# A fixed point theorem for smooth extension maps

- Nirattaya Khamsemanan
^{1}Email author, - Robert F Brown
^{2}, - Catherine Lee
^{3}and - Sompong Dhompongsa
^{4}

**2014**:97

https://doi.org/10.1186/1687-1812-2014-97

© Khamsemanan et al.; licensee Springer. 2014

**Received: **26 December 2013

**Accepted: **31 March 2014

**Published: **16 April 2014

## Abstract

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 $\varphi :A\to A$ be a smooth map and $f:(X,A)\to (X,A)$ be a smooth map whose restriction to *A* is *ϕ*. If $p\in A$ is an isolated fixed point of *f* that is a transversal fixed point of *ϕ*, that is, the linear transformation $d{\varphi}_{p}-{I}_{A}:{T}_{p}A\to {T}_{p}A$ is nonsingular, then the fixed point index of *f* at *p* satisfies the inequality $|i(X,f,p)|\le 1$. 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 $X\setminus A$. Examples demonstrate that these results do not hold if the maps are not smooth.

**MSC:**55M20, 54C20.

## Keywords

## 1 Introduction

It has been known at least since the work of Shub and Sullivan in 1974 [1] that the values of the fixed point index of smooth maps are more restricted than they are for continuous functions in general. In [2] it is proved that, given integers *r* and *s*, there is a map $f:X\to X$ of a manifold with boundary *∂X* that restricts to $f{|}_{\partial X}=\varphi :\partial X\to \partial X$ and an isolated fixed point *p* of *f* such that the fixed point indices are $i(\partial X,\varphi ,p)=r$ and $i(X,f,p)=s$. On the other hand, it is proved in that paper that if $f:(X,\partial X)\to (X,\partial X)$ is smooth and *p* is a transverse fixed point of *ϕ*, then either $i(X,f,p)=0$ or $i(X,f,p)=i(\partial X,\varphi ,p)$. A consequence of this result is that, under appropriate hypotheses on a smooth map *f*, it must have fixed points on $X\setminus \partial X$, 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 $(n-1)$-dimensional submanifold of the interior of *X*. As in [2], we shall consider $f:(X,A)\to (X,A)$ to be an *extension* of its restriction $f{|}_{A}=\varphi :A\to A$. Suppose that *p* is a transverse fixed point of *ϕ*, then a simple example will show that the relationship between $i(X,f,p)$ and $i(A,\varphi ,p)$ cannot be as close as it is when $A=\partial X$. However, we will prove that there is still a very strong restriction on the value of $i(X,f,p)$, namely, that $|i(f,X,p)|\le 1$. As a consequence, we obtain a condition on the Lefschetz number $L(f)$ of *f* that implies the existence of fixed points of *f* in $X\setminus A$. We demonstrate by an example that the same Lefschetz number condition is not sufficient to imply the existence of fixed points in $X\setminus A$ for maps $f:(X,A)\to (X,A)$ in general.

## 2 The index of fixed points of smooth extension maps

The index theorem of [2] is the following.

**Theorem 1** *Let* *X* *be a compact smooth* *n*-*manifold with boundary* *∂X*. *Given a smooth map* $\varphi :\partial X\to \partial X$ *and a smooth map* $f:(X,\partial X)\to (X,\partial X)$ *extending* *ϕ*, *suppose that* $p\in \partial X$ *is an isolated fixed point of* *f* *and that* $d{\varphi}_{p}-{I}_{\partial X}:{T}_{p}(\partial X)\to {T}_{p}(\partial X)$ *is a nonsingular linear transformation*. *Then either* $i(X,f,p)=0$ *or* $i(X,f,p)=i(\partial X,\varphi ,p)$.

We will modify the proof of Theorem 1 that was given in [2] 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 $(n-1)$-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 $f:({S}^{2},{S}^{1})\to ({S}^{2},{S}^{1})$ where ${S}^{2}$ is viewed as the complex plane ℂ compactified at infinity, ${S}^{1}$ is the unit circle, $f(z)={z}^{2}$ for $z\in \mathbb{C}$ and $f(\mathrm{\infty})=\mathrm{\infty}$. Let $\varphi :{S}^{1}\to {S}^{1}$ be the restriction of *f*. Then $i({S}^{1},\varphi ,1)=-1$, whereas $i({S}^{2},f,1)=1$.

**Theorem 2** *Let* *X* *be a smooth* *n*-*manifold*, *with or without boundary*, *and let* *A* *be an* $(n-1)$-*dimensional smooth submanifold of the interior of* *X*. *Let* $\varphi :A\to A$ *be a smooth map with smooth extension* $f:(X,A)\to (X,A)$. *Suppose that* $p\in A$ *is an isolated fixed point of* *f* *and that* $d{\varphi}_{p}-{I}_{A}:{T}_{p}A\to {T}_{p}A$ *is a nonsingular linear transformation*. *Then* $|i(X,f,p)|\le 1$.

*Proof* If the linear transformation $d{f}_{p}-I:{T}_{p}X\to {T}_{p}X$ is nonsingular, then $i(X,f,p)=\pm 1$, see [5]. Therefore, we assume that the linear transformation $d{f}_{p}-I:{T}_{p}X\to {T}_{p}X$ 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 ${\mathbb{R}}^{n}$ such that *p* is the origin in ${\mathbb{R}}^{n}$ and the smooth submanifold *A* is identified with the subspace ${\mathbb{R}}^{n-1}$.

*f*at

*p*, we need to determine the degree of −

*G*restricted to a sphere around

**0**. More specifically, for $\u03f5>0$ sufficiently small, we may consider the map ${E}_{\u03f5}:{S}^{n-1}\to {\mathbb{R}}^{n}-\{\mathbf{0}\}$ defined by

*G*is a ${C}^{1}$ function with value

**0**at the point

*p*, we know from the definition of the derivative that

*x*coordinates into

*y*coordinates, has the following form:

*B*. Define

- (i)The point${\mathbf{x}}^{+}=(0,\dots ,0,-\frac{B}{{(1+{B}^{2})}^{\frac{1}{2}}},\frac{1}{{(1+{B}^{2})}^{\frac{1}{2}}})$
is the unique point of ${S}_{+}$ with the property that $d{G}_{p}({\mathbf{x}}^{+})=\mathbf{0}$. From the inverse function theorem, there is a small neighborhood ${U}^{+}$ of the point ${\mathbf{x}}^{+}$ such that $d{G}_{p}{|}_{{S}_{+}}$ is a nonsingular diffeomorphism of ${U}^{+}$ onto a neighborhood of

**0**in the hyperplane ${y}_{n}=0$ and ${(d{G}_{p}{|}_{{S}_{+}})}^{-1}(d{G}_{p}({U}^{+}))={U}^{+}$. - (ii)The point${\mathbf{x}}^{-}=(0,\dots ,0,-\frac{B}{{(1+{B}^{2})}^{\frac{1}{2}}},-\frac{1}{{(1+{B}^{2})}^{\frac{1}{2}}})$

is the unique point of ${S}_{-}$ with the property that $d{G}_{p}({\mathbf{x}}^{-})=\mathbf{0}$. Similarly, from the inverse function theorem, there is a small neighborhood ${U}^{-}$ of the point ${\mathbf{x}}^{-}$ such that $d{G}_{p}{|}_{{S}_{-}}$ is a nonsingular diffeomorphism of ${U}^{-}$ onto a neighborhood of **0** in the hyperplane ${y}_{n}=0$ and ${(d{G}_{p}{|}_{{S}_{-}})}^{-1}(d{G}_{p}({U}^{-}))={U}^{-}$.

- (1)There is a unique point ${\mathbf{x}}_{\u03f5}^{+}\in {S}_{+}$ such that the ${y}_{1},\dots ,{y}_{n-1}$ coordinates of ${P}_{\u03f5}({\mathbf{x}}_{\u03f5}^{+})$ are all 0. Also, since
*p*is an isolated fixed point of $f,{P}_{\u03f5}({\mathbf{x}}_{\u03f5}^{+})\ne \mathbf{0}$. In particular, the ${y}_{n}$ coordinate of ${P}_{\u03f5}({\mathbf{x}}_{\u03f5}^{+})$ is nonzero. Hence we have${P}_{\u03f5}\left({\mathbf{x}}_{\u03f5}^{+}\right)=(0,\dots ,0,{y}_{n}^{{\u03f5}^{+}}),$where ${y}_{n}^{{\u03f5}^{+}}\ne 0$.

- (2)There is a unique point ${\mathbf{x}}_{\u03f5}^{-}\in {S}_{-}$ such that the ${y}_{1},\dots ,{y}_{n-1}$ coordinates of ${P}_{\u03f5}({\mathbf{x}}_{\u03f5}^{-})$ are all 0. Also, since
*p*is an isolated fixed point of $f,{P}_{\u03f5}({\mathbf{x}}_{\u03f5}^{-})\ne \mathbf{0}$. In particular, the ${y}_{n}$ coordinate of ${P}_{\u03f5}({\mathbf{x}}_{\u03f5}^{-})$ is nonzero. Hence we have${P}_{\u03f5}\left({\mathbf{x}}_{\u03f5}^{-}\right)=(0,\dots ,0,{y}_{n}^{{\u03f5}^{-}}),$

where ${y}_{n}^{{\u03f5}^{-}}\ne 0$.

where ${\sigma}^{+}={y}_{n}^{{\u03f5}^{+}}/|{y}_{n}^{{\u03f5}^{+}}|=\pm 1$ and ${\sigma}^{-}={y}_{n}^{{\u03f5}^{-}}/|{y}_{n}^{{\u03f5}^{-}}|=\pm 1$.

For $\mathbf{x}\in {S}_{+}$, either $({y}_{1},\dots ,{y}_{n-1})\ne \mathbf{0}$ or $\mathbf{x}={\mathbf{x}}_{\u03f5}^{+}$. If $({y}_{1},\dots ,{y}_{n-1})\ne \mathbf{0}$, then $H(\mathbf{x},t)\ne \mathbf{0}$ by definition. If $\mathbf{x}={\mathbf{x}}_{\u03f5}^{+}$, then $H(\mathbf{x},t)={P}_{\u03f5}(\mathbf{x})$ which we know is never **0**. A similar argument shows that $H(\mathbf{x},t)\ne \mathbf{0}$ for $\mathbf{x}\in {S}_{-}\mathrm{\setminus}{\mathbb{R}}^{n-1}$.

Since the map ${P}_{\u03f5}^{\sigma}$ takes ${S}_{+}$ into either the upper or lower half-planes with respect to the *y* coordinates and takes ${S}_{-}$ into either the upper or lower half-planes with respect to the ${y}_{1},\dots ,{y}_{n}$ coordinates, the map ${P}_{\u03f5}^{\sigma}$ is homotopic either to a constant map, the suspension of ${P}_{\u03f5}^{\sigma}{|}_{{S}^{n-2}}={P}_{\u03f5}{|}_{{S}^{n-2}}$ or the suspension of ${P}_{\u03f5}^{\sigma}{|}_{{S}^{n-2}}={P}_{\u03f5}{|}_{{S}^{n-2}}$ followed by a reflection about the hyperplane ${y}_{n}=0$. This homotopy tells us that either ${P}_{\u03f5}$ has degree 0 or $deg({P}_{\u03f5}{|}_{{S}^{n-2}})$ or $-deg({P}_{\u03f5}{|}_{{S}^{n-2}})$.

*x*to

*y*coordinates, we know that the

*x*and

*y*coordinates are related by a linear map, call it

*L*, satisfying $L({x}_{i})={y}_{i}$, for $i=1,\dots ,n$. Thus, the map ${L}^{-1}\circ {P}_{\u03f5}$ takes ${x}_{1},\dots ,{x}_{n}$ coordinates to ${x}_{1},\dots ,{x}_{n}$ coordinates and

Thus the index $i(X,f,p)=deg({P}_{\u03f5})$ 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 $X\setminus A$.

**Theorem 3** *Suppose that* *A* *is an* $(n-1)$-*dimensional smooth submanifold of the interior of a compact smooth* *n*-*manifold* *X*. *Given a smooth map* $\varphi :A\to A$ *and a smooth map* $f:(X,A)\to (X,A)$ *extending* *ϕ*, *suppose that the fixed points of* *ϕ* *are* $\{{x}_{1},{x}_{2},\dots ,{x}_{k}\}$, *all of which are transversal*, *that is*, *the linear map* $d{\varphi}_{{x}_{j}}-{I}_{A}$ *is nonsingular for each* ${x}_{j}$. *If the Lefschetz number* $L(f)>k$, *then there must be at least one fixed point in* $X\setminus A$.

*Proof*Suppose that

*f*only has fixed points in

*A*. This means the fixed point set of

*f*is $\{{x}_{1},\dots ,{x}_{k}\}$, the set of fixed points of

*ϕ*. Then, by the Lefschetz-Hopf theorem [6], the Lefschetz number of

*f*is

because $i(X,f,{x}_{j})\le 1$ by Theorem 2. This is contrary to the assumption that $L(f)>k$, so *f* has fixed points in $X\setminus A$. □

A consequence of this theorem is the following.

**Corollary 4** *Let* ${S}^{2}$ *be the complex plane* ℂ *compactified at infinity and* ${S}^{1}$ *be the unit circle*. *Suppose that* $\varphi :{S}^{1}\to {S}^{1}$ *is a smooth map defined by* $\varphi (\zeta )={\zeta}^{k}$ *for some* $k\ge 2$ *and that* $f:({S}^{2},{S}^{1})\to ({S}^{2},{S}^{1})$ *is a smooth extension of* *ϕ*. *If* *f* *is homotopic to the suspension of* *ϕ*, *then there is at least one fixed point in* ${S}^{2}\mathrm{\setminus}{S}^{1}$.

*Proof* Since *f* is homotopic to the suspension of *ϕ*, then *f* is of degree *k* and thus $L(f)=1+k$. However, *ϕ* has only $k-1$ fixed points. Note here that the $k-1$ fixed points of *ϕ* on ${S}^{1}$ 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 $f:({S}^{2},{S}^{1})\to ({S}^{2},{S}^{1})$ homotopic to the suspension of *ϕ* that has no fixed points on ${S}^{2}\mathrm{\setminus}{S}^{1}$.

**Example 1** Let ${S}^{2}=\mathbb{C}\cup \{\mathrm{\infty}\}$ be the complex plane ℂ compactified at infinity and ${S}^{1}$ be the unit circle. Let $\varphi :{S}^{1}\to {S}^{1}$ be defined by $\varphi (\zeta )={\zeta}^{2}$. Let ${B}_{+}^{2}=\{z\in \mathbb{C}:|z|\le 1\}$ and ${B}_{-}^{2}=\{z\in \mathbb{C}:|z|\ge 1\}\cup \{\mathrm{\infty}\}$, so ${S}^{2}={B}_{+}^{2}\cup {B}_{-}^{2}$ and ${B}_{+}^{2}\cap {B}_{-}^{2}={S}^{1}$. For $z\in {B}_{-}^{2}\setminus \{1\}$, there exist $\zeta \in {S}^{1}$ and $t\in [0,1)$ such that $z=t1+(1-t)\zeta $. As in Lemma 6.1 of [7], define $f:{B}_{+}^{2}\to {B}_{+}^{2}$ by setting $f(z)=t1+(1-t)\varphi (\zeta )$ for $z\ne 1$ and $f(1)=1$, then *f* has no fixed points in ${B}_{+}^{2}\setminus {S}^{1}$. In order to extend *f* to a selfmap of ${B}_{-}^{2}$ so that there are no fixed points on ${B}_{-}^{2}\setminus {S}^{1}$, we define $\rho :{S}^{2}\to {S}^{2}$ by $\rho (z)=\rho (r{e}^{i\theta})=\frac{1}{r}{e}^{i\theta}$, $\rho (0)=\mathrm{\infty}$ and $\rho (\mathrm{\infty})=0$, so $\rho ({B}_{-}^{2})={B}_{+}^{2}$ and $\rho ({B}_{+}^{2})={B}_{-}^{2}$. Then, for $z\in {B}_{-}^{2}$, let $f(z)=\rho f\rho (z)$. The only fixed point of *f* is $z=1$ and thus there are no fixed points in ${S}^{2}\setminus {S}^{1}$.

## Declarations

### Acknowledgements

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.

## Authors’ Affiliations

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