The coarse bounded H-complexity conjecture

Let $F$ be a finite PL map, and let coarse bounded sublevel and superlevel $H$-complexity and global $H$-complexity be the homological complexity measures defined for $F$.

Coarse bounded H-complexity conjecture. The coarse bounded sublevel $H$-complexity (respectively, the coarse bounded superlevel $H$-complexity) of $F$ is a lower bound for the global $H$-complexity of $F$.

This conjecture motivates the terminology “coarse”; the preceding theorem establishes the corresponding lower bounds for the coarse bounded sublevel and superlevel $H$-complexities in terms of connected components. Its general status is not resolved in the supplied text.