Fiber bundles and Lie algebras of top spaces (Q459239)

Top spaces (see, for instance, [\textit{M. R. Molaei} and \textit{M. R. Farhangdoost}, Balkan J. Geom. Appl. 14, No. 1, 46--51 (2009; Zbl 1185.22015)]) are generalizations of Lie groups. A top space is a smooth manifold \(T\) with multiplication which is a smooth mapping. Moreover, for each \(t\in T\) there exists a unique element \(t\in T\) called the identity of \(T\) such that \(te(t)=e(t)t=t\). Additionally, it is assumed that for each \(t\in T\) there exists a unique element \(t^{-1}\in T\) such that \(tt^{-1}=tt^{-1}=e(t)\). In the paper top spaces are studied with a finite number of identity elements and their subtops. The author constructs for each top space \(T\) with a finite number of identity elements the principal fiber bundle with the structural group \(e^{-1}(e(t_0))\), where \(t_0\in T\). If \(T\) is a top space with a finite number of identity elements, then \(e^{-1}(e(t))\) are diffeomorphic for all \(t\in T\). A characterization of one-dimensional and two-dimensional top spaces as fiber bundles is given.