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i am currently taking a basic course in homology theory.i am not able to understand the motivation behind the formula of the homology groups.

The best I have been able to find online or in my limited book selection is the brief description "intuitively, the zeroth homology group counts how many disjoint pieces make up the shape and gives that many copies of ℤ, while the other homology groups count different types of holes".now, the approach that we use to find out the homology groups is the concept of "Cycles modulo boundaries" i.e, For a simplicial complex $X$, the $n^{th}$ homology $H_n(X)$ is $Z_n/B_n$, where $Z_n = \ker(d_n : C_n \to C_{n-1})$ is the group of cycles and $B_n = \text{im}(d_{n+1} : C_{n+1} \to C_n)$ is the group of boundaries.

how is the $n^{th}$ homology groups i.e, $H_n(X)$ = $Z_n/B_n$ gives us the idea of presence of n dimensional holes in the space.please give a complete and detailed explanation.

thanks a lot.....

Abhishek Shrivastava
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  • http://math.stackexchange.com/questions/40149/intuition-of-the-meaning-of-homology-groups?rq=1 might be helpful. Further more the first homology group is just the abelisation of the fundamental group (Hurewicz theorem). – Daniel Bernoulli Feb 22 '16 at 14:02
  • I think you have to look at the history of the subject- see the book by Dueudonn'e and the volume "History of Topology" edited by I M James. The notion of homology was originally connected with integration theory. When I was young I was recommended the Introduction to Lefschetz's "Introduction to Topology" , and nothing else in that book. See also my and other answers in the link given by Daniel. Do not assume the development of the subject has assumed a final form given in standard texts! – Ronnie Brown Feb 22 '16 at 17:23

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