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Yes it must be proved in that same book, or any/every basic book, going back to the fundamental proof by Micallef--White: The critical points and the self-intersection points are isolated, as there is …

Compact-open smooth topology, i.e. $C^\infty_\text{loc}$-topology, also the Fréchet topology on sections of the tangent bundle (same for Riemannian metrics). See some textbooks on J-holomorphic curves …

(1) It cannot necessarily be done in general as you suggest, because you don’t have control over the “other Reeb orbits” so there are a priori bad curves that can hit them. It does however work on som …

Assume your 4-manifold is minimal (otherwise there are multiply covered exceptional spheres which potentially give noncompactness). Then it's a computational check to see that: If $d(\alpha)=0$ with $ …

Without elaborating much there are three key points, with the first two laying the bedrock for the third: ECH counts J-curves without caring about most information of the actual branched covers of su …

Eliashberg & Gromov sketched a proof in their paper "Convex symplectic manifolds" (Section 3.4). Written in the 4-dimensional case it says: For $r>0$ define the subspace $X(r)\subset\mathbb{R}^4$ to …

It turns out that the Removable Singularities Theorem and the Monotonicity Lemma do not require compactness, but that the target manifold should have bounded geometry, such as in our case of inserting …

Exercise: The non-symplectic $K3\#n(S^1\times S^3)$ for $n>0$ with the spin-c structure induced from the canonical one on $K3$; it will have $n$-dimensional SW moduli. If you want symplectic examples: …

Just to finalize comments since people are upvoting the question: The canonical reference is Taubes' "Casson's invariant and gauge theory" which makes the statement rigorous and has all the definition …

$\mathbb CP^2\#\overline{K3}$ (trivial SW because it is a connected sum with $b^2_+>0$ for both pieces). This is an example without having to use the Bauer-Furuta invariants (contrast with Kyle's comm …

Spectral flow is the standard way a sign is associated to a point in a zero-dimensional moduli space of curves (I am not sure what you mean by VFC here, it's 0-dimensional). This involves orienting th …

Convergence to cusp curves is the original compactification by Gromov, whereas convergence to stable maps is the compactification by Kontsevich (a cusp curve corresponds to the image of a stable map). …

For your $n=2k$, $\mathbb{C} P^n\#\mathbb{C} P^n$ does not even admit an almost complex structure, so it cannot be symplectic. See also: 1) Goertsches-Konstantis' paper "Almost complex structures on …

This might suffice for you, it is not published and only slightly longer than Etnyre's sketch, but without exercises. This has been shown in the PhD thesis of Noah Daniel Goodman (a student of Etnyre) …

I was there during this IHES conference to scribe the lectures and also give a discussion session on it. So in case it helps, my notes from both of these are available here. Your thought on the reaso …