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We just went over homology with general coefficients in topology and did some of the usual examples ($\mathbb{Z}_2$ for projective space and manifolds being the big examples) which led me to wonder whether there is a "right" set of coefficients for a given space? With a lot of hand waiving I can kind of see why I should expect $\mathbb{Z}_2$ to be "correct" for manifolds and projective space given the binary nature of orientability and the construction of projective space, but I don't feel I have a good grasp of what is going on.

I am starting to think that the issue here is that I do not have a firm grasp on what homology is "counting" in general (beyond of course "cycles modulo boundaries") and why I might want to move to a different group other than comptuational ease.

Apologies for these questions being vague, but I am really just looking to gain some intuition for what is going on. My first question is if the hand waiving I described above can be made more rigorous by way of viewing a group acting on an appropriate object (like $\mathbb{Z}_2$ over $S^n$ generating $\mathbb{RP}^n$)? If so, does such a method generalize to other groups? By this I mean if there is some structure in an appropriate group action on something related to some space should I expect to see corresponding structure when viewing homology groups of that space with that group as the coefficients?

Basically any good way of thinking about what is going on here would be very much appreciated.

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Qiaochu, perhaps I am just not understanding the meaning of the universal coefficient theorem but my understanding is that although the homology over $\mathbb{Z}$ determines the homology over other groups, we can still find structure which is not apparent in the homology over $\mathbb{Z}$ by switching to another group. The example we were presented with is a nonconstant attaching map which has induces the homology of a constant map when viewed over $\mathbb{Z}$ but not over $\mathbb{Z}_m$. –  Chris Janjigian Feb 4 '12 at 18:05
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Your question does not have an answer, for you have not explained what you mean by correct. Correct for what purpose? If you intend to count fixed points of a map $M\to M$ with $M$ a manifold,for example, homology with coefficients in $\mathbb Z_2$ is pretty useless. –  Mariano Suárez-Alvarez Feb 4 '12 at 18:35
    
That is an excellent point and is more or less the question I am trying to ask. I would like to know if there is a higher level way of thinking about the kinds of structure one sees when one changes the coefficients of the homology group and specifically, if I could predict what kind of structure would become apparent by considering some related structure from the space. –  Chris Janjigian Feb 4 '12 at 18:43
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Your choice of coefficients has little to do with the space you're studying. It has much more to do with what you're trying to use homology to study. Why not look at examples where homology with non-integer coefficients come up naturally -- like Stiefel-Whitney classes and obstruction theory? –  Ryan Budney Feb 4 '12 at 18:55

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I am starting to think that the issue here is that I do not have a firm grasp on what homology is "counting" in general (beyond of course "cycles modulo boundaries") and why I might want to move to a different group other than comptuational ease.

(a) One way to visualize homology is to think about cellular homology. Here, we form a chain complex which is free abelian on the number of cells in each dimension, and our differential is determined by the degrees of the various attaching maps. Somehow this measures how twisted up and far away our CW-complex is from being just a wedge of spheres. In the case of $\mathbb{R}P^{n}$, for example, there's only one cell in each dimension, and the degrees of the attaching maps are either $0$ or $2$; in particular, they are always even! So if we take cellular homology with $\mathbb{Z}/2$ coefficients, the maps all become zero. In the case of $\mathbb{C}P^n$, there's too much space in between dimensions for anything to happen, so it "looks" like a wedge of spheres, according to homology. The upshot is that the algebra simplifies greatly for certain choices of coefficients because you can simplify or get rid of differentials in the cellular chain complex.

(b) Often I tend to think of $H_*(-, \mathbb{Z})$ as being singular homology, and other choices of coefficients just a convenient way to organize the algebra (this is justified by the universal coefficient theorem). Picking coefficients in a certain characteristic can either isolate phenomena in that characteristic, or make certain problems vanish. From the cellular perspective it seems to make things vanish (since you're making it closer to a wedge of spheres)... however, it seems that for other invariants you manage to isolate things. This usually has to do with torsion... It's helpful to work through some examples other than just projective space: I recommend lens spaces.

...if there is some structure in an appropriate group action on something related to some space should I expect to see corresponding structure when viewing homology groups of that space with that group as the coefficients?

(a) It's unfortunate that the two different $\mathbb{Z}/2$'s here are named the same. What's important here is that the group $C_2$ is doing the acting and its order is divisible by $2$, and so things will behave differently with coefficients in $\mathbb{Z}/2$ as opposed to $\mathbb{Z}/p$ for odd primes. This generalizes: if you have a finite group $G$ acting on a space, $X$, then things with coefficients in $\mathbb{Z}/p$ will behave differently depending on whether or not $p$ divides $\vert G\vert$. One hint at this is the following: An action of $G$ descends to an action on the singular chain complex with coefficients in whatever (say, field) $k$ you're using, and so you have a bunch of vector spaces with actions of $G$, i.e. a bunch of representations of $G$. Now, it turns out that representation theory for finite groups behaves very differently in characteristic $p$ depending on whether or not $p$ divides the order of the group.

(b) Part (a) can be made even more precise... It turns out that there is a special homology theory for $G$-spaces called (Borel) equivariant homology. I won't go into it, but it has two properties (1) It has a very close relationship with group homology and (2) sometimes the ordinary homology of $X/G$ serves as a good approximation to the equivariant homology of $X$.

And here are some miscellaneous related comments:

  • There is a notion of R-orientability for any ring $R$; see Hatcher. This might help explain some things. In particular, note the sections on Poincare duality, which explains some of the regularity we see when choosing particular coefficients.
  • You don't have to be too exotic in your choice of ring because of the following Exercise: Show that a map $f: X \rightarrow Y$ induces an isomorphism on all homology groups if and only if it induces an isomorphism on rational homology and an isomorphism on homology with coefficients of $\mathbb{Z}/p$ for all $p$. (In some sense this point tells you that all different choices of coefficients can do for you is pinpoint torsion at different primality.)
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This is a very nice writeup. Thank you for taking the time :) –  Chris Janjigian Feb 4 '12 at 18:53
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A very insightful answer indeed. I really like the formulation that $H_*(-, \mathbb{Z})$ is the genuine homology and that the universal coefficient theorem is there to help us organize the algebra. –  Georges Elencwajg Feb 4 '12 at 20:09

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