Although the construction of spectral invariants heavily relies on the analytic theory of pseudoholomorphic curves, and in particular depends on the smooth structure of symplectic manifolds, the invariants themselves are C0-type invariants which are continuous in the C0-Hamiltonian topology introduced in (OhM07) (see Section 6.2 of this book too), and hence can be extended to the topological Hamiltonian flows defined in Section 6.2.

In this chapter, we illustrate several applications of the spectral invariants to the study of symplectic topology. One of the important advantages of spectral invariants over other more direct dynamical invariants of Hofer type is their homotopy-invariance, which enables one to naturally push forward the spectral invariants to the universal covering space of the Hamiltonian diffeomorphism group and sometimes even down to the group itself. This point is highlighted by the striking construction of partial symplectic quasi-states by Entov and Polterovich (EnP06) which is based on the purely axiomatic properties of spectral invariants ρ(H; 1) and the natural operation of taking the asymptotic average in dynamical systems. Their construction was carried out for the monotone case and later extended to the arbitrary compact symplectic manifolds by Usher (Ush10b).

Firstly, we explain the construction of an invariant spectral norm performed in (Oh05d) and its application to problems of symplectic rigidity and of minimality of geodesics in Hofer's geometry. We also explain Usher's applications to Polterovich's and Lalonde and McDuff's minimality conjecture and to the sharp energy–capacity inequality. Secondly, we give a self-contained presentation of Entov and Polterovich's partial symplectic quasi-states and quasimorphisms constructed out of spectral invariants and their applications to symplectic intersection problems. Finally, we return to the study of the group of Hamiltonian homeomorphisms and explain how one can extend all these constructions to the realm of a continuous Hamiltonian category in the sense of Chapter 6.

It appears that these constructions as a whole bear much importance in the development of symplectic topology which is starting to unveil the mystery around what the true meaning of Gromov's pseudoholomorphic curves and Floer homology is from the point of view of pure symplectic topology.

The spectral norm of Hamiltonian Diffeomorphisms

In this section, we explain the construction of an invariant norm of Hamiltonian diffeomorphisms following (Oh05d), which is called the spectral norm.