Algebraic and geometric topology
http://www.msp.warwick.ac.uk/agt/
The journal "Algebraic and Geometric Topology" is published by Mathematical Sciences Publishers, Department of Mathematics, University of California at Berkeley, and the journal's website is hosted by the Mathematics Institute, University of Warwick. It covers all areas of topology. The journal's website provides subscription details, contents lists and abstracts. Access to full text is restricted to subscribers. Students should check whether their university/institution is a subscriber.
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Applied differential geometry : a compendium
http://butler.cc.tut.fi/~bossavit/BackupICM/Compendium.html
This compendium of differential geometry was compiled by Alain Bossavit, Tampere University of Technology, Finland. It lists basic concepts of differential geometry with brief descriptions. The compendium is provided in PDF format (Adobe Acrobat Reader required).
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Bestiary of topological objects
http://neil-strickland.staff.shef.ac.uk/courses/bestiary/bestiary.pdf
This is a resource created by Dr Neil Strickland at the University of Sheffield. It is a collection of topological objects. It gives examples and the construction of those examples. It includes: wedges of circles; the configuration space FkC and BkC; projective spaces; Milnor hypersurfaces; unitary groups; Lens spaces and BCp; Fermat hypersurfaces; two and three-dimensional manifolds; simply connected four-dimensional manifolds; Moore spectra; Eilenberg-MacLane spaces; Wilson spaces; the building for GlnFp; looped spaces; doubly looped spaces; and homology of Eilenberg-MacLane spaces. The document is in PDF format (Adobe Acrobat reader required).
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British Topology Home Page
http://www.maths.gla.ac.uk/~ajb/btop.phtml
The British Topology Home Page is maintained by Andrew Baker of the Department of Mathematics of the University of Glasgow, who describes it as ... "a convenient source of pointers". This resource guide lists topology groups, home pages, seminars and conferences, other topology resources and information about British Topology Meetings. Some photographs of topologists are also available.
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Conferences and meetings on topology and related topics
http://sarah-whitehouse.staff.shef.ac.uk/btconfs.html
This resource provides information on conferences and meetings on topology with links to the conference websites where available. It is maintained by Dr. Sarah Whitehouse in the Department of Mathematics, University of Sheffield.
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Elementary topology : Math 167
http://people.hofstra.edu/Stefan_Waner/RealWorld/pdfs/Topology.pdf
These notes, in PDF format, were written to accompany a course by Stefan Waner, Department of Mathematics, Hofstra University. They cover: sets and relations; functions; metric spaces; closed sets; continuous mappings; topological spaces; continuous functions; category theory; subspaces and products; closed sets, limit points and the pasting lemma; quotient spaces and pushouts; connectedness; compactness; homotopy of paths; the fundamental group of a space; covering spaces; the fundamental group of the circle and other applications; and other fundamental groups.
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Floer homology, dynamics and groups
http://people.uleth.ca/~dave.morris/banff-rigidity/polterovich-survey.pdf
These lecture notes on Floer homology, dynamics and groups were written by Leonid Polterovich, School of Mathematical Sciences, Tel Aviv University, and are made available by Professor Dave Morris, Department of Mathematics and Computer Science, University of Lethbridge, Canada. The notes begin with the Hamiltonian action of finitely generated groups. The second chapter sketches the Floer homology and the third chapter discusses Calabi quasi-isomorphic maps. The notes are available in PDF format (Adobe Acrobat Reader required).
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Foundations : Chapter 1 : Differential topology and cobordism
http://www.dpmms.cam.ac.uk/~kf262/COBOR/L01.pdf
This is the first in a set of lecture notes produced by Dr Konstantin Feldman, Centre for Mathematical Sciences, University of Cambridge, aimed at undergraduate students, year three and above. The notes start with an introduction to topological spaces and include plenty of examples, exercises and problems. They are in PDF format (Adobe Acrobat Reader required).
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Functions of several real variables and general topology : MSM3P10.1
http://web.mat.bham.ac.uk/C.Good/teach/3p10.html
This website was created to accompany a course by Dr Chris Good at the School of Mathematics, University of Birmingham. Topics covered in the lecture notes include: a review of metric spaces; topology; separation axioms; compactness; and the Baire category theorem. Question sheets are also provided. The lecture notes are available as individual PDF files (Adobe Acrobat Reader required) and PostScript files.
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General topology
http://www.math.ku.dk/~moller/e03/3gt/notes/gtnotes.pdf
These notes on the basic notions of topological spaces were written by Jesper M. Moller for a course for undergraduate students at the University of Copenhagen. The first chapter reviews topics in set theory which are required during this course. It concludes with Zorn's lemma. The second chapter introduces topological spaces. Various ways to construct new topological spaces from old ones, such as products or subsets of topological spaces, are discussed. Metric spaces are introduced. Compactness and connectedness are amongst the first important properties that a topological space may have. These notions are introduced at the end of the second chapter, which concludes with the notion of local compactness and Alexandroff compactification. The third chapter discusses the separation axioms, and considers regular and normal spaces (T3 and T4 spaces). T1 and T2 (Hausdorff) spaces are reviewed. It also considers Stone-Cech compactification of a topological space. The notion of a manifold is introduced. Having some background in analysis and metric spaces could be very useful to the reader.
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General topology and real analysis : Course 221
http://www.maths.tcd.ie/~dwilkins/Courses/221/
This is the website of a lecture course on general topology and real analysis given by Brian Wilkins at Trinity College, Dublin. It is aimed at second year undergraduates. The website includes lecture notes for the academic year 2006-2007, which are in PDF format. These cover: sets, functions and countability; metric spaces; normed vector spaces and Banach spaces; topological spaces; compact spaces; the extended real number system; measure spaces; the Lebesgue integral; signed measures and the Radon-Niokodym theorem.
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General topology front for the mathematics arXiv
http://front.math.ucdavis.edu/math.GN
This is the University of California, Davis, front for the general topology page of the arXiv e-print archive. This site allows the user to search or browse e-prints that are contained in the catalogue as well as offering the opportunity to submit e-prints. The latest 12 listings appear along with cross listings and revisions and there is a link to a calendar allowing browsing of listings by date. The e-prints are available for downloading in various formats including DVI, PostScript and PDF.
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Geometry and topology of three-manifolds
http://www.msri.org/publications/books/gt3m/
This is an electronic edition of a set of lecture notes for a course by William P.Thurston in 1978-79 at Princeton University. No attempt has been made to update the contents. The book has been broken down into chapters or portions for downloading and is available in PDF, PostScript or DVI with PostScript figures format. Topics covered include: geometry and three-manifolds, elliptic and hyperbolic geometry, geometric structures on manifolds, hyperbolic Dehn surgery, flexibility and rigidity of geometric structures, Gromov's invariant and the volume of a hyperbolic manifold, the computation of volume, Kleinian groups, algebraic convergence, and deforming Kleinian manifolds by homeomorphisms of the sphere at infinity. The notes are hosted by the Mathematical Sciences Research Institute.
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Geometry of manifolds
http://ocw.mit.edu/OcwWeb/Mathematics/18-965Fall-2004/CourseHome/
The MIT Department of Mathematics provides this course, 18.965 Geometry of Manifolds, Fall 2004, as part of the MIT OpenCourseWare project. The website for this course includes a syllabus, lecture notes and problems. The course "analyzes topics such as the differentiable manifolds and vector fields and forms. It also makes an introduction to Lie groups, the de Rham theorem, and Riemannian manifolds." All lectures are available as PDF files (Adobe Acrobat Reader necessary).
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History of topology
http://www-history.mcs.st-andrews.ac.uk/HistTopics/Topology_in_mathematics.
This resource is part of the MacTutor History of Mathematics archive at the University of St Andrews in Scotland. It describes the development of the theory from Euler's first work in the area to the development of a complete topological theory by Brouwer in 1912. There are links to the biographies of mathematicians who contributed to the theory, and also to references to further reading.
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International Journal of Shape Modeling
http://www.worldscinet.com/ijsm/ijsm.shtml
A free online sample copy is provided for this subscription-based journal covering advanced theories and techniques devised for handling the shape of objects.
Access to the full-text is only available to subscribers. Links to related books and journals are also provided.
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Introduction to compact operators
http://unow.nottingham.ac.uk/resources/resource.aspx?hid=cef3b657-d3d1-56c6
This learning resource provides the basic theory of compact linear operators on Banach spaces for postgraduate students or final year undergraduates. Topics covered include metric spaces; bounded operators and compact operators; spectra and eigenspaces. It is part of the University of Nottingham open courseware initiative (U-NOW) and was written by Dr Joel Feinstein from the School of Mathematical Sciences at the University of Nottingham. The resource is available under a Creative Commons England and Wales Public License as a PDF and mp3 file, and can be viewed or downloaded.
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Introduction to topology
http://ocw.mit.edu/OcwWeb/Mathematics/18-901Fall-2004/CourseHome/
The MIT Department of Mathematics provides this course, 18.901 Introduction to Topology, Fall 2004, as part of the MIT OpenCourseWare project. The website for this course includes a syllabus, readings and lecture notes. The course covers the basics of topology and includes topological spaces, continuous functions, connectedness, compactness, separation axioms, function spaces, metrisation theorems, embedding theorems and fundamental groups. All lectures are in PDF format (Adobe Acrobat Reader necessary).
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Introduction to topology : Maths 353 : Fall 2005
http://www.maths.manchester.ac.uk/~tv/Teaching/Topology%20353/Fall%202005/
This site contains the lecture notes produced by Dr Theodore Voronov of The University of Manchester for an introductory topology course. The notes cover the following material, topological spaces and continuous maps, the induced topology and subspaces, closed sets and the Hausdorff property. The notes then move on to cover manifolds, surfaces and simplicial complexes. Problems and solutions, and past examination papers are also available. The notes are in PDF format (Adobe Acrobat Reader required).
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Manifolds : MA455
http://www.warwick.ac.uk/~masbm/manifolds.html
This website contains the material for a course on manifolds at the University of Warwick by David Mond. The lecture notes are available in PDF and PS files and the website also contains a set of exercises for the course. The course is designed as a complete undergraduate course in manifolds. The notes are very through and comprehensive and include a large number of examples. Some of the topics introduced in this course include: SardÂ’s theorem, the density of transversality, tangent and normal bundles, tubular neighbourhoods, oriented intersection theory, abstract manifolds, differential forms and integration on manifolds.
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Manifolds and homology : MAGIC
http://maths.dept.shef.ac.uk/magic/course.php?id=11
This website was prepared to accompany the course "Manifolds and homology" given by Neil Strickland from the Department of Pure Mathematics of the University of Sheffield, as part of the MAGIC project. MAGIC runs a wide range of postgraduate-level lecture courses in mathematics, using Access Grid videoconferencing technology. This course covers the cohomology of topological spaces, with a heavy emphasis on examples, mainly manifolds. Lecture materials are provided in PDF format.
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Probability tutorials
http://www.probability.net/
This site by Noel Vaillant provides comprehensive introductory knowledge on probability. It contains more real analysis, general topology and measure theory than actual probability. Topics covered include: Dynkin systems, Caratheodory's extension, Stieltjes-Lebesgue measure, measurability, the Lebesgue integral, product spaces, the Fubini theorem, Jensen inequality, Lp spaces, bounded linear functionals in L2, complex measures, the Radon-Nikodym theorem, regular measures, maps of finite variation, Stieltjes integration, differentiation, image measure, the Jacobian formula, Fourier transforms and Gaussian measures. These tutorials are designed as a set of simple exercises, leading gradually to the establishment of deeper results. Theorems, as well as clear definitions are spelt out for future reference. These tutorials do not contain any formal proofs. Instead, they offer the user the means of proving everything. The tutorials are in PDF format (Adobe Acrobat Reader required).
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Riemann surfaces, dynamics and geometry : Math 275
http://math.harvard.edu/~ctm/home/text/class/harvard/275/98/html/
This site contains a PDF version of the Harvard University notes for a course on Riemann surfaces, dynamics and geometry, presented in 1998. There are also links from this page to questions and solutions based on the material covered in the lectures. The lecture notes cover the following material: local holomorphic dynamics; Julia sets; Sullivan's no-wandering-domains theorem; holomorphic families of rational maps; Teichmuller spaces; and hyperbolic manifolds. An image gallery covers curvature flow, the dynamics of blowups on P squared, hyperbolic planes, complex analysis, solving polynomials, Teichmuller spaces, Platonism, fractional dimensions, topology, Kleinian groups, Kleinian movies, quadratic polynomials, complex dynamics, renormalisation, dynamics on K3 surfaces, number theory and billiards.
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Spectral sequences in algebraic topology
http://www.math.cornell.edu/~hatcher/SSAT/SSATpage.html
This website contains the draft of part of the book "Spectral Sequences in Algebraic Topology" by Allen Hatcher of the Department of Mathematics, Cornell University. About 100 pages of this book are available. Topics covered include an introduction to the Serre spectral sequence, the Adams spectral sequence and Eilenberg-Moore spectral sequences. The draft is available as PDF files (Adobe Acrobat Reader required).
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Surfaces : M338_1 : OpenLearn LearningSpace
http://openlearn.open.ac.uk/course/view.php?id=3443
This advanced level unit on a special class of topological spaces called surfaces is from the Open University's OpenLearn resource which provides free access to educational resources for teachers and learners, licensed under a Creative Commons license. This course aims to introduce how to classify surfaces based on homeomorphic criteria. Topics covered in the lecture notes include: topological spaces and homeomorphisms; examples of surfaces; the orientability of surfaces; the Euler characteristic; and edge identifications.
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