# Proving $\partial ^ 2 = 0$ for the case of Morse-Complex with $\mathbb{Z}$ using orientation of the moduli space

I was going through the book Morse theory and Floer homology by Audin-Damian and got stuck where they talk about defining the complex for $\mathbb{Z}$ coefficient.

Assume that $a,b,c$ are critical points of index $k,k-2,k-1$. By definition $$\partial u = \sum_{\text{v is a critical point of 1 dimension less that a}} \eta (u,v) v$$ where $\eta (u,v)$ is the signed sum of the orientations of the elements in the moduli space of flow-lines from $u$ to $v$.

I know that we can compactify the moduli space using broken-flow lines, but what is bugging me is that how does the orientation at the boundary (ie. the broken flow-lines) match up. I need some result of the sort that if $\lambda_1$ is a flowline between $a$ and $c$ and $\lambda_2$ is a flowline between $c$ to $b$ then orientation of the boundary point ($\lambda_1,\lambda_2$) in the oriented 1-manifold with boundary $\bar {\mathcal{M}}(a,b)$ is the product of the orientations of $\lambda_1$ and $\lambda_2$. How do I prove this ?

I tried working it out by keeping track of the co-orientation and orientation but I am getting stuck at moving (orientation) data from the stable manifold of $c$ to the unstable manifold of $c$

• I like the little paper arxiv.org/abs/math/0411465 of Joa Weber for this discussion. May 18, 2018 at 7:34
• You can also look at my thesis, where I do this precisely for a slightly different situation: induced maps in Morse homology. \ May 23, 2018 at 12:24
• If you let me know where you get stuck I can have a look May 23, 2018 at 15:02
• I will try again to work it out and let you know, thanks for the article. May 27, 2018 at 15:15