User paolo ghiggini - MathOverflow

Answer by Paolo Ghiggini for 2Pi and 4Pi rotations in the Pin(1,3) group

2012-12-29T10:17:45Z

I can't recall the computation you need from the top of my head, but I will try to clarify the geometric picture, hoping it will help. From your question and the notation you use I suspect you are a physicist; if I am wrong and I spend too much time explaining standard mathematics, please forgive me.

As you noticed, the condition you gave to verify that $\Lambda_L \in Pin(1,3)$ does not distinguish $\Lambda_L$ from $- \Lambda_L$. Moreover $R(2 \pi)$ and $R(4 \pi)$ are the same element in $O(1,3)$: they both are the identity. What we need to distinguish $R(2 \pi)$ from $R(4 \pi)$ is some "dynamical" information: you should regard them not as elements in $O(1,3)$ but as paths in $O(1,3)$ starting and ending at the identity. The first one will be a path which makes a single turn,while the second one will make two turns. Then you should try to lift these paths to paths in $Pin(1,3)$ starting from the identity: if you do the computation correctly you'll see that the lift of the first path will end at $- \mathbb I$ and the lift of the second path will end at $\mathbb I$.

In mathematical terminology this means that $Pin(1,3)$ is a double cover of $O(1,3)$: that is, there is a surjective map $Pin(1,3) \to O(1,3)$ such that the preimage of an element $L \in O(1,3)$ is the set $\lbrace \Lambda_L, - \Lambda_L \rbrace$. Moreover a non-contractible closed loop in $O(1,3)$ starting and ending at $L$ will lift to a path in $Pin(1,3)$ staring at $\Lambda_L$ and ending at $- \Lambda_L$,

Answer by Paolo Ghiggini for Applications of Floer homology

2012-12-25T23:53:04Z

Lagrangian intersection Floer homology can be used to define topological invariants for knots and 3-manifold. A very successful example is Heegaard Floer homology: although it is now an almost completely combinatorial theory, its first definition used Floer homology.

In short the construction works like this: by more or less standard Morse theory one can encode every closed, connected and oriented 3-manifold by a Heegaard diagram, which is a triple $(\Sigma, \alpha, \beta)$, where $\Sigma$ is a genus $g$ closed surface, and $\alpha$ and $\beta$ are $g$-tuples of pairwise disjoint curves. Then one take the $g$-fold symmetric product $Sym^g(\Sigma)$ with two tori $T_{\alpha}= \alpha_1 \times \ldots \times \alpha_g$ and $T_{\beta}= \beta_1 \times \ldots \times \beta_g$. There is a symplectic form on $Sym^g(\Sigma)$ which makes $T_{\alpha}$ and $T_{\beta}$ Lagrangian submanifolds and a slightly modified version of $HF(T_{\alpha}, T_{\beta})$ is a very useful invariant of the 3-manifold described by $(\Sigma, \alpha, \beta)$.

Answer by Paolo Ghiggini for looking for an algebraic descrption of the non-trivial extension of ZxZ by Z2

2012-11-06T21:42:32Z

What about the group generated by a, b, q with the relations [a,b]=q, [a,q]=[b,q]=0 and q^2=1?

Answer by Paolo Ghiggini for When does a hypersurface have contact-type?

2012-08-24T17:30:32Z

Here is an example of a surface which cannot be made of contact type even after isotopy. The elliptic surface $E(1)$ is obtained by blowing up $\mathbb{C}\mathbb{P}^1$ nine times. It is a symplectic four-manifold with fibre of genus 1 and 12 singular fibres. If we make a fibred connected sum between two copies of E(1) we obtain the elliptic surface $E(2)$, which is a symplectic four-manifold with fibre of genus 1 and 24 singular fibres.

Then $E(2)$ is separated by a torus $T^3$ into two pieces which are diffeomorphic to the complement of a regular fibre in $E(1)$. This shows that the separating torus cannot be made of contact type because, by a result of Chris Wendl, all strong fillings of $T^3$ are diffeomorphic to a blow up of $T^*T^2$.

Answer by Paolo Ghiggini for Maslov Index in heegaard floer homology

2011-10-14T01:38:27Z

The symmetric product doesn't satisfy $2 c_1 =0$, therefore the machinery about graded lagrangians explained in Seidels's book cannot be applied. In Heegaard Floer homology the Maslov index can, and in fact does, depend on the homotopy class of the disc. As Marco said, you should explain us what puzzels you, so that we can give you a better answer.

You might also want to look at Lipshitz's paper "A cylindrical reformulation of Heegaard Floer homology", where he works out an explicit formula for the index.

Comment by Paolo Ghiggini

2012-12-30T00:33:56Z

I don't know if it is the best possible reference, but I've learnt this stuff from the first chapter of Morgan's book "Seiberg-Witten equations and the topology of four-manifolds".

Comment by Paolo Ghiggini

2012-12-23T09:14:26Z

You should look at some standard reference in symplectic topology: I suggest McDuff-Salamon "Holomorphic curves in symplectic topology" or Seidel's book. Then, you can look at Lipshitz's paper "A cylindrical reformulation of Heegaard Floer homology" where he proves the combinatorial formula for the Maslov indexin Heegaard Floer homology.

Comment by Paolo Ghiggini

2012-11-07T00:01:02Z

At a first glance it looks the same group I have described by generators and relations. Am I correct?

Comment by Paolo Ghiggini

2012-11-04T08:35:28Z

Ho do you obtain the symplectic foliation of M? If you are referring to the singular foliation with symplectic leaves which is always associated to a Poisson structure, then in this case M is the unique leaf.

Comment by Paolo Ghiggini

2012-10-10T02:39:07Z

Your claim sounds bizarre: take your favourite 4n-manifold with non-zero signature and remove a ball: then you have a 4n-manifold with non-zero signature, bounding a homotopy sphere, which is however the standard sphere.

Comment by Paolo Ghiggini

2012-10-07T14:30:06Z

You should look at the papers where knot contact homology has been defined: arxiv.org/abs/1109.1542 and its references

Comment by Paolo Ghiggini

2012-10-03T23:42:21Z

Try to look at the book of Gompf and Stipsicz

Comment by Paolo Ghiggini

2012-10-01T14:16:43Z

Unfortunately your first equation goes over the links on the right.