Sarkar's Maslov index formula

Question

I have difficulty understanding Sarkar's maslov index formula in symmetric products from http://arxiv.org/abs/math/0609673.

If $D$ is an $n$-sided region with corner points $p_1,\ldots, p_n$ then it reads like

$\mu(D)= \mu_{p_1}(D)+\mu_{P_n}(D)+ e(D)-g(n-2)/2+\sum_{ 1< i< j \leq n} \partial_j(D) \cdot \partial_i(D)$

where $\mu_{P_i}$ and $e$ are the point and euler measures.

First of all it seems to treat $P_1,P_n$ different from other points. Secondly he says the euler measure of an $n$ sided region is $1-n/4$ which looks different from what Lipshitz says (which takes the accute and obtuse corners into account). Last but not least I don't understand the definition of the last term. It is defined by moving the two sides in 4 different directions in such a way that no endpoint of one is on another and then taking intersection points. It's not clear for me what these 4 directions are and why the result is not always zero.

Answer 1

(1) Yes, $p_1$ and $p_n$ are treated different from other points. If you cyclically permute the indices $\{1,\ldots,n\}$ you do of course get the same answer, but it's not immediately obvious from Sarkar's formula that this is the case.

(2) The Euler measure of a region $R$ equals:$$e(R):=\chi(R)-\frac 14\#\{\text{acute corners}\}+\frac 14\#\{\text{obtuse corners}\}$$Usually one just talks about regions which are disks, so $\chi(R)=1$. Now every connected component of $\Sigma_g\setminus(\eta_1^1\cup\cdots\cup\eta_g^1\cup\cdots\cup\eta_1^k\cup\cdots\eta_g^k)$ has zero obtuse corners. With these two facts in hand one reduces to Sarkar's formula $1-\frac n4$ for an $n$-sided polygon.

(3) The arcs in $\partial_iD$ all lie inside the $\boldsymbol\eta^i$ circles, and they are considered moved in "both directions" along the $\boldsymbol\eta^i$ circles. If you indulge me as I create ugly ascii art, I can give a few examples. Here $|$ denotes $\partial_iD$ and $=$ denotes $\partial_jD$.

The following has intersection $1$: $$\begin{matrix} &|&\cr &|&\cr =&=&=\cr &|&\cr &|&\cr \end{matrix}$$ The following both have intersection $\frac 12$: $$\begin{matrix} &|&&&&|\cr &|&&&&|\cr =&=&&&=&=&=\cr &|&&&&\cr &|&&&&\cr \end{matrix}$$ The following all have intersection $\frac 14$: $$\begin{matrix} &|&&&&|\cr &|&&&&|\cr =&=&&&&=&=\cr &&&&&\cr\cr\cr &&&&&\cr &&&&&\cr &&&&&\cr =&=&&&&=&=\cr &|&&&&|\cr &|&&&&|\cr \end{matrix}$$