Homology groups of $D^2\times S^1$, full torus I know what are the homology groups of a torus $T=S^1\times S^1$, in sense that $$\tilde{H}_1(T)=\mathbb{Z}^2,H_2(T)=\mathbb{Z}$$ but I wonder what happens if we fill it.
What are the homology groups of $X=D^2\times S^1$?
I think all of them are zero but that seems incorrect.

The component $D^2$ is contractible, so $H_*(D^2\times S^1)=H_*(S^1)$, because homology are homotopy invariant.
It is well-known that $H_1(S^1)=\mathbb Z$.
