scholarly journals Givental’s Non-linear Maslov Index on Lens Spaces

2020 ◽  
Author(s):  
Gustavo Granja ◽  
Yael Karshon ◽  
Milena Pabiniak ◽  
Sheila Sandon
Keyword(s):  
Projective Space ◽  
Maslov Index ◽  
Universal Cover ◽  
Lens Spaces ◽  
Real Projective Space ◽  
Arnold Conjecture ◽  
Identity Component ◽  
Contact Rigidity ◽  
Non Linear

Abstract Givental’s non-linear Maslov index, constructed in 1990, is a quasimorphism on the universal cover of the identity component of the contactomorphism group of real projective space. This invariant was used by several authors to prove contact rigidity phenomena such as orderability, unboundedness of the discriminant and oscillation metrics, and a contact geometric version of the Arnold conjecture. In this article, we give an analogue for lens spaces of Givental’s construction and its applications.

scholarly journals THE NONLINEAR MASLOV INDEX AND THE CALABI HOMOMORPHISM

2007 ◽  
Vol 09 (06) ◽  
pp. 769-780 ◽  
Author(s):  
GABI BEN SIMON
Keyword(s):  
Projective Space ◽  
Open Subset ◽  
Complex Projective Space ◽  
Maslov Index ◽  
Universal Cover ◽  
Hamiltonian Diffeomorphisms ◽  
Calabi Invariant ◽  
Nonlinear Maslov Index ◽  
Complex Projective ◽  
Calabi Homomorphism

In this paper, we find that the asymptotic nonlinear Maslov index defined on the universal cover of the group of all contact Hamiltonian diffeomorphisms of the standard (2n - 1)-dimensional contact sphere is a quasimorphism. Then we show our main result: Let M be standard (n - 1)-dimensional complex projective space. We prove that the value of the pullback of the asymptotic nonlinear Maslov index to the universal cover of the group of Hamiltonian diffeomorphisms of M, when evaluated on a diffeomorphism supported in a sufficiently small open subset of M, equals [Formula: see text] times the Calabi invariant of this diffeomorphism.

Immersions and Embeddings Up to Cobordism

1971 ◽  
Vol 23 (6) ◽  
pp. 1102-1115 ◽  
Author(s):  
Richard L. W. Brown
Keyword(s):  
Projective Space ◽  
Compact Manifold ◽  
Real Projective Space ◽  
Binary Expansion ◽  
Simple Argument ◽  
Prove Theorem ◽  
Definition Of

In 1944 Whitney proved that any differentiable n-manifold (n ≧ 2) can be (differentiably) immersed in R2n–1[15] and embedded in R2n [14]. Whitney's results are best possible when n = 2r. (One uses a simple argument involving the dual Stiefel-Whitney classes of real projective space Pn. See [9, pp. 14, 15].) However, there is a widely known conjecture that any R-manifold (n ≧ 2) immerses in R2n–α(n) and embeds in R2n–α(n)+1. Here, α(n) denotes the number of ones in the binary expansion of n. We prove (Theorem 5.1) that every compact manifold is cobordant to a manifold that immerses in (2n – α(n))-space and embeds in (2n – α(n) + 1)-space. (See § 1 for the definition of cobordant manifolds.) It is well known that if the conjecture were true it would be the best possible result.

scholarly journals A wall-crossing formula for degrees of Real central projections

2014 ◽  
Vol 25 (04) ◽  
pp. 1450038 ◽  
Author(s):  
Christian Okonek ◽  
Andrei Teleman
Keyword(s):  
Algebraic Geometry ◽  
Projective Space ◽  
Pole Placement ◽  
Real Projective Space ◽  
Central Projections ◽  
Wall Crossing ◽  
Subspace Problem ◽  
Wall Crossing Formula ◽  
Real Subspace ◽  
Crucial Assumption

The main result is a wall-crossing formula for central projections defined on submanifolds of a Real projective space. Our formula gives the jump of the degree of such a projection when the center of the projection varies. The fact that the degree depends on the projection is a new phenomenon, specific to Real algebraic geometry. We illustrate this phenomenon in many interesting situations. The crucial assumption on the class of maps we consider is relative orientability, a condition which allows us to define a ℤ-valued degree map in a coherent way. We end the article with several examples, e.g. the pole placement map associated with a quotient, the Wronski map, and a new version of the Real subspace problem.

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