A fixed point theorem for smooth extension maps
© Khamsemanan et al.; licensee Springer. 2014
Received: 26 December 2013
Accepted: 31 March 2014
Published: 16 April 2014
Let X be a compact smooth n-manifold, with or without boundary, and let A be an -dimensional smooth submanifold of the interior of X. Let be a smooth map and be a smooth map whose restriction to A is ϕ. If is an isolated fixed point of f that is a transversal fixed point of ϕ, that is, the linear transformation is nonsingular, then the fixed point index of f at p satisfies the inequality . It follows that if ϕ has k fixed points, all transverse, and the Lefschetz number , then there is at least one fixed point of f in . Examples demonstrate that these results do not hold if the maps are not smooth.
It has been known at least since the work of Shub and Sullivan in 1974  that the values of the fixed point index of smooth maps are more restricted than they are for continuous functions in general. In  it is proved that, given integers r and s, there is a map of a manifold with boundary ∂X that restricts to and an isolated fixed point p of f such that the fixed point indices are and . On the other hand, it is proved in that paper that if is smooth and p is a transverse fixed point of ϕ, then either or . A consequence of this result is that, under appropriate hypotheses on a smooth map f, it must have fixed points on , the interior of the manifold X. Those same hypotheses are shown to be insufficient to imply the existence of such interior fixed points if the map f is not smooth. (See also [3, 4].)
We will consider a somewhat different setting, as follows. Let X be a compact smooth n-manifold, with or without boundary, and let A be a smooth -dimensional submanifold of the interior of X. As in , we shall consider to be an extension of its restriction . Suppose that p is a transverse fixed point of ϕ, then a simple example will show that the relationship between and cannot be as close as it is when . However, we will prove that there is still a very strong restriction on the value of , namely, that . As a consequence, we obtain a condition on the Lefschetz number of f that implies the existence of fixed points of f in . We demonstrate by an example that the same Lefschetz number condition is not sufficient to imply the existence of fixed points in for maps in general.
2 The index of fixed points of smooth extension maps
The index theorem of  is the following.
Theorem 1 Let X be a compact smooth n-manifold with boundary ∂X. Given a smooth map and a smooth map extending ϕ, suppose that is an isolated fixed point of f and that is a nonsingular linear transformation. Then either or .
We will modify the proof of Theorem 1 that was given in  in order to obtain a result of this type in the setting of smooth extension maps on a pair consisting of an n-dimensional compact smooth manifold X and an -dimensional smooth submanifold A of the interior of X. Although our index result is similar to Theorem 1, the following example demonstrates that we cannot expect that there will be as close a relationship between the indices of ϕ and of f as there is in Theorem 1. Let where is viewed as the complex plane ℂ compactified at infinity, is the unit circle, for and . Let be the restriction of f. Then , whereas .
Theorem 2 Let X be a smooth n-manifold, with or without boundary, and let A be an -dimensional smooth submanifold of the interior of X. Let be a smooth map with smooth extension . Suppose that is an isolated fixed point of f and that is a nonsingular linear transformation. Then .
Proof If the linear transformation is nonsingular, then , see . Therefore, we assume that the linear transformation is singular. Note that determining the index of a map f at a fixed point p is a local problem, so we may choose a local coordinate system about p in which the smooth manifold X is identified with such that p is the origin in and the smooth submanifold A is identified with the subspace .
- (i)The point
is the unique point of with the property that . From the inverse function theorem, there is a small neighborhood of the point such that is a nonsingular diffeomorphism of onto a neighborhood of 0 in the hyperplane and .
- (ii)The point
is the unique point of with the property that . Similarly, from the inverse function theorem, there is a small neighborhood of the point such that is a nonsingular diffeomorphism of onto a neighborhood of 0 in the hyperplane and .
- (1)There is a unique point such that the coordinates of are all 0. Also, since p is an isolated fixed point of . In particular, the coordinate of is nonzero. Hence we have
- (2)There is a unique point such that the coordinates of are all 0. Also, since p is an isolated fixed point of . In particular, the coordinate of is nonzero. Hence we have
where and .
For , either or . If , then by definition. If , then which we know is never 0. A similar argument shows that for .
Since the map takes into either the upper or lower half-planes with respect to the y coordinates and takes into either the upper or lower half-planes with respect to the coordinates, the map is homotopic either to a constant map, the suspension of or the suspension of followed by a reflection about the hyperplane . This homotopy tells us that either has degree 0 or or .
Thus the index which is either 0 or ±1. □
3 Fixed point theorem for smooth extension maps
We can now use Theorem 2 to establish the existence of fixed points in .
Theorem 3 Suppose that A is an -dimensional smooth submanifold of the interior of a compact smooth n-manifold X. Given a smooth map and a smooth map extending ϕ, suppose that the fixed points of ϕ are , all of which are transversal, that is, the linear map is nonsingular for each . If the Lefschetz number , then there must be at least one fixed point in .
because by Theorem 2. This is contrary to the assumption that , so f has fixed points in . □
A consequence of this theorem is the following.
Corollary 4 Let be the complex plane ℂ compactified at infinity and be the unit circle. Suppose that is a smooth map defined by for some and that is a smooth extension of ϕ. If f is homotopic to the suspension of ϕ, then there is at least one fixed point in .
Proof Since f is homotopic to the suspension of ϕ, then f is of degree k and thus . However, ϕ has only fixed points. Note here that the fixed points of ϕ on are all transversal so that the hypotheses of Theorem 3 hold. □
The following example illustrates the fact that the corollary, and therefore Theorem 3, require the hypothesis that the map f is smooth, by exhibiting a non-smooth map homotopic to the suspension of ϕ that has no fixed points on .
Example 1 Let be the complex plane ℂ compactified at infinity and be the unit circle. Let be defined by . Let and , so and . For , there exist and such that . As in Lemma 6.1 of , define by setting for and , then f has no fixed points in . In order to extend f to a selfmap of so that there are no fixed points on , we define by , and , so and . Then, for , let . The only fixed point of f is and thus there are no fixed points in .
This work has been supported by the Thailand Research Fund MRG5580232 under the mentorship of Professor Robert F. Brown of the Department of Mathematics, UCLA and Professor Sompong Dhompongsa of the Department of Mathematics, Chiang Mai University. The author would also like to thank the referee for all the useful suggestions.
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