M201 Course Description page 1 of 6  J. Simon 2008, all rights reserved 22M:201 Introduction to Algebraic Topology Prof. J. Simon Fall 2008 MWF 9:30 210 MLH Office 1-D MLH [email protected] (319) 335-0768 Class notes etc. may be posted on my web page: http://www.math.uiowa.edu/~jsimon (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 S1 x S1 . This is a typical 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, χ(X) Euler Characteristic Fundamental Group π1(X) First Homology H1(X) Second Homology H2(X) 2nd Homotopy π2(X) S2 2 {1} {0} Z Z S1 x S1 0 Z ⊕ Z Z ⊕ Z Z {0} RP2 1 Z2 Z2 Z2 Z We will develop several ways to do this, with two recurring themes: (1) The algebra keeps track of what kinds of "holes" a space has, and (2) we can analyze a space by seeing how it is built out of simpler pieces. For example, the sphere S2 is the union of two disks joined along their boundaries. If we remove the origin from the plane R2 , 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". The spaces R2 - {one point} and R2 - {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 any method that can 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: If weM201 Course Description page 2 of 6  J. Simon 2008, all rights reserved remove an entire line segment I1={ (x,0) ∈ R2 | 0 ≤ x ≤ 1 } , the resulting space is homeomorphic to what we get when we remove just one point. Also there are different kinds of "holes". The sphere S2 , that is the unit sphere in 3-space { (x,y,z) ∈ R3 | x2+y2+z2 = 1 } surrounds a "hole"; a standard 2-dimensional torus (see figure below) 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, all cubes In and all Euclidean spaces Rn. 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), which we will describe in terms of continuously deforming one map to another. For example, each mapping of a circle into R2 can be continuously deformed to a constant map (i.e. "shrunk to a point"); but there are maps of a circle into R2 - {one point} that are essential, that cannot be deformed to a constant map. This teaches us that R2 and R2 - {one point} are not of the same homotopy type, hence are not homeomorphic. A harder example: R2 - {one point} can be continuously deformed to a circle; we will say that R2 - {one point} and S1 are homotopically equivalent, or have the same homotopy type. But R2 - {2 points} is not homotopy equivalent to S1; which implies that R2 - {one point} and R2 - {2 points}are not homeomorphic. To show that R2 - {two points} is homotopically different from S1 , we invent a "multiplication" on the set of maps of a circle into a space X ; somehow we can take two maps of S1 → X and produce a new map of S1 → 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 group associated with R2 - {one point} is cyclic, whereas the group associated with R2 - {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. two special curves on the torus2-dimensional torus in 3-spaceM201 Course Description page 3 of 6  J. Simon 2008, all rights reserved 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's, and Gauss's theorems, then you've seen important situations where the average behavior of a function on a set can be