I'm reading the book Morse Homology by M. Schwarz, which aims to develop Morse homology in strict analogy with Floer homology. For orientation matters, the book follows the paper
A. Floer and H. Hofer, Coherent orientations for periodic orbit problems in symplectic geometry. Math. Z. 212 (1993), 13-38.
I'm afraid I can't explain the definition of coherent orientations (which occupy an entire chapter in the book). This question is specific to the book (and/or that paper, since the book uses the same notation from that paper, albeit in a different setting).
I got completely confused when the author proves that the Morse complex is indeed a chain complex. The key lemma is the following:
Here $f$ is a Morse function, $\mu$ denotes Morse index; broken trajectories are equivalent if they are the boundary points of the same connected component of the unparametrized moduli space of trajectories $\widehat{\mathcal{M}}_{x,y}^f$; $\tau_\sigma$ denotes the characterisitic sign obtained by comparing a given coherent orientation and the canonical orientation given by the tangent field of the flow line.
Here is the proof given in the book.
I understand that the goal is to verify $(4.6)$, but I don't know why this is related to the diagram below. How do we compute the glued orientation $[\hat{u}_{it}]\#[\hat{v}_{it}]$? The definition of such glued orientations is via abstract determinant bundles of the Fredholm operators associated to the trajectories, and I don't see how it's related to the image of $\frac{\partial}{\partial\rho}$ in $\widehat{\mathcal{M}}_{x,y}^f$... Could someone explain what this proof is doing? I've spent hours thinking and still cannot make sense of this argument...
Thanks for any help!