Banyaga and Hurtubise defined the Morse–Bott–Smale chain complex as a quotient of a large chain complex by introducing five degeneracy relations. However, their five degeneracy relations are in fact redundant. In the present paper, we unify these five conditions into a single degeneracy condition and resolve the issue of the well-definedness of the Morse–Bott–Smale chain complex. This provides an appropriate definition of the Morse–Bott homology and more computable examples. Moreover, we show that our chain complex for a Morse–Smale function is quasi-isomorphic to the usual Morse–Smale–Witten chain complex. As a consequence, we obtain an alternative proof of the Morse Homology Theorem.

Many transformation groups on manifolds are simple, but their universal coverings are not. In the present paper, we study the concept of relatively simple group, that is, a group with the maximum proper normal subgroup. We show that many examples of universal coverings of transformation groups are relatively simple, including the universal covering $\widetilde{\mathrm{Ham}}(M,\omega)$ of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold $(M,\omega)$. Tsuboi constructed a metric space $\mathcal{M}(G)$ for a simple group $G$. We generalize his construction to relatively simple groups, and study their large scale geometric structure. In particular, Tsuboi’s metric space of $\widetilde{\mathrm{Ham}}(M,\omega)$ is not quasi-isometric to the half line for every closed symplectic manifold $(M,\omega)$.