Compare geometric and singular bar constructions

Construct a natural $A_{\infty}$-quasi-isomorphism from the geometric cochain algebra $C^*_{\Gamma}(-)$ to the singular cochain algebra $C^*_{\operatorname{Sing}(-;\mathbb Z)}$, equivalently proving that their associated bar constructions are naturally quasi-isomorphic as differential graded coalgebras.

Background

The paper uses geometric cochains because they represent cohomology classes by immersed surfaces and make intersection calculations explicit. Singular cochains, by contrast, provide the standard cochain model whose bar construction is connected to established algebraic descriptions of link invariants.

A natural comparison between these models would prove that the geometrically computed bar invariants agree with the singular-cochain bar invariants, validating the surface-based calculations as computations of the intrinsic bar-cohomological link invariant.

References

There exists a natural $A_{\infty}$-quasi-isomorphisms

C*_{\Gamma}(-) \longrightarrow C*_{\operatorname{Sing}(-;Z).

Equivalently, the respective bar constructions are naturally quasi-isomorphic as dg coalgebras.

— Bar cohomology of links: beyond Milnor invariants  (2609.11009 - Friedman et al., 10 Sep 2026) in Section 3, subsection “Geometric calculations of the leading-term invariant”