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Register a new userExamples around the strong Viterbo conjecture
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Vinicius Gripp Barros Ramos
Publication date
2020
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English
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A strong version of a conjecture of Viterbo asserts that all normalized symplectic capacities agree on convex domains. We review known results showing that certain specific normalized symplectic capacities agree on convex domains. We also review why all normalized symplectic capacities agree on $S^1$-invariant convex domains. We introduce a new class of examples called monotone toric domains, which are not necessarily convex, and which include all dynamically convex toric domains in four dimensions. We prove that for monotone toric domains in four dimensions, all normalized symplectic capacities agree. For monotone toric domains in arbitrary dimension, we prove that the Gromov width agrees with the first equivariant capacity. We also study a family of examples of non-monotone toric domains and determine when the conclusion of the strong Viterbo conjecture holds for these examples. Along the way we compute the cylindrical capacity of a large class of weakly convex toric domains in four dimensions.
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We present another view dealing with the Arnold-Givental conjecture on a real symplectic manifold $(M, omega, tau)$ with nonempty and compact real part $L={rm Fix}(tau)$. For given $Lambdain (0, +infty]$ and $minNcup{0}$ we show the equivalence of the following two claims: (i) $sharp(Lcapphi^H_1(L))ge m$ for any Hamiltonian function $Hin C_0^infty([0, 1]times M)$ with Hofers norm $|H|<Lambda$; (ii) $sharp {cal P}(H,tau)ge m$ for every $Hin C^infty_0(R/Ztimes M)$ satisfying $H(t,x)=H(-t,tau(x));forall (t,x)inmathbb{R}times M$ and with Hofers norm $|H|<2Lambda$, where ${cal P}(H, tau)$ is the set of all $1$-periodic solutions of $dot{x}(t)=X_{H}(t,x(t))$ satisfying $x(-t)=tau(x(t));forall tinR$ (which are also called brake orbits sometimes). Suppose that $(M, omega)$ is geometrical bounded for some $Jin{cal J}(M,omega)$ with $tau^ast J=-J$ and has a rationality index $r_omega>0$ or $r_omega=+infty$. Using Hofers method we prove that if the Hamiltonian $H$ in (ii) above has Hofers norm $|H|<r_omega$ then $sharp(Lcapphi^H_1(L))gesharp {cal P}_0(H,tau)ge {rm Cuplength}_{F}(L)$ for $F=Z_2$, and further for $F=Z$ if $L$ is orientable, where ${cal P}_0(H,tau)$ consists of all contractible solutions in ${cal P}(H,tau)$.
We prove that any closed connected exact Lagrangian manifold L in a connected cotangent bundle T*N is up to a finite covering space lift a homology equivalence. We prove this by constructing a fibrant parametrized family of ring spectra FL parametrized by the manifold N. The homology of FL will be (twisted) symplectic cohomology of T*L. The fibrancy property will imply that there is a Serre spectral sequence converging to the homology of FL and the product combined with intersection product on N induces a product on this spectral sequence. This product structure and its relation to the intersection product on L is then used to obtain the result. Combining this result with work of Abouzaid we arrive at the conclusion that L -> N is always a homotopy equivalence.
We prove that, for closed exact embedded Lagrangian submanifolds of cotangent bundles, the homomorphism of homotopy groups induced by the stable Lagrangian Gauss map vanishes. In particular, we prove that this map is null-homotopic for all spheres. The key tool that we introduce in order to prove this is the notion of twisted generating function and we show that every closed exact Lagrangian can be described using such an object, by extending a doubling argument developed in the setting of sheaf theory. Floer theory and sheaf theory constrain the type of twisted generating functions that can appear to a class which is closely related to Waldhausens tube space, and our main result follows by a theorem of Bokstedt which computes the rational homotopy type of the tube space.
The paper was withdrawn due to a gap in the proof of Lemma 3.
We generalize the hamiltonian topology on hamiltonian isotopies to an intrinsic symplectic topology on the space of symplectic isotopies. We use it to define the group $SSympeo(M,omega)$ of strong symplectic homeomorphisms, which generalizes the group $Hameo(M,omega)$ of hamiltonian homeomorphisms introduced by Oh and Muller. The group $SSympeo(M,omega)$ is arcwise connected, is contained in the identity component of $Sympeo(M,omega)$; it contains $Hameo(M,omega)$ as a normal subgroup and coincides with it when $M$ is simply connected. Finally its commutator subgroup $[SSympeo(M,omega),SSympeo(M,omega)]$ is contained in $Hameo(M,omega)$.
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