Index parity of closed geodesics and rigidity of Hopf fibrations

Burkhard Wilking

doi:10.1007/pl00005801

Index parity of closed geodesics and rigidity of Hopf fibrations

Inventiones mathematicae ◽

10.1007/pl00005801 ◽

2001 ◽

Vol 144 (2) ◽

pp. 281-295 ◽

Cited By ~ 32

Author(s):

Burkhard Wilking

Keyword(s):

Closed Geodesics ◽

Hopf Fibrations

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Filling words in fundamental groups of Riemann surfaces

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.50.2013.1.1227 ◽

2013 ◽

Vol 50 (1) ◽

pp. 31-50

Author(s):

C. Zhang

Keyword(s):

Riemann Surface ◽

Fundamental Group ◽

Riemann Surfaces ◽

Fundamental Groups ◽

Closed Geodesics ◽

New Words

The purpose of this article is to utilize some exiting words in the fundamental group of a Riemann surface to acquire new words that are represented by filling closed geodesics.

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Cyclic-Homology Chern–Weil Theory for Families of Principal Coactions

Communications in Mathematical Physics ◽

10.1007/s00220-020-03804-2 ◽

2020 ◽

Author(s):

Piotr M. Hajac ◽

Tomasz Maszczyk

Keyword(s):

Cyclic Homology ◽

Quantum Deformation ◽

Homotopy Formula ◽

Standard Quantum ◽

Unital Algebra ◽

Hopf Fibrations ◽

Chain Homotopy ◽

Cartan Model ◽

Extension Algebra ◽

Hochschild Complex

AbstractViewing the space of cotraces in the structural coalgebra of a principal coaction as a noncommutative counterpart of the classical Cartan model, we construct the cyclic-homology Chern–Weil homomorphism. To realize the thus constructed Chern–Weil homomorphism as a Cartan model of the homomorphism tautologically induced by the classifying map on cohomology, we replace the unital subalgebra of coaction-invariants by its natural H-unital nilpotent extension (row extension). Although the row-extension algebra provides a drastically different model of the cyclic object, we prove that, for any row extension of any unital algebra over a commutative ring, the row-extension Hochschild complex and the usual Hochschild complex are chain homotopy equivalent. It is the discovery of an explicit homotopy formula that allows us to improve the homological quasi-isomorphism arguments of Loday and Wodzicki. We work with families of principal coactions, and instantiate our noncommutative Chern–Weil theory by computing the cotrace space and analyzing a dimension-drop-like effect in the spirit of Feng and Tsygan for the quantum-deformation family of the standard quantum Hopf fibrations.

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