In answer to a question of T.
Mackowiak and E. D. Tymchatyn [20] we prove that every atriodic homogeneous
continuum is 1-dimensional. This is accomplished by showing that every atriodic
homogeneous continuum that is not a solenoid and has a decomposable
subcontinuum admits a continuous decomposition to a solenoid and that all elements
of this decomposition are homeomorphic tree-like hereditarily indecomposable
homogeneous continua. It follows from this decomposition theorem that every
tree-like atriodic homogeneous continuum is hereditarily indecomposable. This
decomposition theorem also provides another proof of the author’s theorem [11]
that every indecomposable homogeneous plane continuum is hereditarily
indecomposable.
