Volume growth in the component of fibered twists

Abstract

For a Liouville domain W whose boundary admits a periodic Reeb flow, we can consider the connected component [τ ] ∈ π0(Symp c ( W )) of fibered twists. In this paper, we investigate an entropy-type invariant, called the slow volume growth, of the component [τ ] and give a uniform lower bound of the growth using wrapped Floer homology. We also show that [τ ] has infinite order in π0(Symp c ( W )) if there is an admissible Lagrangian L in W whose wrapped Floer homology is infinite dimensional.We apply our results to fibered twists coming from the Milnor fibers of A k -type singularities and complements of a symplectic hypersurface in a real symplectic manifold. They admit so-called real Lagrangians, and we can explicitly compute wrapped Floer homology groups using a version of Morse-Bott spectral sequences.

Equivariant wrapped Floer homology and symmetric periodic Reeb orbits

“…The degree shift here is the same as that of the non-equivariant part; we refer to the Morse-Bott spectral sequence for the non-equivariant case in [27]. After taking the direct limits, we conclude the following.…”

“…As we will apply the index iteration theory, the homology is equipped with a Z-grading which comes from the Maslov index. Basically, our convention are those of [27]. We refer to [1, Section 3] and [40,Section 4] for more detailed descriptions.…”

“…Suppose that the contact form α on the boundary ∂W is of Morse-Bott type, for the definition we refer to [27]. To have the well-defined Maslov indices of Reeb chords we assume that the Maslov class µ L : π 2 (Σ, L) → Z of the Legendrian L = Fix(ρ) vanishes and that π 1 (Σ, L) = 0.…”

Symplectic homology of complements of smooth divisors

“…Besides the references mentioned throughout the paper, this article shares some important ideas with several works in the literature. A few of these are .…”

A computation of the ring structure in wrapped Floer homology

“…In Section 3.2, we define a class of real Lagrangians in A k -type Milnor fibers which are of particular interest. This includes real Lagrangians considered in [20] where the graded group structure of wrapped Floer homology is computed. Note that A k -type Milnor fibers can be seen as k-fold linear plumbings of cotangent bundles T * S n , see Section 3.3.…”