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 $H_2(X, \mathbb{Z}) = \mathbb{Z} \times \mathbb{Z}.$
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1What does QR mean? (I have upvoted your question, by the way) – Georges Elencwajg Mar 17 '13 at 12:39
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I meant qualifying exam problem. I thought that was abbreviated QR (Qualifying Review?). – dinky Mar 17 '13 at 18:48
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1Maybe, but what's the advantage of using an acronym ? – Georges Elencwajg Mar 17 '13 at 18:56
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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?
rhl
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Hint: Make use of the Kunneth formula for two spaces where one of them has trivial $n$th-degree homology.
Chris Gerig
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This "hint" is much too vague . In particular why introduce an irrelevant (and non-quantified) $n$ ? The purpose of this site is to help people, not confuse them. – Georges Elencwajg Mar 17 '13 at 12:43
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Disagree. In fact, a good hint would simply be "use Kunneth formula"... I just decided to add an extra hint, which is definitely not irrelevant. – Chris Gerig Mar 17 '13 at 23:15