By Eisenhart L. P.

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**Example text**

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.