UI MATH 5400 - Introduction to Algebraic Topology (4 pages)

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Introduction to Algebraic Topology



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Introduction to Algebraic Topology

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Pages:
4
School:
University of Iowa
Course:
Math 5400 - General Topology & Introduction to Smooth Manifolds

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22M 201 Introduction to Algebraic Topology Prof J Simon Fall 2005 MWF 9 30 118 MLH Office 1 D MLH jsimon math uiowa edu 319 335 0768 Office hours will be set in a few days for now please see me after class to make an appointment Introduction This course will introduce some of the basic ideas of algebraic topology using algebra to answer topological questions such as Are these two spaces homeomorphic Are these two mappings very similar to each other homotopic that will be made precise in the course Does a mapping f X X have a fixed point Typically the algebra can tell us that two spaces or maps are different from each other we need more direct analysis to show they are similar But sometimes if we restrict our attention to a particular set of spaces then the algebra can provide a perfect classification The general method is to associate various algebraic objects e g numbers groups rings vector spaces to topological spaces in such a way that similar spaces have equivalent algebraic objects Thus for example a hard problem of trying to show two spaces are not homeomorphic might be changed to an easier problem of trying to show that two groups are not isomorphic Topologists study the shapes of sets they spend half their time deciding what that means and the other half doing it The most fundamental insight into the shape of a set is to count the number of components You have studied many other topological properties of spaces To distinguish one space from another one we might ask if the space is compact locally connected separable metric etc If we were trying to distinguish two smooth manifolds we could ask about their dimensions But what if the two spaces are both compact connected 2 dimensional manifolds say a 2 sphere vs a torus 1 1 S x S This is the task of Algebraic Topology Our goal is to develop ways of associating topologically invariant numbers groups etc to the spaces that will distinguish them For example 2 S 1 1 S xS X Euler Characteristic 2 0 Fundamental Group



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