Math 443 Algebraic topology.

Spring,
2005.

Lecture notes.

This page will
give a summary of each lecture as the semester progresses. Here are last
year's lecture notes.

## April 18

Continuing the description of the cohomology of the special orthogonal group, I defined the reflection map from*RP*to

^{n-1}*SO(n)*and defined Hopf algebras (see section 3.C of Hatcher). Used Hopf algebras to prove that an even dimensional sphere cannot be a H-space.

## April 15

Gave an expository lecture in which I described several things without any proofs. These included cup products (see section 3.2 of Hatcher) and the Pontryagin product (page 287 of Hatcher). Also described the homology and cohomology of the classical groups, i.e., the orthogonal group (see section 3D of Hatcher), the special orthogonal group, the unitary group, the special unitary group and the symplectic group.## April 13

Proved the the universal coefficient theorems for homology and cohomology and the Kunneth Theorem, whihc describes the homology of a Cartesian product of spaces in terms of that of its factors.## April 11

Stated the universal coefficient theorems for homology and cohomology and computed Tor and Ext in some simple cases.The following homework problems are due on April 27, the day of our last meeting. Page 205, #5, 7, and 11; page 229, #11, 13, and 14.

## April 1

Defined Tor and Ext, the derived functors of tensor product and Hom.## March 30

Stated the cellular boundary formula (see page 141) and used the cellular chain complex to find the homology of oriented and unoriented surfaces and of a certain acyclic space related to the icosahedron.Computed the homology of real projective spaces and lens spaces. Defined the Euler characteristic and showed that it is a topological invariant.

## March 28

Defined cellular homology and showed that it is isomorphic to singular homology for any CW-complex*X*(Theorem 2.35).

## March 25

Discussed the lens space problem, group rings and tensor products. The following homework assignment is due on Monday, April 11: page 155, #2, 8, 9a, 9b, 9c, 9d, and 12.## March 23

Reviewed previous homework assignment.## March 18

Discussed the degree of a map from*S*to itself and used it to prove that an even dimensional sphere cannot have a tangent vector field that is nonzero at every point (Theorem 2.28).

^{n}## March 16

Computed the homology of a surface of genus*g*using the the Mayer-Vietoris sequence. Stated and proved the 5-lemma.

## March 14

Proved the excision axiom (Theorem 2.20) using barycentric subdivision, omitting the clever proof of Prop. 2.21. Proved the Mayer-Vietoris theorem by the same method.## March 4

Proved the homotopy axiom using the prism operator. Stated the Mayer-Vietoris theorem and used the Mayer-Vietoris sequence to computed the homolgy of the n-sphere.## March 2

Defined singular homology and oporoved the dimension axion, i.e., computed the homology pf a point. Defined relative homology an dproved the exactness axiom.## February 28

Discussed categories, functors and natural transformations. Introduced the cellular chain complex of a CW-complex.## February 21

Sketched the proof that a short exact sequence of chain complexes induces a long exact sequence of homoology groups. Stated the Eilenberg-Steenrod axioms for homology. Introduced categories and functors.## February 18

Introduced homological algebra by discussing chain complexes, chain homotopies and tensor products.## February 16

Constructed the universal covering space and discussed Theorem 1.38, which says there is a one-to-one correspondance between isomophism classes of path connected covering spaces of X and subgroups of the fundamental group of X.## February 14

Discussed Propositions 1.33 and 1.34.## February 11

Discussed Propositions 1.30, 1.31 and 1.32 of Hatcher, all of which concern covering spaces.## February 9

Proved the Van Kampen theorem and introduced covering spaces, the subject of Section 1.3 of Hatcher.## February 7

Used the still unproved Van Kampen theorem to compute the fundamental group of a a path connected graph (using the maximal tree), and a surface of genus*g*.

## February 4

Defined the amalgamated free product in group theory and stated the simplest case of the Van Kampen theorem. Used it to compute the fundamental group of a one point union of circles and a torus.## February 2 (ii)

Proved that the fundamental group of a higher dimensional sphere is trivial (1.14) and showed that the fundamental groups respects Cartesian products. Defined free groups.The following homework assignment is due on Friday, February 11: pages 38-39: #5, 6, 8, 10 and 16.

## February 2 (i)

Proved the Borsuk-Ulam Theorem (1.10) and the Ham Sandwich Theorem, which is not in the book and says that given any three measurable sets in R3, one can find a plane which simultaneously bisects each of them..## January 31

Gave a topological proof of the Fundamental Theorem of Algebra (1.8), and proved the Brouwer Fixed Point Theorem (1.9). Defined higher homotopy groups and showed they are abelian.## January 28

Proved that the fundamental group of a circle is isomorphic to the integers.## January 26

Defined the fundamental group of a pointed space. See Chapter 1 of Hatcher. Our next meeting is at 1:00 Friday. The homework assignment is due February 2.## January 25

Defined mapping cylinders, mapping cones, CW-complexes , pointed spaces, smash products and joins. Most of this can be found in Chapter 0 of Hatcher.## January 24

Defined the*n*-sphere

*S*, the

^{n}*n*-ball

*B*,

^{n}*n*-dimensional real projective space

*, and the*

**R**P^{n}*n*-dimensional special orthogonal group

*SO(n)*. Showed that

*SO(3)*is homeomorphic to

*.*

**R**P^{3}You should read Chapter 0 of Hatcher and do problems #1, 2, 9, 17 and 18 by Wednesday February 2.