In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle. It is homeomorphic to the Cartesian product
S
1
×
D
2
of the disk and the circle, endowed with the product topology. A standard way to visualize a solid torus is as a toroid, embedded in 3-space. However, it should be distinguished from a torus, which has the same visual appearance: the torus is the two-dimensional space on the boundary of a toroid, while the solid torus includes also the compact interior space enclosed by the torus.
The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary. The boundary is homeomorphic to
S
1
×
S
1
, the ordinary torus.
Since the disk
D
2
is contractible, the solid torus has the homotopy type of a circle,
S
1
. Therefore the fundamental group and homology groups are isomorphic to those of the circle:
π
1
(
S
1
×
D
2
)
≅
π
1
(
S
1
)
≅
Z
,
H
k
(
S
1
×
D
2
)
≅
H
k
(
S
1
)
≅
{
Z
if
k
=
0
,
1
,
0
otherwise
.
