A fixed point theorem for smooth extension maps
Let X be a compact smooth n-manifold, with or without boundary, and let A be an (n - 1)-dimensional smooth submanifold of the interior of X. Let φ: A→A be a smooth map and f: (X, A)→(X, A) be a smooth map whose restriction to A is φ. If p ş̌ A is an isolated fixed point of f that is a transversal fi...
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| Main Authors: | , , , |
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| Format: | Journal |
| Published: |
2018
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| Online Access: | https://www.scopus.com/inward/record.uri?partnerID=HzOxMe3b&scp=84899815709&origin=inward http://cmuir.cmu.ac.th/jspui/handle/6653943832/45783 |
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| Institution: | Chiang Mai University |
| Summary: | Let X be a compact smooth n-manifold, with or without boundary, and let A be an (n - 1)-dimensional smooth submanifold of the interior of X. Let φ: A→A be a smooth map and f: (X, A)→(X, A) be a smooth map whose restriction to A is φ. If p ş̌ A is an isolated fixed point of f that is a transversal fixed point of φ, that is, the linear transformation dφ p - I A : T p A→T p A is nonsingular, then the fixed point index of f at p satisfies the inequality |i(X, f , p)| ≤ 1. It follows that if φ has k fixed points, all transverse, and the Lefschetz number L(f) > k, then there is at least one fixed point of f in X \ A. Examples demonstrate that these results do not hold if the maps are not smooth. © 2014 Khamsemanan et al.; licensee Springer. |
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