Homology conjecture for affine and linear fixing cycles

Let $X$ be a simplicial manifold. A fixing cycle is a $(3n-2)$-cycle $\phi\in Z_{3n-2}(Y,\pi^*({\mathcal D}))$ satisfying

$$\|\pi\|_*(\Omega^{n-1}\frown[\phi])=[X],$$

where $n$ is the dimension of $X$, ${\mathcal D}$ is its orientation presheaf, and $\Omega$ is the relevant cocycle. Fixing-cycle homology conjecture. All affine fixing cycles are homologous, and all linear fixing cycles are homologous.

Fixing cycles are designed as a combinatorial analogue of the fundamental class of the associated space arising from a smoothing. The conjecture would make the fixing-cycle contribution to the Pontrjagin-class formula independent of the chosen fixing cycle; its resolution is not specified in the supplied text.