I was studying simplicial homology in Hatcher's Algebraic topology book.In one paragraph book says following: Some obvious general questions arise: Are the groups $H_n(X)$ independent of the choice of simplicial complex structure on X? In other words, if two complexes are homeomorphic, do they have isomorphic homology groups? More generally, do they have isomorphic homology groups if they are merely homotopy equivalent? To answer such questions and to develop a general theory it is best to leave the rather rigid simplicial realm and introduce the singular homology groups.

My Question is what is the answers of the above mentioned questions?? any help would be appreciated.


The answers are, broadly speaking, "yes": homology is a property that's invariant within a homoeomorphism class, i.e., if $X$ and $Y$ are homeomorphic, they end up having the same homology groups, and the same is true if they're merely homotopy equivalent.

These claims can be proved (with limitations) for simiplicial objects, but the proofs are not very pretty or general, which is why Hatcher says "let's move on to other kinds of homology theory" (which will produce, for "nice" shapes at least, the same groups). Indeed, one of the big theorems is that if two homology theories produces the same values on certain basic shapes, and they both have 5 specific properties, then they produce the same values on a wide category of shapes. (The 5 properties are known as the "Eilenberg Steenrod Axioms.")

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