We prove that for regular contact forms there exists a bijective correspondence between the $C^0$ limits of sequences of smooth strictly contact isotopies and the limits with respect to the contact distance of their corresponding Hamiltonians.
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 grou
p $Hameo(M,omega)$ 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 $Hameo(M,omega)$.
Let $f:M to mathbb{R}$ be a Morse-Bott function on a compact smooth finite dimensional manifold $M$. The polynomial Morse inequalities and an explicit perturbation of $f$ defined using Morse functions $f_j$ on the critical submanifolds $C_j$ of $f$ s
how immediately that $MB_t(f) = P_t(M) + (1+t)R(t)$, where $MB_t(f)$ is the Morse-Bott polynomial of $f$ and $P_t(M)$ is the Poincare polynomial of $M$. We prove that $R(t)$ is a polynomial with nonnegative integer coefficients by showing that the number of gradient flow lines of the perturbation of $f$ between two critical points $p,q in C_j$ coincides with the number of gradient flow lines between $p$ and $q$ of the Morse function $f_j$. This leads to a relationship between the kernels of the Morse-Smale-Witten boundary operators associated to the Morse functions $f_j$ and the perturbation of $f$. This method works when $M$ and all the critical submanifolds are oriented or when $mathbb{Z}_2$ coefficients are used.
Using a Hodge decomposition of symplectic isotopies on a compact symplectic manifold $(M,omega)$, we construct a norm on the identity component in the group of all symplectic diffeomorphisms of $(M,omega)$ whose restriction to the group $Ham(M,omega)
$ of hamiltonian diffeomorphisms is bounded from above by the Hofer norm. Moreover, $Ham(M,omega)$ is closed in $Symp(M,omega)$ equipped with the topology induced by the extended norm. We give an application to the $C^0$ symplectic topology. We also discuss extensions of Ohs spectral distance.
