  Let $M$ be a $3\times 3$ integer matrix which is expanding in the sense that
each of its eigenvalues is greater than $1$ in modulus and let $\mathcal{D}
\subset \mathbb{Z}^3$ be a digit set containing $|\det M|$ elements. Then the
unique nonempty compact set $T=T(M,\mathcal{D})$ defined by the set equation
$MT=T+\mathcal{D}$ is called an integral self-affine tile if its interior is
nonempty. If $\mathcal{D}$ is of the form $\mathcal{D}=\{0,v,\ldots, (|\det
M|-1)v\}$ we say that $T$ has a collinear digit set. The present paper is
devoted to the topology of integral self-affine tiles with collinear digit
sets. In particular, we prove that a large class of these tiles is homeomorphic
to a closed $3$-dimensional ball. Moreover, we show that in this case $T$
carries a natural CW complex structure that is defined in terms of the
intersections of $T$ with its neighbors in the lattice tiling $\{T+z\,:\, z\in
\mathbb{Z}^3\}$ induced by $T$. This CW complex structure is isomorphic to the
CW complex defined by the truncated octahedron.
