Definitions and mathematical statement

For $n\ge2$, the group \[ \operatorname{Spin}^c(n)= (\operatorname{Spin}(n)\times\mathrm U(1))/\langle(-1,-1)\rangle \] maps naturally to $\mathrm{SO}(n)$. A $\operatorname{Spin}^c$ structure on an oriented manifold is a lift of its oriented orthonormal frame bundle through this map, with the corresponding stable-tangent and boundary conventions when used in a cobordism category.

Problem 8.5 asks to "formulate and find $\operatorname{Spin}^c$ TQFTs." A usual formal interpretation, after selecting dimension $n$ and coefficient field $F$, would define a $\operatorname{Spin}^c$ cobordism category with closed decorated $(n-1)$-manifolds as objects and compatible decorated $n$-cobordisms as morphisms. It would then construct symmetric monoidal functors \[ Z:\operatorname{Cob}^{\operatorname{Spin}^c}_n \longrightarrow\operatorname{Vect}_F, \] assigning state spaces and cobordism maps that respect identities, gluing and disjoint union. This is explanatory rather than a source-fixed axiom system: the record does not specify dimension, target category, finite-dimensionality, linearity conventions, anomaly treatment, or the exact handling of boundary structures.

Both the formulation and the construction are requested. The source does not prescribe which $\operatorname{Spin}^c$ invariants should be recovered, whether nontriviality or detection of the $\operatorname{Spin}^c$ structure is required, or what distinguishes an acceptable example from a theory that forgets the extra structure. Those criteria must be recovered from context rather than answered by an arbitrarily chosen trivial theory.

Short English statement

Specify a suitable notion of field theory for manifolds equipped with $\operatorname{Spin}^c$ structures and construct examples. The source gives no dimension or target axioms and no required invariant or nontriviality criterion. Its supplied classification is open-triage.

Sources

• Local UnsolvedMath registry, permanent ID $10400151$, exact source code catalogue entry, Problem 8.5. The extracted "spin c" is read as $\operatorname{Spin}^c$. No individual proposer or proposed year is supplied.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 8.5, PDF page 124: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Bold printed headings distinguish 231 numbered items from surrounding references.
• Extraction pdf-font-structure, accessed 2026-07-29. Source evidence NEEDS_REVIEW presents an open problem and requires a current primary-literature audit. Supplied research classification OPEN-TRIAGE repeats the formulation-and-construction request without further references.
• Dataset attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. Source rights NEEDS_REVIEW: publicly accessible author/publisher copy, redistribution terms requiring review.