Associativity of orientations of determinant bundles in Floer homology

Question

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.

Also in Schwarz's book Morse homology, he mentions how one can prove associativity, and it's using an homotopy argument, however I would like to understand why this method wouldn't work.

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

Answer 1

The issue here are the isomorphisms in your suggested proof. There are choices and conventions involved. To give names to the isomorphisms clarifies what needs to be checked.

$$ \iota_{K,L}:Det(K\sharp_{\rho}L)\to Det(K)\otimes Det(L). $$ You need to check that you've set things up so that: $$ \iota_{K\sharp_{\rho_0}L,M}\circ (Id_{K} \otimes \iota_{L,M})=\iota_{K,L\sharp_{\rho_1}M}\circ (\iota_{K,L}\otimes Id_M). $$ If you get your conventions wrong there is a sign lurking in here. As shameless promotion you can look at the proof of Lemma 20.2.1. on page 380 of Monopoles and Three Manifolds by Kronheimer and myself. The proof is actually on the pages before.