I am trying to compute the relative homology groups $H_n(X, T)$ where $X$ is the solid torus $S^1 × D^2$ and $T$ is the subspace $S^1 × S^1$.
I have computed the homology groups of the solid torus ( $\mathbb{Z}$ for $n = 0,1$ and trivial otherwise) and of the torus ( $\mathbb{Z}\oplus\mathbb{Z}$ for $n = 1 $ , $\mathbb{Z}$ for $n = 0, 2$ and trivial otherwise) but I am struggling to using the long exact sequence of a pair $(X, T)$.
I am not sure what the generators of the homology groups of $X$ will be. Any help / example of similar solution would be appreciated.
Best Answer
Using LES: $$ H_2(X)\to H_2(X,A)\to H_1(A)\to H_1(X)\to H_1(X,A)\to H_0(A)\to H_0(X)\ldots $$ As $H_2(X)=0$, the second arrow is an injection. The third arrow, after the obvious identifications, is $(a,b)\mapsto b$ which has kernel $\{(a,0)\}$ and hence $H_2(X,A)\simeq \Bbb Z$. It is generated by the meridian $2$-cell that has boundary in $A$. The last map is an isomorphism, so the second-last is a zero map; but the third map is onto, so the fourth map is zero as well: it follows that $H_1(X,A)=0$. (Intuitively, the only candidate for $H_1(X,A)$ is a circle going around the solid torus, but this can be homotoped to the boundary).