Avraham (Rami) Aizenbud

אברהם (רמי) איזנבוד

My Group




Tuesday 12:15-14:00, Room 261.

Office hours.

Tuesday 14:15-15:00, Room 122.

If you would like to come to the office hours or at some other time please e-mail me.


1. General information

The course is part of the course “Basic topics in geometry 1”. The second part “Analysis on manifold” by Dmitry Novikov.

The two parts will depend on each other and the students are expected to attend both parts (unless they know the material of one of them).

The course is of M.Sc. level. It includes some basic geometry facts every mathematician is expected to know. Math students are strongly recommended to attend, CS or physics students wishing to broaden their mathematical background are also welcome.

2. Overview

We will discuss homotopy and homology theory. The course is split into two units. The first one contains the most elementary facts of those theories, together with their detailed proofs. The second will contains more advanced material of both theories, with sometimes more sketchy proofs. The last lecture will be an overview of more advance topics

3. Prerequisite

The students are expected to know basic general topology and group theory.

4. Chronological list of topics

4.1. Basic Homotopy theory.

Lecture 1.
Motivation and overview.
Basic Homotopy theory: homotopy, homotopy category, homotopy equivalence, pointed topological space. [GH, 1,2], [FF, 1], [HAT, 0].
Operation with spaces: product, bouquet, quotient, smash product, suspension, join, loop space,(mapping) cylinder and (mapping) cone. [GH, 7], [FF, 1], [HAT, 0]
Lecture 2.
fundamental group π1: definition, homotopy invariance, coverings, universal covering (existence and uniqueness), relation between coverings and π1, examples. [GH, 4-6], [FF, 4,5], [HAT, 1.1,1.3].
Lecture 3.
π1 of a bouquet product and suspension, Seifert-van Kampen theorem, equivalent definitions of π1, fundamental groupoid. [HAT, 1.2].
Lecture 4.
Higher homotopy groups πn (basic facts): definition, commutativity, homotopical groups and co-groups. πn of products, coverings and loop spaces, difficulties of computation of πn of bouquets and suspensions. Weak homotopy equivalence of topological spaces, examples. [GH, 7], [FF, 6], [HAT, 4.1].
Simplicial complexes: definition, realization. Whitehead theorem: weak homotopy equivalence of simplicial complexes implies their homotopy equivalence. Barycentric subdivision. Any topological space is weak homotopy equivalent to a simplicial complex. [HAT, 2.1, 4.1].

4.2. Basic Homology theory.

Lecture 5.
Euler theorem, Euler characteristic of a simplicial complex.
Homologies of a simplicial complex: definitions, examples. [GH, 10], [HAT, 2.1].
Lecture 6.
Axiomatic approach to homologies: Definition, Barratt-Puppe sequence, relative homologies. Some corollaries and equivalent axioms: Mayer-Vietoris theorem, excision theorem, Hn of bouquet, long exact sequence of a triple, examples, uniqueness, Generalized Homology theories, problems with Hn of loop space. [GH, 16-17], [FF, 12], [HAT, 2.2, 2.3].
Lecture 7.
Singular homologies: definition, proof of axioms. [GH, 14-15], [FF, 11], [HAT, 2.1].

4.3. Advance Homotopy theory.

Lecture 9.
πi(Sn); i n. [FF, 9], [HAT, 4.2].
CW complexes: definition, cellular approximation, CW aproximation, Whitehead theorem, computation of π1 and of homologies of CW complexes, obstacles to computation of πn of CW complexes. [GH, 21], [FF, 3], [HAT, 0, 4.1].
Lecture 10.
Simplicial sets. Definition, realisation, Kan condition. combinatorial description of homotopy classes of maps between realisations of Kan simplicial sets.
long exact sequence of (Serre) fibration. Examples. [FF, 7,8], [HAT, 4.2].
Eilenberg-MacLane spaces [FF, 2], [HAT, 4.2].
Lecture 11.
relative homotopy groups and long exact sequence a pair. [FF, 8], [HAT, 4.1].
Excision and corolaries: Hurewicz theorem, Freudenthal suspension theorem, stable homotopy groups [FF, 9], [HAT, 4.2].

4.4. Advanced Homology theory.

Lecture 12.
Kunneth theorem. [GH, 29], [HAT, 3.2,3.B].
Universal coefficient theorem [GH, 29], [FF, 15], [HAT, 3.1, 3.A]
Cohomology: definition, cup product, duality to homologies. [GH, 23, 24], [FF, 14], [HAT, 3.1].
Cohomology with compact support and Borel-Moore homology. [GH, 26], [HAT, 3.3].
Lecture 13.
  Cech (co-)homology. [HAT, 3.3].
Lecture 14.
Orientation and Poincare duality [GH, 22, 26], [HAT, 3.3].
relation to Eilenberg-MacLane spaces [FF, 2], [HAT, 4.3]

4.5. Advanced topics.

Lecture 15.
Sheaf cohomology.
Spectral sequences.
the stable homotopy category and spectra.
Alexander duality
Cohomology operations
Bott periodicity theorem

5. Textbooks

The literature for the course is [GHFFHAT]. The course will follow a “convex combination” of [GH] and [FF]. We will use [HAT] as a source of examples, problems and additional information.

[GH] is the easiest one of the three, but it doesn’t cover all of the required information. [FF] contains almost everything we will need, but omits too many details in some proofs. Also, the order of the topics in the course will be something between [GH] and [FF]. Additionally, [FF] is highly recommended for its illustrations. Finally, [HAT] is the most detailed of these three books, but it is too big to serve as a textbook for a first course in algebraic topology.

[GH]    Greenberg and Harper, Algebraic topology: a first course.

[FF]    Fomenko and Fuks, Homotopic topology.

[HAT]    Hatcher, algebraic topology.

6. E-mail list

To join/un-join the course e-mail list send me an e-mail (from the address you wish to join/un-join) with subject “join/un-join me to geom-5775”. To send a message to the course mailing list send me an e-mail with subject “e-mail to geom-5775 – the subject of your message”.

7. How to get credit for the course?

The homework will be 30% of the grade and the final exam 70%. If you fill that you know same of the material well enough and you do not need to attend part of the course, you can come to me and convince me in that. If this is the case, you will be excused from the corresponding part of the homework, and accordingly the wight of the exam will increase.

8. Lecture notes

Lecture 1; Exercise Session 1

Lecture 2

Lecture 3

Lecture 4; Exercise Session 4

Lecture 5

Lecture 6

Lecture 7

Lecture 8

Lecture 9

Lecture 10

Lecture 11

Lecture 12

Lecture 13

9. Homework

Homework 1, Solution

Homework 2, Solution a, Solution b

Homework 3, Solution

Homework 4, Solution

Homework 5, Solution a, Solution b

Homework 6, Solution a, , Solution b

Homework 7, Solution a, Solution b

Homework 8, Solution

Homework 9, Solution a, Solution b

Homework 10, Solution

Homework 11, Solution

Homework 12, Solution

Homework 13, Solution


1. The notes and the solutions are writen by the students, they might contain some mistakes. Students that wish to upload corrected versions are wellcome to send them to me.