12/11/19, 12:20-2:20 p.m.: Final Exam (in Kolthoff Hall 134)
12/11/19, 12:20-2:20 p.m.: Final Exam (in Kolthoff Hall 134)
12/9/19: Review before the Final. We will discuss a selection of
problems from the Fall 2019 (and, time permitting, Spring 2019)
Preliminary Written Exam(s) (Part A). See
https://math.umn.edu/requirements/previous-written-exams. See
also my rewording of Problem 4 on the Fall 2019 Prelim on the class
News web page. Also, Problem 3 on the Spring 2019 and problems on
smooth manfiolds on earlier prelims are definitely not on our
syllabus.
12/6/19: Homework is due. Homology and Homotopy. Topological
manifolds. Triangulation of manifolds. Classification of
surfaces. [Class notes. Shastri: Section 4.5 (skip the proof),
5.1 (through Example 5.1.3; Theorem 5.1.13 (just the wording)), 5.2
(first page), and 5.3 (first page, Theorem 5.3, the wordings of
Theorems 5.3.16 and 5.3.17)]
12/4/19: The Mayer-Vietoris sequence, more computations. The
suspension theorem. Invariance of dimension revisited. Simplicial
homology. Example of computation. [Class notes. Shastri:
Sections 4.2 (after Corollary 4.2.20), The Wikipedia article on the
basics
of Simplicial
Homology, then Theorems 4.3.13-4.3.15, from Section 4.3 (B). Take
it for granted that the simplicial chain complex in Shastri is
isomorphic to the one in Wikipedia.]
12/2/19: Singular homology: reduced homology, basic properties (long
exact sequence, excision), some examples. [Class
notes. Shastri: Section 4.2 (Basic Properties of Singular Homology
through Corollary 4.2.20)]
11/29/19: Thanksgiving Break. Have a happy Thanksgiving!
11/27/19: Snow day: no classes. More time to think about life and
homework!
11/25/19: Homework is due at the beginning of class. Singular
homology: homotopy invariance, some more trivial
computations. [Class notes. Shastri: Section 4.2 (Basic
Properties of Singular Homology, I and III)]
11/22/19: Singular simplices and chains. Singular homology: definition
and the first computation. [Class notes. Shastri: Section 4.2
through Remark 4.2.8; Example 4.2.12 (i)]
11/20/19: Basic homological algebra, continued: functoriality of
homology, chain homotopy, the snake lemma, the long exact sequence,
the Five lemma. [Class notes. Shastri: Section 4.1 through the
end, skipping 4.1.14-4.1.20]
11/18/19: Homework is due at the beginning of class. The fundamental
groups of the Klein bottle and the torus. Basic homological algebra:
homology of a complex, example. [Class notes. Shastri: Section
4.1 through Definition 4.1.5]
11/15/19: The amalgamated products and Seifert-van Kampen
theorem. Examples. [Class notes. Shastri: Section 3.6 for the
case Gij = H for all i,j and Section 3.7]
11/13/19: Some group theory: free products. [Class
notes. Shastri: Section 3.6 for the case Gij = {e} for
all i,j]
11/11/19: Homework is due at the beginning of class. Group actions and
covering spaces. [Class notes. Shastri: Section 3.5 through
Example 3.5.9(ii)]
11/8/19: Galois theory for covering spaces: complete classification of
covering projections. The existence of a universal covering.
[Class notes. Shastri: Section 3.4 (Theorems 3.4.12 and
3.4.17)]
11/6/19: Deck transformations and the Galois group. Universal
coverings. [Class notes. Shastri: Section 3.4 from
Definition 3.4.5 through Remark 3.4.11, as well as Remark 3.4.13
through Remark 3.4.16]
11/4/19: Homework is due at the beginning of class. Equivalence of
covering spaces. [Class notes. Shastri: Section 3.4 through
Remark 3.4.4]
11/1/19: The sufficient condition of the Lifting theorem. [Class
notes. Shastri: Theorem 3.3.12 and Corollary 3.3.13]
10/30/19: Coverings and fundamental groups. [Class
notes. Shastri: Sections 3.3 through 3.3.12 (the necessary
condition)]
10/28/19: Homework is due at the beginning of class. The path and
homotopy lifting properties. [Class notes. Shastri: Section
3.2 from 3.2.2]
10/25/19: Covering spaces. The unique lifting property. [Class
notes. Shastri: Sections 3.1-3.2 through 3.2.1]
10/23/19: Invariance of domain. [Class notes. Shastri: Section
2.9 starting from 2.9.16]
10/21/19: Homework is due at the beginning of class. Sperner's
lemma. Brouwer's fixed point theorem. Invariance of
dimension. [Class notes. Shastri: Section 2.9 from 2.9.9
through 2.9.15]
10/18/19: Simplicial approximation. [Class notes. Shastri:
Section 2.9 up to 2.9.9]
10/16/19: Barycentric subdivision as a refinement. [Class
notes. Shastri: Section 2.8 from 2.8.6]
10/14/19: Homework is due at the beginning of class. Barycentric
subdivision. Simplicial approximation. Sperner's lemma. Invariance of
domain. [Class notes. Shastri: Section 2.8 through 2.8.4]
10/11/19: More on geometric realization. Triangulations of some
spaces. [Class notes. Shastri: Section 2.7 (through Example
2.7.13)]
10/9/19: Simplicial complexes. Geometric realization. [Class
notes. Shastri: Sections 2.6 and 2.7 through 2.7.1.]
10/7/19: Homework is due at the beginning of class. Cellular maps. The
cellular approximation theorem. Applications to homotopy groups of
spheres and skeleta of CW complexes. [Class notes. Shastri:
Section 2.5]
10/4/19: Products of cell complexes. [Class notes. Shastri:
Section 2.3]
10/2/19: The topology of CW complexes. [Class notes. Shastri:
Section 2.2 from 2.2.12]
9/30/19: Homework is due at the beginning of class. Projective spaces
as CW complexes. [Class notes. Shastri: Section 2.2.11]
9/27/19: Cell complexes. [Class notes. Shastri: Section 2.2
through 2.2.11(iv)]
9/25/19: The topology of convex polytopes. The Euler characteristic of
a polytope. Polyhedral complexes. Platonic solids. [Class
notes. Shastri: Section 2.1 from 2.1.12 through the end (just skim
through the basic concepts and facts)]
9/23/19: Homework is due at the beginning of class. Intro to convex
polytopes: convex hulls, standard and geometric simplices. [Class
notes. Shastri: Section 2.1 through 2.1.12 (just skim through the
basic concepts and facts)]
9/20/19: Some typical constructions, continued: adjunction space,
mapping cylinder and cone, useful strong deformation retractions. The
idea of the homotopy cofiber and the cofiber sequence. [Class
notes. Shastri: Section 1.5 (from 1.5.7 through 1.5.13, plus
Remark 1.5.16)]
9/18/19: Some typical constructions: the cone. [Class
notes. Shastri: Section 1.5 (before 1.5.7)]
9/16/19: Homework is due at the beginning of class. Function spaces
and quotient spaces. Relative homotopy and SDRs. [Class
notes. Shastri: Sections 1.3-1.4]
9/13/19: Brouwer's fixed point theorem. The Seifert-van Kampen theorem
(simplified version). Computation of the fundamental group of
Sn, n > 1. [Class notes. Shastri: Section 1.2 from
1.2.28 through the end]
9/11/19: The degree of a map S1 --->
S1. Computation of the fundamental group of the
circle. [Class notes. Shastri: Section 1.2 from 1.2.17 through
1.2.27]
9/9/19: The fundamental group. [Class notes. Shastri: Section
1.2 through 1.2.16]
9/6/19: Relation between homotopy equivalence and homeomorphism. The
Poincaré conjecture. Invariance of domain. The Möbius band
and the cut-and-paste technique. [Class
notes. Shastri: Section 1.1 from Example 1.1.5]
9/4/19: Introduction. Syllabus handed out. The fundamental problem of
Algebraic Topology. Homeomorphism, homotopy, and homotopy
equivalence. The homotopy category of spaces. Contractible spaces.
[Syllabus.
Class notes. Shastri: Section 1.1 through Example 1.1.14
(skipping Q. Ia: Homotopy Lifting Property through Q. IIa: Homotopy
Extension Property)]
Last modified: (2019-12-08 13:55:51 CST)