Math 443 Algebraic topology.
Spring,
2006.
Lecture notes.
This page will
give a summary of each lecture as the semester progresses. It is
written in reverse chronological order, so you see the most recent
notes first. However this means that helpful links appear when a
topic is first mentioned chronologically, meaning the last time
it appears below.
Here are last
year's lecture notes.
April 26
Defined the homotopy lifting property, fibrations and Serre fibrations. Proved that every fiber bundle is a Serre fibration. Defined Steifel manifolds.We will meet twice this Friday in consideration for not meeting next Wednesday.
April 24
Proved the Hurewicz theorem and defined fiber bundles.The following homework problems are due on May 5: pages 389-391: 1, 8 ,15, 21, 22, 23.
April 21
Constructed the classifying space of a group G, with special attention to the case G=U(1). Discussed the connection with group cohomology.April 19
Discussed Eilenberg-Mac Lane spaces and cohomology operations.April 17
Discussed the Whitehead theorem, the homotopy excision theorem, cellular approximation, CW approximation and the construction of Moore spaces.April 14
Discussed the Hopf fibration, the Hopf maps and the Freudenthal suspension theorem and the Hopf invariant.April 12
Discussed clock problems as an illustration of Poincaré duality and the signs in the cup product. Began discussing homotopy theory (Chapter 4) by talking about the table of homotopy groups of spheres shown on page 339. Stated the Hurewicz theorem.April 10
Proved the Künneth theorem, which describes the homology of the tensor product of two chain complexes of free abelian groups. Stated the Poincaré duality theorem and mentioned the duality between cup products and the intersection pairing.April 5 and 7
Proved the universal coefficient theorem.April 3
Defined homology and cohomology with coefficients. Discussed the failure of tensor product and the Hom functor to preserve exactness. Defined projective and injective modules.March 31
Defined the degree of a map from Sn to itself (page 134) 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). Also discussed the problem ofvector fields on spheres.March 29
Computed the cellular homology of some spaces and defined the Euler characteristic.March 27
Sketched the proof of Proposition 2.21. Defined the cellular chain complex of a CW-complex X and showed that its homology is the singular homology of X.March 24
Derived Theorem 2.20 (the excision theorem) and the Mayer-Vietoris sequence from Proposition 2.21 and defined barycentric subdivision.March 22
Proved the 5-lemma. Stated Theorem 2.20 (the excision axiom) and Prop.2.21.March 20
Used the Mayer-Vietoris sequence to compute the homology of a surface of genus g and stated the 5-lemma.March 10, two sessions
Stated the Eilenberg-Steenrod axioms for homology and used them to compute the homology of the n-sphere. Stated the Mayer-Vietoris theorem and used it to compute the homogy of the 2-dimensional torus. Also defined categories, functors and natural transformations.The following homework is due March 24. Page 131: 3, 4, 5, 8, and 9.
March 8
Defined simplicial homology and computed it for some simple examples including the projective plane. Defined singular homology and computed it for a point.March 6
Defined the tensor product of chain complexes, introduced the standard n-simplex and discussed Δ-complexes (page 103).March 3, second session
Proved that chain homotopic maps induce the same map in homology, and sketched the proof that a short exact sequence of chain complexes leads a long exact sequence of homology groups.March 3, first session
A geometric interlude. Showed how to construct coverings of surfaces with free abelian groups (of ranks 1, 2 and 3) of deck transformations. Here is a picture of the embedding I described of a surface of genus 3 into a 3-dimensional torus.March 1
Finished the construction of the universal covering space. Began discussing homology by introducing chain complexes and chain homotopies.February 27
Almost finished the construction of the universal covering space.February 15
Proved 1.34 (the unique lifting property) and began the construction of the universal covering space, described on pages 64-65.The following homework is due March 1: Page 79-83: 10, 13, 14, 18, 19.
Our next meeting will be on February 27.
February 13
Proved 1.31, 1.32 and 1.33, all of which concern covering spaces.February 10, two sessions
Gave an example of a covering that proves that the free group on 2 generators has a subgroup isomorphic to a free group on n for any n>2. Proved the Van Kampen Theorem as in the book. Described the analogy between covering space theory and Galois theory.February 8
Used the still unproved Van Kampen theorem to compute the fundamental group of a surface of genus g and a finite path connected graph using the maximal tree.The following homework is due February 15: Page 52-53: 1, 3 ,4, 5, 9.
February 6
Discussed the fundamental group of the Klein bottle (see example 1.42 in the book) and stated the van Kampen Theorem (1.20) in full generality. Applied it to torus viewed as the union of a disk and the compliment of a point.February 3, second session
Stated the simplest case of the van Kampen Theorem. Defined the free product of two groups. Used the VKT to find the fundamental groups of the real projective plane, the one point union of two circles, and the Klein bottle. The latter is closely related to the infinite dihedral group.February 3, first session
Proved the Ham Sandwich Theorem, Prop. 1.12, which identifies the fundamental group of a Cartesian product, and Prop. 1.14, which says that higher dimensional spheres are simply connected. Used 1.12 to find the fundamental group of the torus.February 1
Proved the Borsuk-Ulam Theorem (1.10) and its corollary, the Ham Sandwich Theorem. Here is a paper about the history of the latter.The following homework is due Wednesday, February 8. Do problems #5, 6, 8, 10 and 16 on pages 38-39.
January 30
Finished the proof of Theorem 1.7, which says that the fundamental group of the circle is the integers. Used it to prove the Fundamental Theorem of Algebra (1.8), an the Brouwer Fixed Point Theorem (1.9).January 27
Started to prove Theorem 1.7, which says that the fundamental group of the circle is the integers. Defined covering spaces and proved Proposition 1.30, the homotopy lifting property for coverings.January 25
Discussed real projective spaces and showed that SO(3) is homeomorphic to RP3. The real projective plane cannot be embedded in R3, which means it is not homeomorphic to a subspace of R3, but Boy's surface is the image of a revealing map from RP2 to R3.January 23
Defined mapping cylinders, mapping cones and CW-complexes, using the 2-dimensional torus as an example. Also introduced real projective n-space.January 20
Finished the definition of the fundamental group and stated theorems about it for SO(2) and SO(3). Defined homotopy, homotopy equivalence, contractibility, and retracts.You should read Chapter 0 of Hatcher and do problems #1, 2, 9, 17 and 18 by Friday, January 27.
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