Abstract
We consider a locally path-connected compact metric space K with finite first Betti number
$\textrm {b}_1(K)$
and a flow
$(K, G)$
on K such that G is abelian and all G-invariant functions
$f\,{\in}\, \text{\rm C}(K)$
are constant. We prove that every equicontinuous factor of the flow
$(K, G)$
is isomorphic to a flow on a compact abelian Lie group of dimension less than
${\textrm {b}_1(K)}/{\textrm {b}_0(K)}$
. For this purpose, we use and provide a new proof for Theorem 2.12 of Hauser and Jäger [Monotonicity of maximal equicontinuous factors and an application to toral flows. Proc. Amer. Math. Soc.147 (2019), 4539–4554], which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map
$p\colon K\to L$
between locally connected compact spaces K and L that we obtain by characterizing the local connectedness of K in terms of the Banach lattice
$\textrm {C}(K)$
.
Co-rank and Betti number of a group
