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Math 443. Algebraic topology . Spring , 2002.
Lecture notes.

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January 17

Introduced the Euler characteristic and Euler's formula V-E+F=2 .  Used it to show that the complete graph on 5 points and the houses and uttilities graph are both nonplanar.  The Kuratowski theorem says that every nonplanar graph is a supergraph of one of these two.  Discussed the problem of enumerating Platonic and Archimedean solids .

Proved Euler's formula, giving the 13th of the Seventeen Proofs .  Gave an example of a polyhedron homeomorphic to a torus and found that its Euler characteristic (ie V-E+F) is 0.   Proved in general that the Euler characteristic of a union of two spaces is the the sum of the ones for the subspaces less the one for the intersection.  Used this to show that the Euler characteristic of a surface of genus g is 2-2g.  Showed my model of a great dodecahedron and explained how it is the continuous image of a surface of genus 4. In passing I mentioned the 59 stellations of the icosahedron .

Defined covering spaces (see Chapter 5 of our textbook, Greenberg and Harper) and orbit spaces, giving several examples of each.  When a group of homeomorphism G acts on a space X, the map from X to its orbit space X/G may or may not be a covering.  It will not be if the group action has any fixed points, meaning that some point in X is unchanged by some homeomorphism is G other than the identity map. If G is finite and there are no fixed points, then the Euler characteristic of X (if it is a polyhedron) is |G| times the Euler characteristic of X/G, where |G| denotes the order of G .  This formula is useful when X and X/G are surfaces.  Also talked about duality of polyhedra, which is a special case of Poincare duality.

Triangulations are defined on page 136 Greenberg and Harper.  I am using a less restrictive definition that is limited to surfaces but does not require all faces to have 3 sides.

Discussed the real projective plane and the Klein bottle , giving several descriptions of each.  The projective plane admits an immersion (smooth locally one-to-one map) into R^3 known as Boy's surface .  There is also a map that failes to be smooth at 6 points known as the Roman surface , which is related to the cuboctahedron .  Another map to R^3 which I did not dicuss is the crosscap .  The projective plane contains the  Möbius band as a subspace.

In addition to the standard immersion  of the Klein bottle depicted in the department's logo , there is the figure-8 immersion . It can be obtained by gluing two Möbius bands together along their boundaries.

The projective plane, Möbius band and Klein bottle are each nonorientable.  This means they do not admit global notions of clockwise and counterclockwise.  Any surface that contains a Möbius band is nonorientable, otherwise it is orientable.

I gave two extra credit problems due Tuesday February 5:

1. Give an explicit geometric description of Boy's surface, preferqably with some kind of triangluation.   "Explicit" means that you can make everyone in the class understand it.

 
2. Give an explicit description of an action of the group Z/2 on the torus whose orbit space is the Klein bottle.  (This was solved by Nick Record.)

Defined homotopy   ( animated illustration ), relative homotopy, homotopy equivalence   ( animiated illustration ), contractibility, and the fundamental group of a space.  This material is in the first four chapters of the text, which you should read by February 7.  An online reference for this material is Ran Levi 's Geometric Topology ( pdf file ). Ran got his doctorate here in 1993.

Showed that the fundamental group really is a group and mentioned several applications including knot theory (for more links see Brian Sanderson's home page ) and the fundamental theorem of algebra, which has a topological proof . Defined higher homotopy groups and mentioned the Hopf map as an interesting example; see the links to it on the home page .

Gave two proofs that higher homotopy groups are abelian.  The first involved an explicit homotopy.  The second required the definition of loop spaces (Chapter 7 of Greenberg/Harper), H-spaces, the functoriality of homotopy groups (meaning that a continuous map of spaces induces a homomorphism of homotopy groups), the fact that the homotopy group of a Cartesian product is the Cartesian product of the homotopy group s of the factors, and the fact that a group is abelian iff its multiplication map is a homomorphism.

Finished the determination of the fundamental group of the circle.  Strickland's illustration of path lifting is very helpful in understanding the proof of Lemmas 4.1 and 4.2.  The proof generalizes to other spaces such as the torus and the Klein bottle.  Mentioned (not by name) the Brouwer fixed point theorem (see also the Math fun facts link for an amusing interpretation), which says that any continuous map of the n-ball to itself must have a fixed point.

Our next topic will be the Van Kampen theorem, which is not covered in the book.  It is covered in Ran Levi's notes and in pages 40-46 of Hatcher's book .

Explained how each Platonic triangulation of the torus can be derived as a quotient of one of the three tessellations of the plane by the action of a group of translations.  Discussed the classification theorem for surfaces .  (This link is from Zig Fiedorowicz' topology course .) Stated the van Kampen theorem after defining amalgamated free products of groups.  The proof of the van Kampen theorem can be found in pages 41-46 of Hatcher's book and on pages 426-432 of Munkres book .  The classification theorem is proved by Munkres on pages 462-471.

Explained how the van Kampen theorem can be used to find the fundamental group of many surfaces and did three quarters of the proof of it.  Our next meeting will be on February 28.

Finished the proof of the van Kampen theorem, following Hatcher's treatment.  Introduced coverings and stated the main theorems about them and compared it with Galois theory.

Proved Theorems 5.1 and 5.2. in Greenberg and Harper.

Proved Theorems 5.3, 5.8  and 6.1. in Greenberg and Harper.

Finished Chapter 6, proving the existence of a simply connected cover under certain hypotheses (Thoerem 6.7).

Introduced homology by stating the Eilenberg-Steenrod axioms and the Mayer-Vietoris sequence.  Used the latter to compute the homology of the n-sphere.

Defined chain complexes and chain homomorphisms. Proved that a short exact sequence of them leads to a long exact sequence of homology groups. Defined the standard n-simplex and singular homology.

Computed the homology of a point (9.4) (thereby proving the dimension axiom) and the 0th homology group of a general space (9.6).  Defined chain homotopy (10.5) and showed that chain homotopic maps induce the same homomorphism in homology (10.6).

Discussed the topological structure of SO(4) (the 4-dimensional special orthogonal group, see the handout), defining fiber bundles in the process. Defined augmentation and related concepts and outlined the proof of the homotopy axiom. Began discussing the relation between the fundamental group and the first homology group.

Distributed a handout about the 5-lemma problem. Micah proved Theorem 10.13 and Niles proved Theorem 11.4.  The homotopy axiom follows from these.

Proved Theorem 12.1, which says that the first homology group of a path connected space is the abelianization of its fundamental group.

Proved the Excision Theorem (15.1) and the Mayer-Vietoris sequence (Chapter 17).

Defined spherical complexes (page 114) and illustrated the use of the Mayer-Vietoris sequence to compute their homology.  Examples included real and complex projective spaces.

Defined CW-complexes and the cellular chain complex associated with one.  Also defined Betti numbers and Euler characteristic. 

  

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This page was last revised on April 30, 2002, by Doug Ravenel .