ALGEBRAIC TOPOLOGY
AVRAHAM AIZENBUD
Lectures.
First term: Tuesday 11:1513:00, Room 155; Second term: Tuesday 10:1512:00, Room 155; .
T.A.
Shachar Carmenli  First term: Wednesday 13:1515:00, Room 155; Second term: Monday 11:1513:00, Room 155; .
Oﬃce hours.
by appointment..
Contents
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.


(a)
 Motivation and overview.

(b)
 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 π_{1}: deﬁnition, homotopy invariance, coverings, universal covering
(existence and uniqueness), relation between coverings and π_{1}, examples. [GH, 46],
[FF, 4,5], [HAT, 1.1,1.3].
Lecture 4. π_{1} of a bouquet product and suspension, Seifertvan Kampen theorem, equivalent
deﬁnitions of π_{1}, fundamental groupoid. [HAT, 1.2].
Lecture 5. Higher homotopy groups π_{n} (basic facts): deﬁnition, commutativity,
homotopical groups and cogroups. π_{n} of products, coverings and loop
spaces, diﬃculties of computation of π_{n} 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.


(a)
 Euler theorem, Euler characteristic of a simplicial complex.

(b)
 Homologies of a simplicial complex: deﬁnitions, examples. [GH, 10], [HAT,
2.1].

Lecture 89.
 Axiomatic approach to homologies: Deﬁnition, BarrattPuppe sequence, relative
homologies. Some corollaries and equivalent axioms: MayerVietoris theorem,
excision theorem, H_{n} of bouquet, long exact sequence of a triple, examples,
uniqueness, Generalized Homology theories, problems with H_{n} of loop space. [GH,
1617], [FF, 12], [HAT, 2.2, 2.3].

Lecture 1011.
 Singular homologies: deﬁnition, proof of axioms. [GH, 1415], [FF, 11], [HAT,
2.1].
4.3. Advance Homotopy theory.

Lecture 12.
 π_{i}(S^{n}); i ≤ n. [FF, 9], [HAT, 4.2].

Lecture 13.
 CW complexes: deﬁnition, cellular approximation, CW aproximation,
Whitehead theorem, computation of π_{1} and of homologies of CW
complexes, obstacles to computation of π_{n} of CW complexes. [GH, 21],
[FF, 3], [HAT, 0, 4.1].

Lecture 1415.


(a)
 Simplicial sets. Deﬁnition, realisation, Kan condition. combinatorial
description of homotopy classes of maps between realisations of Kan
simplicial sets.

(b)
 long exact sequence of (Serre) ﬁbration. Examples. [FF, 7,8], [HAT, 4.2].

(c)
 EilenbergMacLane spaces [FF, 2], [HAT, 4.2].

Lecture 1617.


(a)
 relative homotopy groups and long exact sequence a pair. [FF, 8], [HAT,
4.1].

(b)
 Excision and corolaries: Hurewicz theorem, Freudenthal suspension
theorem, stable homotopy groups [FF, 9], [HAT, 4.2].
4.4. Advanced Homology theory.

Lecture 1819.


(a)
 Kunneth theorem. [GH, 29], [HAT, 3.2,3.B].

(b)
 Universal coeﬃcient theorem [GH, 29], [FF, 15], [HAT, 3.1, 3.A]

(c)
 Cohomology: deﬁnition, cup product, duality to homologies. [GH, 23, 24],
[FF, 14], [HAT, 3.1].

(d)
 Cohomology with compact support and BorelMoore homology. [GH, 26],
[HAT, 3.3].

Lecture 2021.
 Cech (co)homology. [HAT, 3.3].

Lecture 2223.


(a)
 Orientation and Poincare duality [GH, 22, 26], [HAT, 3.3].

(b)
 relation to EilenbergMacLane spaces [FF, 2], [HAT, 4.3]
4.5. Advanced topics.

Lecture 2425.


(a)
 Sheaf cohomology.

(b)
 Spectral sequences.

(c)
 the stable homotopy category and spectra.

(d)
 Alexander duality

(e)
 Cohomology operations

(f)
 Bott periodicity theorem

(g)
 Ktheory

(h)
 Bordisms
5. Textbooks
The literature for the course is [GH, FF, HAT].
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.
6. Email list
To join/unjoin the course email list send Shachar an email (from the address you wish to
join/unjoin) with subject “join/unjoin me to geom5779”. To send a message to the course
mailing list send Shachar an email with subject “email to geom5779 – 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.
