Knowledge and understanding: understanding the most important
homology and cohomology theories in algebraic topology, knowing the
main applications of these theories, and acquiring basic notions of
homological algebra.

Applying knowledge and understanding: computing homology and
cohomology groups of topological spaces, solving problems in algebraic
topology by using homology theories.

Learning skills: reading and understanding advanced texts in algebraic
topology concerning homology or cohomology theories.

Making judgements: being able to decide whether a given problem can
be dealt with the tools introduced in the course.

Communication skills: presenting in a correct and appropriate manner
definitions and theorems included in a homology or a cohomology theory.

General topology, basic algebra.

Simplicial homology: simplices and simplicial complexes; simplicial
homology groups; zero-dimensional homology groups; homology of
cones.

Singular homology: basic homological algebra; singular homology groups;
topological invariance; zero-dimensional homology groups; homology of a
point; homotopy invariance; long exact homology sequence; relative
homology; excision; Mayer-Vietoris exact sequence; Eilenberg–Steenrod
axioms; homology of spheres; fixed point theorem of Brouwer; degree of
maps between spheres; generalized Jordan curve theorem and invariance
of domain; isomorphism between singular and simplicial homology; Euler
characteristic; homology with coefficients; the Tor functor; the universal
coefficient theorem for homology.

Cellular homology: CW complexes; cellular homology groups;
equivalence with singular homology; cellular boundary formula;
classification and homology of surfaces.

Cohomology: cohomology groups; the Ext functor; universal coefficient
theorem for cohomology; Poincaré duality.

Main textbooks:

J. R. Munkres, Elements of Algebraic Topology. Addison-Wesley Publishing
Company 1984

A. Hatcher, Algebraic Topology. Cambridge University Press 2002
(http://www.math.cornell.edu/~hatcher/AT/ATpage.html)

Simplicial homology: simplices and simplicial complexes; simplicial
homology groups; zero-dimensional homology groups; homology of
cones.

Singular homology: basic homological algebra; singular homology groups;
topological invariance; zero-dimensional homology groups; homology of a
point; homotopy invariance; long exact homology sequence; relative
homology; excision; Mayer-Vietoris exact sequence; Eilenberg–Steenrod
axioms; homology of spheres; fixed point theorem of Brouwer; degree of
maps between spheres; generalized Jordan curve theorem and invariance
of domain; isomorphism between singular and simplicial homology; Euler
characteristic; homology with coefficients; the Tor functor; the universal
coefficient theorem for homology.

Cellular homology: CW complexes; cellular homology groups;
equivalence with singular homology; cellular boundary formula;
classification and homology of surfaces.

Cohomology: cohomology groups; the Ext functor; universal coefficient
theorem for cohomology; Poincaré duality.

During the lesson, theoretical aspects and exercises are presented using
the blackboard. Students are strongly invited to actively take part to the
lessons. Regularly I assign to the students some exercises as homework.

Teacher's notes and homework assignments can be found on the
MOODLE platform.

The final exam is composed of two parts: a written test and an oral exam. The written test consists of two questions: the presentation of a basic definition or a basic result discussed during the classes, and a simple computation of homology groups. A score of at least 16/30 grants access to the oral test which must be taken during the same exam session in which the written test is passed. Submitting any written test replaces any previously submitted one. The oral exam consists of a verbal discussion about the topics covered in the course. It is based on two questions and lasts on average 30-45 minutes. The final score is based on both written and oral tests. The grading system is as follows: (18-23): The student can explain in detail the basic constructions and definitions; they know the statements and the ideas of the proofs of the main theorems; they can reproduce simple computations presented during the lessons. (24-28): The student can explain all constructions and definitions proficiently; they know the statements of all theorems introduced during the course; they can explain the proofs of the theorems with a good level of independence; they can perform computations using various homological techniques. (29-30 with honours): The student can explain all the constructions and definitions proficiently; they knows the statements of all the theorems introduced during the course; he can explain the proofs of the theorems with an excellent level of independence; they can solve simple problems using homological and cohomological technics. To ensure the access to aids at the exam from students with disabilities, specific learning disorder (SLD), or special educational needs (SEN), please contact the University's Disability Service or SLD Service in advance.