Reconstruction of $f(T)$-gravity in the absence of matter
  We derive an exact $f(T)$ gravity in the absence of ordinary matter in
Friedmann-Robertson-Walker (FRW) universe, where $T$ is the teleparallel
torsion scalar. We show that vanishing of the energy-momentum tensor
$\mathcal{T}^{\mu \nu}$ of matter does not imply vanishing of the teleparallel
torsion scalar, in contrast to general relativity, where the Ricci scalar
vanishes. The theory provides an exponential (inflationary) scale factor
independent of the choice of the sectional curvature. In addition, the obtained
$f(T)$ acts just like cosmological constant in the flat space model.
Nevertheless, it is dynamical in non-flat models. In particular, the open
universe provides a decaying pattern of the $f(T)$ contributing directly to
solve the fine-tuning problem of the cosmological constant. The equation of
state (EoS) of the torsion vacuum fluid has been studied in positive and
negative Hubble regimes. We study the case when the torsion is made of a scalar
field (tlaplon) which acts as torsion potential. This treatment enables to
induce a tlaplon field sensitive to the symmetry of the spacetime in addition
to the reconstruction of its effective potential from the $f(T)$ theory. The
theory provides six different versions of inflationary models. The real
solutions are mainly quadratic, the complex solutions, remarkably, provide
Higgs-like potential.
