Handlebody

Remarks

The orientability condition fixes the way each $1$-handle is attached, and then a handlebody of genus $g$ is unique up to diffeomorphism. Its boundary $\partial V$ is the closed orientable surface of genus $g$, and $V$ is homotopy equivalent to a wedge of $g$ circles. Equivalently, for $g \ge 1$, $V$ is the boundary connected sum of $g$ solid tori $S^1 \times D^2$. Also, $V$ is a closed regular neighbourhood of a connected finite graph in $\R^3$ with Euler characteristic $1 - g$. Example: the solid torus $S^1 \times D^2$ is a handlebody of genus $1$. It is the $3$-ball $D^3$ with a $1$-handle $D^1 \times D^2$ attached along two disjoint discs of $\partial D^3$, in an orientation-compatible way. Non-example: the solid Klein bottle is obtained from $D^3$ by attaching one $1$-handle so that the result is non-orientable. It has the right handles but is not orientable, so it is not a handlebody. In higher-dimensional topology, handlebody often means any manifold with a handle decomposition, with no restriction on dimension or on the indices of the handles.