By W.V.D. Hodge

Arithmetic textual content, 282 pages.

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**Extra info for The Theory and Applications of Harmonic Integrals. Second Edition **

**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.