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U where ''U''(''n'') is the Unitary Group and ''O''(''n'') the Orthogonal Group . After Vladimir Arnold it is denoted by Λ(''n''). TOPOLOGY Its Fundamental Group is Infinite Cyclic , with a distinguished generator given by the square of the Determinant of a Unitary Matrix , as a mapping to the Unit Circle . Its first Homology Group is therefore also infinite cyclic, as is its first cohomology group. Arnold showed that this leads to a description of the Maslov index, introduced by V. P. Maslov . For a Lagrangian Submanifold ''M'' of ''V'', in fact, there is a mapping M which classifies its Tangent Space at each point (cf. Gauss Map ). The Maslov index is the pullback via this mapping, in H of the distinguished generator of H THE MASLOV INDEX A path of Symplectomorphism s of a symplectic vector space may be assigned a Maslov index; it will be an integer if the path is a loop, and a half-integer in general. If this path arises from trivializing the Symplectic Vector Bundle over a periodic orbit of a Hamiltonian vector field on a Symplectic Manifold or the Reeb Vector Field on a Contact Manifold , it is known as the Conley-Zehnder Index . It computes the spectral flow of the Cauchy-Riemann -type operators that arise in Floer Homology . It appeared originally in the study of the WKB Approximation and appears frequently in the study of Quantization and in Symplectic Geometry and topology. It can be described as above in terms of a Maslov index for linear Lagrangian submanifolds. REFERENCE
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