Math 443 Algebraic topology.
Spring,
2004.
Lecture notes.
This page will
give a summary of each lecture as the semester progresses. Here are last
year's lecture notes.
April 19
Sketched the proof of Theorem 3.14, which concerns the commutativity of cup products. Began discussing the Künneth theorem.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 15
Introduced the cup product and described some applications of including the Hopf invariant.April 13
Discussed cohomology with coefficients and the Universal Coefficient Theorem for cohomology (3.2). Introduced the Ext group.April 6
Discussed homology with coefficients and the Universal Coefficient Theorem (3A.3) for homology. Introduced the Tor group.April 5
Defined CW-complexes and cellular homology, proving it is isomorphic to singular homology (Theorem 2.35). Defined the Euler chartacteristic if a finite CW-complex and showed it is a topological invaraiant. Introduced homology with coefficients. Homework due April 13: Page 155, #2, 8, 9 and 12.April 2
Discussed the degree of a map from Sn 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).March 30
Proved the Mayer-Vietoris theorem and used the Mayer-Vietoris sequence and used it to compute the homology of complex projectives spaces and surfaces of arbitrary genus. Stated the 5-lemma. Homework problem due next time: Find an analog of the 5-lemma in which the conclusion is that the middle map is trivial.March 25
Sketched the proof of the excision theorem using barycentric subdivision.March 23
Proved the homotopy and exactness axioms for singular homology. Stated the excisions theorem and the Eilenberg-Steerod axioms. Defined categories and functors.March 4
Defined singular homology and chain homotopy.March 2
Defined Δ-homology and computed it for two of the examples shown on page 102 of Hatcher; the third example, the Klein bottle, is a homework problem. Defined singular homology and computed it for a point. The following homework is due on March 23, the day of our next meeting after March 4: Page 131, #3, 4, 5, 8, 9.February 26
Used the Mayer-Vietoris sequence to compute the homology of the n-sphere. Defined Δ-complexes, referring to the examples shown on page 102 of Hatcher.February 24
Finished the discussion of covering spaces and began discussing homology, the subject of Chapter 2.February 19
Finished the construction of 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 17
Discussed Propositions 1.33 and 1.34 and began the construction of the universal covering space.February 12
Discussed Propositions 1.30, 1.31 and 1.32 of Hatcher, all of which concern covering spaces.The following homework problems are due on February 24: pages 52-53, #1, 3, 4, 5, and 9.
February 10
Finished the proof of the Van Kampen theorem and introduced covering spaces, the subject of Section 1.3 of Hatcher.February 5
Gave some more applications of the Van Kampen theorem (VKT), including the fundamental groups of the complements of a pair of linked and unlinked circles in R3. Then I proved the first half of the VKT, the statement that the homomorphism to the fundamental group of a union from the free product of the fundamental groups of its subspaces is onto.February 3
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, a path connected graph (using the maximal tree), and a surface of genus g.January 29
Proved 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. . Discussed free groups and free products in preparation for stating the Van Kampen theorem.The following homework assignment is due on February 5: pages 38-39: #5, 6, 8, 10 and 16.
January 27
Gave a topological proof of the Fundamental Theorem of Algebra (1.8), and proved the Brouwer Fixed Point Theorem (1.9). Proved the Borsuk-Ulam Theorem (1.10). Proved that the fundamental group of a higher dimensional sphere is trivial (1.14).January 22
Proved that the fundamental group of the circle is isomorphic to the integers (Thoerme 1.7) and the Brouwer Fixed Point Theorem (1.9).January 20
Introduced the fundamental group, the subject of Chapter 1.January 15
Defined cell complexes. Discussed real and complex projective spaces as cell complexes. Mentioned the Hopf map; see the home page for some links to help visualize it. Defined products, quotients, cones, mapping cones, suspensions, joins, wedge sums, and smash products. Most of these are discussed on pages 8-10 of Hatcher.January 14
Defined homotopy and homotopy equivalence. You should read Chapter 0 of Hatcher and do problems #1, 2, 9, 17 and 18 by Tuesday January 20. Our next meeting is tomorrow at 2:00.
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