show/hide this revision's text 5 Added a formal writeup.

EDIT 1/5/2010: I was a little dissatisfied with the informality of what I wrote below, so here is a somewhat more formal writeup. The statement is somehow geometrically obvious, but the proof is still a bit nice.

EDIT: I initially claimed the sequence below is short exact, which is false; it is right exact. It is fixed below, with some explanation, and a bit of a geometric explanation of what's going on.

I know little about the theory of linear independence over $\mathbb{Q}$, but I'll attempt an answer to this part of the question:

Are there relationships between the (co)homology groups of the covering and the residues?

The answer is "yes." Let $f$ be a meromorphic function on $\mathbb{C}$, and for convenience let's assume that it has poles $z_1, ..., z_n$ of order $1$, and no other singularities. Let $g$ be an antiderivative of $f$. Then the Riemann surface of $f$ is $M_f=\mathbb{C}-\{z_1, ..., z_n\}$ and the Riemann surface of $g$, which we will denote by $M_g$, is a covering space of $M_f$ with covering map $\pi: M_g\to M_f$. Let $V\subset \mathbb{C}$ be the $\mathbb{Q}$-vector space spanned by $\operatorname{Res}_{z_1}(f), ..., \operatorname{Res}_{z_n}(f)$. Then there is a short exact sequence $$H_1(M_g, \mathbb{Q})\overset{H_1(\pi)}{\longrightarrow} H_1(M_f, \mathbb{Q})\overset{\int}{\longrightarrow} V\to 0,$$ where the map $\int$ is given as follows. Namely, $\int: H_1(M_f, \mathbb{Q})\to V$ is given by $[\gamma]\mapsto \frac{1}{2\pi i}\int_\gamma f~\operatorname{dz}$.

Let's elucidate the connection to the linear independence of $\operatorname{Res}_{z_1}(f), ..., \operatorname{Res}_{z_n}(f)$ over $\mathbb{Q}$. $H_1(M_f, \mathbb{Q})$ is a $\mathbb{Q}$-vector space with a basis of cycles $[\lambda_1], ..., [\lambda_n]$ corresponding to the punctures $z_1, ..., z_n$. Then the map $\int$ sends $[\lambda_i]$ to $\operatorname{Res}_{z_i}(f)$. So the image of $H_1(M_g, \mathbb{Q})$ in $H_1(M_f, \mathbb{Q})$ is precisely the vector space of relations between the residues of $f$.

Added: We can extend this right exact sequence into a longer sequence. In particular, by covering space theory we have that $\pi_1(M_g)\to \pi_1(M_f)$ is an injection. It is easy to see that the commutator subgroup of $\pi_1(M_f)$ is contained in the image of $\pi_1(M_g)$. By the Hurewicz theorem $$H_1(M_g, \mathbb{Q})\simeq \pi_1(M_g)^{Ab}\underset{\mathbb{Z}}{\otimes} \mathbb{Q}.$$ So the kernel of the map $H_1(M_g, \mathbb{Q})\to H_1(M_f, \mathbb{Q})$ is given by the image of $[\pi_1(M_f), \pi_1(M_f)]$ (which is contained in $\pi_1(M_g)$) in $H_1(M_g, \mathbb{Q})$. One can extend the exact sequence further back by looking at quotients of commutators in this manner.

This first extension has a geometrical interpretation. Namely, let $h$ be a meromorphic function whose poles have the same locations as those of $f$, but whose residues are linearly independent over $\mathbb{Q}$. Then the antiderivative of $h$, denoted $s$ has Riemann surface $M_s$, which is a covering space over $M_f$, with covering map $\pi': M_s\to M_f$. By the properties of covering spaces, $\pi'$ factors through $\pi$, and it is not hard to see that $\pi_1(M_s)$ is exactly the commutator subgroup of $\pi_1(M_f)$. Then the sequence $$H_1(M_s, \mathbb{Q})\overset{H_1(\pi')}{\longrightarrow} H_1(M_g, \mathbb{Q})\overset{H_1(\pi)}{\longrightarrow} H_1(M_f, \mathbb{Q})\overset{\int}{\longrightarrow} V\to 0,$$ is exact, and coincides with the sequence described above.

I don't know if the continuing left extensions of this sequence have similar geometric interpretations. Also, it would be nice to have a naturally arising description of this sequence, rather than the somewhat ad hoc one I've given.
show/hide this revision's text 4 Fixed an error and expanded on the answer.

EDIT: I initially claimed the sequence below is short exact, which is false; it is right exact. It is fixed below, with some explanation, and a bit of a geometric explanation of what's going on.

I know little about the theory of linear independence over $\mathbb{Q}$, but I'll attempt an answer to this part of the question:

The answer is "yes." Let $f$ be a meromorphic function on $\mathbb{C}$, and for convenience let's assume that it has poles $z_1, ..., z_n$ of order $1$, and no other singularities. Let $g$ be an antiderivative of $f$. Then the Riemann surface of $f$ is $M_f=\mathbb{C}-\{z_1, ..., z_n\}$ and the Riemann surface of $g$, which we will denote by $M_g$, is a covering space of $M_f$ with covering map $\pi: M_g\to M_f$. Let $V\subset \mathbb{C}$ be the $\mathbb{Q}$-vector space spanned by $\operatorname{Res}_{z_1}(f), ..., \operatorname{Res}_{z_n}(f)$. Then there is a short exact sequence $$0\to H_1(M_g, $H_1(M_g, \mathbb{Q})\overset{H_1(\pi)}{\longrightarrow} H_1(M_f, \mathbb{Q})\overset{\int}{\longrightarrow} V\to 0,$$

Let's elucidate the connection to the linear independence of $\operatorname{Res}_{z_1}(f), ..., \operatorname{Res}_{z_n}(f)$ over $\mathbb{Q}$. $H_1(M_f, \mathbb{Q})$ is a $\mathbb{Q}$-vector space with a basis of cycles $[\lambda_1], ..., [\lambda_n]$ corresponding to the punctures $z_1, ..., z_n$. Then the map $\int$ sends $[\lambda_i]$ to $\operatorname{Res}_{z_i}(f)$. So the image of $H_1(M_g, \mathbb{Q})$ in $H_1(M_f, \mathbb{Q})$ is precisely the vector space of relations between the residues of $f$.

That's the summary; it is all relatively clear from the Cauchy integral formula and the properties of covering spaces

Added: We can extend this right exact sequence into a longer sequence. But it's worth getting some intuition for what's going onIn particular, so I'll try to briefly describe my (inexpert) intuition. I'm sure by covering space theory we have that many readers are intimately familiar with what follows$\pi_1(M_g)\to \pi_1(M_f)$ is an injection. I'll work from It is easy to see that the covering space/fundamental group point of view; think of homology commutator subgroup of the Abelianization, through which integration factors.

Namely, let $\mathcal{H}$ be \pi_1(M_f)$ is contained in the sheaf image of holomorphic functions over $\mathbb{C}$, with etale space $H$ and surjection \pi_1(M_g)$. By the Hurewicz theorem $\pi: H\to $H_1(M_g, \mathbb{C}$. Then mathbb{Q})\simeq \pi_1(M_g)^{Ab}\underset{\mathbb{Z}}{\otimes} \mathbb{Q}.$$So the Riemann surface of a function on an open region kernel of $\mathbb{C}$ is the connected component of $H$ containing any germ of that function. The map $d: H_1(M_g, \mathcal{H}\to mathbb{Q})\to H_1(M_f, \mathcal{H}$ mathbb{Q})$ is an endomorphism given by the image of this sheaf, and thus induces an endomorphism $d: H\to H$, which a covering map [\pi_1(M_f), \pi_1(M_f)]$ (this is true as a holomorphic function which is determined by its germ at any point).

Now, there are two natural maps contained in $H\to \pi_1(M_g)$) in $H_1(M_g, \mathbb{C}$. mathbb{Q})$. One is the surjection $\pi$; the other is the evaluation map $e$, sending a germ $[f]_x$, of can extend the function $f$ exact sequence further back by looking at the point $x$, to $f(x)$. Let quotients of commutators in this manner.

This first extension has a geometrical interpretation. Namely, let $M_f$ h$ be a meromorphic function whose poles have the Riemann surface same locations as those of $f$, and let but whose residues are linearly independent over $M_g$ be the covering space through \mathbb{Q}$. Then the map antiderivative of $d$; namely, the h$, denoted $s$ has Riemann surface of an antiderivative of $f$.

Then M_s$, which is a loop in covering space over $M_f$, with covering map $\pi': M_s\to M_f$lifts to a path . By the properties of covering spaces, $\gamma: [0, 1] \to M_g$ with \pi'$ factors through $\pi(\gamma(0))=\pi(\gamma(1))$ \pi$, and $e(\gamma(1))-e(\gamma(0))=2\pi i \operatorname{Res}(f)$, where it is not hard to see that $\operatorname{Res}(f)$ \pi_1(M_s)$ is the sum of exactly the residues commutator subgroup of $f$ contained in \pi_1(M_f)$. Then the loop (weighted by winding number). What does this look like geometrically? Well, at any pole of $f$, sequence $M_f$ contains a puncture; locally$H_1(M_s, \mathbb{Q})\overset{H_1(\pi')}{\longrightarrow} H_1(M_g, \mathbb{Q})\overset{H_1(\pi)}{\longrightarrow} H_1(M_f, \mathbb{Q})\overset{\int}{\longrightarrow} V\to 0,$$is exact, and coincides with the covering space sequence described abovea neighborhood of this puncture looks like a helix.A single loop around the puncture lifts to a path that moves up one "level" of the helix;

I don't know if the "height" continuing left extensions of this path is the residue of the polesequence have similar geometric interpretations.

In particularAlso, a loop in $M_f$ lifts it would be nice to have a loop in $M_g$ exactly when the corresponding linear combination of residues naturally arising description of $f$ is zero. And this gives intuition for the claim. Verifying that the sequenceis short exact is not difficult with this intuition in hand, and if anyone would like I can be more formal about itrather than the somewhat ad hoc one I've given.
show/hide this revision's text 3 Fixed an error in the definition of $\int$

I know little about the theory of linear independence over $\mathbb{Q}$, but I'll attempt an answer to this part of the question:

Are there relationships between the (co)homology groups of the covering and the residues?

The answer is "yes." Let $f$ be a meromorphic function on $\mathbb{C}$, and for convenience let's assume that it has poles $z_1, ..., z_n$ of order $1$, and no other singularities. Let $g$ be an antiderivative of $f$. Then the Riemann surface of $f$ is $M_f=\mathbb{C}-\{z_1, ..., z_n\}$ and the Riemann surface of $g$, which we will denote by $M_g$, is a covering space of $M_f$ with covering map $\pi: M_g\to M_f$. Let $V\subset \mathbb{C}$ be the $\mathbb{Q}$-vector space spanned by $\operatorname{Res}_{z_1}(f), ..., \operatorname{Res}_{z_n}(f)$. Then there is a short exact sequence $$0\to H_1(M_g, \mathbb{Q})\overset{H_1(\pi)}{\longrightarrow} H_1(M_f, \mathbb{Q})\overset{\int}{\longrightarrow} V\to 0,$$ where the map $\int$ is given as follows. Namely, $\int: H_1(M_f, \mathbb{Q})\to V$ is given by $[\gamma]\mapsto \frac{1}{2\pi i}\int_\gamma * f$ where $*$ is the Hodge star (this is the pairing from Poincare duality)f~\operatorname{dz}$.

Let's elucidate the connection to the linear independence of $\operatorname{Res}_{z_1}(f), ..., \operatorname{Res}_{z_n}(f)$ over $\mathbb{Q}$. $H_1(M_f, \mathbb{Q})$ is a $\mathbb{Q}$-vector space with a basis of cycles $[\lambda_1], ..., [\lambda_n]$ corresponding to the punctures $z_1, ..., z_n$. Then the map $\int$ sends $[\lambda_i]$ to $\operatorname{Res}_{z_i}(f)$. So $H_1(M_g, \mathbb{Q})$ is precisely the vector space of relations between the residues of $f$.

That's the summary; it is all relatively clear from the Cauchy integral formula and the properties of covering spaces. But it's worth getting some intuition for what's going on, so I'll try to briefly describe my (inexpert) intuition. I'm sure that many readers are intimately familiar with what follows. I'll work from the covering space/fundamental group point of view; think of homology of the Abelianization, through which integration factors.

Namely, let $\mathcal{H}$ be the sheaf of holomorphic functions over $\mathbb{C}$, with etale space $H$ and surjection $\pi: H\to \mathbb{C}$. Then the Riemann surface of a function on an open region of $\mathbb{C}$ is the connected component of $H$ containing any germ of that function. The map $d: \mathcal{H}\to \mathcal{H}$ is an endomorphism of this sheaf, and thus induces an endomorphism $d: H\to H$, which a covering map (this is true as a holomorphic function is determined by its germ at any point).

Now, there are two natural maps $H\to \mathbb{C}$. One is the surjection $\pi$; the other is the evaluation map $e$, sending a germ $[f]_x$, of the function $f$ at the point $x$, to $f(x)$. Let $M_f$ be the Riemann surface of $f$, and let $M_g$ be the covering space through the map $d$; namely, the Riemann surface of an antiderivative of $f$.

Then a loop in $M_f$ lifts to a path $\gamma: [0, 1] \to M_g$ with $\pi(\gamma(0))=\pi(\gamma(1))$ and $e(\gamma(1))-e(\gamma(0))=2\pi i \operatorname{Res}(f)$, where $\operatorname{Res}(f)$ is the sum of the residues of $f$ contained in the loop (weighted by winding number). What does this look like geometrically? Well, at any pole of $f$, $M_f$ contains a puncture; locally, the covering space above a neighborhood of this puncture looks like a helix. A single loop around the puncture lifts to a path that moves up one "level" of the helix; the "height" of this path is the residue of the pole.

In particular, a loop in $M_f$ lifts to a loop in $M_g$ exactly when the corresponding linear combination of residues of $f$ is zero. And this gives intuition for the claim. Verifying that the sequence is short exact is not difficult with this intuition in hand, and if anyone would like I can be more formal about it.
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