Hello.

I am a beginning graduate student who wants to study Heegaard Floer Homologies.

I am now reading the paper

http://front.math.ucdavis.edu/1301.4919 Errata to 'A cylindrical reformulation of Heegaard Floer homology'

and I am now stuck on the proof of Lemma 4.1'(p.5).

The statement is about that there is a certain map with some conditions representing a positive homology class in $\pi_2 ({\overrightarrow{x}, \overrightarrow{y}})$ in his cylindrical setting.

After gluing domains away from the $\overrightarrow{x} \cup \overrightarrow{y}$, he made special arrangement for gluing near intersection points.

The point I could not understand is the following.

For given point $p \in \overrightarrow{x} \cup \overrightarrow{y}$, in a neighborhood of that point there are 4 domains and their coefficients(intersection number of homology class and base points $\times$ a disc).

And the arrangement of these coefficients are given by

$\{ n, n+k, n+l , n+k+l\}$ or $\{n+1, n+k, n+l, n+k+l\}$ or other variants that 1 goes to other parts.

I could not understand why coefficients are given like this.

I think I miss something easy, but I could not see the point.

Please give me any hint or answer to understand this.

Thank you.

Hello.

I am a beginning graduate student who wants to study Heegaard Floer Homologies.

I am now reading the paper

https://arxiv.org/abs/1301.4919 Errata to 'A cylindrical reformulation of Heegaard Floer homology'

and I am now stuck on the proof of Lemma 4.1'(p.5).

The statement is about that there is a certain map with some conditions representing a positive homology class in $\pi_2 ({\overrightarrow{x}, \overrightarrow{y}})$ in his cylindrical setting.

After gluing domains away from the $\overrightarrow{x} \cup \overrightarrow{y}$, he made special arrangement for gluing near intersection points.

The point I could not understand is the following.

For given point $p \in \overrightarrow{x} \cup \overrightarrow{y}$, in a neighborhood of that point there are 4 domains and their coefficients(intersection number of homology class and base points $\times$ a disc).

And the arrangement of these coefficients are given by

$\{ n, n+k, n+l , n+k+l\}$ or $\{n+1, n+k, n+l, n+k+l\}$ or other variants that 1 goes to other parts.

I could not understand why coefficients are given like this.

I think I miss something easy, but I could not see the point.

Please give me any hint or answer to understand this.

Thank you.