Author | Dimca, Alexandru. author |
---|---|

Title | Sheaves in Topology [electronic resource] / by Alexandru Dimca |

Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2004 |

Connect to | http://dx.doi.org/10.1007/978-3-642-18868-8 |

Descript | XVI, 240 p. online resource |

SUMMARY

Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds, a great geometrical idea due to R. Thom and H. Whitney. These sheaves, generalizing the local systems that are so ubiquitous in mathematics, have powerful applications to the topology of such singular spaces (mainly algebraic and analytic complex varieties). This introduction to the subject can be regarded as a textbook on "Modern Algebraic Topology'', which treats the cohomology of spaces with sheaf coefficients (as opposed to the classical constant coefficient cohomology). The first five chapters introduce derived categories, direct and inverse images of sheaf complexes, Verdier duality, constructible and perverse sheaves, vanishing and characteristic cycles. They also discuss relations to D-modules and intersection cohomology. The final chapters apply this powerful tool to the study of the topology of singularities, of polynomial functions and of hyperplane arrangements. Some fundamental results, for which excellent sources exist, are not proved but just stated and illustrated by examples and corollaries. In this way, the reader is guided rather quickly from the A-B-C of the theory to current research questions, supported in this by a wealth of examples and exercises

CONTENT

1 Derived Categories -- 1.1 Categories of Complexes C*(A) -- 1.2 Homotopical Categories K*(A) -- 1.3 The Derived Categories D*(A) -- 1.4 The Derived Functors of Hom -- 2 Derived Categories in Topology -- 2.1 Generahties on Sheaves -- 2.2 Derived Tensor Products -- 2.3 Direct and Inverse Images -- 2.4 The Adjunction Triangle -- 2.5 Local Systems -- 3 Poincarรฉ-Verdier Duality -- 3.1 Cohomological Dimension of Rings and Spaces -- 3.2 The Functor f! -- 3.3 Poincarรฉ and Alexander Duality -- 3.4 Vanishing Results -- 4 Constructible Sheaves, Vanishing Cycles and Characteristic Varieties -- 4.1 Constructible Sheaves -- 4.2 Nearby and Vanishing Cycles -- 4.3 Characteristic Varieties and Characteristic Cycles -- 5 Perverse Sheaves -- 5.1 t-Structures and the Definition of Perverse -- 5.2 Properties of Perverse -- 5.3 D-Modules and Perverse -- 5.4 Intersection Cohomology -- 6 Applications to the Geometry of Singular Spaces -- Singularities, Milnor Fibers and Monodromy -- Topology of Deformations -- Topology of Polynomial Functions -- Hyperplane and Hypersurface Arrangements -- References

Mathematics
Algebraic geometry
Functions of complex variables
Algebraic topology
Mathematics
Algebraic Topology
Algebraic Geometry
Several Complex Variables and Analytic Spaces