  After summarising the physical approach leading to twisted homotopy and after
developing the cohomological approach further with respect to our previous work
we propose a third alternative approach to twisted homotopy based on group
theoretic considerations. In this approach the fundamental group $\Pi (m) $
isomorphic to Z which describes homotopic loops on the punctured plane$ R^2/(0)
$ is enhanced in a special way to the continuous SO(2) group . This is
performed by letting the parameter of the group $ m \rightarrow \lambda $ while
keeping its generator unchanged .It is shown that such non-trivial procedure
has the effect of introducing well defined self-interactions among loops which
are at the basis of twisted homotopy where the angle $ \lambda $ plays the role
of the self coupling constant.
  KEYWORDS: Homotopy, Group Theory, Quantum Mechanics
  MSC:55Q35; PACS:02.20.Fh ; 03.65.Fd
