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I have been reading the paper "Coherent Orientations for periodic orbits problems in symplectic geometry" by Floer and Hofer, trying to understand how we can orient the moduli spaces that appear in Floer Homology in a way that is coherent.

So to do this we first fix on the problem on considering the Fredholm Operators of the form $L=\partial_s+J_0\partial_t + S_L(s,t)$ that are asymptotically stable and we try to see how the orientations of their determinant bundles induce an orientation on $L\#_{\rho} K$ for a specific choice of $\rho$.

Now the part that I am having some trouble with is the associativity part. They say that to prove this it's analogous to what was done to proving that the induced orientation doesn't depend on the choice of constant asymptotically operator, provided that they have the same limit operators. This is done by using an Homotopy argument. Now I am not sure how one can do something similar to this to prove associativity.

My idea was to just use the fact that $\text{Det}((L\#_{\rho_0} K)_{\rho_1} M) \cong \text{Det}(L\#_{\rho_0} K)\otimes \text{Det}(M)\cong (\text{Det}(L)\otimes \text{Det}(K))\otimes \text{Det}(M)$ and then use the associativity of the tensor product. However here the argument is not similar to the previous one, hence I am not sure this is correct. Also this would be a lot simpler, and hence I would assume that if this was correct it would be mentioned that this works.

So I would like to understand if there is anything wrong with this argument. Any enlightenment is appreciated, thanks in advance.

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