Lectures on algebraic topology. Transl. from the Russian by Ekaterina Pervova.

*(English)*Zbl 1094.55001
EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society Publishing House (ISBN 3-03719-023-X/pbk). vi, 99 p. (2006).

This is a brief introduction to algebraic topology, consisting of two parts:

1. Elements of homology theory

This chapter deals with ordinary homology and cohomology, starting with a section categories and functors and ending with a section about Poincaré duality. In between are the usual items which are appearing more or less in each course on algebraic topology. The uniqueness of axiomatically defined homology is introduced in its “weak” form, i.e., if a natural transformation between two homology theories is an isomorphism on points, then it is an isomorphism for all finite polyhedra.

2. Elements of homotopy theory

Here the classical results about the fundamental group are indicated in considerable more detail, in comparison to many other subjects in both chapters. Bundles and coverings, higher homotopy groups and the exact homotopy sequence are the objectives of the last sections of the book.

There are many exercises in both parts of the book. At the end there is a section “Answers, hints, solutions”.

The intention of the author is seemingly not so much to present detailed proofs but to give the reader an idea about what is going on, referring to some other places in the literature for further reading and deeper insights.

1. Elements of homology theory

This chapter deals with ordinary homology and cohomology, starting with a section categories and functors and ending with a section about Poincaré duality. In between are the usual items which are appearing more or less in each course on algebraic topology. The uniqueness of axiomatically defined homology is introduced in its “weak” form, i.e., if a natural transformation between two homology theories is an isomorphism on points, then it is an isomorphism for all finite polyhedra.

2. Elements of homotopy theory

Here the classical results about the fundamental group are indicated in considerable more detail, in comparison to many other subjects in both chapters. Bundles and coverings, higher homotopy groups and the exact homotopy sequence are the objectives of the last sections of the book.

There are many exercises in both parts of the book. At the end there is a section “Answers, hints, solutions”.

The intention of the author is seemingly not so much to present detailed proofs but to give the reader an idea about what is going on, referring to some other places in the literature for further reading and deeper insights.

##### MSC:

55-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic topology |