Download Applicable Geometry (1977)(en)(207s) by Heinrich W Guggenheimer PDF

By Heinrich W Guggenheimer

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Hint. Use the decomposition w(λ) = (λ − λ1 )d1 . . 27) and the Bezout Lemma. Now we show a property that will be of use in the next. 11) Proposition. Let K be a field and M a finitely generated K -module, thus a vector space over K . Let next M ⊂ M be a linear subspace and A: M → M an endomorphism such that A(M ) ⊂ M . Then A induces an endomorphism A of the quotient space M := M/M and we have tr A = tr A + tr A, where A denotes A|M . Proof. Let d = dimK M and dimK M = d ≤ d. Take any basis e1 , .

LEFSCHETZ–HOPF FIXED POINT THEORY f −1 (0) ⊂ U ⊂ cl U ⊂ U . By the Localization property deg (f, U ) = deg (f, U ) and we may consider U instead of U . Let δ = inf{ f(x) : x ∈ bd U }. By the compactness of cl U , δ > 0. By Sard’s Theorem there exists a δ/2 homotopy ft from f0 = fcl U to a map f1 with f1−1 (0) finite and Df1 an isomorphism at each point x ∈ f1−1 (0). Now ft (x) > δ − δ/2 > 0 for all x ∈ bd U . Hence Homotopy and Additivity Properties imply deg (f) = deg (f1 ) = deg (f1 , xi), xi where the summation runs over the finite set f1−1 (0).

I=1 Of the most interest for us will be the following rings: the ring of integral numbers Z, the fields Q, R, C, of rational, real, and complex numbers correspondingly. 3. THE LEFSCHETZ NUMBER 39 We also use the ring Zm of remainders modulo m ∈ N, in particular the field Zp , p-prime. The following property of the trace follows by direct computation of matrices. 2) Proposition. Let A, B ∈ Md×d (R). 3) Exercise. 2). As a direct consequence we get. 4) Corollary. The trace is an invariant of the endomorphism A: M → M.

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