1© J. Simon 2005, all rights reserved22M:201Introduction to Algebraic TopologyProf. J. SimonFall 2005MWF 9:30118 MLHOffice 1-D MLH [email protected] (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 answertopological questions such as: Are these two spaces homeomorphic? Are these two mappings verysimilar to each other ("homotopic" - that will be made precise in the course)? Does a mapping f:X→Xhave a fixed point? Typically, the algebra can tell us that two spaces or maps are different from eachother; we need more direct analysis to show they are similar. (But sometimes, if we restrict ourattention to a particular set of spaces, then the algebra can provide a perfect classification.) Thegeneral method is to associate various algebraic objects (e.g. numbers, groups, rings, vector spaces) totopological spaces in such a way that similar spaces have equivalent algebraic objects. Thus, forexample, a hard problem of trying to show two spaces are not homeomorphic might be changed to aneasier 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 theother half doing it. The most(?) fundamental insight into the "shape" of a set is to count the number ofcomponents. You have studied many other topological properties of spaces: To distinguish one spacefrom another one, we might ask if the space is compact? locally connected? separable? metric? etc. Ifwe were trying to distinguish two smooth manifolds, we could ask about their dimensions. But whatif the two spaces are both compact, connected, 2-dimensional manifolds, say a 2-sphere vs. a torusS1 x S1 . This is the task of Algebraic Topology. Our goal is to develop ways of associatingtopologically invariant numbers, groups, etc. to the spaces that will distinguish them. For example,χ(X) EulerCharacteristicFundamentalGroup π1(X)H1(X) FirstHomologyH2(X) SecondHomologyπ2(X) SecondHomotopyS22{1}{0}ZZS1 x S10Z ⊕ ZZ ⊕ ZZ{0}We will develop several different ways to do this, with two recurring themes: (1) We will try tounderstand a space by seeing how it can be built out of simpler pieces. For example, the sphere S2 isthe union of two disks joined along boundaries. (2) We will try to describe the "holes" in a space.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 provethat. Both are separable metric spaces and they are identical locally - that is, each point has aneighborhood homeomorphic to an open disk. So whatever method we might seek to distinguish thespaces topologically must be "aware of" the entire spaces, not just isolated parts. Furthermore, ourintuition that the "number of holes" is just the number of points removed cannot be trusted completely:2© J. Simon 2005, all rights reservedIf we remove an entire line segment I1={ (x,0) ∈ R2 | 0 ≤ x ≤ 1 } , the resulting space ishomeomorphic 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)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 ofshapes. These include intervals, the real line, and in fact all cubes In and all Euclidean spaces Rn.The key idea in distinguishing the numbers and kinds of "holes" is homotopy: the ability tocontinuously deform one space to another (e.g. a simpler looking subspace), and the ability tocontinuously deform one mapping to another. For example, R2 - {one point} can be continuouslydeformed to a circle; R2 - {2 points} cannot. We will say that these two spaces are homotopicallyequivalent, or have the same homotopy type.Every mapping of a circle into R2 can be continuously deformed to a constant map (i.e. "shrunk to apoint"); but there are maps of a circle into R2 - {one point} that are essential, that cannot be deformedto a constant map. This teaches us that R2 and R2 - {one point} are not the same homotopy type, henceare not homeomorphic.To distinguish R2 - {one point} from R2 - {two points}, we get fancier: We can invent a notion of"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, thegroup associated with R2 - {one point} is cyclic, whereas the group associated with R2 - {2 points} isnot generated by any one element. These are the sorts of ideas involved in the fundamental group of aspace 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 froman interval [0,1] where the two endpoints are sent to the same place.two special curves on the torus2-dimensional torus in 3-space3© J. Simon 2005, all rights reservedAnother basic idea related to "holes" is the notion of one set being the boundary of another. A circle inthe plane is the boundary of a disk; the 2-dimensional torus (above) in R3 is the boundary of a solidtorus. If our whole world were just the 2-dimensional torus, then we would have circles that do notbound 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 canbe described by its behavior on the boundary of the set. (e.g. if F is a vector field on R3 , then theintegral of the divergence of F over some domain is equal to the flux of F through the boundary of thedomain.) If you were careful about stating those calculus theorems, you know there were subtle issuesof orientation: the set and the boundary components had to be oriented consistently. By thinking aboutthings that are "capable of being boundaries" (we