Avraham (Rami) Aizenbud

אברהם (רמי) איזנבוד

My Group




First term: Tuesday 11:15-13:00, Room 155; Second term: Tuesday 10:15-12:00, Room 155; .


Shachar Carmenli - First term: Wednesday 13:15-15:00, Room 155; Second term: Monday 11:15-13:00, Room 155; .

Office hours.

by appointment..


1. General information

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, The last lecture(s) 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].
Lecture 2.
Operation with spaces: product, bouquet, quotient, smash product, suspension, join, loop space,(mapping) cylinder and (mapping) cone. [GH, 7], [FF, 1], [HAT, 0]
Lecture 3.
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 4.
π1 of a bouquet product and suspension, Seifert-van Kampen theorem, equivalent definitions of π1, fundamental groupoid. [HAT, 1.2].
Lecture 5.
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].
Lecture 6.
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 7.
Euler theorem, Euler characteristic of a simplicial complex.
Homologies of a simplicial complex: definitions, examples. [GH, 10], [HAT, 2.1].
Lecture 8-9.
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 10-11.
Singular homologies: definition, proof of axioms. [GH, 14-15], [FF, 11], [HAT, 2.1].

4.3. Advance Homotopy theory.

Lecture 12.
πi(Sn); i n. [FF, 9], [HAT, 4.2].

Lecture 13.
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 14-15.
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 16-17.
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 18-19.
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 20-21.
  Cech (co-)homology. [HAT, 3.3].
Lecture 22-23.
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 24-25.
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]. All the material of the course (except some of the advance topics) appears in [HAT]. However, the order of the topics will be more similar to [GH] and [FF].

[GH] is the easiest one of the three, but it doesn’t cover all of the required information. [FF] contains most of what we will need, but omits too many details in some proofs. Additionally, [FF] is highly recommended for its illustrations.

[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 Shachar an e-mail (from the address you wish to join/un-join) with subject “join/un-join me to geom-5779”. To send a message to the course mailing list send Shachar an e-mail with subject “e-mail to geom-5779 – the subject of your message”.

7. How to get credit for the course?

The homework will be 100% of the grade. If you fill that you know some 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 whight of the rest of the homework will increase.

The course is an year course, but the credit is separate for each term.

8. Lecture notes

9. Homework

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.