Geometric formulas for Smale invariants of codimension two immersions |
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Authors: | Tobias Ekholm András Szücs |
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Institution: | a Department of Mathematics, Uppsala University, Box 480, SE-751 06 Uppsala, Sweden b Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/C, Budapest, Hungary 1117 |
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Abstract: | We give three formulas expressing the Smale invariant of an immersion f of a (4k−1)-sphere into (4k+1)-space. The terms of the formulas are geometric characteristics of any generic smooth map g of any oriented 4k-dimensional manifold, where g restricted to the boundary is an immersion regularly homotopic to f in (6k−1)-space.The formulas imply that if f and g are two non-regularly homotopic immersions of a (4k−1)-sphere into (4k+1)-space then they are also non-regularly homotopic as immersions into (6k−1)-space. Moreover, any generic homotopy in (6k−1)-space connecting f to g must have at least ak(2k−1)! cusps, where ak=2 if k is odd and ak=1 if k is even. |
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Keywords: | 57R42 57R45 |
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