part for this abstract perturbative structure to achieve transversality, and an algebraic construction part which organizes the end result of previous two steps into choice-independent algebraic invariants obtained from moduli spaces. Here independence of abstract choices made with given a fixed admissible geometric choice is due to mechanisms/constructions in the last two parts above, and independence of geometric choices made is shown by considering similar moduli spaces interpolating two different geometric choices and interpolating between two different such interpolations.

Polyfold theory in a nutshell
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Polyfold theory are developed and elaborated in Hofer-Wysocki-Zehnder’s series of papers, [@PolyfoldI], [@PolyfoldII], [@PolyfoldIII], [@Polyfoldinteg], [@Polyfoldanalysis], [@PolyfoldGW], [@PolyfoldSurveyNew] and [@PFI]. Good places to learn polyfolds are Hofer’s updated survey [@PolyfoldSurveyNew], polyfold Gromov-Witten paper [@PolyfoldGW] and book [@PFI].

To be concrete and have a feeling of polyfold notions and ingredients, let us look at Gromov-Witten moduli spaces following [@PolyfoldGW]: Let $(Q,\omega)$ be a closed symplectic manifold and choose a compatible almost complex structure $J$, then consider the moduli space $\overline{\mathcal{M}}(Q,\omega, J)$ of finite energy stable marked noded $J$-holomorphic curves (such a $J$-holomorphic curve is an isomorphism class of a $J$-holomorphic map from a marked noded Riemann surface to $(Q,J)$, where an isomorphism is a diffeomorphism of 