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Complex Ball Quotients and Line Arrangements in the Projective Plane (MN-51)$
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Paula Tretkoff

Print publication date: 2016

Print ISBN-13: 9780691144771

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691144771.001.0001

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Topological Invariants and Differential Geometry

Topological Invariants and Differential Geometry

Chapter:
(p.6) Chapter One Topological Invariants and Differential Geometry
Source:
Complex Ball Quotients and Line Arrangements in the Projective Plane (MN-51)
Author(s):

Paula Tretkoff

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691144771.003.0002

This chapter deals with topological invariants and differential geometry. It first considers a topological space X for which singular homology and cohomology are defined, along with the Euler number e(X). The Euler number, also known as the Euler-Poincaré characteristic, is an important invariant of a topological space X. It generalizes the notion of the cardinality of a finite set. The chapter presents the simple formulas for computing the Euler-Poincaré characteristic (Euler number) of many of the spaces to be encountered throughout the book. It also discusses fundamental groups and covering spaces and some basics of the theory of complex manifolds and Hermitian metrics, including the concept of real manifold. Finally, it provides some general facts about divisors, line bundles, and the first Chern class on a complex manifold X.

Keywords:   topological invariant, differential geometry, Euler number, fundamental group, covering space, complex manifold, Hermitian metric, divisor, line bundle, first Chern class

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