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W. W. Comfort - Continuous cross sections on abelian groups equipped with the Bohr topology



W. W. COMFORT, Wesleyan University
Continuous cross sections on abelian groups equipped with the Bohr topology


All groups here are Abelian. A closed subgroup H of a topological group G is a ccs-subgroup if there is a continuous cross section from G/H to G--that is, a continuous function $\Gamma$such that $\pi\circ\Gamma=\textrm{id}\vert _{G/H}$ (with $\pi\colon G\rightarrow
G/H$ the natural homomorphism).

The symbol $G^{\char93 }$ denotes G with its Bohr topology, i.e., the topology induced by $\textrm{Hom}(G,{\Bbb T})$.

A topological group H is an absolute ccs-group $({\char93 })$ [resp., an absolute retract $({\char93 })$] if His a ccs-subgroup [resp., is a retract] in every group of the form $G^{\char93 }$ containing H as a (necessarily closed) subgroup. One then writes $H\in\textrm{ACCS}({\char93 })$ [resp., $H\in\textrm{AR}({\char93 })$].


Theorem 1. Every ccs-subgroup H of a group of the form $G^{\char93 }$ is a retract of $G^{\char93 }$ (and $G^{\char93 }$ is homeomorphic to $(G/H)^{\char93 }\times H^{\char93 }$); hence $\textrm{ACCS}({\char93 }) \subseteq \textrm{AR}
({\char93 })$.


Theorem 2.  $H^{\char93 }\in\textrm{ACCS}({\char93 })$ [resp., $H^{\char93 }\in\textrm{AR}({\char93 })$] iff $H^{\char93 }$ is a ccs-subgroup of its divisible hull $\bigl({\rm div}\,(H)\bigr)^{\char93 }$ [resp., $H^{\char93 }$ is a retract of $\bigl({\rm div}\,(H)\bigr)^{\char93 }$].


Theorem 3. (a)  Every cyclic group is in $\textrm{ACCS}({\char93 })$.
(b)  The classes
$\textrm{ACCS}({\char93 })$ and $\textrm{AR}({\char93 })$ are closed under finite products.


Theorem 4. Not every Abelian group is in $\textrm{ACCS}({\char93 })$.

Question [van Douwen, 1990]. Is every Abelian group in $\textrm{AR}({\char93 })$?

* Presented in Kingston by this co-author.


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