TOPOLOGY QUALIFYING EXAM SYLLABUS Mathematics 205A. Text: J. Munkres, Topology. Supplementary texts W. Sutherland, Introduction to Metric and Topological Spaces. A. Hatcher Algebraic Topology. Part 1: Point Set Topology Topologies, bases, closed sets, limit points, various concepts of connectedness and compactness, countability and separation properties, subspace, product and quotient topologies. Part 2: Fundamental groups and covering spaces. Homotopy, Fundamental group, retractions, deformation retractions, homotopy type, covering spaces, the fundamental group of the circle, Seifert-Van Kampen Theorem. Mathematics 205B. Texts: J. Munkres, Topology. A. Hatcher Algebraic Topology. Part 1: Fundamental groups and covering spaces. Seifert-Van Kampen Theorem (continued), Classication of Covering Spaces. Part 2: Homology Simplicial and chain complexes, simplicial and singular homology, and appli- cations: homotopy invariance, relative homology, excision, long exact sequence of a pair, Mayer-Vietoris sequences, Brouwers Fixed Point Theorem, Invariance of domain, Jordan-Brouwer Separation Theorem. Mathematics 205C. Text: J. Lee, Introduction to Smooth Manifolds. Supple- mentary text: J. Milnor, Topology from the Di/erentiable Viewpoint. Topological manifolds, smooth structures, smooth maps and their di/erentials, the tangent bundle, Lie groups and their Lie algebras, partitions of unity, vector elds, Lie brackets, submersions, immersions, embeddings, submanifolds, embed- ding and approximation theorems, Lie group actions, Sards theorem, ows, degree theory. 1

TOPOLOGY QUALIFYING EXAM SYLLABUS …mathdept.ucr.edu/pdf/syllabi/grad/205ABC_Syl.pdfTOPOLOGY QUALIFYING EXAM SYLLABUS Mathematics 205A. Text: ... Hatcher Algebraic Topology. Part

Download PDF Report

Upload
nguyenthuy
View
218
Download
4

Embed Size (px)

Citation preview

TOPOLOGY QUALIFYING EXAM SYLLABUS

Mathematics 205A. Text: J. Munkres, Topology. Supplementary texts W.Sutherland, Introduction to Metric and Topological Spaces. A. Hatcher AlgebraicTopology.

Part 1: Point Set TopologyTopologies, bases, closed sets, limit points, various concepts of connectedness

and compactness, countability and separation properties, subspace, product andquotient topologies.Part 2: Fundamental groups and covering spaces. Homotopy, Fundamental group,retractions, deformation retractions, homotopy type, covering spaces, the funda-mental group of the circle, Seifert-Van Kampen Theorem.

Mathematics 205B. Texts: J. Munkres, Topology. A. Hatcher Algebraic Topology.

Part 1: Fundamental groups and covering spaces.Seifert-Van Kampen Theorem (continued), Classi�cation of Covering Spaces.

Part 2: HomologySimplicial and chain complexes, simplicial and singular homology, and appli-

cations: homotopy invariance, relative homology, excision, long exact sequence ofa pair, Mayer-Vietoris sequences, Brouwer�s Fixed Point Theorem, Invariance ofdomain, Jordan-Brouwer Separation Theorem.

Mathematics 205C. Text: J. Lee, Introduction to Smooth Manifolds. Supple-mentary text: J. Milnor, Topology from the Di¤erentiable Viewpoint.Topological manifolds, smooth structures, smooth maps and their di¤erentials,

the tangent bundle, Lie groups and their Lie algebras, partitions of unity, vector�elds, Lie brackets, submersions, immersions, embeddings, submanifolds, embed-ding and approximation theorems, Lie group actions, Sard�s theorem, �ows, degreetheory.