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By Barmak J.A., Minian E.G.

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8. Uniqueness of the degree. 29) Theorem. There exists a unique function subordinating to each dcompact map f: U → E, U an open subset of the Euclidean space E, an integer deg(f, U ) which satisfies the properties: Localization, Units, Additivity, Homotopy Invariance, Multiplicativity. Proof. We first show that the above properties determine the degree deg (f) for special cases. Case 1. Let f: R → R be given by f(x) = ax + b for a, b ∈ R, a = 0. We show that deg (f) = sgn (a). 1. DEGREE OF A MAP 27 Assume that moreover a > 0.

16) Theorem. Let X be a finite complex and K a field and L(f; K ) the Lefschetz number with respect to K . Then the Lefschetz numbers derived in homology and cohomology are equal. Moreover, the Lefschetz number L(f; Q) with respect to the field of rational numbers is an integer and we have L(f; K ) = L(f; Q) if char (K ) = 0, the reminder mod p of L(f; Q) if char (K ) = p. 17) Remark. We underlined the field of rational numbers for two reasons: it is the smallest field of characteristic zero and traditionally the Lefschetz number is defined by use of it.

0 ... 1 ... . . 0 0 ⎤ 0 0 0 .. 0 0 λ1 .. 0 . 0 λr 0 0 .. 1 λr 0 .. 0 1 λr .. 0 0 1 .. ... ... . 0 0 0 .. 0 0 ... 4) r 1 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ di = d. By d tr A = tr B = d1 λ1 + · · · + dr λr = λj . j=1 On the other hand the characteristic polynomials χA (λ) = χB (λ) are equal. This gives χA (λ) = (λ − λ1 )d1 . . (λ − λr )dr by the property of the determinant. This shows that λj are all the eigenvalues of A and dj their multiplicities, and consequently proves the statement.

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2-Dimension from the Topological Viewpoint by Barmak J.A., Minian E.G.


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