User paolo ghiggini - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T21:42:11Z http://mathoverflow.net/feeds/user/18516 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/117468/2pi-and-4pi-rotations-in-the-pin1-3-group/117497#117497 Answer by Paolo Ghiggini for 2Pi and 4Pi rotations in the Pin(1,3) group Paolo Ghiggini 2012-12-29T10:17:45Z 2012-12-29T10:17:45Z <p>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.</p> <p>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 $\mathbb I$.</p> <p>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 $- \Lambda_L$,</p> http://mathoverflow.net/questions/117213/applications-of-floer-homology/117215#117215 Answer by Paolo Ghiggini for Applications of Floer homology Paolo Ghiggini 2012-12-25T23:53:04Z 2012-12-25T23:53:04Z <p>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.</p> <p>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 $(\Sigma, \alpha, \beta)$.</p> http://mathoverflow.net/questions/111683/looking-for-an-algebraic-descrption-of-the-non-trivial-extension-of-zxz-by-z2/111686#111686 Answer by Paolo Ghiggini for looking for an algebraic descrption of the non-trivial extension of ZxZ by Z2 Paolo Ghiggini 2012-11-06T21:42:32Z 2012-11-06T21:42:32Z <p>What about the group generated by a, b, q with the relations [a,b]=q, [a,q]=[b,q]=0 and q^2=1?</p> http://mathoverflow.net/questions/105370/when-does-a-hypersurface-have-contact-type/105402#105402 Answer by Paolo Ghiggini for When does a hypersurface have contact-type? Paolo Ghiggini 2012-08-24T17:30:32Z 2012-08-24T17:30:32Z <p>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.</p> <p>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 $T^*T^2$.</p> http://mathoverflow.net/questions/78017/maslov-index-in-heegaard-floer-homology/78087#78087 Answer by Paolo Ghiggini for Maslov Index in heegaard floer homology Paolo Ghiggini 2011-10-14T01:38:27Z 2011-10-14T01:38:27Z <p>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.</p> <p>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.</p> <hr> http://mathoverflow.net/questions/117468/2pi-and-4pi-rotations-in-the-pin1-3-group/117497#117497 Comment by Paolo Ghiggini Paolo Ghiggini 2012-12-30T00:33:56Z 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 &quot;Seiberg-Witten equations and the topology of four-manifolds&quot;. http://mathoverflow.net/questions/117064/maslov-index-of-a-holomorphic-disk Comment by Paolo Ghiggini Paolo Ghiggini 2012-12-23T09:14:26Z 2012-12-23T09:14:26Z You should look at some standard reference in symplectic topology: I suggest McDuff-Salamon &quot;Holomorphic curves in symplectic topology&quot; or Seidel's book. Then, you can look at Lipshitz's paper &quot;A cylindrical reformulation of Heegaard Floer homology&quot; where he proves the combinatorial formula for the Maslov indexin Heegaard Floer homology. http://mathoverflow.net/questions/111683/looking-for-an-algebraic-descrption-of-the-non-trivial-extension-of-zxz-by-z2/111692#111692 Comment by Paolo Ghiggini Paolo Ghiggini 2012-11-07T00:01:02Z 2012-11-07T00:01:02Z At a first glance it looks the same group I have described by generators and relations. Am I correct? http://mathoverflow.net/questions/111433/symplectic-manifold-with-totally-geodesic-foliation Comment by Paolo Ghiggini Paolo Ghiggini 2012-11-04T08:35:28Z 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. http://mathoverflow.net/questions/109235/exotic-spheres-signature Comment by Paolo Ghiggini Paolo Ghiggini 2012-10-10T02:39:07Z 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. http://mathoverflow.net/questions/108965/what-is-knot-contact-homology Comment by Paolo Ghiggini Paolo Ghiggini 2012-10-07T14:30:06Z 2012-10-07T14:30:06Z You should look at the papers where knot contact homology has been defined: <a href="http://arxiv.org/abs/1109.1542" rel="nofollow">arxiv.org/abs/1109.1542</a> and its references http://mathoverflow.net/questions/108747/topology-of-k3-as-a-sum-of-two-abelian-fibrations Comment by Paolo Ghiggini Paolo Ghiggini 2012-10-03T23:42:21Z 2012-10-03T23:42:21Z Try to look at the book of Gompf and Stipsicz http://mathoverflow.net/questions/7357/photon-propagator/7372#7372 Comment by Paolo Ghiggini Paolo Ghiggini 2012-10-01T14:16:43Z 2012-10-01T14:16:43Z Unfortunately your first equation goes over the links on the right.