Algebraic topology question (Qual) I am having trouble with a QR problem.  I would appreciate some help. Construct a connected $CW$-complex $X$ with $H_0(X, \mathbb{Z}) =
\mathbb{Z}, H_1(X, \mathbb{Z}) = \mathbb{Z}\times \mathbb{Z}/10\mathbb{Z}$ and $H_2(X, \mathbb{Z}) = \mathbb{Z}
\times \mathbb{Z}.$
Can you first find a connected space with $$H_1(X,Z) = \mathbb{Z}_{10}$$ and $$H_2 = 0$$

How about a connected space with $$H_1 = \mathbb{Z}$$ and $$H_2 = \mathbb{Z} \times \mathbb{Z}$$

Then maybe you can find a way to combine these spaces?
Hint: Make use of the Kunneth formula for two spaces where one of them has trivial $n$th-degree homology.