Interior fixed points of unit-sphere-preserving Euclidean maps

Abstract

Schirmer proved that there is a class of smooth self-maps of the unit sphere in Euclidean n-space with the property that any smooth self-map of the unit ball that extends a map of that class must have at least one fixed point in the interior of the ball. We generalize Schirmer’s result by proving that a smooth self-map of Euclidean n-space that extends a self-map of the unit sphere of that class must have at least one fixed point in the interior of the unit ball.

MSC:55M20, 54C20.

1 Introduction

For spaces X, Y and subsets $V\subseteq X$, $W\subseteq Y$, a map $f:X\to Y$ is an extension of a map $\varphi :V\to W$ if $f(x)=\varphi (x)$ for all $x\in V$. We denote by $B^n$ the unit ball in ${\mathbb{R}}^n$, by $S^{n-1}$ its boundary and by $int(B^n)$ its interior.

If $f:B^1\to B^1$ is an extension of $\varphi :S^0\to S^0=\{-1,1\}$ and ϕ has no fixed points, then f must have an interior fixed point, that is, a fixed point in $int(B^1)$. However, if ϕ has a fixed point, then there need not be any interior fixed points.

If $n=2$, the situation is more complicated. Of course the Brouwer fixed point theorem implies that a map $f:B^2\to B^2$ must have at least one interior fixed point if it is an extension of a map $\varphi :S^1\to S^1$ that has no fixed points. But it was proved in [1] (see also [2]) that if the extension f is smooth, it may still be required to have interior fixed points for certain maps ϕ that have many fixed points. Representing the points of $S^1$ by complex numbers, let $\varphi ={\varphi }_d:S^1\to S^1$, for an integer d, be the power map defined by ${\varphi }_d(z)=z^d$. If $d\ge 2$ and $f:B^2\to B^2$ is a smooth extension of ${\varphi }_d$, then f has at least one interior fixed point. It is also demonstrated in [1] that interior fixed points of extensions need not exist if $d\le 1$ or if f is not smooth. Schirmer generalized this interior fixed point result to smooth extensions $f:B^n\to B^n$ for $n\ge 2$ to show in Example 4.7 of [3] that if f is a smooth extension of a ‘sparse’ map $\varphi :S^{n-1}\to S^{n-1}$, a generalization of ${\varphi }_d$ that is defined below, of degree d such that ${(-1)}^{n}d\ge 2$, then f must have at least one interior fixed point.

Returning to the case $n=1$, if we extend the map $\varphi :S^0\to S^0$ without fixed points to a map $f:{\mathbb{R}}^1\to {\mathbb{R}}^1$, there still must be a fixed point of f in $int(B^1)$. The reason for the interior fixed points of the extension $f:B^1\to B^1$ of the map of $S^0$ without fixed points, namely that $(-1,1)$ and $(1,-1)$, lie in different components of $B^1\times B^1\setminus \mathrm{\Delta }$, where $\mathrm{\Delta }=\{(x,x):x\in B^1\}$, applies also to the extension $f:{\mathbb{R}}^1\to {\mathbb{R}}^1$ since those points are also in different components of $B^1\times {\mathbb{R}}^1\setminus \mathrm{\Delta }$.

On the other hand, the reason for the presence of fixed points in $int(B^n)$ for smooth extensions of certain maps of $S^{n-1}$ demonstrated in [3] is considerably more subtle. Therefore, it is reasonable to ask whether such fixed points would persist if, instead of smooth extensions $f:B^n\to B^n$ of $\varphi :S^{n-1}\to S^{n-1}$, we consider extensions that are smooth Euclidean maps, that is, maps $f:{\mathbb{R}}^n\to {\mathbb{R}}^n$. Thus, we ask whether there still must be fixed points of f in $int(B^n)$ if we allow f to map points of $int(B^n)$ outside of $B^n$.

We will prove that the interior fixed points do persist, even in this more general setting. As in the case of self-maps of balls, the interior fixed points of Euclidean maps are detected by means of a theorem that relates the index of a fixed point of $\varphi :S^{n-1}\to S^{n-1}$ to its index as a fixed point of an extension. We will therefore devote Section 2 to a discussion of the properties of the fixed point index that we will use. In Section 3, we prove that a smooth extension $f:{\mathbb{R}}^2\to {\mathbb{R}}^2$ of a power map ${\varphi }_d:S^1\to S^1$ for $d\ge 2$ must have at least one interior fixed point. Section 4 then contains the proof that Schirmer’s result generalizes to smooth extensions $f:{\mathbb{R}}^n\to {\mathbb{R}}^n$ of sparse maps $\varphi :S^{n-1}\to S^{n-1}$ that satisfy the same degree restrictions.

2 The fixed point indices

Before extending the results of [1, 2] and [3] to the case of a smooth Euclidean map $f:({\mathbb{R}}^n,S^{n-1})\to ({\mathbb{R}}^n,S^{n-1})$ extending $\varphi :S^{n-1}\to S^{n-1}$, we need to define the relevant fixed point indices. We will consider the restriction $f:(B^n,S^{n-1})\to ({\mathbb{R}}^n,S^{n-1})$. Since our goal is to establish conditions for the existence of fixed points on the interior of $B^n$, the behavior of the function outside of $B^n$ is not relevant. Therefore, we will make use of the indices $i(B^n,f,p)$ and $i(S^{n-1},\varphi ,p)$ of an isolated fixed point $p\in S^{n-1}$. We do so by generalizing the approach used in [1] (see also [4]).

For an isolated fixed point $p\in S^{n-1}$, we can choose a small enough neighborhood U so that it contains only this fixed point and no other. We then may write f in this neighborhood of p in terms of a local coordinate system in which $U\cap B^n$ is contained in the upper half-space

$${\mathbb{R}}_{+}^n=\{(x_1,x_2,\dots ,x_n)\in {\mathbb{R}}^n|x_n\ge 0\}$$

in such a way that p is the origin 0 in this setting and $U\cap S^{n-1}$ is contained in the subspace

$${\mathbb{R}}^{n-1}=\{(x_1,x_2,\dots ,x_{n-1},0)\in {\mathbb{R}}^n\}.$$

In order to calculate the index of p in each space, we consider the map $F:U\to {\mathbb{R}}^n$ defined by

$$F(x_1,x_2,\dots ,x_n)=\{\begin{array}{cc}(x_1,x_2,\dots ,x_n)-f(x_1,x_2,\dots ,x_n),\hfill & \text{if }{x}_n\ge 0,\hfill \\ (x_1,x_2,\dots ,x_n)-f(x_1,x_2,\dots ,x_{n-1},0),\hfill & \text{if }{x}_n<0.\hfill \end{array}$$

Note that F sends the origin, lower half-plane and ${\mathbb{R}}^{n-1}$ to itself respectively. Also, $F(z)\ne z$ for $z\ne 0$. The index $i(S^{n-1},\varphi ,p)$ is equal to $i({\mathbb{R}}^{n-1},F,\mathbf{0})$ in this setting as in the traditional definition of the index. Moreover, $i(B^n,f,p)$ is identified with $i({\mathbb{R}}^n,F,\mathbf{0})$, which can be computed as the degree of the map $\rho \circ F:S^{n-1}\to S^{n-1}$ where $\rho :{\mathbb{R}}^n\mathrm{\setminus }\mathbf{0}\to S^{n-1}$ is the retraction defined by $\rho (z)=z/|z|$.

3 Unit-circle-preserving maps of the plane

Brown, Greene and Schirmer proved

Theorem 1 [1, 2]

Let $f:B^2\to B^2$ be a smooth map with a finite number of fixed points such that $f(\zeta )={\zeta }^k$ for all $\zeta \in S^1$ for some $k\ge 2$, where $B^2$ is the closed two-dimensional ball with the boundary $S^1$. If π is a fixed point of f that lies in $S^1$, then either $i(B^2,f,\pi )=0$ or $i(B^2,f,\pi )=-1$.

The contractibility of $B^2$ implies the following corollary.

Corollary 2 [2]

Suppose $f:B^2\to B^2$ is a smooth map such that $f(\zeta )={\zeta }^k$ for all $\zeta \in S^1$, for some $k\ge 2$. Then there exists $z\in int(B^2)$ such that $f(z)=z$.

We will extend Theorem 1 to maps $f:(B^2,S^1)\to ({\mathbb{R}}^2,S^1)$ by modifying the proof of Theorem 1 in [2]. Corollary 2 will then extend to maps $f:({\mathbb{R}}^2,S^1)\to ({\mathbb{R}}^2,S^1)$.

Theorem 3 Let $f:(B^2,S^1)\to ({\mathbb{R}}^2,S^1)$ be a smooth map with a finite number of fixed points such that $f(\zeta )={\zeta }^k$ for all $\zeta \in S^1$ for some $k\ge 2$. If π is a fixed point of f that lies in $S^1$, then either $i(B^2,f,\pi )=0$ or $i(B^2,f,\pi )=-1$.

Proof Let π be a fixed point of f in $S^1$. We can write this fixed point in the polar coordinates $(r,\theta )$ as $(1,{\theta }_0)$. We will introduce new coordinates on a neighborhood U of π as follows:

$$x_1=\theta -{\theta }_0,x_2=1-r.$$

In the new coordinate setting, the fixed point π is the origin and $U\cap S^1$ corresponds to the $x_1$-axis near 0, and the portion of the interior of the unit ball in U is contained in the upper half-plane. Consider the following map (as described in Section 2) in the new coordinate setting:

$$F(x_1,x_2)=\{\begin{array}{cc}(x_1,x_2)-f(x_1,x_2),\hfill & \text{if }{x}_2\ge 0,\hfill \\ (x_1,x_2)-f(x_1,0),\hfill & \text{if }{x}_2<0.\hfill \end{array}$$

Now write $F=(F_1,F_2)$ and define $g(x_1)=F_1(x_1,0)$. Since $f(\zeta )={\zeta }^k$ for $\zeta \in S^1$, we have

$$g(x_1)=F_1(x_1,0)=x_1-k{x}_1=(1-k)x_1.$$

Since the map f is defined to be smooth on $B^2$, the map F is smooth on the upper half-plane. Let $F^{+}$ denote the restriction of F to the upper half-plane. We will see that smoothness is only required in a neighborhood of the fixed point at 0. Since we are assuming that $k\ge 2$, then

$$\frac{d}{dx}{F}_1(x_1,0)=g^{\prime }(x_1)=1-k<0$$

and the smoothness of $F^{+}$ implies that

$$\frac{\partial F_1^{+}(x_1,x_2)}{\partial x_1}<0$$

for $(x_1,x_2)$ in an ϵ-neighborhood of the origin, for $ϵ>0$ sufficiently small and for $x_2\ge 0$. Let Γ be a circle of radius $ϵ/2$ about the origin.

Let ${\mathrm{\Gamma }}^{+}$ and ${\mathrm{\Gamma }}^{-}$ denote the half-circles above and below the $x_1$-axis respectively. Since F takes the lower half-plane to itself, we know that F maps ${\mathrm{\Gamma }}^{-}$ to the lower half-plane. Calculating the fixed point index of f at the origin in ${\mathbb{R}}^2$ is equivalent to finding the winding number of $F(\mathrm{\Gamma })$ around 0. Thus, we need to understand $F({\mathrm{\Gamma }}^{+})$. Since ${\mathrm{\Gamma }}^{+}$ lies in the upper half-plane, we only consider $F^{+}$. Assuming F has only a finite number of fixed points, we can choose ϵ small enough so that only one point on Γ or its interior that F maps to the origin is the origin itself. Therefore, we can homotope the restriction of $F^{+}$ to ${\mathrm{\Gamma }}^{+}$ in ${\mathbb{R}}^2\mathrm{\setminus }0$ to the restriction of $F^{+}$ to the curve ${\mathrm{\Gamma }}_{\delta }^{+}$ for $\delta >0$ given by

$${\mathrm{\Gamma }}_{\delta }^{+}(t)=(\frac{ϵ}{2}(2t-1),\delta (1-{(1-2t)}^2)),$$

where $0\le t\le 1$.

We write the restriction of $F^{+}$ to ${\mathrm{\Gamma }}_{\delta }^{+}$ in coordinates as

$$F^{+}({\mathrm{\Gamma }}_{\delta }^{+}(t))=(F_1^{+}({\mathrm{\Gamma }}_{\delta }^{+}(t)),F_2^{+}({\mathrm{\Gamma }}_{\delta }^{+}(t)))=({\varphi }_{\delta }(t),{\psi }_{\delta }(t)).$$

The key idea of the proof is that for δ sufficiently small, the smoothness of $F^{+}$ and the fact that

$$\frac{\partial F_1^{+}(x_1,x_2)}{\partial x_1}<0\text{implies that}\frac{d}{dt}{\varphi }_{\delta }(t)<0$$

for all t. This tells us that the $x_1$-coordinate of the curve $F^{+}({\mathrm{\Gamma }}_{\delta }^{+}(t))$ is a strictly monotone function of t. In particular, the curve $F^{+}({\mathrm{\Gamma }}_{\delta }^{+}(t))$ only crosses the $x_2$-axis once. This implies the desired result since the winding number of $F(\mathrm{\Gamma })$ can then only be either 0 or −1.

Notice that it is never specified that f maps $B^2$ into itself. In considering the map $F(z)=z-f(z)$, although it is assumed that F maps the exterior of the disc to the exterior of the disc, the proof allows the image of the interior of the disc under F to lie anywhere in ${\mathbb{R}}^2$. □

Let $\overline{f}:B^2\to {\mathbb{R}}^2$ be the restriction of $f:({\mathbb{R}}^2,S^1)\to ({\mathbb{R}}^2,S^1)$ to $B^2$. Since $B^2$ is contractible, the sum of the indices of $\rho \overline{f}:B^2\to B^2$ equals one, and therefore $\rho \overline{f}(x)=x$ for some $x\in int(B^2)$. But then $f(x)=x$ as well, so $f:({\mathbb{B}}^2,S^1)\to ({\mathbb{R}}^2,S^1)$ has a fixed point in the interior of $B^2$. Therefore, we can extend Corollary 2 as the following result.

Corollary 4 Let $f:({\mathbb{R}}^2,S^1)\to ({\mathbb{R}}^2,S^1)$ be a smooth map such that $f(\zeta )={\zeta }^k$ for all $\zeta \in S^1$ for some $k\ge 2$. Then there exist $z\in int(B^2)$ such that $f(z)=z$.

4 Interior fixed points of a map $f:({\mathbb{R}}^n,S^{n-1})\to ({\mathbb{R}}^n,S^{n-1})$

Definition 1 ([3], p.34)

A smooth map $\varphi :S^{n-1}\to S^{n-1}$ with finitely many fixed points is transversely fixed if $d{\varphi }_p-I:T_p(S^{n-1})\to T_p(S^{n-1})$ is a nonsingular linear map for each fixed point p. For $F=\{p_1,\dots ,p_r\}$ a fixed point class of ϕ, let

$$i(F)=\sum _{j=1}^{r}i(S^{n-1},\varphi ,p_j).$$

The transverse Nielsen number $N_{⋔}(\varphi )$ is defined by

$$N_{⋔}(\varphi )=\sum _{F\in \mathfrak{F}}|i(F)|,$$

where $\mathfrak{F}$ is the set of fixed point classes of ϕ.

A smooth map $\varphi :S^{n-1}\to S^{n-1}$ is sparse if it is transversely fixed and it has $N_{⋔}(\varphi )$ fixed points.

In [3], p.45 Schirmer obtained the following result.

Theorem 5 Let $\varphi :S^{n-1}\to S^{n-1}$ be a sparse map of degree d and suppose $f:(B^n,S^{n-1})\to (B^n,S^{n-1})$ is a smooth map extending ϕ. If ${(-1)}^{n}d\ge 2$, then f must have a fixed point in $int(B^n)$.

We will extend this result as a consequence of the following

Theorem 6 Given a smooth map $\varphi :S^{n-1}\to S^{n-1}$ and a smooth map $f:(B^n,S^{n-1})\to ({\mathbb{R}}^n,S^{n-1})$ extending ϕ, suppose that $p\in S^{n-1}$ is an isolated fixed point of f and that $d{\varphi }_p-I:T_p(S^{n-1})\to T_p(S^{n-1})$ is a nonsingular linear transformation. Then either $i(B^n,f,p)=0$ or $i(B^n,f,p)=i(S^{n-1},\varphi ,p)$.

Proof The following proof is a modified version of Theorem 5.1 in [1]. We again write f in a small ball that contains $p\in S^{n-1}$ as described in Section 2 and the map F is also as defined there. Moreover, for $\epsilon >0$ small enough, let

$$D_{\epsilon }(x_1,\dots ,x_n)=\{\begin{array}{cc}(\epsilon x_1,\epsilon x_2,\dots ,\epsilon x_n)-f(\epsilon x_1,\epsilon x_2,\dots ,\epsilon x_n),\hfill & \text{if }{x}_n\ge 0,\hfill \\ (\epsilon x_1,\epsilon x_2,\dots ,\epsilon x_n)-f(\epsilon x_1,\epsilon x_2,\dots ,\epsilon x_{n-1},0),\hfill & \text{if }{x}_n<0.\hfill \end{array}$$

This then means that the index $i(B^n,f,p)=i(B^n,D_{\epsilon },\mathbf{0})$ is the degree of $\rho \circ D_{\epsilon }:S^{n-1}\to S^{n-1}$, where $\rho (x)=x/|x|$ for $x\in {\mathbb{R}}^n\mathrm{\setminus }\mathbf{0}$.

Note that

$$|F(\epsilon x_1,\epsilon x_2,\dots ,\epsilon x_n)-d{F}_p(\epsilon x_1,\epsilon x_2,\dots ,\epsilon x_n)|\le o(\epsilon )$$

because F is a $C^1$ function. We also have

$$|d{F}_p(\epsilon x_1,\epsilon x_2,\dots ,\epsilon x_n)|\ge C\epsilon$$

for some $C>0$ independent of ε and $(x_1,x_2,\dots ,x_n)$ due to the fact that $d{F}_p=-(d{f}_p-I)$ is nonsingular by hypothesis.

Since $d{F}_p$ is a nonsingular linear map, this last degree is easily seen to be 0, or ±1. But the images of $D_{\epsilon }$ of the upper and lower hemisphere are each contained entirely in either the lower or upper half-space. This means that $D_{\epsilon }$ is of degree 0 or $D_{\epsilon }$ is homotopic in ${\mathbb{R}}^n\mathrm{\setminus }\mathbf{0}$ to the suspension of $D_{\epsilon }{|}_{S^{n-2}}$. □

We can use Theorem 6 to extend Theorem 5 to the case $f:({\mathbb{R}}^n,S^{n-1})\to ({\mathbb{R}}^n,S^{n-1})$. The following is a modified version of part of [3]. Note that despite the fact that the case $n=2$ is solved in the previous section, the new material presented below extends the solution to all the cases.

Suppose we have $\varphi :S^{n-1}\to S^{n-1}$ and a smooth map $f:({\mathbb{R}}^n,S^{n-1})\to ({\mathbb{R}}^n,S^{n-1})$ extending ϕ. A fixed point class F of f is called a common fixed point class of f and ϕ if there exists an essential fixed point class $\overline{F}$ of ϕ which is contained in F.

We will again consider the restriction $f:(B^n,S^{n-1})\to ({\mathbb{R}}^n,S^{n-1})$. In the notation of [3], p.39, let

$$\begin{array}{c}u(F)=max\{0,\sum \left(i(S^{n-1},\varphi ,\overline{F})\right|\overline{F}\subset F\text{ and }i(S^{n-1},\varphi ,\overline{F})>0)\},\hfill \\ l(F)=min\{0,\sum \left(i(S^{n-1},\varphi ,\overline{F})\right|\overline{F}\subset F\text{ and }i(S^{n-1},\varphi ,\overline{F})<0)\}.\hfill \end{array}$$

Then F is a transversally common fixed point class of f and ϕ if

$$l(F)\le i(F)\le u(F).$$

Definition 2 The boundary transversal Nielsen number of $f:(B^n,S^{n-1})\to ({\mathbb{R}}^n,S^{n-1})$ is

$$N(f;B^n,S_{⋔}^{n-1})=N_{⋔}(\varphi )+N(f)-N(f,{\varphi }_{⋔}),$$

where $N(f,{\varphi }_{⋔})$ is the number of essential and transversally common fixed point classes of f and ϕ.

Suppose that $\varphi :S^{n-1}\to S^{n-1}$ has degree d. Then $f:B^n\to {\mathbb{R}}^n$ has one fixed point class F with $i(F)=1$, and so $l(F)\le 0<i(F)$. If $n=2$, then ϕ has $|1-d|$ essential fixed point classes, each of the same index, and $\sum i(S^1,\varphi ,\overline{F})=L(\varphi )=1-d$. Hence, $i(F)\le u(F)$ if and only if $d\le 0$, and

$$N(f,{\varphi }_{⋔})=\{\begin{array}{cc}1\hfill & \text{if }d\le 0,\hfill \\ 0\hfill & \text{if }d>0.\hfill \end{array}$$

If $n\ge 3$, then ϕ has one fixed point class $\overline{F}$ with $i(S^{n-1},\varphi ,\overline{F})=1+{(-1)}^{n-1}d$ and so

$$N(f,{\varphi }_{⋔})=\{\begin{array}{cc}1\hfill & \text{if }{(-1)}^{n-1}d\ge 0,\hfill \\ 0\hfill & \text{if }{(-1)}^{n-1}d<0.\hfill \end{array}$$

Note that this formula is still true for the case $n=2$.

If $\varphi :S^{n-1}\to S^{n-1}$ is a sparse map of degree d then $N_{⋔}(\varphi )=|1-{(-1)}^{n}d|$ for all d and all $n\ge 2$.

Since $N(f)=1$, for the case that ${(-1)}^{n}d\le 0$, the boundary transversal Nielsen number is

$$\begin{array}{rcl}N(f;B^n,S_{⋔}^{n-1})& =& N_{⋔}(\varphi )+N(f)-N(f,{\varphi }_{⋔})\\ =& |1-{(-1)}^{n}d|+1-1\\ =& |1-{(-1)}^{n}d|.\end{array}$$

As for the case that ${(-1)}^{n}d>0$, the boundary transversal Nielsen number is

$$\begin{array}{rcl}N(f;B^n,S_{⋔}^{n-1})& =& N_{⋔}(\varphi )+N(f)-N(f,{\varphi }_{⋔})\\ =& |1-{(-1)}^{n}d|+1-0\\ =& |1-{(-1)}^{n}d|+1.\end{array}$$

Hence, we have just proven the following

Proposition 7 If $f:(B^n,S^{n-1})\to ({\mathbb{R}}^n,S^{n-1})$ is a smooth extension of a sparse map $\varphi :S^{n-1}\to S^{n-1}$ of degree d, with $n\ge 2$, then the boundary transversal Nielsen number is

$$N(f;B^n,S_{⋔}^{n-1})=\{\begin{array}{cc}|1-{(-1)}^{n}d|\hfill & \mathit{\text{if }}{(-1)}^{n}d\le 0,\hfill \\ |1-{(-1)}^{n}d|+1\hfill & \mathit{\text{if }}{(-1)}^{n}d>0.\hfill \end{array}$$

As defined in [1], p.2, the extension Nielsen number $N(f|\varphi )$ is a lower bound for the number of fixed points on $S^{n-1}$ of continuous extensions of a continuous map ϕ. It is equal to the number of essential (classical) fixed point classes F of f with $F\cap S^{n-1}=\mathrm{\varnothing }$. A fixed point class F is representable on $S^{n-1}$ if there exists a subset $F^{\prime }\subset F\cap S^{n-1}$ with $i(B^n,f,F)=i(S^{n-1},\varphi ,F^{\prime })$. The smooth extension number $N^1(f|\varphi )$ is the number of essential (classical) fixed point classes F of f which are not representable on $S^{n-1}$. It is a lower bound for the number of fixed points in $S^{n-1}$ of a smooth extension of a smooth and transversally fixed map ϕ.

Proposition 8 If $\varphi :S^{n-1}\to S^{n-1}$ is sparse, then

$$\begin{array}{c}{N}^1(f|\varphi )=N(f;B^n,S_{⋔}^{n-1})-N_{⋔}(\varphi ),\hfill \\ N(f|\varphi )=N(f;B^n,S^{n-1})-N(\varphi ).\hfill \end{array}$$

Proof Our proof is modeled on the proofs of Proposition 4.3 and Corollary 4.4 in [3]. For any essential fixed point class F of f, since ϕ is sparse, $F\cap S^{n-1}$ contains $u(F)$ fixed points p such that $i(S^{n-1},\varphi ,p)=1$ and $l(F)$ fixed points p such that $i(S^{n-1},\varphi ,p)=-1$. This means that F is representable on $S^{n-1}$ if and only if $l(F)\le i(F)\le u(F)$. By the definitions of all the Nielsen numbers involved, we have the result stated above for $N^1(f|\varphi )$.

The result for $N(f|\varphi )$ can be obtained in a similar manner by using Corollary 2.6 from [1] along with the fact that all fixed point classes of a sparse map are essential. □

By the definitions of $N(f,\varphi )$ in [5] and the definition of $N^1(f|\varphi )$ defined early, we obtain

Proposition 9 If $\varphi :S^{n-1}\to S^{n-1}$ is sparse, then the number of essential fixed point classes of f which are common but not transversally common is

$$N^1(f|\varphi )-N(f|\varphi )=N(f,\varphi )-N(f,{\varphi }_{⋔}).$$

We are now ready to prove the following Theorem.

Theorem 10 Let $n\ge 2$, and let $\varphi :S^{n-1}\to S^{n-1}$ be a sparse map of degree d and suppose $f:(B^n,S^{n-1})\to ({\mathbb{R}}^n,S^{n-1})$ is a smooth map extending ϕ. If ${(-1)}^{n}d\ge 2$, then f must have a fixed point in $int(B^n)$.

Proof Since ϕ has degree d and it is sparse, by definition

$$N(f|\varphi )=\{\begin{array}{cc}1\hfill & \text{if }d\ne {(-1)}^n,\hfill \\ 0\hfill & \text{if }d={(-1)}^n\hfill \end{array}$$

and

$$\begin{array}{rcl}{N}^1(f|\varphi )-N(f|\varphi )& =& N(f;B^n,S_{⋔}^{n-1})-N_{⋔}(\varphi )-N(f|\varphi )(\text{from Proposition 8})\\ =& N(f;B^n,S_{⋔}^{n-1})-|1-{(-1)}^{n}d|-N(f|\varphi ).\end{array}$$

Applying Proposition 7, we have

$$N^1(f|\varphi )-N(f|\varphi )=\{\begin{array}{cc}0\hfill & \text{if }{(-1)}^{n}d\le 1,\hfill \\ 1\hfill & \text{if }{(-1)}^{n}d\ge 2.\hfill \end{array}$$

Thus, every smooth extension over $B^n$ of a sparse map of $S^{n-1}$ of degree d with ${(-1)}^{n}d\ge 2$ has a fixed point on the interior of $B^n$. □