Erratum to “Floer homology of families I”

One of the things I would like to do with this blog is to post updates to my papers.  To start this off on an embarrassing note, I recently learned from the paper “Fredholm theory and transversality for the parametrized and for the S^1-invariant symplectic action” by Bourgeois and Oancea that there is a mistake in several places in my paper “Floer homology of families I”.  Fortunately the mistake is easy to fix.

The mistake first appears in the beginning of section 2.2, in the review of the definition of the continuation map to prove that the Morse homology of a Morse-Smale pair (f_0,g_0) on a manifold X is isomorphic to the Morse homology of another Morse-Smale pair (f_1,g_1) on X.  The paper claims that one can first choose an arbitrary homotopy \{f_t\} from f_0 to f_1, and then choose a generic homotopy \{g_t\} from g_0 to g_1, to arrange that the moduli spaces of “continuation paths” (flow lines of the vector field (2-2) in the paper) counted by the continuation map are cut out transversely.  This is wrong.  If there is some point p\in X which is a critical point of every function f_t, then there will be a “continuation path” from p to p for every homotopy of metrics \{g_t\}, which means that transversality fails if the Morse index of p as a critical point of f_1 is greater than its Morse index as a critical point of f_0.  Oops.  This mistake is not a big deal for the paper, and one can fix it by choosing the path \{(f_t,g_t)\} to be generic, instead of just the path \{g_t\}.  However for the more general constructions in the rest of the paper I want to fix this a different way, as I will explain.

The more significant mistake for my paper is that Proposition 3.4 is false, for essentially the same reason.  (The gap in the argument is in the last sentence of the proof of Lemma 10.1(a), where the key condition “noncritical” cannot be guaranteed.)  Likewise, the claim in Section 6.1 that one can obtain transversality by choosing a generic fiberwise metric g^Z (“the proof of this is the same as the proof of Proposition 3.4…”) is false.

There is a simple way to fix all of this.  In Section 6 (the simpler construction of the spectral sequence using Morse homology on B, which is the most likely to be of interest to anyone), one simply has to choose the connection \nabla to be generic.  In fact it suffices to choose an arbitrary fiberwise metric such that you have Morse-Smale pairs on the fibers over the critical points of f^B, and then choose a generic connection \nabla.  In Section 3 and following sections that concern the more general construction of the spectral sequence using singular cubes on B, in addition to choosing a generic fiberwise metric over the singular cube, you also have to choose a generic connection over the singular cube (instead of taking the pullback of some fixed connection \nabla on Z\to B).  As in the simpler construction in Section 6, if you use generic connections like this, then you can use a fixed generic fiberwise metric on Z\to B, provided that you strengthen Definition 3.3 (of “admissible cube”) to require Morse-Smale fiberwise data over the center of each face.  Finally, going back to the original definition of continuation maps, you can fix the homotopy \{f_t\} provided that you repace \partial_t in equation (2-2) by the covariant derivative with respect to a generic nontrivial connection on [0,1]\times X \to [0,1].

Anyway, I’m sorry I didn’t say things right in the paper, and it’s too bad that it is too late to fix it, but hopefully the above clears up the story.

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