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A.Y. 2020 / 2021

First semester
Type of Learning Activity 
Programme specific subjects
Study Path 
[PDS0-2018 - Ord. 2018] common
Teaching language 


Learning objectives 

Knowledge and understanding: understanding the most important homology and coomology theories in algebraic topology, knowing the main applications of these


General topology, basic algebra


Algebraic topology: simplicial and singular homology, cellular homology, cohomology

Teaching format 

During the lesson theoretical aspects and exercises are presented by using the blackboard. Students are strongly invited to actively take part to the lessons. Regularly I assign to the students some exercises as homework and then solutions are discussed in class.

Extended Programme 

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 Brower; 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 funtor; universal coefficient theorem for cohomology; Poincaré duality.

End-of-course test 

Six possible dates, each consisting of an oral exam. The student has to be able to explain the constructions and the definitions introduced during the course and to present the proof of the theorems which I explained during the lessons

Other information 

Homework assignments can be found on the MOODLE platform

Any changes to the methods described here, which become necessary to
ensure the application of the safety protocols related to the COVID19
emergency, will be communicated on the websites of the Department of
Mathematics and Geoscience - DMG and of the Study Program in


main textbooks:

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

A. Hatcher, Algebraic Topology. Cambridge University Press 2002 (

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