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 1 X 1 Z Z H1 X First Homology 0 Z Z H2 X Second Homology Z Z 2 X Second Homotopy Z 0 We will develop several different ways to do this with two recurring themes 1 We will try to 2 understand a space by seeing how it can be built out of simpler pieces For example the sphere S is the union of two disks joined along boundaries 2 We will try to describe the holes in a space 2 If we remove the origin from the plane R we obtain a space with a hole whatever that means We could go on deleting points from the plane and obtain spaces with any number of holes 2 2 The spaces R one point and R 2 points are not homeomorphic But it is not so easy to prove that Both are separable metric spaces and they are identical locally that is each point has a neighborhood homeomorphic to an open disk So whatever method we might seek to distinguish the spaces topologically must be aware of the entire spaces not just isolated parts Furthermore our intuition that the number of holes is just the number of points removed cannot be trusted completely 1 J Simon 2005 all rights reserved 1 2 If we remove an entire line segment I x 0 R 0 x 1 the resulting space is homeomorphic to what we get when we remove just one point 2 Also there are different kinds of holes The sphere S that is the unit sphere in 3 space two special curves3 on the 2 2torus 2 x y z R x y z 1 surrounds a hole a standard 2 dimensional torus see figure also surrounds a hole but the holes are different in some fundamental way Spaces with no holes what we might call solid spaces are the simplest objects in this world of shapes These include intervals the real line and in fact all cubes In and all Euclidean spaces Rn 2 dimensional torus in 3 space The key idea in distinguishing the numbers and kinds of holes is homotopy the ability to continuously deform one space to another e g a simpler looking subspace and the ability to 2 continuously deform one mapping to another For example R one point can be continuously 2 deformed to a circle R 2 points cannot We will say that these two spaces are homotopically equivalent or have the same homotopy type 2 Every mapping of a circle into R can be continuously deformed to a constant map i e shrunk to a 2 point but there are maps of a circle into R one point that are essential that cannot be deformed 2 2 to a constant map This teaches us that R and R one point are not the same homotopy type hence are not homeomorphic 2 2 To distinguish R one point from R two points we get fancier We can invent a notion of 1 multiplication on the set of maps of a circle into a space X somehow we can take two maps of S 1 X and produce a new map of S X that combines in a meaningful way the original two maps Once we have a way to combine maps we actually can make them into a group In this sense the 2 2 group associated with R one point is cyclic whereas the group associated with R 2 points is not generated by any one element These are the sorts of ideas involved in the fundamental group of a space and its natural companion covering spaces For this combining we don t actually work with maps from a circle into X we work with maps from an interval 0 1 where the two endpoints are sent to the same place J Simon 2005 all rights reserved 2 Another basic idea related to holes is the notion of one set being the boundary of another A circle in the plane is the boundary of a disk the 2 dimensional torus above in R3 is the boundary of a solid torus If our whole world were just the 2 dimensional torus then we would have circles that do not bound any disks If you have studied vector calculus in particular Green s Stokes and Gauss theorems then you ve seen important situations where the average behavior of a function on a set can be described by its behavior on the boundary of the set e g if F is a vector field on R3 then the integral of the divergence of F over some domain is equal to the flux of F through the boundary of the domain If you were careful about stating those calculus theorems you know there were subtle issues of orientation the set and the boundary components had to be oriented consistently By thinking about things that are capable of being boundaries we call them cycles we are led to develop homology theories A space contains cycles …


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