By the **Arnold Conjecture**, I mean the following statement:

Let $M$ be a closed symplectic manifold, and $\phi:M\to M$ a Hamiltonian symplectomorphism with nondegenerate fixed points. Then $ \# \operatorname{Fix} (\phi) \geq \dim H_\bullet (M;\mathbb Q)$.

The proof is via defining the "Floer homology" $HF(M;H)$ of $M$ with respect to any $1$-periodic Hamiltonian $H:M\times\mathbb S^1\to\mathbb R$, which is then shown to be independent of $H$ (and thus can be written as $HF(M)$). There are some technicalities involved in this definition (Novikov rings, transversality, bubbling, etc.), which for the purposes of this question I would like to ignore.

The Arnold Conjecture is proven by exhibiting an isomorphism $HF(M)=H(M;\mathbb Q)$. My question is about whether this can be proven without "$\mathbb S^1$-localization". I know of the following two proofs:

Floer's original proof is as follows. We use the Hamiltonian $H(x,t)=\epsilon\cdot f(x)$ for some Morse--Smale function $f:M\to\mathbb R$. Then the generators of the Floer chain complex correspond to the critical points of $f$, so it remains to identify the differential with the Morse--Smale differential. Floer exploits an $\mathbb S^1$-symmetric in the problem to show that any part of the Floer differential which does

*not*come from a Morse--Smale flow line must carry a free $\mathbb S^1$-action. Since we only count zero-dimensional moduli spaces for the differential, this is enough since the only $0$-manifold with a free action of $\mathbb S^1$ is the empty set.The PSS isomorphism (Piunikhin-Salamon-Schwarz). I won't write the details on this, but it is another approach to the isomorphism, which also uses the freeness of the action of $\mathbb S^1$ on "bad" moduli spaces to show they do not contribute to certain maps (

**EDIT**: this is wrong, the PSS approach yields an isomorphism without appealing to $S^1$-localization; see Thomas Kragh's answer below). See, for example, these notes by Salamon: http://www.math.ethz.ch/~salamon/PREPRINTS/floer.pdf

Both of these proofs use **$\mathbb S^1$-localization**, meaning some statement to the effect that a free $\mathbb S^1$ action on a zero-dimensional moduli space implies the fundamental class is zero.

Are there approaches which genuinely avoid $\mathbb S^1$-localization?

In fact, I have heard of two such approaches, but so far have not understood either of them:

Katrin Wehrheim in research statement http://math.mit.edu/~katrin/slides/research.pdf (see section 5.6) says she and Albers and Fish have such an approach, summarized in some slides http://www-math.mit.edu/∼katrin/slides/arnold.pdf, but the paper ("Polyfold Proof of the Weak Arnold Conjecture") remains unreleased as far as I can tell (probably the slides contain something of an outline of their strategy, but I have yet to read them carefully).

Fukaya--Oh--Ohta--Ono in Remark 31.18 of their recent paper http://arxiv.org/abs/1209.4410 say there is such an approach, but it is based on the machinery of their books "Lagrangian intersection Floer theory: anomaly and obstruction I - II" which are beyond my current knowledge.

Is there a down-to-earth explanation of how these approaches to the isomorphism $HF(M)=H(M;\mathbb Q)$ manage to avoid $\mathbb S^1$-localization arguments (which seem crucial for both Floer's approach and the PSS approach)? Do the approaches of Albers--Fish--Wehrheim and Fukaya--Oh--Ohta--Ono genuinely get around the need for $\mathbb S^1$-localization, or is the same difficulty hiding somewhere else?