13 votes

The Floer Equation is Elliptic

This is actually a system of first-order PDEs, of $2n$ (the dimension of $M$) equations. To see that it is elliptic, let us consider the symplest case of $M={\mathbb R}^{2n}$ with the standard ...
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  • 47.7k
10 votes

Why is embedded contact homology so powerful?

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 ...
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10 votes
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Reference Request: "Neck Stretching Procedure" (In Symplectic Field Theory)

Neck-stretching is a deformation of an almost complex structure in a neighbourhood of a hypersurface. In the Eliashberg-Givental-Hofer paper, neck-stretching is called "splitting along a contact ...
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  • 6,230
9 votes
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Lagrangian intersection Floer homology: understanding some assumptions

When you try and prove that $d^2=0$ ($d$ being the Floer differential) you need to look at the boundary of the moduli space of index 2 J-holomorphic strips with one boundary on $L_0$, one on $L_1$. ...
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  • 6,230
9 votes
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Is the instanton homology for webs and foams a categorified Chern-Simons?

It’s not clear what you mean by categorify here: the J# invariant doesn’t have a grading in their definition. Just taking dimension to get a number then doesn’t seem to yield a polynomial invariant. ...
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  • 62.1k
7 votes

Reference Request: "Neck Stretching Procedure" (In Symplectic Field Theory)

You'll find an extensive discussion with applications of the "neck stretching" technique in Jonathan Evans' thesis, Symplectic topology of some Stein and rational surfaces (chapter 5).
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6 votes

Transversality and $C^l$, $C^{\infty}$ spaces of almost complex structures

There are several different issues happening here. The space of $C^\infty$ is not a Banach space because the topology is generated by a countable family of semi-norms not by one norm. You might try to ...
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  • 2,894
6 votes
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Invariance of morse homology, doubt in proof in book "Morse Theory and Floer homology"

Oh, I happen to know the guy who wrote that PDF. As to your questions. Yes, that's the idea. In $V \times A$, $F = f_0$ so the critical points are in one-to-one correspondence with those of $f_0$. ...
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  • 438
6 votes

What does Yang-Mills and mass gap problem has to do with mathematics?

Mathematical physics is the study of physical questions from the point of view of full mathematical rigor. Physical questions are phrased as well-defined mathematical problems, to be attacked with ...
6 votes
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Manifold of mappings between $M$ and $N$, with non-compact source $M$

Let $M$ and $N$ be Riemannian manifolds. In general, the space of Sobolev mappings $W^{k,p}(M,N)$ should not be defined as a completion of smooth mappings even if $k=1$ and manifolds are compact. The ...
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6 votes
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Monopole Floer Homology vs. Heegaard-Floer theory

Ozsvath and Szabo constructed HF as a topological interpretation of SWF, and they noticed many links between the two. Roughly speaking, their Euler characteristics are the same, and the analog of the ...
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5 votes

How is Chern-Simons theory related to Floer homology?

I'm far from an expert, and I apologize if this is too basic / philosophical / vague. In instanton Floer homology, the functional $CS(A)$ plays the role of the potential energy function for a $4$d ...
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5 votes
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Choice of a family of almost complex structures when defining Floer Homology

For a lot of things, you can work with a generic time-independent $J_0$ as you suggest; for instance, Audien & Damien work in this context in their book (so for most of the "fundamental" ...
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4 votes
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Bubbling off of a pseudo holomorphic sphere on surface with cylindrical ends

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 ...
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4 votes
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Computing the Fredholm index in Floer theory

The idea is to show that the kernel & cokernel consist of smooth sections, and thus are independent of $p$. Since the Fredholm index is the difference between the dimensions of these, it doesn't ...
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  • 1,202
4 votes
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$\mathbb{R}\mathbb{P}^n$ is monotone in $\mathbb{C}\mathbb{P}^n$, Lagrangian floer cohomology

The relative homology long exact sequence puts this group in between $H_2(\mathbb{CP}^n)=\mathbb{Z}$ and $\mathbb{Z}/2$. It maps surjectively to $\mathbb{Z}/2$. Let D be a generator for relative ...
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  • 6,230
4 votes

Background needed to understand modern research on knot homology theories

I'm not an expert in Floer homology or Khovanov homology, but if that's your goal I don't think you need quite as wide a background as suggested in the other current answer (though admittedly that ...
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4 votes
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The Floer Equation is Elliptic

First of all, ellipticity is defined in terms of the principal symbol of an operator, and the Hamiltonian term is zeroth order in the derivatives of u, so let's assume wlog that we're taking about J-...
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  • 6,230
4 votes
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The singular cohomology embeds into the symplectic cohomology

There is a Morse-Bott spectral sequence computing the symplectic cohomology of affine varieties which are complements of normal crossing divisors in smooth projective varieties. For simplicity, let's ...
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  • 2,950
3 votes

Ozsváth-Szabó's contact invariant on the Brieskorn sphere $\Sigma(2,3,6m+1)$

Brieskorn-Pham calculated the signature of the smoothing of the Milnor fiber of that Brieskorn singularity (in this case the smoothing of the complex singularity $x^2+y^3+z^{6m+1}=0$ or the symplectic ...
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  • 773
3 votes
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Ozsváth-Szabó's contact invariant on the Brieskorn sphere $\Sigma(2,3,6m+1)$

(Note: I might have screwed up orientations, so take everything with a grain of salt.) I will write an argument for the case $m=1$, showing that the reduced contact invariant $c^{\rm red}(\xi)$ is non-...
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  • 8,884
3 votes
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Intuition about bubbling off a ghost bubble

Expanding/fixing my comment: I preface with history, Parker-Wolfson's paper came before Kontsevich's compactification with the notion of "stable map". A "stable" ghost bubble necessarily contains at ...
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3 votes

Lagrangian Floer (co)homology, Novikov coverings and exact symplectic manifolds

The index of a critical point of the action functional cannot be defined. This is where Floer theory departs from the usual Morse theory, and Floer discusses it in the intro to his original paper. I ...
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  • 326
3 votes

Heegard diagrams for three-manifolds

Chapter four of "Knots, Links, Braids and 3-Manifolds" by Prasolov and Sossinsky gives a highly readable (and nicely illustrated) introduction to three-manifolds via Heegaard splittings. ...
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  • 19.9k
3 votes
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Associativity of orientations of determinant bundles in Floer homology

The issue here are the isomorphisms in your suggested proof. There are choices and conventions involved. To give names to the isomorphisms clarifies what needs to be checked. $$ \iota_{K,L}:Det(K\...
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  • 2,894
3 votes

Dismissing pseudoholomorphic curves in embedded contact homology

Just some context of "what it means to choose generic Morse–Bott perturbation": So when doing Morse–Bott theory what you are secretly doing in the Morse–Bott picture is counting cascades (...
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  • 101
3 votes

Moduli space of flat connection over homology 3-sphere

Let $M$ be your homology $3$-sphere. First, (as suggested by @MoisheKohan on MSE), note that your space is the quotient of $\mathrm{Hom}(\pi_1(M), SU(2))$ by the conjugation action of $SU(2)$. Indeed, ...
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2 votes

Dismissing pseudoholomorphic curves in embedded contact homology

(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 ...
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2 votes

Heegard diagrams for three-manifolds

You are probably familiar with definitions and theorems. But I prefer to write those for completeness. And also excuse for a paint-like drawing. I hope that they will be useful. A handlebody of genus $...
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  • 1,212
2 votes

Linearization of the Floer equation

$\mathcal{F}^{H,J}_u$ is exactly the Floer operator (what you have called $\mathcal{F}$ in your post) locally near the cylinder $u$ in $\mathcal{P}^{1,p}(x^-,x^+)$, after we use the underlying metric &...
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