Editing Solid torus (section)

==Topological properties==

The solid torus is a [[connected space|connected]], [[compact space|compact]], [[Orientation (mathematics)|orientable]] 3-dimensional [[manifold]] with boundary.  The boundary is homeomorphic to <math>S^1 \times S^1</math>, the ordinary [[torus]].

Since the disk <math>D^2</math> is [[contractible]], the solid torus has the [[homotopy]] type of a circle, <math>S^1</math>.<ref>{{citation|title=Nilpotence and Periodicity in Stable Homotopy Theory|volume= 128 |series= Annals of mathematics studies|first=Douglas C.|last=Ravenel|publisher=[[Princeton University Press]]|year=1992|isbn= 9780691025728 |page=2|url=https://books.google.com/books?id=RA18_pxdPK4C&pg=PA2}}.</ref> Therefore the [[fundamental group]] and [[Homology (mathematics)|homology]] groups are [[isomorphism|isomorphic]] to those of the circle:
<math display=block>\begin{align}
  \pi_1\left(S^1 \times D^2\right) &\cong \pi_1\left(S^1\right) \cong \mathbb{Z}, \\
    H_k\left(S^1 \times D^2\right) &\cong H_k\left(S^1\right) \cong \begin{cases}
      \mathbb{Z} & \text{if } k = 0, 1, \\
      0          & \text{otherwise}. 
    \end{cases}
\end{align}</math>