Topological geons with self-gravitating phantom scalar field

A topological geon is the quotient manifold $M\mathrm{/}{Z}_2$ where $M$ is a static spherically symmetric wormhole having the reflection symmetry with respect to its throat. We distinguish such asymptotically flat solutions of the Einstein equations according to the form of the time-time metric function by using the quadrature formulas of the so-called inverse problem for self-gravitating spherically symmetric scalar fields. We distinguish three types of geon spacetimes and illustrate them by simple examples. We also study possible observational effects associated with bounded geodesic motion near topological geons.