6 problems

Omega-bounded Lefschetz fixed-point conjecture. Let $f\colon M \to M$ be a (continuous) map of an $c9$-bounded manifold. If the Lefschetz number $b7_f$ is non-zero, then $f$ has a…

Lefschetz homology conjecture. Let $M$ be an $c9$-bounded manifold. Then $H_p(M,\mathbb Q)$ is finite dimensional for each $p$ (and of course vanish above the dimension of $M$ (Mil…

Omega-bounded Lefschetz-space conjecture. $c9$-bounded $n$-manifolds have finite-dimensional (singular) homology over the rationals, and are Lefschetz spaces, i.e., any self-map wi…

Euler-characteristic fixed-point conjecture. An $c9$-bounded $n$-manifold $M$ with $c7(M)\neq 0$ has the ffp for flows.

Non-metric three-dimensional hairy ball conjecture. Any simply-connected $c9$-bounded non-metric $3$-manifold has the fpp for flows.

Cytoplasmic decomposition conjecture. Any separable $2$-manifold with countable fundamental group has a cytoplasmic decomposition.