I am currently studying a bit of homology theory (on topological spaces). Let $H_n(X)$ denote the singular homology groups of the topological space $X$, then as you know we can define the singular homology with coefficients in the abelian group $G$ by
$$H_n(X;G)=H_n(S_*(X)\otimes G)$$
Now, I understand the definitions and computations, proofs etc., but I would like to understand the following:
- Why is this interesting to study? Does it make computations easier somewhere (for some groups $G$)? Does it allow us to prove interesting theorems we can't prove using $G=\mathbb{Z}$?
- What are interesting examples of use of homology with coefficients?
- Is the theory of "degree modulo $2$" (treated e.g. in Milnor's book Topology from the differentiable viewpoint) in truth just degree theory on $H_n(X;\mathbb{Z}_2)$?
Best Answer
You can find the construction of homology with general coefficients and the universal coefficient theorem in Hatcher's Algebraic Topology, which is available free from his website.
The answer to your third question is yes.
The answer to the second part of your first question is yes, especially in the case that we take $G$ to be a field, most often finite or $\mathbb{Q}$, or $\mathbb{R}$ in differential topology. Homology over a field is simple because $\operatorname{Tor}$ always vanishes, so you get e.g. an exact duality between homology and cohomology. Homology with $\mathbb{Z}_2$ coefficients is also the appropriate theory for many questions about non-orientable manifolds-their top $\Bbb{Z}$-homology is zero, but their top $\Bbb{Z}_2$ homology is $\Bbb{Z}_2$, which leads to the degree theory in Milnor you were mentioning.
Cohomology with more general coefficients than $\mathbb{Z}$ is even more useful than homology. For instance it leads to the result that if a manifold $M$ has any Betti number $b_i(M)<b_i(N)$, where $b_i$ is the rank of the free part of $H_i$, there's no map $M\to N$ of non-zero degree. This has lots of quick corollaries-for instance, there's no surjection of $S^n$ onto any $n$-manifold with nontrivial lower homology! Edit: This is obviously false, and I no longer have any idea whether I meant anything true.
But in the end $H_*(X;G)$ is more of a stepping stone than anything else; it gets you thinking about how much variety there could be in theories satisfying the axioms of homology. It turns out there's almost none-singular homology with coefficients in $G$ is the only example-but if we rid ourselves of the "dimension axiom" $$H_*(\star)=\left\{\begin{matrix}\mathbb{Z},*=0\\0,*>0\end{matrix}\right.$$ then we get a vast collection of "generalized (co)homology theories," beginning with K-theory, cobordism, and stable homotopy, which really do contain new information. In some cases, so much new information that we can't actually compute them yet!