Math 443. Algebraic topology . Spring, 2003.
Lecture notes.
This page will
give a summary of each lecture as the semester progresses. Here are last year's lecture notes.
April 29
Defined cohomology with compact support and proved the Poincaré duality theorem.April 25
Discussed orientability and proved Theorem 3.26 and Lemma 3.27, which guarantee the existence of a fundamental class in the top dimensional homology group of an orientable manifold. Defined the cap product and used it to state the Poincaré duality theorem precisely.April 24
Discussed the duality between cup products in cohomology and the intersection pairing in homology.April 22
Discussed the homology and cohomology of groups (which is not in the book), and began discussing Poincaré duality by defining manifolds.April 18
Proved the homological form of the general Künneth theorem, 3B.6. Described a triangulation of S3 with a free action of the group Z/n .April 17
Described the cohomology rings for torii and projective spaces. Proved the Künneth theorem in cohomology for the case when the cohomology of one of the factors is free and finitely generated over the coefficient ring.The following homework problems are due on April 29, the day of our last meeting. Page 205, #5, 7, and 11; page 229, #11, 13, and 14.
April 15
Described the computation of cup products for surfaces (Examples 3.7 and 3.8), defined the Hopf invariant and sketched the proof of Theorem 3.14, which concerns the commutativity of cup products.April 11
Defined natural transformations and stated the Eilenberg-Steenrod axioms for homology and cohomology, Eilenberg-Mac Lane spaces, and introduced cup products.April 10
Discussed cohomology with coefficients and the Universal Coefficient Theorem (3.2). Also discussed categories and functors; see pp 162-165. For more information on this subject, see Mac Lane's book Categories for the Working Mathematician.April 8
Discussed homology with coefficients and the Universal Coefficient Theorem for homology (3A.2).April 1
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). Defined the cellular chain complex of a CW-complex X and showed that its homology is the same as the singular homology of X. Discussed Sn, CPn, and RPn as examples of CW-complexes and computed the cellular homology of each. Showed (Theorem 2.44) that the Euler-Poincaré characteristic of a finite CW-complex X, by definition the alternating sum of the number of cells in each dimension, is equal to the alternating sum of the dimensions of its homology groups.Our next meeting will on April 8, and the following problems are due then. Page 155, #2, 8, 9 and 12.
March 28
This was an expository lecture on some future topics. I demonstrated the belt and coffee cup tricks and explained how they are related to π1(SO(3)). Explained why π1(SO(3)) = Z/2. Defined higher homotopy groups, stated the Bott Periodicity theorem (see page 384 of Hatcher) and described the Hopf map from S3 to S2; see "Hopf fibration" section of the course home page.March 27
Discussed barycentric subdivision and derived the Mayer-Vietoris sequence from the excision theorem. Mentioned the the 5-lemma (its proof is a simple diagram chase) and used it in the proof of Theorem 2.27, which says that simplicial and singular homology agree.March 25
Discussed the long exact homology sequence for a pair of spaces (2.13) and the Excision Theorem (2.20).March 7
Defined chain homotopy and showed that homotopic maps induce the same homomorphism in singular homology.Our next meeting will on March 25, and the following problems are due then. Page 131, #3, 4, 5, 8 and 9.
March 6
Defined simplices, barycentric coordinates, and Δ-complexes. Defined the simplical homology of a Δ-complex and the singular homology of a topological space.March 4
Introduced homology without defining it. Stated the Mayer-Vietoris sequence (page 149) and used it to compute H*(Sn). Defined chain complexes (page 106) and stated Theorem 2.16, which says that a short exact sequence of chain complexes induces a long exact sequence of homology groups.February 25
Discussed more examples of covering spaces and the analogy with Galois theory. Also discussed the classifying space of a group.February 21
Constructed the universal covering space of a semi-locally simply connected space and discussed group actions and orbit spaces.February 20
Discussed Propositions 1.30, 1.31 and 1.32 of Hatcher, all of which concern covering spaces.February 18
Finished the proof of the Van Kampen theorem and introduced covering spaces, the subject of Section 1.3 of Hatcher.The following homework problems are due on March 4: pages 52-53, #1, 3, 4, 5, and 9.
February 14
Gave some more applications of the Van Kampen theorem (VKT), including the fundamental groups of connected graphs and 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 13
Stated the Van Kampen theorem (VKT), which is a powerful tool for calculating the fundamental group, and gave some simple examples. This entails defining the free product in group theory. VKT is the subject of Section 1.2 of Hatcher. Our next meeting is at 2:00 tomorrow.January 30
Proved the Borsuk-Ulam Theorem (1.10) and its corollary, the Ham Sandwich Theorem (not in the book), which says that given any three measurable sets in R^3, one can find a plane which simultaneously bisects each of them. Proved that the fundamental group of a higher dimensional sphere is trivial (1.14) and that the fundamental group of a Cartesian product of spaces is the Cartesian product of the fundamental groups of its factors (1.12).The following homework assignment is due on February 13, the day of our next meeting: pages 38-39: #5, 6, 8, 10 and 16.
January 28
Proved that the fundamental group of the circle is the integers (Theorem 1.7), gave a topological proof of the Fundamental Theorem of Algebra (1.8), and proved the Brouwer Fixed Point Theorem (1.9).January 23
Discussed the homotopy extension property (see pages 14-17 of Hatcher) and introduced the fundamental group, the subject of Chapter 1.January 21
Discussed real and complex projective spaces as cell complexes. Defined products, quotients, cones, mapping cones, suspensions, joins, wedge sums, and smash products. Most of these are discussed on pages 8-10 of Hatcher. Our next meeting is Thursday at 2:00. We will not meet Friday as previously announced.January 16
Defined the mapping cylinder and showed that it is homotopy equivalent to the target space. Defined retracts and deformation retracts. Defined cell complexes and gave several examples including 2 ways of constructing the n-sphere as a cell complex. Our next meeting is Tuesday at 2:00.January 15
Defined homotopy and homotopy equivalence. You should read Chapter 0 of Hatcher and do problems #1, 2, 9, 17 and 18 by Tuesday January 21. Our next meeting is tomorrow at 2:00.
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