notion of a portable manifold has been introduced in [@BIP] for smooth open manifolds. The definition there is specific for smooth category and a bit stronger than Def. 6.4 (a definition similar to 6.4 is also mentioned in [@BIP]).

The class of portable manifolds comprises the euclidean spaces $\R^n$, the manifolds of the form $M\t\R^n$, or the manifolds admitting an exhausting Morse function with finite numbers of critical points such that all their indices are less that $\frac{1}{2}\dim M$. In particular, every three-dimensional handlebody is a portable manifold.

If $M$ is portable that $\H_c(M)$ is bounded.

The proof is a consequence of Prop. 5.2, and is completely analogous to that for diffeomorphisms (Theorem 1.7 in [@BIP]).

If $M^o$ is portable then $\hdm$ is bounded.

The proof follows from Theorems 1.5 and 6.6. In contrast, for diffeomorphism groups we have the following

Let $M$ be a smooth manifold with boundary and let $\D^{\p}(M)$ be the subgroup of all $f\in\D(M)$ such that there exists a compactly supported isotopy $f_t$ with $f_0=\operatorname{id}$ and $f_1=f$ satisfying $f_t|_{\pm}=\operatorname{id}$ for all $t$. Then $\D^{\p}(M)$ is an unbounded group.

Choose a chart at $p\in\pm$. Then there is the epimorphism $$\D^{\p}(M)\ni f\mapsto \operatorname{Jac}_p(f)\in\R_+,$$ where $\operatorname{Jac}_p(f)$ is the Jacobian 