The positive fundamental group of Sp(2) and Sp(4)

Abstract

In this paper, we examine the homotopy classes of positive loops in Sp (2) and Sp(4). We show that two positive loops are homotopic if and only if they are homotopic through positive loops. The Behavior of Positive Paths and Lifting LemmasA useful tool for describing the movement of eigenvalues along a positive path is the splitting number. The notion of splitting number arises from Krein theory, described in [2] and [3], and is explained further in Lalonde and McDuff [6]. They define the non-degenerate Hermitian symmetric form β on C 2n by β(v, w) = −iw T Jv where J is the standard 2n × 2n block matrix with the identity in the lower left box and minus the identity in the upper right box. They prove theHence, for any simple eigenvalue λ ∈ S 1 we may define σ(λ) = ±1 where β(v, v) ∈ σ(λ)R + . Using properties of β, we can check that σ(λ) = −σ(λ). As an illustration, when n = 1, the matrix J = 0 −1 1 0

Degenerate maxima in Hamiltonian systems

“…This result allows us to then construct loops of Hamiltonian diffeomorphisms with fixed nondegenerate global maxima. Combining these constructions with results of Slimowitz [11] we obtain:…”

“…In the event that Q t is symmetric, but only positive semidefinite (i.e., Q t could have eigenvalues of zero for certain values of t), the path is called semipositive. In [11] Slimowitz proves the following:…”

Fukaya 𝐴_{∞}-structures associated to Lefschetz fibrations. IV

“…One can conjecture that statements similar to (3.45) should hold more generally for spaces of nonnegative paths which are "sufficiently long" (meaning that the rotation number increases by a sufficient amount to force transitions between elliptic and hyperbolic elements). This idea is in tune with the results of [15,32].…”

“…References. Nonnegative paths and the rotation number quasimorphism generalize to linear symplectic groups (see [15,32] and [2,24,3], respectively). The original definition of the rotation number as (3.5), due to Poincaré, applies to orientation-preserving homeomorphisms of the circle (for a basic exposition, see [9]; and for a proof of the quasimorphism property with the optimal bound (3. with θ ∈ (0, π) =⇒ tr(gγ) = −2α sin(θ) ≤ 0, so the angle of rotation θ can't decrease along a nonnegative path (and its derivative vanishes exactly when that of the path vanishes).…”

“…References. The discussion above is taken from [15,32] (see more specifically [32, Lemma 2.2]), and we will continue to fundamentally follow those references, while making changes and adding material as required for our intended applications. Specifically, Lemma 3.8 below is a special case of [15,Proposition 2.4] (our proof fills in some technical details); and Lemmas 3.11-3.13 can be viewed as modified versions of [32,Theorem 3.3] (note however that [32, Theorem 3.6] is incorrect).…”