The graded braid complex followed by its inverse

The cochain totalization of Bᵢ ⊗ Bᵢ' is supported in degrees −1, 0 and 1. Its terms are Ui, Z ⊞ (Ui{−2} ⊞ Ui), and Ui{−2}. A sign change in the volume coordinate makes the first differential's Ui block the identity; the second differential restricts to the identity on Ui{−2}. The three-term cancellation criterion then contracts this complex to the regular graded bimodule Z in degree zero: TauCeti.zigzagBraidComplexTensorInverseHomotopyEquiv. The component comparisons use the braid complexes of TauCeti.zigzagBraidComplex and the cochain totalization signs.

References

• M. Khovanov and P. Seidel, Quivers, Floer cohomology, and braid group actions, J. Amer. Math. Soc. 15 (2002) (the complexes R_i and R_i' of the zigzag algebra A_m).
• R. S. Huerfano and M. Khovanov, A category for the adjoint representation, J. Algebra 246 (2001) (zigzag algebras of arbitrary graphs and the same complexes).

noncomputable def TauCeti.zigzagBraidComplexTensorInverseHomotopyEquiv (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (i : V) : HomotopyEquiv (zigzagBraidInverseComplexTensor k G i) (zigzagBraidUnitComplex k G)

The graded braid complex followed by its inverse is homotopy equivalent to the regular graded zigzag bimodule in cochain degree zero.