Lectures on Algebraic Topology (EMS Series of Lectures in by Sergey V. Matveev

By Sergey V. Matveev

Algebraic topology is the learn of the worldwide houses of areas through algebra. it truly is a big 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 idea, and theoretical physics. This e-book offers an advent to the elemental recommendations and techniques of algebraic topology for the newbie. It provides parts of either homology idea and homotopy conception, and contains quite a few purposes. The author's purpose is to depend on the geometric technique via beautiful to the reader's personal instinct to assist realizing. the various illustrations within the textual content additionally serve this function. gains make the textual content diversified from the normal literature: first, particular awareness is given to supplying specific algorithms for calculating the homology teams and for manipulating the elemental teams. moment, the ebook comprises many routines, all of that are provided with tricks or options. This makes the booklet compatible for either lecture room use and for self sustaining examine. A e-book of the eu Mathematical Society (EMS). dispensed in the Americas via the yankee Mathematical Society.

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Exercise 37. What is the total number of windings around the marked point that is performed by the closed path shown in Figure 14? Figure 14. A closed path in an annulus determines a map from a circle to a circle. 24 1 Elements of homology theory Theorem 10. Maps f, g : S 1 → S 1 are homotopic ⇐⇒ deg f = deg g. Proof. The part ⇒ follows from the definition of the degree and from Theorem 8, which states that homotopic maps between two polyhedra induce identical homomorphisms of their respective homology groups.

Transforming it to the canonic form (and removing superfluous rows and columns), we get the matrix 02 06 , which yields H1 (X) = Z2 ⊕ Z6 . a1 a3 a4 a5 a6 4 5 7 6 3 a4 a5 1 8 2 a1 a6 a3 1 1 1 1 _1 2 1 1 _1 2 2 1 Figure 27. Choosing generators and writing down a relation matrix of the first homology group. Exercise 55. Calculate the first homology group of the Klein bottle. Along with simplicial and cellular homologies, one can use homologies of other types, for instance, singular ones. The difference between the singular homology and the simplicial and cellular ones is in the method of assignment of a chain complex to a given space.

For each n the n-dimensional chain group Cn (X) is the free Abelian group freely generated (in the natural sense) by all the n-dimensional cells. Its elements are formal linear combinations of the form k1 a1 + · · · + km am , where a1 , . . , am are all the n-dimensional cells of X. To describe the boundary homomorphisms, we need the notion of the incidence coefficient of cells. Definition. Let a be an (n − 1)-dimensional cell of a cell complex X, and let ϕ : ∂D n → X (n−1) be the gluing map of an n-dimensional cell b.

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