Equality of the Chern-class slant-product homomorphisms

Let $\eta$ be the complex line bundle over $H^1(\Gamma;U(1))\times\check{\mathcal M}_{\Gamma}$ obtained from the equivariant ground-state line bundle, and let $\xi$ denote the comparison line bundle defined in the surrounding text. Slant product with their first Chern classes gives homomorphisms

$$c_1(\eta)/,\ c_1(\xi)/:H_1(\check{\mathcal M}_{\Gamma};\mathbb Z)\to H_1(\Gamma;\mathbb Z).$$

Equality of the Chern-class slant-product homomorphisms. The homomorphisms $c_1(\eta)/$ and $c_1(\xi)/$ coincide. This asserts that the analytically constructed line bundle and the comparison bundle determine the same homological invariant, but the supplied text gives no evidence that the assertion has been proved or refuted.