Creating a topological space a $1$ and a $2$-dimensional holes w.r.t homology I am trying to figure out how to create a topogical space with homology modules 
$$H_0X\cong H_1X \cong H_2X\cong \mathbb{Z}.$$
Intuitively, $H_0$is the number of path connected components, so this shouldn't be a problem. Similarly, using geometric intuition, one realizes that $H_i$ are roughly speaking the number of $i$-dimensional holes, and thus is led to the conclusion that "glueing" together the spaces $S^{2}$ and $S^1$, $X=S_1\cup_\alpha S_2$.
I am however left here with tho problems:


*

*How do I give explicitly the map $\alpha$ along which I attach $S^2, S^1$?

*How do I compute the Homology modules of the attached space? I only have access to a Theorem giving me an exact sequence for attaching $n-$discs, I don't know how to "transform" this to the case of $n$-spheres, which is the case I am in.


I would appreciate any help, hints or references to solutions.
Kindly
afightingchance

Just take the wedge sum $S^2\vee S^1$ (as the simpliest version of glueing) since 
$$H_*(X\vee Y)\simeq H_*(X)\oplus H_*(Y)$$
for $*>0$. For the proof look here.
Since $S^2\vee S^1$ is connected then $H_0(S^2\vee S^1)\simeq\mathbb{Z}$ and you're done.
