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While studying homology, why does one take free abelian groups over the chain complex, how does that group is related to the space, of which we are calculating homology group.

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 Dear Girish. I don't want to discourage you from asking questions here. However, I find this question rather poorly formulated. What kind of answer do you want??? Your question is "why does one do this?". Well... there isn't really a good answer. You could improve your question by formulating it e.g. in the form "why does one do this instead of that?" – André Henriques Nov 8 2011 at 16:40
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 I think there's a nice question underlying this, which maybe you'd like to edit your question to: What was Poincare's motivation for considering the free abelian group generated by simplices in a simplicial complex, and why is the resulting chain complex a natural place to look for invariants? Maybe as a question it's better for math.stackexchange.com , but anyway it's a nice question whose answer one doesn't learn in introductory algebraic topology courses- the answer has to do perhaps with the idea that a homology class is somehow an "abstraction" of an immersed closed submanifold. – Daniel Moskovich Nov 8 2011 at 16:57
Perhaps take a look at the first few sections of Dieudonne's History of Algebraic and Differential Topology where he discusses Poincare's work leading up to Poincare Duality. – Ryan Budney Nov 8 2011 at 18:23
Dear Andre, I apologize for not putting a precise question. Daniel has helped me for that by suggesting the right words, thank you Daniel. Also thanks Ryan for the source. – Girish Nov 8 2011 at 20:33

closed as off topic by Andrew Stacey, Andreas Blass, Simon Thomas, Qiaochu Yuan, Charles Matthews Nov 8 2011 at 19:01

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