Austin-Braam approach uses the multicomplexes of de Rham complex on critical submanifolds to describe Bott-Morse theory.


For more details, see the follows:
<https://link.springer.com/chapter/10.1007/978-3-0348-9217-9_8>


Could we construct the similar approaches for the Floer type theory? For example, if the one-parameter family of Hamiltonian $H\_t$ on the symplectic manifold $M$ has degenerate orbit, could we do the similar things for such Floer homology?


I have tried to find such approaches for the Floer type theory, but I couldn't. If someone knows, please notice me.



Is there an analogy of Austin-Braam approach to Bott-Morse type Hamiltonian Floer homology?
First, Austin and Braam *did* already apply their machine to a Floer-type theory: it was just instanton homology and not Hamiltonian Floer homology [here](https://www.sciencedirect.com/science/article/pii/0040938395000046). Their machine uses $\Bbb R$ coefficients, but you can work over $\Bbb Z$ by using an appropriate model for singular homology which includes chain-level fiber product maps. There is a brief discussion of this in [Section 1 here](https://www.sciencedirect.com/science/article/pii/0040938395000097); other authors have later taken this up.


All their machine actually needs is the data of a Morse--Bott flow category, meaning


* A collection of closed (oriented) smooth manifolds $X, Y, \dots$ to serve as the critical submanifolds.
* For each pair $X, Y$, a compact smooth manifold with corners $\mathcal C(X,Y)$ together with a smooth map to $X \times Y$. (There is often technical difficulty establishing smoothness on the boundary / corner strata, and the requirement of smoothness here can often be slightly relaxed.) For their construction, you need these moduli spaces through dimension $\dim \mathcal C(X,Y) \le \dim X + \dim Y + 1$.
* A transversality assumption: $\mathcal C(X,Y) \to Y$ and $Y \leftarrow \mathcal C(Y, Z)$ are transverse.
* The fundamental point is that these satisfy the boundary relation $$\partial \mathcal C(X,Z) = \sum\_Y \mathcal C(X,Y) \times\_Y \mathcal C(Y,Z).$$


Given this, Austin and Braam provide you a chain complex, which you can then take the homology of.


If you wish to apply this machine to Hamiltonian Floer theory, "all you need to do" is guarantee that you can set things up so that you get compact smooth moduli spaces satisfying the right boundary relation. **I am not an expert in Hamiltonian Floer theory, do the rest is speculation on my part.**


The main part of this I would be concerned about is compactness (eg, can you avoid sphere bubbles?) This gets harder in the Morse--Bott setting, because you need larger-dimensional moduli spaces (you cannot get away with just 1-dimensional moduli spaces). If you assume $\langle c\_1, \pi\_2\rangle = 0$ --- or more generally that the minimal Chern number has $2N > \dim X + \dim Y + 1$ for all $X, Y$ among your critical manifolds --- I think you should be fine.


There is also the appearance of orbifold structure on your moduli spaces. I think this occurs for two reasons: the appearance of multiply-covered curves (which arises in positive-genus settings unlike ours) and the appearance of stable maps from wedges of spheres with some additional symmetry. I think these should be ruled out by the assumption I made above that implies no sphere bubbling.


