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A simple approach to Brouwer degree based on differential forms. (English) Zbl 1082.47052

This article deals with Brouwer–Hopf degree theory for mappings in \({\mathbb R}^n\), based on the Kronecker integral \[ i_K[f,\partial D] = \frac1{\mu_{n-1}} \, \int_{\partial D} \| f\| ^{-n} \left(\sum_{j=1}^n (-1)^{j-1} f_j \, df_1 \wedge \ldots \wedge \widehat{df_j} \wedge \ldots \wedge df_n\right), \] where \(\mu_{n-1}\) is \((n-1)\)-dimensional measure of the unit sphere. The author proves that this integral can be rewritten as the Heinz integral \[ i_H[f,D] = \int_D c(\| f(x)\| ) \, J_f(x) \, dx, \] where \(J_f(x)\) is the Jacobian of \(f\), \(c(\cdot)\) is a continuous function with support in \(]0,\min_{\partial D} \, \| f(\cdot)\| [\). He then states five basic properties of the Brouwer–Hopf degree (excision, existence, homotopy invariance, Rouché, and additivity). The author’s approach is a simple and clear realization of some old arguments offered by J. Hadamard.

MSC:

47H11 Degree theory for nonlinear operators
55M25 Degree, winding number
58C30 Fixed-point theorems on manifolds
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[1] Mawhin, Poincare s early use of Analysis situs in nonlinear differential equations : Variations around the theme of Kronecker s integral Scientiae, Philosophia 4 pp 103– (2000)
[2] Fu, hrer Ein elementarer analytischer Beweis zur Eindeutigket des Abbildungs - grades im Rn, Math Nachr 54 pp 259– (1972) · Zbl 0246.55007 · doi:10.1002/mana.19720540117
[3] Siegberg, Some historical remarks concerning degree theory Monthly, Amer Math pp 125– (1981) · Zbl 0463.55002 · doi:10.2307/2321135
[4] Hatziafratis, On the integral giving the degree of a map and a Rouche type theorem, Anal Anwend 16 pp 239– (1997) · Zbl 0882.58006 · doi:10.4171/ZAA/761
[5] Brouwer, On continuous vector distributions on surfaces, Wetensch Proc 11 pp 850– (1909)
[6] Poincare, Me moire sur les courbes de finies par une e quation diffe rentielle I, Math Pures Appl pp 375– (1881)
[7] Brouwer, Ueber Abbildungen von Mannigfaltigkeiten, Math Ann 71 pp 97– (1912) · JFM 42.0417.01 · doi:10.1007/BF01456931
[8] Heinz, An elementary analytic theory of the degree of mapping in n - dimensional space, Math Mech 8 pp 231– (1959) · Zbl 0085.17105
[9] Johnson, The problem of the invariance of dimension in the growth of modern topology Part II Exact, Arch History Sci 25 pp 85– (1981) · Zbl 0532.55001 · doi:10.1007/BF02116242
[10] Fucik, Spectral Analysis of Nonlinear Operators, Notes Math No pp 346– (1973) · Zbl 0324.47036
[11] Amann, On the uniqueness of the topological degree, Math Z pp 130– (1973) · Zbl 0249.55004
[12] Mawhin, Simple proofs of various fixed point andexistence theorems based on ex - terior calculus to appear Topology from the Differentiable Viewpoint Virginia Press Char - lottesville A theory of degree of mappings based upon infinitesimal analysis, Math Nachr Math 73 pp 548– (1965)
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