By Sergey V. Matveev

Algebraic topology is the research of the worldwide homes of areas through algebra. it really is an immense department of recent arithmetic with a large measure of applicability to different fields, together with geometric topology, differential geometry, sensible research, differential equations, algebraic geometry, quantity conception, and theoretical physics. This booklet presents an advent to the elemental options and strategies of algebraic topology for the newbie. It provides components of either homology conception and homotopy conception, and contains a number of purposes. The author's purpose is to depend upon the geometric technique by way of attractive to the reader's personal instinct to assist knowing. the various illustrations within the textual content additionally serve this function. positive factors make the textual content various from the traditional literature: first, specified realization is given to supplying particular algorithms for calculating the homology teams and for manipulating the elemental teams. moment, the ebook comprises many workouts, all of that are provided with tricks or strategies. This makes the booklet compatible for either school room use and for self reliant learn. A ebook of the ecu Mathematical Society (EMS). allotted in the Americas by way of the yankee Mathematical Society.

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**Extra info for Lectures on Algebraic Topology (EMS Series of Lectures in Mathematics)**

**Example text**

Then assigning to the pair (t, τ ) the pair (α(t, τ ), β(t, τ )) defines a map Cfg : St1 × Sτ1 → Sα1 × Sβ1 between two tori. e. to the number 4deg(Cfg ). The proof of this claim is given by the following observations. 1. e. when the point (t, τ ) belongs to the pre-image of one of the points (0, 0), (0, π ), (π, 0), (π, π ). 32 1 Elements of homology theory β g(τ ) α f (t) Figure 24. Assignment to the pair (t, τ ) of the pair (α, β). 2. The points (0, 0), (0, π ), (π, 0), (π, π ) are regular values of the map Cfg .

Many theorems that are quite complicated in the simplicial theory become very simple in the singular one. For instance, the theorem on isomorphism of the homology groups of homeomorphic spaces is trivial in the frame of the singular theory. 15 Lefschetz fixed point theorem 47 disadvantages of the singular homology groups are the difficulty of calculating them and the psychological discomfort of operating with infinitely generated groups. 15 Lefschetz fixed point theorem Let f : K → K be a simplicial map from a simplicial complex K to itself.

Its homology groups are denoted by Hn (K; G) and are called the homology groups of K with coefficients in G. The further construction of the homology theory with coefficients in G does not differ from the case of integer coefficients.