The goal of algebraic topology is to assign algebraic data (like groups or rings) to topological spaces, in the hopes of using that data to distinguish spaces. The fundamental group is a good example: if X and Y are path-connected spaces with pi1(X) not isomorphic to pi1(Y), then you know that X and Y are not homeomorphic; in fact, they are not homotopy equivalent.
In Math 564, we will study another example: homology. Homology consists of a sequence of abelian groups, with the nth group Hn(X) having something to do with n-dimensional properties of X. For example, if S2 denotes the 2-sphere, then H2(S2) = Z, essentially because the 2-sphere has a "2-dimensional hole", while H3(S2) = 0, because the 2-sphere doesn't have any 3-dimensional holes. Similarly, if S1 is the circle, then H1(S1) = Z.
There are several ways of defining homology groups, each with its own advantages and disadvantages; we will look at simplicial homology, singular homology, and cellular homology, as well as an axiomatic approach that unifies all three. We will study various nice properties of homology groups, like the long exact sequence of a pair and the Mayer-Vietoris sequence. We will also discuss some applications, like the Borsuk-Ulam theorem and the Lefschetz fixed point theorem.
Along the way, we will discuss CW complexes, and we'll do a bit of category theory.
After that (probably in Math 565), we will study cohomology, which is related to homology, but different enough to be interesting in its own right. One goal here is the cup product: in addition to being a sequence of groups, there is a product in cohomology that makes it into a ring. Another goal is Poincaré duality, which describes additional structure present in the homology and cohomology groups of manifolds.
Both homology and cohomology are "homotopy invariants," meaning that if two spaces are homotopy equivalent, then their (co)homology groups are isomorphic. Homotopy theory is the general study of homotopy equivalence, and homotopy groups are the other main tool we will discuss. The fundamental group is the first of these. Compared to homology groups, say, they tend to be harder to compute--no one knows all of the homotopy groups of the sphere S2, for instance--but also more powerful--the Whitehead theorem says that if a map between CW complexes induces an isomorphism on every homotopy group, it is a homotopy equivalence.
Go back to the Math 566 home page.
Go to John Palmieri's home page.
Last modified: Fri Sep 15 12:58:12 PDT 2000