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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th-cmuir.6653943832-457832018-01-24T06:17:29Z A fixed point theorem for smooth extension maps Nirattaya Khamsemanan Robert F. Brown Catherine Lee Sompong Dhompongsa 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. 2018-01-24T06:17:29Z 2018-01-24T06:17:29Z 2014-01-01 Journal 16871812 16871820 2-s2.0-84899815709 10.1186/1687-1812-2014-97 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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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. |
| format |
Journal |
| author |
Nirattaya Khamsemanan Robert F. Brown Catherine Lee Sompong Dhompongsa |
| spellingShingle |
Nirattaya Khamsemanan Robert F. Brown Catherine Lee Sompong Dhompongsa A fixed point theorem for smooth extension maps |
| author_facet |
Nirattaya Khamsemanan Robert F. Brown Catherine Lee Sompong Dhompongsa |
| author_sort |
Nirattaya Khamsemanan |
| title |
A fixed point theorem for smooth extension maps |
| title_short |
A fixed point theorem for smooth extension maps |
| title_full |
A fixed point theorem for smooth extension maps |
| title_fullStr |
A fixed point theorem for smooth extension maps |
| title_full_unstemmed |
A fixed point theorem for smooth extension maps |
| title_sort |
fixed point theorem for smooth extension maps |
| publishDate |
2018 |
| url |
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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