Graded-fusion 2-categories and quantum homotopy invariants of 4-manifolds

Abstract: We introduce 3-group-graded extensions of fusion 2-categories, where 3-groups are modeled by 2-crossed modules. From this data, we derive a state-sum invariant of 4-manifolds equipped with a homotopy class of maps to a homotopy 3-type, or equivalently of flat 3-bundles over 4-manifolds. This invariant is nontrivial and generalizes the Douglas-Reutter invariant of 4-manifolds. To construct our invariant, we encode homotopy classes of maps to the classifying space of a 3-group $σ$ via $σ$-colorings of triangulations, and we generalize Pachner's theorem in this context.

• The paper constructs a novel framework for developing quantum invariants for 4-manifolds equipped with a homotopy class of maps to a 2-crossed module modeling a 3-group, using $\sigma$-fusion 2-categories.
• The fundamental contribution of the paper is the Homotopy Classification Theorem and Colored Pachner Theorem, proving that the resulting invariants are genuine in the sense of topologically stable.
• The developed invariant detects genuinely new information about 4-manifolds, showing non-trivial phantom map detection.

This paper develops a framework for constructing quantum invariants of closed oriented smooth 4-manifolds equipped with a homotopy class of maps to a connected homotopy 3-type. The construction generalizes the Douglas–Reutter state-sum invariant of 4-manifolds (Billingham, 2018) by replacing fusion 2-categories with new objects called $\sigma$-fusion 2-categories, graded by a 2-crossed module modeling a 3-group. The main result is that the resulting state sum is a genuine invariant of such "σ-manifolds," equivalently of 4-manifolds with flat 3-bundles.

Background and motivation

The Turaev–Viro–Barrett–Westbury invariant of 3-manifolds is defined as a state sum over triangulations using quantum 6j-symbols from a spherical fusion category, and extends to homotopy quantum field theories (HQFTs) with target $K(G,1)$ via $G$-graded fusion categories, and further to targets that are connected homotopy 2-types modeled by crossed modules, following Sözer–Virelizier. In dimension four, Douglas and Reutter constructed invariants of closed 4-manifolds from spherical fusion 2-categories, subsuming the Crane–Yetter–Kauffman, Yetter–Dijkgraaf–Witten, Mackaay, and Cui invariants. The present work completes the pattern at the next categorical level: it adapts the Douglas–Reutter state-sum machinery to 4-manifolds with maps into the classifying space of a 2-crossed module

$$\sigma = (L \xrightarrow{\delta} E \xrightarrow{\partial} H,\ \omega),$$

where the Peiffer lifting $\omega\colon E\times E\to L$ encodes the nontrivial braiding data of the associated 3-group. A σ-manifold $(W,g)$ consists of a closed oriented smooth 4-manifold together with a homotopy class of maps $g\colon W\to B\sigma$; this data is equivalent to a flat 3-bundle over $W$.

Homotopy classification via colorings

A central technical contribution is an unpointed version of Faria Martins' fundamental crossed complex machinery. The author defines the fundamental 2-crossed module $\Pi_3(K)$ of a canonical simplicial complex $K$, built from triad homotopy groupoids of the form $K(G,1)$0, where the decomposition $K(G,1)$1 comes from a choice of distinguished vertex on each triangle. The main classification result states:

$K(G,1)$2

i.e., unpointed homotopy classes of maps correspond canonically to gauge-equivalence classes of morphisms of 2-crossed modules ("σ-colorings"). Gauge equivalence is formulated through quadratic derivations acting on colorings; the paper constructs a gauge groupoid $K(G,1)$3 whose morphisms are quadratic derivations, proves it is indeed a groupoid (with explicit inverses), and shows gauge equivalence of colorings matches homotopy of their geometric realizations, using the Quillen model structure on 2-crossed modules of groups due to Cabello–Garzón for cofibrant replacement when $K(G,1)$4 is not free.

Graded-fusion 2-categories

The algebraic input is a categorification of Sözer–Virelizier's crossed-module-graded monoidal categories. A σ-graded linear monoidal 2-category has objects graded by $K(G,1)$5, 1-morphisms by $K(G,1)$6, and 2-morphisms by $K(G,1)$7, with the tensor product compatible with the $K(G,1)$8-action and—crucially—the interchange 2-isomorphism carrying degree $K(G,1)$9, so the Peiffer lifting controls the Gray-type interchange. A σ-fusion 2-category is then a finite σ-semisimple such category with duals, simple unit, and suitable finiteness; its neutral component recovers a fusion 2-category up to idempotent and additive completion, though a σ-fusion 2-category need not itself be a fusion 2-category. Two canonical examples are developed: the linearization $G$0 of any 2-crossed module, which is always spherical σ-fusion, and pushforwards along surjective morphisms of 2-crossed modules with finite kernels.

Dimensions are treated carefully: dimensions of simple objects and simple 1-morphisms in the 1-subcategory $G$1 are shown invertible in $G$2, the trace pairing is nondegenerate, and weighted dimensions and categorical dimensions are introduced. When $G$3 is an algebraically closed field of characteristic zero, every spherical σ-fusion 2-category automatically has invertible dimensions, so the hypothesis is mild.

The state sum and colored Pachner moves

Given a spherical σ-fusion 2-category $G$4 with invertible dimensions, a simplicial skeleton $G$5, and a σ-triangulation $G$6 of $G$7, the invariant is the state sum

$G$8

summing over $G$9-states labeling edges by homogeneous simple objects and triangles by homogeneous simple 1-morphisms compatible with the coloring $$\sigma = (L \xrightarrow{\delta} E \xrightarrow{\partial} H,\ \omega),$$0. The weights $$\sigma = (L \xrightarrow{\delta} E \xrightarrow{\partial} H,\ \omega),$$1 are built from colored 10j-symbols, which generalize the Douglas–Reutter 10j-symbols by incorporating the grading constraints from $$\sigma = (L \xrightarrow{\delta} E \xrightarrow{\partial} H,\ \omega),$$2. The main theorem asserts this scalar depends only on the equivalence class of the σ-manifold.

Invariance rests on a Colored Pachner Theorem: two combinatorial σ-manifolds are equivalent if and only if they are related by finite sequences of σ-colored Pachner moves. This is proved by combining the Homotopy Classification Theorem with a generator decomposition of the gauge groupoid—any quadratic derivation factors into generating derivations supported on single basis simplices—and an explicit five-step sequence of colored moves realizing each generator. The state sum is then checked against the three colored Pachner move types ((1,5), (2,4), (3,3)), with proofs reducing to the Douglas–Reutter arguments plus the dimension formulas established earlier. Independence from the choice of ordered simplices and simplicial skeleton is also verified directly.

Nontriviality

The paper demonstrates the invariant detects genuinely new information. Taking $$\sigma = (L \xrightarrow{\delta} E \xrightarrow{\partial} H,\ \omega),$$3, so $$\sigma = (L \xrightarrow{\delta} E \xrightarrow{\partial} H,\ \omega),$$4, and $$\sigma = (L \xrightarrow{\delta} E \xrightarrow{\partial} H,\ \omega),$$5, one has $$\sigma = (L \xrightarrow{\delta} E \xrightarrow{\partial} H,\ \omega),$$6. For a suitable σ-fusion 2-category constructed as a pushforward of a linearization, the trivial class yields $$\sigma = (L \xrightarrow{\delta} E \xrightarrow{\partial} H,\ \omega),$$7 (coinciding with the untwisted Yetter–Dijkgraaf–Witten invariant), while any nontrivial class $$\sigma = (L \xrightarrow{\delta} E \xrightarrow{\partial} H,\ \omega),$$8 gives $$\sigma = (L \xrightarrow{\delta} E \xrightarrow{\partial} H,\ \omega),$$9, since no lift to the contractible $\omega\colon E\times E\to L$0 can exist. Notably, all maps $\omega\colon E\times E\to L$1 here are phantom maps—they induce trivial maps on all homotopy groups because $\omega\colon E\times E\to L$2—so the invariant distinguishes homotopy classes invisible to ordinary homotopy-group-based invariants. This is the strongest concrete evidence in the paper that the construction carries substantive content beyond the trivial-target case, where it reduces exactly to the Douglas–Reutter invariant of the neutral component.

Limitations and open questions

Several points remain open or conditional. The extension of the scalar invariant to a full 4-dimensional HQFT—with functorial assignments of modules to colored 3-manifolds and linear maps to cobordisms—is anticipated but not carried out; the module attached to a colored closed 3-manifold is defined en route to the boundary-gluing lemma, but the HQFT structure is left for future work. The conjectured recovery of Mochida's Hopf-$\omega\colon E\times E\to L$3-algebra invariants and Bridges–Cui's Hopf $\omega\colon E\times E\to L$4-triplet invariants as special cases (for $\omega\colon E\times E\to L$5) relies on deloopings of representation categories yielding spherical σ-fusion 2-categories, which is asserted only as an expectation. Proofs of invariance under the colored (2,4) and (3,3) Pachner moves, and of the analogous coloring lemmas, are stated to follow the Douglas–Reutter proofs and are "left to the reader" or "left as an exercise," so the complete verification of some steps is not written out in full detail. Finally, the theory is developed over an arbitrary nonzero commutative ring, but the automatic invertibility of dimensions—and hence unconditional applicability—is only established over algebraically closed fields of characteristic zero.

Conclusion

The paper establishes a coherent bridge between higher gauge theory (flat 3-bundles, 2-crossed modules) and categorified quantum topology (state sums from fusion 2-categories). Its two structural results—the unpointed Homotopy Classification Theorem and the Colored Pachner Theorem—are of independent combinatorial-topological interest, and the phantom-map example confirms the resulting invariants are nontrivial. The natural open problems are the construction of the surrounding 4-dimensional HQFT and the precise identification of previously known 4-dimensional invariants within this framework.