.

# On approximation of maps into
real algebraic homogeneous spaces

Jacek Bochnak  Wojciech Kucharz

###### Abstract

Let $X$ be a real algebraic variety (resp. nonsingular real algebraic variety)
and let $Y$ be a homogeneous space for some linear real algebraic group. We
prove that a continuous (resp. $\mathcal{C}^{\infty}$) map $f\colon X\to Y$
can be approximated by regular maps in the $\mathcal{C}^{0}$ (resp.
$\mathcal{C}^{\infty}$) topology if and only if it is homotopic to a regular
map. Taking $Y=\mathbb{S}^{p}$, the unit $p$-dimensional sphere, we obtain
solutions of several problems that have been open since the 1980’s and which
concern approximation of maps with values in the unit spheres. This has
several consequences for approximation of maps between unit spheres. For
example, we prove that for every positive integer $n$ every
$\mathcal{C}^{\infty}$ map from $\mathbb{S}^{n}$ into $\mathbb{S}^{n}$ can be
approximated by regular maps in the $\mathcal{C}^{\infty}$ topology. Up to now
such a result has only been known for five special values of $n$, namely,
$n=1,2,3,4$ or $7$.

Key words and phrases Real algebraic variety, regular map, approximation,
homotopy, real algebraic group, homogeneous space, unit sphere.

Mathematics subject classification (2020) 14P05, 14P25, 14P99.

##  1 Introduction

In the present paper, we study approximation of continuous or
$\mathcal{C}^{\infty}$ maps by real regular maps, that is, real algebraic
morphisms. Our results concern maps with values in homogeneous spaces for
linear real algebraic groups. Special attention is paid to maps into unit
spheres. The main result, Theorem 1.1, and its consequences provide answers to
some approximation problems that have been open since the 1980’s.

Throughout this work, by an _algebraic variety_ we always mean a
quasiprojective variety. To be precise, a _real algebraic variety_ is a ringed
space with structure sheaf of $\mathbb{R}$-algebras of $\mathbb{R}$-valued
functions, which is isomorphic to a Zariski locally closed subset of real
projective $n$-space $\mathbb{P}^{n}(\mathbb{R})$, for some $n$, endowed with
the Zariski topology and the sheaf of regular functions. This is compatible
with [3], which contains a detailed exposition of real algebraic geometry. We
use the analogous definition of a _complex algebraic variety_ , replacing
$\mathbb{R}$ by $\mathbb{C}$. Recall that each real algebraic variety in the
sense used here is actually affine, that is, isomorphic to an algebraic subset
of $\mathbb{R}^{n}$, for some $n$, see [3, Proposition 3.2.10 and Theorem
3.4.4]. Morphisms of algebraic varieties are called _regular maps_ (in some of
our references they are called _entire rational maps_ , see [4, 5, 48, 49]).

A _real algebraic group_ is a real algebraic variety $G$ endowed with the
structure of a group such that the group operations $G\times G\to G$,
$(a,b)\mapsto ab$, and $G\to G$, $a\mapsto a^{-1}$ are regular maps.
_Morphisms of real algebraic groups_ are regular maps that are group
homomorphisms. A real algebraic group is said to be _linear_ if it is
isomorphic to a Zariski closed subgroup of the general linear group
$\mathrm{GL}_{n}(\mathbb{R})$, for some $n$. Obviously, the familiar subgroups
of $\mathrm{GL}_{n}(\mathbb{R})$, the orthogonal group $\mathrm{O}(n)$ and
special orthogonal group $\mathrm{SO}(n)$, are linear real algebraic groups. A
_complex algebraic group_ , linear or not, is defined analogously, replacing
$\mathbb{R}$ by $\mathbb{C}$. Clearly, each real or complex algebraic group
$G$ is a nonsingular algebraic variety of pure dimension. Moreover, if $G$ is
linear, then each Zariski closed subgroup of $G$ is a linear algebraic group.

Any complex algebraic variety $V$ carries the obvious underlying structure
$\real{V}$ of a real algebraic variety, called the _realification_ of $V$ (for
example, $\real{(\mathbb{C}^{n})}=\mathbb{R}^{2n}$). If $G$ is a complex
algebraic group, then its realification $\real{G}$ is a real algebraic group
with the same group operations as in $G$. If $H$ is a Zariski closed subgroup
of $G$, then $\real{H}$ is a Zariski closed subgroup of $\real{G}$. Moreover,
if the group $G$ is linear, then so is the group $\real{G}$; indeed, it
suffices to note that the real algebraic group
${}_{\mathbb{R}}\mathrm{GL}_{n}(\mathbb{C})$ is isomorphic to the image of the
real regular embedding

${}_{\mathbb{R}}\mathrm{GL}_{n}(\mathbb{C})\to\mathrm{GL}_{2n}(\mathbb{R}),\quad
A+\sqrt{-1}\,B\mapsto\begin{pmatrix}A&-B\\\ B&A\end{pmatrix},$

where $A$, $B$ are real $n$-by-$n$ matrices. Consequently, each Zariski closed
subgroup of ${}_{\mathbb{R}}\mathrm{GL}_{n}(\mathbb{C})$ is a linear real
algebraic group. In particular, the unitary group $\mathrm{U}(n)$ and special
unitary group $\mathrm{SU}(n)$ are linear real algebraic groups, being Zariski
closed subgroups of ${}_{\mathbb{R}}\mathrm{GL}_{n}(\mathbb{C})$.

The general linear group $\mathrm{GL}_{n}(\mathbb{H})$, where $\mathbb{H}$ is
the (skew) field of quaternions, can also be viewed as a linear real algebraic
group via the standard embedding into $\mathrm{GL}_{4n}(\mathbb{R})$.
Therefore the symplectic subgroup $\mathrm{Sp}(n)$ of
$\mathrm{GL}_{n}(\mathbb{H})$ is a linear real algebraic group.

Let $G$ be a real algebraic group. A _$G$ -space_ (or a _$G$ -variety_) is a
real algebraic variety $Y$ on which $G$ acts, the action $G\times Y\to Y$,
$(a,y)\mapsto a\cdot y$ being a regular map. A _homogeneous space_ for $G$ is
a $G$-space on which $G$ acts transitively. Note that each homogeneous space
for $G$ is a nonsingular real algebraic variety of pure dimension.

Besides the Zariski topology, every real algebraic variety is endowed with the
Euclidean topology determined by the standard metric on $\mathbb{R}$. Unless
explicitly stated otherwise, all topological notions relating to real
algebraic varieties will refer to the Euclidean topology.

Given real algebraic varieties $X$ and $Y$, we denote by $\mathcal{R}(X,Y)$
the set of all regular maps from $X$ into $Y$. We regard $\mathcal{R}(X,Y)$ as
a subset of the space $\mathcal{C}^{0}(X,Y)$ of all continuous maps endowed
with the $\mathcal{C}^{0}$ (that is, compact-open) topology. We say that a
continuous map $f\colon X\to Y$ can be _approximated by regular maps in the
$\mathcal{C}^{0}$ topology_ if, for every neighborhood $\mathcal{U}$ of $f$ in
$\mathcal{C}^{0}(X,Y)$, there is a regular map $g\colon X\to Y$ which belongs
to $\mathcal{U}$. If both varieties $X$ and $Y$ are nonsingular, then
$\mathcal{R}(X,Y)$ is a subset of the space $\mathcal{C}^{\infty}(X,Y)$ of all
$\mathcal{C}^{\infty}$ maps endowed with the $\mathcal{C}^{\infty}$ topology
(see [29, p. 36] or [57, p. 311] for the definition of this topology and note
that in [29] it is called the weak $\mathcal{C}^{\infty}$ topology); therefore
the concept of approximation of a $\mathcal{C}^{\infty}$ map $f\colon X\to Y$
by regular maps in the $\mathcal{C}^{\infty}$ topology is well-defined.

Our main result is the following.

######  Theorem 1.1.

Let $X$ be a real algebraic variety (resp. nonsingular real algebraic variety)
and let $Y$ be a homogeneous space for some linear real algebraic group. Then,
for a continuous (resp. $\mathcal{C}^{\infty}$) map $f\colon X\to Y$, the
following conditions are equivalent:

  1. (a)

$f$ can be approximated by regular maps in the $\mathcal{C}^{0}$ (resp.
$\mathcal{C}^{\infty}$) topology.

  2. (b)

$f$ is homotopic to a regular map.

The implication (a)$\Rightarrow$(b) holds because $X$ deformation retracts to
some compact subset $K\subseteq X$ [3, Corollary 9.3.7], and any two
continuous maps from $K$ into $Y$ that are sufficiently close in the
$\mathcal{C}^{0}$ topology are homotopic (the latter assertion is valid if $Y$
is an arbitrary $\mathcal{C}^{\infty}$ manifold). The proof of
(b)$\Rightarrow$(a) is given in Section 4 and depends on the techniques
developed in Sections 2 and 3. With notation as in Theorem 1.1, if $X$ is a
compact nonsingular real algebraic curve, then, by the recent result of
Benoist and Wittenberg [2, Theorem A(3)], each $\mathcal{C}^{\infty}$ map from
$X$ into $Y$ can be approximated by regular maps in the $\mathcal{C}^{\infty}$
topology. Some special cases of Theorem 1.1 have been anticipated since the
1980’s, however, the results obtained heretofore have been very incomplete.

As immediate consequences of Theorem 1.1 we get the following two corollaries.

######  Corollary 1.2.

Let $X$ be a real algebraic variety (resp. nonsingular real algebraic variety)
and let $Y$ be a homogeneous space for some linear real algebraic group. Then
every continuous (resp. $\mathcal{C}^{\infty}$) null homotopic map from $X$ to
$Y$ can be approximated by regular maps in the $C^{0}$ (resp.
$\mathcal{C}^{\infty}$) topology. ∎

######  Corollary 1.3.

Let $Y$ be a homogeneous space for some linear real algebraic group. Then
every continuous (resp. $\mathcal{C}^{\infty}$) map $Y\to Y$ that is homotopic
to the identity map can be approximated by regular maps in the
$\mathcal{C}^{0}$ (resp. $\mathcal{C}^{\infty})$ topology. ∎

It should be mentioned that no direct proofs of these corollaries, which do
not make use of Theorem 1.1, are available.

In the rest of this section, we discuss the impact of Theorem 1.1 within the
framework described in the following example.

######  Example 1.4.

Here are some homogeneous spaces of interest.

  1. (i)

For every nonnegative integer $n$ the unit $n$-sphere

$\mathbb{S}^{n}=\\{(x_{0},\ldots,x_{n})\in\mathbb{R}^{n+1}:x_{0}^{2}+\cdots+x_{n}^{2}=1\\}$

is a homogeneous space for the real orthogonal group $\mathrm{O}(n+1)$.

  2. (ii)

Let $\mathbb{F}$ stand for one of the fields $\mathbb{R}$, $\mathbb{C}$ or
$\mathbb{H}$. Denote by $\mathbb{G}_{r}(\mathbb{F}^{n})$ the Grassmannian of
$r$-dimensional $\mathbb{F}$-vector subspaces of $\mathbb{F}^{n}$, regarded as
a real algebraic variety (see [3, pp. 72, 73, 352]). Clearly,
$\mathbb{G}_{r}(\mathbb{F}^{n})$ is a homogeneous space for the group
$\mathrm{GL}_{n}(\mathbb{F})$ viewed as a linear real algebraic group.

  3. (iii)

Denote by $\mathbb{V}_{r}(\mathbb{F}^{n})$ the Stiefel manifold of all
orthonormal $r$-frames in $\mathbb{F}^{n}$, regarded as a real algebraic
variety. Clearly, $\mathbb{V}_{r}(\mathbb{F}^{n})$ is a homogeneous space for
the linear real algebraic group $\mathrm{O}(n)$ if $\mathbb{F}=\mathbb{R}$,
$\mathrm{U}(n)$ if $\mathbb{F}=\mathbb{C}$, and $\mathrm{Sp}(n)$ if
$\mathbb{F}=\mathbb{H}$.

  4. (iv)

Every real algebraic group $G$ is a homogeneous space for $G$ under action by
left translations.

In view of Theorem 1.1 and Example 1.4(i), we get at once the following result
on maps into the unit $p$-sphere $\mathbb{S}^{p}$.

######  Corollary 1.5.

Let $X$ be a real algebraic variety (resp. nonsingular real algebraic variety)
and let $p$ be a positive integer. Then, for a continuous (resp.
$\mathcal{C}^{\infty}$) map $f\colon X\to\mathbb{S}^{p}$, the following
conditions are equivalent:

  1. (a)

$f$ can be approximated by regular maps in the $\mathcal{C}^{0}$ (resp.
$\mathcal{C}^{\infty}$) topology.

  2. (b)

$f$ is homotopic to a regular map.∎

Until now Corollary 1.5 with $X$ compact and $\dim X\geq p$ has only been
known for a few special values of $p$, namely, for $p=1,2$ or $4$ (see [4])
and $p=3$ or $7$ (see [1]). Of course, if $\dim X<p$, then the set of regular
maps $\mathcal{R}(X,\mathbb{S}^{p})$ is dense in the space
$\mathcal{C}^{0}(X,\mathbb{S}^{p})$ (resp.
$\mathcal{C}^{\infty}(X,\mathbb{S}^{p})$). Indeed, $\mathbb{S}^{p}$ with one
point removed is biregularly isomorphic to $\mathbb{R}^{p}$ via the
stereographic projection. Moreover, for $\dim X<p$, every continuous (resp.
$\mathcal{C}^{\infty}$) map from $X$ into $\mathbb{S}^{p}$ can be approximated
in the space $\mathcal{C}^{0}(X,\mathbb{S}^{p})$ (resp.
$\mathcal{C}^{\infty}(X,\mathbb{S}^{p})$) by maps that are not surjective, and
hence the density assertion follows from the Weierstrass approximation
theorem.

The question which continuous or $\mathcal{C}^{\infty}$ maps are homotopic to
regular ones has been investigated in numerous works
[3,5,8–10,14,15,25–27,37,42,44,48–50,55,58]. By applying Theorem 1.1 or
Corollary 1.5, some of these results can now be translated into the results on
approximation by regular maps. Next we give the first example of this
principle.

######  Theorem 1.6.

For every positive integer $n$ the set of regular maps
$\mathcal{R}(\mathbb{S}^{n},\mathbb{S}^{n})$ is dense in the space of
$\mathcal{C}^{\infty}$ maps
$\mathcal{C}^{\infty}(\mathbb{S}^{n},\mathbb{S}^{n})$.

###### Proof.

Since each $\mathcal{C}^{\infty}$ map $\mathbb{S}^{n}\to\mathbb{S}^{n}$ is
homotopic to a regular one (see [58, Theorem 1] for $n$ odd, and [5, Corollary
4.2] for $n$ arbitrary), the conclusion follows at once from Corollary 1.5. ∎

Up to now Theorem 1.6 has only been known for $n=1,2$ or $4$ (see [4]) and
$n=3$ or $7$ (see [1]).

The following conjecture, although plausible, is wide open.

######  Conjecture I.

Let $Y$ be a homogeneous space for some linear real algebraic group. Then, for
every positive integer $n$, the set of regular maps
$\mathcal{R}(\mathbb{S}^{n},Y)$ is dense in the space of
$\mathcal{C}^{\infty}$ maps $\mathcal{C}^{\infty}(\mathbb{S}^{n},Y)$. In
particular, $\mathcal{R}(\mathbb{S}^{n},\mathbb{S}^{p})$ is dense in
$\mathcal{C}^{\infty}(\mathbb{S}^{n},\mathbb{S}^{p})$ for every pair $(n,p)$
of positive integers.

By [4, Corollary 2.7] (see also [52, Theorems 2.2 and 11.1]), Conjecture I is
valid for the homogeneous space $Y=\mathbb{G}_{r}(\mathbb{F}^{n})$ defined in
Example 1.4(ii). Recall that $\mathbb{G}_{1}(\mathbb{F}^{2})$ is biregularly
isomorphic to the unit sphere $\mathbb{S}^{d(\mathbb{F})}$,
$d(\mathbb{F})=\dim_{\mathbb{R}}\mathbb{F}$. In particular, Conjecture I holds
if $Y$ is one of the spheres $\mathbb{S}^{1}$, $\mathbb{S}^{2}$ or
$\mathbb{S}^{4}$. Presently Conjecture I for maps between unit spheres remains
open, but it is supported further by Theorem 5.6.

In Section 6 we obtain some results supporting Conjecture I for maps with
values in the classical linear real algebraic groups $\mathrm{O}(m)$,
$\mathrm{SO}(m)$, $\mathrm{U}(m)$, $\mathrm{SU}(m)$, $\mathrm{Sp}(m)$ or in
the Stiefel manifold $\mathbb{V}_{r}(\mathbb{F}^{m})$ (see Propositions 6.1
and 6.3, and Theorem 6.2).

In general, the equivalent conditions (a) and (b) in Corollary 1.5 are not
satisfied. Indeed, suppose given a compact nonsingular real algebraic variety
$X$ such that every regular map from $X$ into $\mathbb{S}^{p}$ is null
homotopic. Then, in view of Corollary 1.5, a $\mathcal{C}^{\infty}$ map
$f\colon X\to\mathbb{S}^{p}$ can be approximated by regular maps if and only
if it is null homotopic. Of course, assuming that $\dim X=p$, there are always
$\mathcal{C}^{\infty}$ maps from $X$ into $\mathbb{S}^{p}$ that are not null
homotopic. By [6, Theorem 2.1], for “most” orientable nonsingular algebraic
hypersurfaces $X$ in $\mathbb{P}^{2n+1}(\mathbb{R})$, every regular map from
$X$ into $\mathbb{S}^{2n}$ is null homotopic. Moreover, according to [10,
Theorems 1.2 and 7.3], for “most” nonsingular real cubic curves $C$ in
$\mathbb{P}^{2}(\mathbb{R})$, every regular map from $C^{2n}$ into
$\mathbb{S}^{2n}$ is null homotopic. For real cubic curves one can also
substitute other real algebraic curves, see [15, Theorem 1.3] and [11,
Corollary 2]. We explicitly state the following.

######  Theorem 1.7.

Let $X$ be a compact connected nonsingular real algebraic variety of odd
dimension $k<2n$. Then a $\mathcal{C}^{\infty}$ map $f\colon
X\times\mathbb{S}^{2n-k}\to\mathbb{S}^{2n}$ can be approximated by regular
maps in the $\mathcal{C}^{\infty}$ topology if and only if it is null
homotopic.

###### Proof.

By [6, Theorem 2.4], each regular map from $X\times\mathbb{S}^{2n-k}$ into
$\mathbb{S}^{2n}$ is null homotopic, and hence the conclusion follows from
Corollary 1.5. ∎

The behavior of regular maps into odd-dimensional spheres is entirely
different.

######  Theorem 1.8.

Let $X$ be a compact connected oriented nonsingular real algebraic variety of
odd dimension $k$. Then either

  1. (i)

the set $\mathcal{R}(X,\mathbb{S}^{k})$ is dense in the space
$\mathcal{C}^{\infty}(X,\mathbb{S}^{k})$, or

  2. (ii)

the closure of $\mathcal{R}(X,\mathbb{S}^{k})$ in
$\mathcal{C}^{\infty}(X,\mathbb{S}^{k})$ coincides with the set

$\\{f\in\mathcal{C}^{\infty}(X,\mathbb{S}^{k}):\deg(f)\in 2\mathbb{Z}\\},$

where $\deg(f)$ is the topological degree of the map $f$.

###### Proof.

By [5, Corollary 2.5], the set

$\\{m\in\mathbb{Z}:m=\deg(g),\ g\in\mathcal{R}(X,\mathbb{S}^{k})\\}$

is equal either to $\mathbb{Z}$ or $2\mathbb{Z}$, so the statement follows by
combining Hopf’s theorem with Corollary 1.5. ∎

No example of an odd-dimensional variety $X$ for which condition (ii) in
Theorem 1.8 holds is known, which raises the possibility that the following
conjecture might be true.

######  Conjecture II.

Let $X$ be a compact connected nonsingular real algebraic variety and let $k$
be a positive odd integer. If $\dim X=k$, then the set of regular maps
$\mathcal{R}(X,\mathbb{S}^{k})$ is dense in the space of
$\mathcal{C}^{\infty}$ maps $\mathcal{C}^{\infty}(X,\mathbb{S}^{k})$.

This conjecture is known to be true for $k=1$ (see [4, Corollary 1.5]), but
remains open for all other odd values of $k$. Example 5.8 shows that the
assumption $\dim X=k$ in Conjecture II cannot be replaced by $\dim X\geq k$.

According to the celebrated Nash–Tognoli theorem, every compact
$\mathcal{C}^{\infty}$ manifold $M$ is diffeomorphic to a nonsingular real
algebraic variety $X$, called an _algebraic model_ of $M$, see [47, 54] and
[3, Theorem 14.1.10]. Actually, $M$ admits an uncountable family of pairwise
nonisomorphic algebraic models, provided $\dim M\geq 1$, see [8]. The behavior
of regular maps from $X$ into unit spheres, as $X$ runs through the class of
algebraic models of $M$, has been investigated in [5–7,10,37]. The following
is a new result in this direction.

######  Theorem 1.9.

Let $M$ be a compact connected $\mathcal{C}^{\infty}$ manifold of dimension
$m$. Then there exists an algebraic model $X$ of $M$ such that the set of
regular maps $\mathcal{R}(X,\mathbb{S}^{m})$ is dense in the space of
$\mathcal{C}^{\infty}$ maps $\mathcal{C}^{\infty}(X,\mathbb{S}^{m})$.

###### Proof.

By [5, Proposition 4.5], there exists an algebraic model $X$ of $M$ such that
each $\mathcal{C}^{\infty}$ map from $X$ into $\mathbb{S}^{m}$ is homotopic to
a regular map. Hence the conclusion follows from Corollary 1.5. ∎

Recall that a compact oriented $\mathcal{C}^{\infty}$ manifold is said to be
an _oriented boundary_ if it is the boundary, with the induced orientation, of
a compact oriented $\mathcal{C}^{\infty}$ manifold with boundary.

######  Theorem 1.10.

Let $M$ be a compact connected oriented $\mathcal{C}^{\infty}$ manifold of
dimension $m$. Assume that either

  1. (i)

$m\equiv 2\pmod{4}$, or

  2. (ii)

$m\equiv 0\pmod{4}$ and the disjoint union of two copies of $M$ is an oriented
boundary.

Then there exists an algebraic model $X$ of $M$ such that each regular map
from $X$ into $\mathbb{S}^{m}$ is null homotopic. Moreover, a
$\mathcal{C}^{\infty}$ map $f\colon X\to\mathbb{S}^{m}$ can be approximated by
regular maps in the $\mathcal{C}^{\infty}$ topology if and only if it is null
homotopic.

###### Proof.

The first assertion is proved in [37, Corollary 2.3], and the second follows
from Corollary 1.5. ∎

Let us comment on the boundary assumption in (ii). For each integer $k\geq 4$
there exists a compact connected oriented $\mathcal{C}^{\infty}$ manifold of
dimension $4k$, which is not an oriented boundary, but the disjoint union of
its two copies is; there is no such a manifold in dimensions $4,8$ and $12$.
This statement follows from the description of the torsion elements in the
oriented cobordism ring, see [56, p. 309].

We believe that in Theorem 1.10 (with $m\equiv 0\pmod{4}$) the boundary
condition is not only sufficient, but also necessary.

######  Conjecture III.

Let $M$ be a compact connected oriented $\mathcal{C}^{\infty}$ manifold of
dimension $4k$. Assume that there exists an algebraic model $X$ of $M$ such
that each regular map from $X$ into $\mathbb{S}^{4k}$ is null homotopic. Then
the disjoint union of two copies of $M$ is an oriented boundary.

This conjecture is known to be true if $\dim M=4$, see [6, Theorem 3]. Recall
that if $\dim M=4$, then $M$ is an oriented boundary exactly when its
signature $\sigma(M)$ is zero. If $\sigma(M)\neq 0$ and $X$ is an algebraic
model of $M$, then each $\mathcal{C}^{\infty}$ map $f\colon
X\to\mathbb{S}^{4}$ with $\deg(f)$ divisible by $6\sigma(M)$ can be
approximated by regular maps in the $\mathcal{C}^{\infty}$ topology, see [6,
Theorem 5.6].

Along with approximation by regular maps, investigated further in
[3,4,7,10,12,13,15,32–35,42,45], one can also consider approximation by
stratified-regular (= regulous = continuous hereditarily rational = continuous
rational, the last equality holds assuming nonsingularity of varieties) maps
[36, 38, 39, 41, 42, 59] or by piecewise-regular maps [40]. There are
remarkable similarities and differences in these three cases.

We have already indicated how the paper is organized. It should be added that
the inspiration for Sections 2 and 3, in which we develop the main technical
tools, comes from complex geometry, above all Gromov’s paper [28] and the
related works of Forstnerič and others, see [24] and the references therein.

The paper comes with the Appendix written by János Kollár.

##  2 Malleable real algebraic varieties

The notions and results of this section will be generalized in the next one.
Nevertheless, it seems to be beneficial to discuss first the key special case.

In what follows we work with vector bundles, which are always
$\mathbb{R}$-vector bundles. Let $Y$ be a real algebraic variety. Given a
vector bundle $p\colon E\to Y$ over $Y$, with total space $E$ and bundle
projection $p$, we sometimes refer to $E$ as a vector bundle over $Y$. If $y$
is a point in $Y$, we let $E_{y}\coloneqq p^{-1}(y)$ denote the fiber of $E$
over $y$ and write $0_{y}$ for the zero vector in $E_{y}$. As usual, we call
the set $Z(E)\coloneqq\\{0_{y}\in E_{y}:y\in Y\\}$ the zero section of $E$.

The general theory of algebraic vector bundles over real algebraic varieties
is discussed in [3, Section 12.1]. For each algebraic vector bundle $E$ over
$Y$ there exists an algebraic vector bundle $E^{\prime}$ over $Y$ such that
the direct sum $E\oplus E^{\prime}$ is algebraically trivial, see [3, Theorem
12.1.7]. Moreover, if $E_{1}$ is an algebraic vector subbundle of $E$, then
there exists an algebraic vector subbundle $E_{2}$ of $E$ such that
$E_{1}\oplus E_{2}=E$. This is the case since $E$ can be regarded as an
algebraic vector subbundle of the product vector bundle
$Y\times\mathbb{R}^{n}$, for some $n$, and hence as $E_{2}$ one can take the
orthogonal complement (with respect to the standard scalar product on
$\mathbb{R}^{n}$) of $E_{1}$ in $E$.

Assuming that $Y$ is a nonsingular real algebraic variety, we write $TY$ for
the tangent bundle to $Y$ and $T_{y}Y$ for the tangent space to $Y$ at $y\in
Y$.

######  Definition 2.1.

Let $Y$ be a nonsingular real algebraic variety.

  1. (i)

A _spray_ for $Y$ is a triple $(E,p,s)$, where $p\colon E\to Y$ is an
algebraic vector bundle over $Y$ and $s\colon E\to Y$ is a regular map such
that $s(0_{y})=y$ for all $y\in Y$.

  2. (ii)

A spray $(E,p,s)$ for $Y$ is said to be _dominating_ if the derivative

$d_{0_{y}}s\colon T_{0_{y}}E\to T_{y}Y$

maps the subspace $E_{y}=T_{0_{y}}E_{y}$ of $T_{0_{y}}E$ onto $T_{y}Y$, that
is,

$d_{0_{y}}s(E_{y})=T_{y}Y$

for all $y\in Y$.

  3. (iii)

The variety $Y$ is called _malleable_ if it admits a dominating spray.

The simplest malleable variety is $Y=\mathbb{R}^{n}$ which admits a dominating
spray $(E,p,s)$, where $p\colon E\coloneqq Y\times\mathbb{R}^{n}\to Y$ is the
product vector bundle and $s\colon E\to Y$ is defined by $s(y,v)=y+v$ for all
$(y,v)\in E$. More substantial and useful examples are provided by Examples
2.5 and 2.6, and above all by Propositions 2.7 and 2.8.

In the following two lemmas we describe some basic properties of dominating
sprays.

######  Lemma 2.2.

Any malleable nonsingular real algebraic variety $Y$ admits a dominating spray
$(E,p,s)$ such that $p\colon E\to Y$ is an algebraically trivial vector
bundle.

###### Proof.

Suppose that $(\tilde{E},\tilde{p},\tilde{s})$ is a dominating spray for $Y$.
Choose an algebraic vector bundle
$\tilde{p}^{\prime}\colon\tilde{E}^{\prime}\to Y$ such that the direct sum
$p\colon E\coloneqq\tilde{E}\oplus\tilde{E}^{\prime}\to Y$ is an algebraically
trivial vector bundle. Defining $s\colon E\to Y$ by
$s(v,v^{\prime})=\tilde{s}(v)$, we get a dominating spray $(E,p,s)$ for $Y$. ∎

######  Lemma 2.3.

Let $Y$ be a malleable nonsingular real algebraic variety and let $(E,p,s)$ be
a dominating spray for $Y$. Then there exists a dominating spray
$(\hat{E},\hat{p},\hat{s})$ for $Y$ such that $\hat{p}\colon\hat{E}\to Y$ is
an algebraic vector subbundle of $p\colon E\to Y$, $\hat{s}=s|_{\hat{E}}$, and
the restriction

$d_{0_{y}}\hat{s}|_{\hat{E}_{y}}\colon\hat{E}_{y}\to T_{y}Y$

of the derivative $d_{0_{y}}\hat{s}\colon T_{0_{y}}\hat{E}\to T_{y}Y$ is a
linear isomorphism for all $y\in Y$.

###### Proof.

Let $\alpha\colon E\to TY$ be the morphism of algebraic vector bundles over
$Y$ defined by

$\alpha(v)=d_{0_{p(v)}}s(v)\quad\text{for all }v\in E.$

Since $\alpha$ is a surjective morphism, its kernel $\operatorname{Ker}\alpha$
is an algebraic vector subbundle of $E$. It follows that
$E=\hat{E}\oplus\operatorname{Ker}\alpha$ is the direct sum for some algebraic
vector subbundle $\hat{p}\colon\hat{E}\to Y$ of $p\colon E\to Y$. This
completes the proof. ∎

As a consequence of Lemma 2.3, we obtain the following.

######  Proposition 2.4.

Any malleable nonsingular real algebraic variety $Y$ admits a dominating spray
of the form $(TY,\pi,\sigma)$, where $\pi\colon TY\to Y$ is the tangent bundle
to $Y$.

###### Proof.

With notation as in Lemma 2.3, the vector bundle $\hat{E}$ is isomorphic to
the tangent bundle $TY$, which completes the proof. ∎

Next we identify some malleable real algebraic varieties.

######  Example 2.5.

Let $G$ denote either the real orthogonal group $\mathrm{O}(n)$ or the real
special orthogonal group $\mathrm{SO}(n)$. Then each homogeneous space $Y$ for
$G$ is malleable. This assertion can be established by direct computation as
follows.

Let $\mathrm{Mat}_{n}(\mathbb{R})\cong\mathbb{R}^{n^{2}}$ be the
$\mathbb{R}$-algebra of all $n$-by-$n$ matrices with real entries. Let $0$ and
$1$ stand for the zero matrix and the identity matrix in
$\mathrm{Mat}_{n}(\mathbb{R})$. Using the determinant function $\det$, we
obtain a Zariski open neighborhood

$\Omega\coloneqq\\{x\in\mathrm{Mat}_{n}(\mathbb{R}):\det(1+x)\neq 0\\}$

of the zero matrix. We identify the tangent space to $G$ at $1$ with the
vector subspace $V$ of $\mathrm{Mat}_{n}(\mathbb{R})$ comprising all skew
symmetric matrices; $V\cong\mathbb{R}^{d}$ where $d=n(n-1)/2$. Thus

$\varphi\colon V\to G\cap\Omega,\quad v\mapsto(1-v)(1+v)^{-1}$

is a well-defined biregular isomorphism with $\varphi(0)=1$. We get a
dominating spray $(E,p,s)$ for $Y$, where $p\colon E\coloneqq Y\times V\to Y$
is the product vector bundle and $s\colon E\to Y$ is defined by
$s(y,v)=\varphi(v)\cdot y$ for all $(y,v)\in E$.

In particular, for unit spheres we get the following.

######  Example 2.6.

For every positive integer $n$ the unit sphere $\mathbb{S}^{n}$ is a malleable
real algebraic variety. Indeed, this is the case by Example 2.5 because
$\mathbb{S}^{n}$ is a homogeneous space for the real orthogonal group
$\mathrm{O}(n+1)$. Actually, as observed by János Kollár, one obtains directly
a dominating spray $(T\mathbb{S}^{n},\pi,\sigma)$ for $\mathbb{S}^{n}$, where
$\pi\colon T\mathbb{S}^{n}\to\mathbb{S}^{n}$ is the tangent bundle to
$\mathbb{S}^{n}$ and $\sigma\colon T\mathbb{S}^{n}\to\mathbb{S}^{n}$
restricted to the tangent space $T_{p}\mathbb{S}^{n}$ to $\mathbb{S}^{n}$ at
$p$, viewed as a hyperplane in $\mathbb{R}^{n+1}$ passing through $p$, is the
inverse of stereographic projection from $-p$ to $T_{p}\mathbb{S}^{n}$.

For the proof of Theorem 1.1 it is essential to generalize Example 2.5. First
we recall some terminology and notation which is compatible with [16, 21, 51].
Let $V$ be an algebraic _$\mathbb{R}$ -variety_, that is, a complex algebraic
variety defined over $\mathbb{R}$ (equivalently, $V$ can be viewed as a
reduced quasiprojective scheme over $\mathbb{R}$). The set of real points of
$V$ will be denoted by $V(\mathbb{R})$. We emphasize the distinction between
algebraic $\mathbb{R}$-varieties and real algebraic varieties. An _algebraic
$\mathbb{R}$-group_ is an algebraic $\mathbb{R}$-variety $\Gamma$ endowed with
the structure of a group such that the identity element of $\Gamma$ is in
$\Gamma(\mathbb{R})$, and the group operations are regular maps defined over
$\mathbb{R}$. An algebraic $\mathbb{R}$-group is said to be _linear_ if, for
some positive integer $n$, it is isomorphic over $\mathbb{R}$ to a Zariski
closed subgroup of $\mathrm{GL}_{n}(\mathbb{C})$ defined over $\mathbb{R}$. If
$\Gamma$ is an algebraic $\mathbb{R}$-group, then $\Gamma(\mathbb{R})$ is a
real algebraic group. Moreover, if $\Gamma$ is a linear algebraic
$\mathbb{R}$-group, then $\Gamma(\mathbb{R})$ is a linear real algebraic
group. Conversely, each linear real algebraic group $G$ is up to isomorphism
of the form $\Gamma(\mathbb{R})$ for some linear algebraic $\mathbb{R}$-group
$\Gamma$. Indeed, we may assume that $G$ is a Zariski closed subgroup of
$\mathrm{GL}_{n}(\mathbb{R})$ and take as $\Gamma$ the Zariski closure of $G$
in $\mathrm{GL}_{n}(\mathbb{C})$, see [51, Lemma 2.2.4]. In view of this
discussion we can make use of Chevalley’s paper [21] in the proof of
Proposition 2.7 below.

######  Proposition 2.7.

Let $G$ be a linear real algebraic group. Then each homogeneous space $Y$ for
$G$ is a malleable variety.

###### Proof.

Let $d=\dim G$ and let $G^{0}$ be the irreducible component of the variety $G$
that contains the identity element $1$ of $G$. By a theorem of Chevalley [21,
Theorem 2] or [16, Corollary 7.12], $G^{0}$ is a unirational variety, and
hence there exist a Zariski open subset $U$ of $\mathbb{R}^{d}$ and a regular
map $\psi\colon U\to G^{0}$ such that the image $\psi(U)$ is Zariski dense in
$G^{0}$. Clearly, at some point $a\in U$ the derivative
$d_{a}\psi\colon\mathbb{R}^{d}\to T_{\psi(a)}G^{0}$ is a linear isomorphism.
Using a translation in $\mathbb{R}^{d}$ we may assume that $a=0\in U$. The map
$\varphi\colon U\to G$, $v\mapsto\psi(v)\psi(0)^{-1}$ is regular,
$\varphi(0)=1$, and the derivative of $\varphi$ at $0$ is a linear
isomorphism. As noted by János Kollár in no. A.2 of the Appendix, taking the
map

$h\colon\mathbb{R}^{d}\to\mathbb{R}^{d},\quad h(v)=\frac{cv}{1+\left\lVert
v\right\rVert^{2}},$

where $c$ is a small positive real number, we obtain a regular map
$g\coloneqq\varphi\circ h\colon\mathbb{R}^{d}\to G$ such that $g(0)=1$ and the
derivative of $g$ at $0$ is a linear isomorphism. Now we obtain a dominating
spray $(E,p,s)$ for $Y$, where $p\colon E\coloneqq Y\times\mathbb{R}^{d}\to Y$
is the product vector bundle and $s\colon E\to Y$ is defined by
$s(y,v)=g(v)\cdot y$ for all $(y,v)\in E$. ∎

In an earlier version of Proposition 2.7 we only used the map
$s(y,v)=\varphi(v)\cdot y$ defined on the Zariski open subset $Y\times U$ of
$E$. This was sufficient for all our main results, but Kollár’s observation
helped us to improve the presentation.

Prompted by some comments by Olivier Benoist, we next discuss a generalization
of Proposition 2.7.

Let $Y$ be a real algebraic variety and let $G$ be a linear real algebraic
group acting on $Y$. The $G$-space $Y$ is called _good_ if $Y$ is nonsingular
and for each point $y\in Y$ the derivative of the map $G\to Y$, $a\mapsto
a\cdot y$ at the identity element of $G$ is surjective. If $Y$ is homogeneous
for $G$, then the $G$-space $Y$ is good. The converse is not true in general.
Indeed, let $V$ be an algebraic $\mathbb{R}$-variety, with $V(\mathbb{R})$
nonempty, and let $\Gamma$ be a linear algebraic $\mathbb{R}$-group acting on
$V$. Then the real algebraic group $\Gamma(\mathbb{R})$ acts on the real
algebraic variety $V(\mathbb{R})$. Clearly, if the action of $\Gamma$ on $V$
is transitive, then the $\Gamma(\mathbb{R})$-space $V(\mathbb{R})$ is good,
but it can happen, as is well-known, that $V(\mathbb{R})$ is not homogeneous
for $\Gamma(\mathbb{R})$.

######  Proposition 2.8.

Let $G$ be a linear real algebraic group. Then each good $G$-space $Y$ is a
malleable variety.

###### Proof.

The same as that of Proposition 2.7. ∎

Proposition 2.8 is of interest in view of Theorems 4.2 and 4.3.

##  3 Malleable submersions

This section is devoted to developing techniques used in the proof of Theorem
1.1. Of independent interest are Theorem 3.9 and Corollary 3.10.

######  Notation 3.1.

Throughout the present section $X$, $Z$ are nonsingular real algebraic
varieties, and $h\colon Z\to X$ is a regular map which is surjective and
submersive. Furthermore, $V(h)$ denotes the algebraic vector subbundle of the
tangent bundle $TZ$ to $Z$ defined by

$V(h)_{z}=\operatorname{Ker}(d_{z}h\colon T_{z}Z\to T_{h(z)}X)\quad\text{for
all }z\in Z.$

Observe that $V(h)_{z}$ is the tangent space to the fiber $h^{-1}(h(z))$.

Let $U$ be an open subset of $X$. A map $f\colon U\to Z$ is called a _section_
of $h\colon Z\to X$ if $h(f(x))=x$ for all $x\in U$. A section which is a
$\mathcal{C}^{\infty}$ map is called a _$\mathcal{C}^{\infty}$ section_. We
say that a section $g\colon U\to Z$ is _regular_ if there exist a Zariski open
neighborhood $\tilde{U}\subseteq X$ of $U$ and a section
$\tilde{g}\colon\tilde{U}\to Z$ such that $\tilde{g}|_{U}=g$ and $\tilde{g}$
is a regular map. Obviously, every regular section is a $\mathcal{C}^{\infty}$
section. We say that a $\mathcal{C}^{\infty}$ section $f\colon U\to Z$ can be
_approximated by regular sections in the $\mathcal{C}^{\infty}$ topology_ if,
for every neighborhood $\mathcal{U}$ of $f$ in the space
$\mathcal{C}^{\infty}(U,Z)$ of $\mathcal{C}^{\infty}$ maps, there exists a
regular section $g\colon U\to Z$ that belongs to $\mathcal{U}$. By a _homotopy
of $\mathcal{C}^{\infty}$ sections_ we mean a continuous map $F\colon
U\times[0,1]\to Z$ such that $F_{t}\colon U\to Z$, $x\mapsto F(x,t)$ is a
$\mathcal{C}^{\infty}$ section for every $t\in[0,1]$. Two
$\mathcal{C}^{\infty}$ sections $f_{0},f_{1}\colon U\to Z$ are said to be
_homotopic through $\mathcal{C}^{\infty}$ sections_ if there exists a homotopy
$F\colon U\times[0,1]\to Z$ of $\mathcal{C}^{\infty}$ sections with
$F_{0}=f_{0}$ and $F_{1}=f_{1}$. To study approximation of
$\mathcal{C}^{\infty}$ sections by regular sections we need several notions
and auxiliary results.

######  Definition 3.2.

Let $h\colon Z\to X$ be the submersion of Notation 3.1.

  1. (i)

A _spray_ for $h\colon Z\to X$ is a triple $(E,p,s)$, where $p\colon E\to Z$
is an algebraic vector bundle over $Z$ and $s\colon E\to Z$ is a regular map
such that

$s(E_{z})\subseteq h^{-1}(h(z))\quad\text{and}\quad s(0_{z})=z\quad\text{for
all }z\in Z.$

  2. (ii)

A spray $(E,p,s)$ for $h\colon Z\to X$ is said to be _dominating_ if the
derivative

$d_{0_{z}}s\colon T_{0_{z}}E\to T_{z}Z$

maps the subspace $E_{z}=T_{0_{z}}E_{z}$ of $T_{0_{z}}E$ onto $V(h)_{z}$, that
is,

$d_{0_{z}}s(E_{z})=V(h)_{z}$

for all $z\in Z$.

  3. (iii)

The submersion $h\colon Z\to X$ is called _malleable_ if it admits a
dominating spray.

Significant examples of malleable submersions are provided by combining
Proposition 2.7 and Lemma 4.1. Note that if $X$ is reduced to a point, then
Definition 3.2 coincides with Definition 2.1. The next two lemmas are
generalizations of Lemmas 2.2 and 2.3.

######  Lemma 3.3.

If the submersion $h\colon Z\to X$ is malleable, then it admits a dominating
spray $(E,p,s)$ such that $p\colon E\to Z$ is an algebraically trivial vector
bundle.

###### Proof.

Suppose that $(\tilde{E},\tilde{p},\tilde{s})$ is a dominating spray for
$h\colon Z\to X$. Choose an algebraic vector bundle
$\tilde{p}^{\prime}\colon\tilde{E}^{\prime}\to Z$ such that the direct sum
$p\colon E\coloneqq\tilde{E}\oplus\tilde{E}^{\prime}\to Z$ is an algebraically
trivial vector bundle. Defining $s\colon E\to Z$ by
$s(v,v^{\prime})=\tilde{s}(v)$, we get a dominating spray $(E,p,s)$ for
$h\colon Z\to X$. ∎

######  Lemma 3.4.

Suppose that $(E,p,s)$ is a dominating spray for the submersion $h\colon Z\to
X$. Then there exists a dominating spray $(\hat{E},\hat{p},,\hat{s})$ for
$h\colon Z\to X$ such that $\hat{p}\colon\hat{E}\to Z$ is an algebraic vector
subbundle of $p\colon E\to Z$, $\hat{s}=s|_{\hat{E}}$, and the restriction

$d_{0_{z}}\hat{s}|_{\hat{E}_{z}}\colon\hat{E}_{z}\to V(h)_{z}$

of the derivative $d_{0_{z}}\hat{s}\colon T_{0_{z}}\hat{E}\to T_{z}Z$ is a
linear isomorphism for all $z\in Z$.

###### Proof.

Let $\alpha\colon E\to V(h)$ be a morphism of algebraic vector bundles over
$Z$ defined by

$\alpha(v)=d_{0_{p(v)}}s(v)\quad\text{for all }v\in E.$

Since $\alpha$ is a surjective morphism, its kernel $\operatorname{Ker}\alpha$
is an algebraic vector subbundle of $E$. It follows that
$E=\hat{E}\oplus\operatorname{Ker}\alpha$ is the direct sum for some algebraic
vector subbundle $\hat{p}\colon\hat{E}\to Z$ of $p\colon E\to Z$. This
completes the proof. ∎

######  Notation 3.5.

Suppose that $(E,p,s)$ is a dominating spray for the submersion $h\colon Z\to
X$. Let $U$ be an open subset of $X$ and let $f\colon U\to Z$ be a
$\mathcal{C}^{\infty}$ section of $h\colon Z\to X$. Denote by $p_{f}\colon
E_{f}\to U$ the pullback of the vector bundle $p\colon E\to Z$ under the map
$f\colon U\to Z$. Recall that $p_{f}\colon E_{f}\to U$ is a
$\mathcal{C}^{\infty}$ vector bundle over $U$, where

$E_{f}\coloneqq\\{(x,v)\in U\times E:f(x)=p(v)\\},\quad p_{f}(x,v)=x.$

We define

$s_{f}\colon E_{f}\to Z,\quad s_{f}(x,v)=s(v).$

######  Lemma 3.6.

Using Notation 3.5, assume that

$d_{0_{z}}s|_{E_{z}}\colon E_{z}\to V(h)_{z}$

is a linear isomorphism for all $z\in Z$. Then $s_{f}\colon E_{f}\to Z$ maps
diffeomorphically an open neighborhood of the zero section $Z(E_{f})$ in
$E_{f}$ onto an open neighborhood of $f(U)$ in $Z$.

###### Proof.

Let $x\in U$. The zero vector in the fiber $(E_{f})_{x}$ is $(x,0_{f(x)})$,
where $0_{f(x)}$ is the zero vector in the fiber $E_{f(x)}$. Since
$s_{f}(x,0_{f(x)})=s(0_{f(x)})=f(x)$, it follows that $s_{f}$ induces a
diffeomorphism between $Z(E_{f})$ and $f(U)$. Moreover, the derivative

$d_{(x,0_{f(x)})}s_{f}\colon T_{(x,0_{f(x)})}E_{f}\to T_{f(x)}Z$

is an isomorphism because

$d_{0_{z}}s|_{E_{z}}\colon E_{z}\to V(h)_{z}$

is an isomorphism for all $z\in Z$. Consequently, $s_{f}$ is a local
diffeomorphism at the point $(x,0_{f(x)})$. Now the lemma follows from [20,
(12.7)]. ∎

######  Lemma 3.7.

Suppose that $(E,p,s)$ is a dominating spray for the submersion $h\colon Z\to
X$. Let $U$ be an open subset of $X$ and let $F\colon U\times[0,1]\to Z$ be a
homotopy of $\mathcal{C}^{\infty}$ sections of $h\colon Z\to X$. Let $U_{0}$
be an open subset of $X$ whose closure $\overline{U_{0}}$ is compact and
contained in $U$. Let $t_{0}$ be a point in $[0,1]$. Then there exist a
neighborhood $I_{0}$ of $t_{0}$ in $[0,1]$ and a continuous map $\xi\colon
U_{0}\times I_{0}\to E$ such that

  1. (3.7.1)

$p(\xi(x,t))=F(x,t_{0})$ for all $(x,t)\in U_{0}\times I_{0}$,

  2. (3.7.2)

$\xi(x,t_{0})=0_{F(x,t_{0})}$ for all $x\in U_{0}$,

  3. (3.7.3)

$s(\xi(x,t))=F(x,t)$ for all $(x,t)\in U_{0}\times I_{0}$,

  4. (3.7.4)

for every $t\in I_{0}$ the map $U_{0}\to E^{0}$, $x\mapsto\xi(x,t)$ is of
class $\mathcal{C}^{\infty}$.

###### Proof.

By Lemma 3.4, we may assume without loss of generality that

$d_{0_{z}}s|_{E_{z}}\colon E_{z}\to V(h)_{z}$

is an isomorphism for all $z\in Z$. Defining $f\colon U\to Z$ by
$f(x)=F(x,t_{0})$, in view of Lemma 3.6, there exist an open neighborhood
$M\subseteq E_{f}$ of the zero section $Z(E_{f})$ and an open neighborhood
$N\subseteq Z$ of $f(U)$ such that the restriction $\sigma\colon M\to N$ of
$s_{f}\colon E_{f}\to Z$ is a diffeomorphism. Since $\overline{U_{0}}$ is a
compact subset of $U$, we can choose an open neighborhood $I_{0}$ of $t_{0}$
in $[0,1]$ such that $F_{t}(U_{0})\subseteq N$ for all $t\in I_{0}$.
Therefore, for each $t\in I_{0}$, there exists a unique $\mathcal{C}^{\infty}$
map $\zeta_{t}\colon U_{0}\to E_{f}$ satisfying $\zeta_{t}(U_{0})\subseteq M$
and $F_{t}(x)=\sigma(\zeta_{t}(x))$ for all $x\in U_{0}$. Writing
$\zeta_{t}(x)$ as $\zeta_{t}(x)=(\alpha_{t}(x),\xi_{t}(x))$, where
$\alpha_{t}\colon U_{0}\to U$ and $\xi_{t}\colon U_{0}\to E$ are
$\mathcal{C}^{\infty}$ maps, we get

$f(\alpha_{t}(x))=p(\xi_{t}(x))\quad\text{and}\quad s(\xi_{t}(x))=F_{t}(x).$

By Definition 3.2(i), $s(\xi_{t}(x))\in h^{-1}(h(p(\xi_{t}(x))))$, and hence

$h(s(\xi_{t}(x)))=h(f(\alpha_{t}(x)))=\alpha_{t}(x).$

On the other hand,

$h(s(\xi_{t}(x)))=h(F_{t}(x))=x.$

Consequently, $\alpha_{t}(x)=x$. It follows that $\zeta_{t}\colon U_{0}\to
E_{f}$ is a $\mathcal{C}^{\infty}$ section, over $U_{0}$, of the vector bundle
$p_{f}\colon E_{f}\to U$. Clearly, $\zeta_{t_{0}}(U_{0})\subseteq Z(E_{f})$.
Furthermore, the map

$\zeta\colon U_{0}\times I_{0}\to E_{f},\quad(x,t)\mapsto\zeta_{t}(x)$

is continuous. Note that $\zeta(x,t)=(x,\xi(x,t))$, where

$\xi\colon U_{0}\times I_{0}\to E,\quad(x,t)\mapsto\xi_{t}(x)$

is a continuous map with $p(\xi(x,t))=f(x)=F(x,t_{0})$ for all $(x,t)\in
U_{0}\times I_{0}$. By construction, the map $\xi$ satisfies conditions
(3.7.1)–(3.7.4). ∎

######  Lemma 3.8.

Assume that the submersion $h\colon Z\to X$ is malleable. Let $U$ be an open
subset of $X$ and let $F\colon U\times[0,1]\to Z$ be a homotopy of
$\mathcal{C}^{\infty}$ sections of $h\colon Z\to X$. Let $U_{0}$ be an open
subset of $X$ whose closure $\overline{U_{0}}$ is compact and contained in
$U$. Then there exist a dominating spray $(E,p,s)$ for $h\colon Z\to X$ and a
continuous map $\xi\colon U_{0}\times[0,1]\to E$ such that $p\colon
E=Z\times\mathbb{R}^{n}\to Z$ is the product vector bundle and
$\xi(x,t)=(F(x,0),\eta(x,t))$ for all $(x,t)\in U_{0}\times[0,1]$, where the
map $\eta\colon U_{0}\times[0,1]\to\mathbb{R}^{n}$ satisfies

  1. (3.8.1)

$\eta(x,0)=0$ for all $x\in U_{0}$,

  2. (3.8.2)

$s(F(x,0),\eta(x,t))=F(x,t)$ for all $(x,t)\in U_{0}\times[0,1]$,

  3. (3.8.3)

for every $t\in[0,1]$ the map $U_{0}\to\mathbb{R}^{n}$, $x\mapsto\eta(x,t)$ is
of class $\mathcal{C}^{\infty}$.

###### Proof.

By Lemma 3.3, the submersion $h\colon Z\to X$ admits a dominating spray
$(\tilde{E},\tilde{p},\tilde{s})$ such that
$\tilde{p}\colon\tilde{E}=Z\times\mathbb{R}^{m}\to Z$ is the product vector
bundle. In view of Lemma 3.7 and the compactness of the interval $[0,1]$ (see
the Lebesgue lemma for compact metric spaces [19, p. 28, Lemma 9.11]), there
exists a partition $0=t_{0}<t_{1}<\cdots<t_{k}=1$ of $[0,1]$ such that for
each $i=1,\ldots,k$ there exists a continuous map $\xi^{i}\colon
U_{0}\times[t_{i-1},t_{i}]\to\tilde{E}$ with the following properties:

  * •

$\xi^{i}(x,t)=(F(x,t_{i-1}),\eta^{i}(x,t))$ for all $(x,t)\in
U_{0}\times[t_{i-1},t_{i}]$,

  * •

$\eta^{i}(x,t_{i-1})=0$ for all $x\in U_{0}$,

  * •

$\tilde{s}(F(x,t_{i-1}),\eta^{i}(x,t))=F(x,t)$ for all $(x,t)\in
U_{0}\times[t_{i-1},t_{i}]$,

  * •

for every $t\in[t_{i-1},t_{i}]$ the map $U_{0}\to\mathbb{R}^{m}$,
$x\mapsto\eta^{i}(x,t)$ is of class $\mathcal{C}^{\infty}$.

For $i=1,\ldots,k$ we define recursively a dominating spray
$(E(i),p^{(i)},s^{(i)})$ for $h\colon Z\to X$ by

$(E(i),p^{(i)},s^{(i)})=(\tilde{E},\tilde{p},\tilde{s})\quad\text{if }i=1,$

while for $i\geq 2$ we require

$p^{(i)}\colon E(i)=Z\times(\mathbb{R}^{m})^{i}\to Z$

to be the product vector bundle and set

$s^{(i)}\colon E(i)\to Z,\quad
s^{(i)}(z,v_{1},\ldots,v_{i})=s^{(1)}(s^{(i-1)}(z,v_{1},\ldots,v_{i-1}),v_{i}),$

where $z\in Z$ and $v_{1},\ldots,v_{i}\in\mathbb{R}^{m}$.

In particular, $(E,p,s)\coloneqq(E(k),p^{(k)},s^{(k)})$ is a dominating spray
for $h\colon Z\to X$. Note that $p\colon E=Z\times\mathbb{R}^{n}\to Z$ is the
product vector bundle with $\mathbb{R}^{n}=(\mathbb{R}^{m})^{k}$. Now,
consider a map

$\xi\colon U_{0}\times[0,1]\to E,\quad\xi(x,t)=(F(x,0),\eta(x,t)),$

where $\eta\colon U_{0}\times[0,1]\to\mathbb{R}^{n}=(\mathbb{R}^{m})^{k}$ is
defined by

$\eta(x,t)=(\eta^{1}(x,t),0,\ldots,0)$

for all $(x,t)\in U_{0}\times[t_{0},t_{1}]$, and

$\eta(x,t)=(\eta^{1}(x,t_{1}),\ldots,\eta^{i-1}(x,t_{i-1}),\eta^{i}(x,t),0,\ldots,0)$

for all $(x,t)\in U_{0}\times[t_{i-1},t_{i}]$ with $i=2,\ldots,k$. One readily
checks that $\eta$ is a well-defined continuous map satisfying
(3.8.1)–(3.8.3). ∎

The main result of this section is the following.

######  Theorem 3.9.

Assume that the submersion $h\colon Z\to X$ is malleable. Let $U$ be an open
subset of $X$ and let $f\colon U\to Z$ be a $\mathcal{C}^{\infty}$ section of
$h\colon Z\to X$ that is homotopic through $\mathcal{C}^{\infty}$ sections to
a regular section. Then $f$ can be approximated by regular sections of
$h\colon Z\to X$ in the $\mathcal{C}^{\infty}$ topology.

###### Proof.

Let $F\colon U\times[0,1]\to Z$ be a homotopy of $\mathcal{C}^{\infty}$
sections such that $F_{0}$ is a regular section and $F_{1}=f$. Let $U_{0}$ be
an open subset of $X$ such that its closure $\overline{U_{0}}$ is compact and
contained in $U$, and let $(E,p,s)$, $\xi\colon U_{0}\times[0,1]\to E$,
$\xi(x,t)=(F(x,0),\eta(x,t))$, ${\eta\colon
U_{0}\times[0,1]\to\mathbb{R}^{n}}$ be as in Lemma 3.8. In particular, we have

$s(F(x,0),\eta(x,1))=F(x,1)=f(x)\quad\text{for all }x\in U_{0}.$

By the Weierstrass approximation theorem, there exists a regular map
$\beta\colon X\to\mathbb{R}^{n}$ such that the restriction $\beta|_{U_{0}}$ is
arbitrarily close to the $\mathcal{C}^{\infty}$ map $\eta_{1}\colon
U_{0}\to\mathbb{R}^{n}$, $x\mapsto\eta(x,1)$ in the $\mathcal{C}^{\infty}$
topology. Then

$g\colon U\to Z,\quad x\mapsto s(F(x,0),\beta(x))$

is a regular map such that $g|_{U_{0}}$ is close to $f|_{U_{0}}$ in the
$\mathcal{C}^{\infty}$ topology. Moreover, in view of Definition 3.2(i),
$g\colon U\to Z$ is a section of $h\colon Z\to X$. The proof is complete since
$U_{0}$ is chosen in an arbitrary way. ∎

It is worthwhile to point out the following special case of Theorem 3.9.

######  Corollary 3.10.

Assume that the submersion $h\colon Z\to X$ is malleable. Let $f\colon X\to Z$
be a $\mathcal{C}^{\infty}$ section of $h\colon Z\to X$ that is homotopic
through $\mathcal{C}^{\infty}$ sections to a regular section. Then $f$ can be
approximated by regular sections of $h\colon Z\to X$ in the
$\mathcal{C}^{\infty}$ topology. ∎

##  4 Proofs of Theorem 1.1 and related results

To begin with we discuss approximation of maps with values in a malleable real
algebraic variety. In order to make use of Theorem 3.9 or Corollary 3.10 we
need the following observation.

######  Lemma 4.1.

Let $X,Y$ be nonsingular real algebraic varieties. Assume that the variety $Y$
is malleable. Then the canonical projection

$h\colon X\times Y\to X,\quad(x,y)\mapsto x$

is a malleable submersion.

###### Proof.

Let $(E,p,s)$ be a dominating spray for $Y$. We obtain a dominating spray
$(\tilde{E},\tilde{p},\tilde{s})$ for $h\colon X\times Y\to X$ setting

$\displaystyle\tilde{E}=\\{((x,y),v)\in(X\times Y)\times E:y=p(v)\\},$
$\displaystyle\tilde{p}\colon\tilde{E}\to X\times
Y,\quad((x,y),v)\mapsto(x,y),$ $\displaystyle\tilde{s}\colon\tilde{E}\to
X\times Y,\quad((x,y),v)\mapsto(x,s(v)).\qed$

######  Theorem 4.2.

Let $X$ be a nonsingular real algebraic variety and let $Y$ be a malleable
nonsingular real algebraic variety. Then, for a $\mathcal{C}^{\infty}$ map
$f\colon X\to Y$, the following conditions are equivalent:

  1. (a)

$f$ can be approximated by regular maps in the $\mathcal{C}^{\infty}$
topology.

  2. (b)

$f$ is homotopic to a regular map.

###### Proof.

It suffices to prove (b)$\Rightarrow$(a). To this end let $\Phi\colon
X\times[0,1]\to Y$ be a homotopy such that $\Phi_{0}$ is a regular map and
$\Phi_{1}=f$. We may assume that $\Phi$ is a $\mathcal{C}^{\infty}$ map, see
[43, Proposition 10.22]. By Lemma 4.1, the canonical projection $h\colon
X\times Y\to X$ is a malleable submersion. Clearly,

$F\colon X\times[0,1]\to X\times Y,\quad(x,t)\mapsto(x,\Phi(x,t))$

is a homotopy of $\mathcal{C}^{\infty}$ sections of $h\colon X\times Y\to X$.
Therefore, according to Corollary 3.10, the $\mathcal{C}^{\infty}$ section

$X\to X\times Y,\quad x\mapsto(x,f(x))$

can be approximated by regular sections in the $\mathcal{C}^{\infty}$
topology, which implies that (a) holds. ∎

The following variant of Theorem 4.2 can be derived from Lemma 3.8.

######  Theorem 4.3.

Let $X$ be a real algebraic variety (possibly singular) and let $Y$ be a
malleable nonsingular real algebraic variety. Then, for a continuous map
$f\colon X\to Y$, the following conditions are equivalent:

  1. (a)

$f$ can be approximated by regular maps in the $\mathcal{C}^{0}$ topology.

  2. (b)

$f$ is homotopic to a regular map.

###### Proof.

It suffices to prove (b)$\Rightarrow$(a). Suppose that (b) holds, and let
$\gamma\colon X\to Y$ be a regular map homotopic to $f$. We may assume that
$X,Y$ are Zariski closed subsets of $\mathbb{R}^{k},\mathbb{R}^{l}$,
respectively. Then there exists a Zariski open neighborhood
$\Omega\subseteq\mathbb{R}^{k}$ of $X$ and a regular map
$\tilde{\gamma}\colon\Omega\to\mathbb{R}^{l}$ with
$\tilde{\gamma}|_{X}=\gamma$, see [3, p. 62]. By the Tietze extension theorem,
there exists a continuous map $\tilde{f}\colon\Omega\to\mathbb{R}^{l}$ with
$\tilde{f}|_{X}=f$. Now, let $\rho\colon T\to Y$ be a $\mathcal{C}^{\infty}$
tubular neighborhood of $Y$ in $\mathbb{R}^{l}$, where $T$ is an open
neighborhood of $Y$ in $\mathbb{R}^{l}$ and $\rho$ is a $\mathcal{C}^{\infty}$
retraction. Choose an open neighborhood $U\subseteq\Omega$ of $X$ such that
$\tilde{f}(U)\subseteq T$ and $\tilde{\gamma}(U)\subseteq T$. By [29, p. 44,
Theorem 2.2], we can also choose a $\mathcal{C}^{\infty}$ map $U\to T$
arbitrarily close to $\tilde{f}|_{U}\colon U\to T$ in the strong
$\mathcal{C}^{0}$ topology; such a map is homotopic to $\tilde{f}|_{U}\colon
U\to T$. Therefore, for the proof of (a), we may assume that the map
$\tilde{f}|_{U}\colon U\to T$ is of class $\mathcal{C}^{\infty}$. Clearly, the
maps $\varphi,\psi\colon U\to Y$, defined by

$\varphi(x)=\rho(\tilde{\gamma}(x)),\quad\psi(x)=\rho(\tilde{f}(x))\quad\text{for
all }x\in U,$

are of class $\mathcal{C}^{\infty}$ and satisfy $\varphi|_{X}=\gamma$,
$\psi|_{X}=f$. Shrinking $U$, the variety $X$ is a continuous retract of $U$
($X$ is locally contractible [3, Theorem 9.3.6], so it is a Euclidean
neighborhood retract by Borsuk’s theorem [19, p. 537, Theorem E.3]).
Consequently, the $\mathcal{C}^{\infty}$ maps $\varphi,\psi$ are homotopic,
and hence there exists a $\mathcal{C}^{\infty}$ homotopy $\Phi\colon
U\times[0,1]\to Y$ with $\Phi_{0}=\varphi$ and $\Phi_{1}=\psi$ [43,
Proposition 10.22]. Let $U_{0}\subseteq\mathbb{R}^{k}$ be an open subset whose
closure $\overline{U_{0}}$ is compact and contained in $U$.

Consider the canonical projection $h\colon\mathbb{R}^{k}\times
Y\to\mathbb{R}^{k}$ and the homotopy of its $\mathcal{C}^{\infty}$ sections

$F\colon U\to\mathbb{R}^{k}\times Y,\quad x\mapsto(x,\Phi(x,t)).$

According to Lemma 4.1, $h\colon\mathbb{R}^{k}\times Y\to\mathbb{R}^{k}$ is a
malleable submersion. Hence, by Lemma 3.8, there exist a dominating spray
$(E,p,s)$ for $h\colon\mathbb{R}^{k}\times Y\to\mathbb{R}^{k}$ and a
continuous map ${\xi\colon U_{0}\times[0,1]\to E}$ such that $p\colon
E=(\mathbb{R}^{k}\times Y)\times\mathbb{R}^{n}\to\mathbb{R}^{k}\times Y$ is
the product vector bundle and $\xi(x,t)=(F(x,0),\eta(x,t))$, where the
continuous map $\eta\colon U_{0}\times[0,1]\to\mathbb{R}^{n}$ satisfies

$s(F(x,0),\eta(x,t))=F(x,t)\quad\text{for all }(x,t)\in U_{0}\times[0,1].$

In particular,

$s(F(x,0),\eta(x,1))=F(x,1)=(x,\psi(x))\quad\text{for all }x\in U_{0}.$

By the Weierstrass approximation theorem, there exists a regular map
$\beta\colon\mathbb{R}^{k}\to\mathbb{R}^{n}$ such that the restriction
$\beta|_{U_{0}}$ is arbitrarily close to the $\mathcal{C}^{\infty}$ map
$\eta_{1}\colon U_{0}\to\mathbb{R}^{n}$, $x\mapsto\eta(x,1)$ in the
$\mathcal{C}^{0}$ topology. Then

$\alpha\colon U\to\mathbb{R}^{k}\times Y,\quad x\mapsto s(F(x,0),\beta(x))$

is a $\mathcal{C}^{\infty}$ section of $h\colon\mathbb{R}^{k}\times
Y\to\mathbb{R}^{k}$ such that $\alpha|_{U_{0}}$ is close in the
$\mathcal{C}^{0}$ topology to the section

$U_{0}\to\mathbb{R}^{k}\times Y,\quad x\mapsto(x,\psi(x)).$

We have $\alpha(x)=(x,\theta(x))$, where $\theta\colon U\to Y$ is a
$\mathcal{C}^{\infty}$ map. Since $F(x,0)=(x,\varphi(x))$ and
$\varphi|_{X}=\gamma$ is a regular map, the restriction
$g\coloneqq\theta|_{X}\colon X\to Y$ is a regular map. By construction,
$g|_{X\cap U_{0}}$ is close to the continuous map $\psi|_{X\cap
U_{0}}=f|_{X\cap U_{0}}$. The proof is complete since $U_{0}$ is chosen in an
arbitrary way. ∎

Now our main theorem follows immediately.

###### Proof of Theorem 1.1.

By Proposition 2.7, the variety $Y$ is malleable. Hence the conclusion follows
from Theorems 4.2 and 4.3. ∎

Theorem 1.1 holds, in particular, for maps with values in an arbitrary linear
real algebraic group $G$. The assumption that $G$ is linear cannot be omitted.
Indeed, each linear real algebraic group is up to isomorphism of the form
$\Gamma(\mathbb{R})$ for some linear algebraic $\mathbb{R}$-group $\Gamma$
(see Section 2). In the category of algebraic $\mathbb{R}$-groups the linear
ones can be characterized as follows.

######  Theorem 4.4.

Let $k$ be a positive integer and let $\Gamma$ be an irreducible algebraic
$\mathbb{R}$-group. Then the following conditions are equivalent:

  1. (a)

$\Gamma$ is a linear algebraic $\mathbb{R}$-group.

  2. (b)

Every continuous null homotopic map from $\mathbb{S}^{k}$ into
$\Gamma(\mathbb{R})$ can be approximated by regular maps in the
$\mathcal{C}^{0}$ topology.

###### Proof.

By Theorem 1.1, (a) implies (b). To prove the reversed implication suppose
that (a) does not hold. Then, by Chevalley’s theorem [22] (see [23] for a
modern treatment), there exists a surjective morphism of algebraic
$\mathbb{R}$-groups $\varphi\colon\Gamma\to A$, where $A$ is an Abelian
$\mathbb{R}$-variety, $\dim A\geq 1$. Since $\varphi$ is defined over
$\mathbb{R}$, its restriction $\varphi(\mathbb{R})\colon\Gamma(\mathbb{R})\to
A(\mathbb{R})$ is a regular map of real algebraic varieties. The image
$\varphi(\mathbb{R})(\Gamma(\mathbb{R}))$ is Zariski dense in $A(\mathbb{R})$,
the map $\varphi$ being surjective. Hence there exists a point
$a\in\Gamma(\mathbb{R})$ at which the derivative
$d_{a}\varphi(\mathbb{R})\colon T_{a}\Gamma(\mathbb{R})\to
T_{\varphi(\mathbb{R})(a)}A(\mathbb{R})$ is surjective. Therefore, in view of
the rank theorem for $\mathcal{C}^{\infty}$ maps, we can find a continuous
null homotopic map $f\colon\mathbb{S}^{k}\to\Gamma(\mathbb{R})$ such that the
composite map $\varphi(\mathbb{R})\circ f\colon\mathbb{S}^{k}\to
A(\mathbb{R})$ is not constant. It follows that $\varphi(\mathbb{R})\circ f$
cannot be approximated by regular maps in the $\mathcal{C}^{0}$ topology
because each regular map from $\mathbb{S}^{k}$ into $A(\mathbb{R})$ is
constant, see [46, Corollary 3.9]. Consequently, (b) does not hold. In other
words, (b) implies (a). ∎

##  5 Further results on regular maps into unit spheres

The first result of this section can be viewed as a generalization of Theorem
1.6.

######  Theorem 5.1.

Let $X$ be a compact connected oriented nonsingular real algebraic variety of
dimension $n$. Then the set of regular maps $\mathcal{R}(X,\mathbb{S}^{n})$ is
dense in the space of $\mathcal{C}^{\infty}$ maps
$\mathcal{C}^{\infty}(X,\mathbb{S}^{n})$ if and only if there exists a regular
map from $X$ into $\mathbb{S}^{n}$ of topological degree $1$.

###### Proof.

Suppose that $f\colon X\to\mathbb{S}^{n}$ is a regular map with $\deg(f)=1$.
By [5, Corollary 4.2], for every integer $d$ there exists a regular map
$\varphi_{d}\colon\mathbb{S}^{n}\to\mathbb{S}^{n}$ with $\deg(\varphi_{d})=d$.
Since the composite map $\varphi_{d}\circ f$ is regular and
$\deg(\varphi_{d}\circ f)=d$, it follows from Hopf’s theorem that each
$\mathcal{C}^{\infty}$ map from $X$ into $\mathbb{S}^{n}$ is homotopic to a
regular map. Therefore the set $\mathcal{R}(X,\mathbb{S}^{n})$ is dense in the
space $\mathcal{C}^{\infty}(X,\mathbb{S}^{n})$ by Corollary 1.5. The converse
is obvious. ∎

Here is an illuminating example, which itself is a generalization of Theorem
1.6.

######  Example 5.2.

Let $n,k$ be two positive integers and let $\mathbb{S}^{n}_{2k}$ be the Fermat
$n$-sphere of degree $2k$,

$\mathbb{S}^{n}_{2k}=\\{(x_{0},\ldots,x_{n})\in\mathbb{R}^{n+1}:x_{0}^{2k}+\cdots+x_{n}^{2k}=1\\}.$

Clearly, $\mathbb{S}^{n}_{2k}$ is a nonsingular real algebraic variety
diffeomorphic to the unit $n$-sphere ${\mathbb{S}^{n}=\mathbb{S}^{n}_{2}}$. If
$k$ is odd, then the set of regular maps
$\mathcal{R}(\mathbb{S}^{n}_{2k},\mathbb{S}^{n})$ is dense in the space of
$\mathcal{C}^{\infty}$ maps
$\mathcal{C}^{\infty}(\mathbb{S}^{n}_{2k},\mathbb{S}^{n})$. Indeed, the
regular map

$f\colon\mathbb{S}^{n}_{2k}\to\mathbb{S}^{n},\quad(x_{0},\ldots,x_{n})\mapsto(x_{0}^{k},\ldots,x_{n}^{k})$

is a homeomorphism, and hence $\deg(f)=1$ if $\mathbb{S}^{n}_{2k}$ is oriented
in the standard way. Therefore the assertion follows from Theorem 5.1.∎

The following is a variant of Theorem 5.1 for nonorientable varieties.

######  Theorem 5.3.

Let $X$ be a compact connected nonorientable nonsingular real algebraic
variety of dimension $n$. Then either

  1. (i)

the set $\mathcal{R}(X,\mathbb{S}^{n})$ is dense in the space
$\mathcal{C}^{\infty}(X,\mathbb{S}^{n})$, or

  2. (ii)

the closure of $\mathcal{R}(X,\mathbb{S}^{n})$ in
$\mathcal{C}^{\infty}(X,\mathbb{S}^{n})$ coincides with the set of all
$\mathcal{C}^{\infty}$ null homotopic maps from $X$ into $\mathbb{S}^{n}$.

###### Proof.

By Hopf’s theorem, there are exactly two homotopic classes of
$\mathcal{C}^{\infty}$ maps from $X$ into $\mathbb{S}^{n}$. One of them is
represented by a constant map (which obviously is a regular map). Therefore
the proof is complete in view of Corollary 1.5. ∎

Next we give a relevant example.

######  Example 5.4.

Let $k$ be a positive integer and let $X_{k}$ be the blowup of the $2$-sphere
$\mathbb{S}^{2}$ at $k$ points. Clearly, $X_{k}$ is a compact connected
nonsingular real algebraic surface, which is a $\mathcal{C}^{\infty}$
nonorientable surface of genus $k$. According to [4, Theorem 1.7], the set
$\mathcal{R}(X_{k},\mathbb{S}^{2})$ is dense in
$\mathcal{C}^{\infty}(X_{k},\mathbb{S}^{2})$ for all $k\geq 1$. Furthermore,
if $k$ is odd, then for every algebraic model $X$ of $X_{k}$ the set
$\mathcal{R}(X,\mathbb{S}^{2})$ is dense in
$\mathcal{C}^{\infty}(X,\mathbb{S}^{2})$ by [6, Theorem 2]. However, if $k$ is
even, then there exists an algebraic model $Y$ of $X_{k}$ such that the
closure of $\mathcal{R}(Y,\mathbb{S}^{2})$ in
$\mathcal{C}^{\infty}(Y,\mathbb{S}^{2})$ consists precisely of all
$\mathcal{C}^{\infty}$ null homotopic maps from Y into $\mathbb{S}^{2}$, see
[6, Theorem 3.3].

It is convenient to bring into play the homotopy groups. Let $n$ be a positive
integer. As a base point in the unit $n$-sphere $\mathbb{S}^{n}$ we choose
$s_{n}=(1,0,\ldots,0)$. For any given real algebraic variety $Y$ with base
point $y_{0}\in Y$, let $\pi^{\mathrm{alg}}_{n}(Y,y_{0})$ denote the subset of
the $n$th homotopy group $\pi_{n}(Y,y_{0})$ comprising the homotopy classes
represented by regular maps from $\mathbb{S}^{n}$ into $Y$ that preserve the
base points. We write $\pi^{\mathrm{alg}}_{n}(\mathbb{S}^{p})$,
$\pi_{n}(\mathbb{S}^{p})$ instead of
$\pi^{\mathrm{alg}}_{n}(\mathbb{S}^{p},s_{p})$,
$\pi_{n}(\mathbb{S}^{p},s_{p})$, respectively. It is an open problem whether
$\pi^{\mathrm{alg}}_{n}(\mathbb{S}^{p})$ is a subgroup of
$\pi_{n}(\mathbb{S}^{p})$ for all pairs of positive integers $(n,p)$, see [3,
p. 366], [50] and Proposition 5.5 for partial results.

Theorem 1.1 implies the following.

######  Proposition 5.5.

Let $Y$ be a homogeneous space for a linear real algebraic group $G$ and let
$y_{0}$ be a point in $Y$. Assume that
$\pi^{\mathrm{alg}}_{n}(Y,y_{0})=\pi_{n}(Y,y_{0})$ for some positive integer
$n$. Then the set of regular maps $\mathcal{R}(\mathbb{S}^{n},Y)$ is dense in
the space of $\mathcal{C}^{\infty}$ maps
$\mathcal{C}^{\infty}(\mathbb{S}^{n},Y)$.

###### Proof.

Let $f\colon\mathbb{S}^{n}\to Y$ be a $\mathcal{C}^{\infty}$ map. Choose an
element $a\in G$ such that $a\cdot f(s_{n})=y_{0}$. Since
$\pi^{\mathrm{alg}}_{n}(Y,y_{0})=\pi_{n}(Y,y_{0})$, the $\mathcal{C}^{\infty}$
map $g\colon\mathbb{S}^{n}\to Y$, $x\mapsto a\cdot f(x)$ is homotopic to a
regular map, and hence, by Theorem 1.1, it can be approximated by regular maps
in the $\mathcal{C}^{\infty}$ topology. Consequently, the map $f$ can be
approximated by regular maps in the $\mathcal{C}^{\infty}$ topology because
$f(x)=a^{-1}\cdot g(x)$ for all $x\in\mathbb{S}^{n}$. ∎

In particular, for every pair $(n,p)$ of positive integers, the set
$\mathcal{R}(\mathbb{S}^{n},\mathbb{S}^{p})$ is dense in the space of
$\mathcal{C}^{\infty}$ maps
$\mathcal{C}^{\infty}(\mathbb{S}^{n},\mathbb{S}^{p})$ if and only if
$\pi^{\mathrm{alg}}_{n}(\mathbb{S}^{p})=\pi_{n}(\mathbb{S}^{p})$. The
following result is another generalization of Theorem 1.6 and provides and
additional support for Conjecture I.

######  Theorem 5.6.

Let $(n,p)$ be a pair of positive integers. Then the set of regular maps
$\mathcal{R}(\mathbb{S}^{n},\mathbb{S}^{p})$ is dense in the space of
$\mathcal{C}^{\infty}$ maps
$\mathcal{C}^{\infty}(\mathbb{S}^{n},\mathbb{S}^{p})$ in each of the following
five cases:

  1. (i)

$p=1,2$ or $4$.

  2. (ii)

$n-p\leq 3$.

  3. (iii)

$4\leq n-p\leq 5$ with possible exception for the pairs $(9,5)$, $(7,3)$,
$(11,6)$, $(10,5)$ and $(8,3)$.

  4. (iv)

The homotopy group $\pi_{n}(\mathbb{S}^{p})$ is finite cyclic of odd order,
and $p$ is odd with $n\leq 2p-2$.

  5. (v)

$n=p+13$, where $p$ is odd and $p\geq 15$.

###### Proof.

(i) This is proved in [4, Theorem 1.1] and has already been mentioned in
Section 1.

(ii) The case $n=p$ is contained in Theorem 1.6. For $1\leq n-p\leq 3$, one
has $\pi^{\mathrm{alg}}_{n}(\mathbb{S}^{p})=\pi_{n}(\mathbb{S}^{p})$ by [50,
Corollaries 1.2, 1.3 and 1.4]. The case $n<p$ is trivial.

(iii) One has $\pi_{p+4}(\mathbb{S}^{p})=0$ for $p\geq 6$ and
$\pi_{p+5}(\mathbb{S}^{p})=0$ for $p\geq 7$ (see [30, pp. 331, 332]), which
together with (i) completes the proof.

(iv) In this case,
$\pi^{\mathrm{alg}}_{n}(\mathbb{S}^{p})=\pi_{n}(\mathbb{S}^{p})$ by [5,
Theorem 2.1 and p. 163].

(v) One has $\pi_{p+13}(\mathbb{S}^{p})=\mathbb{Z}/3$ if $p\geq 15$ (see [53,
p. 188]), so the assertion follows from (iv). ∎

Recall a basic notion from algebraic topology. An _H-space_ is a pointed
topological space $X$ with base point $e$, together with a continuous map
(H-space multiplication) ${\mu\colon X\times X\to X}$ such that $\mu(e,e)=e$,
and the maps $X\to X$ defined by

$x\mapsto\mu(x,e)\quad\text{and}\quad x\mapsto\mu(e,x)$

are homotopic to the identity map of $X$ through homotopies that keep the base
point $e$ fixed. For every positive integer $n$, the group operation in the
homotopy group $\pi_{n}(X)\coloneqq\pi_{n}(X,e)$ is induced by the H-space
multiplication $\mu$, see [19, p. 443, Theorem 4.1].

For our purpose relevant examples of H-spaces are real algebraic groups (in
particular, the unit sphere $\mathbb{S}^{3}$ with the multiplication of the
quaternions of norm $1$) and the unit sphere $\mathbb{S}^{7}$, the latter with
the multiplication of the octonions of norm $1$.

######  Proposition 5.7.

Let $Y$ be either a real algebraic group or the unit sphere $\mathbb{S}^{7}$.
Then, for every positive integer $n$, the subset $\pi^{\mathrm{alg}}_{n}(Y)$
is a subgroup of the homotopy group $\pi_{n}(Y)$.

###### Proof.

In the case under consideration, the H-space multiplication $Y\times Y\to Y$,
$(a,b)\mapsto ab$ and the inverse operation $Y\to Y$, $a\mapsto a^{-1}$ are
regular maps. The assertion follows since the group operation in $\pi_{n}(Y)$
is induced by the H-space multiplication. ∎

The following example has already been alluded to in connection with
Conjecture II in Section 1.

######  Example 5.8.

Let $(n,p)$ be a pair of integers, $n>p\geq 1$. Then there exist an algebraic
model $X$ of the $n$-dimensional torus $(\mathbb{S}^{1})^{n}$ and a
$\mathcal{C}^{\infty}$ map $f\colon X\to\mathbb{S}^{p}$ such that $f$ is not
homotopic to any regular map from $X$ into $\mathbb{S}^{p}$.

This assertion can be proved as follows. Let
$h\colon(\mathbb{S}^{1})^{p}\to\mathbb{S}^{p}$ be a $\mathcal{C}^{\infty}$ map
of topological degree $1$. Then the induced homomorphism in homology
${h_{*}\colon H_{p}((\mathbb{S}^{1})^{p};\mathbb{Z}/2)\to
H_{p}(\mathbb{S}^{p};\mathbb{Z}/2)}$ is an isomorphism. Let $w$ be a generator
of the cohomology group $H^{p}(\mathbb{S}^{p};\mathbb{Z}/2)\cong\mathbb{Z}/2$;
so $w$ corresponds via the Poincaré duality to a point in $\mathbb{S}^{p}$.
Clearly, if $[(\mathbb{S}^{1})^{p}]\in
H_{p}((\mathbb{S}^{1})^{p};\mathbb{Z}/2)$ is the fundamental class of
$(\mathbb{S}^{1})^{p}$, then the Kronecker index $\langle
w,h_{*}([(\mathbb{S}^{1})^{p}])\rangle$ is nonzero. Therefore the assertion
holds by [14, Theorem 2.8] (with $K=(\mathbb{S}^{1})^{p}$,
$L=(\mathbb{S}^{1})^{n-p}$, $Y=\mathbb{S}^{p}$).

Consequently, the set of regular maps $\mathcal{R}(X,\mathbb{S}^{p})$ is not
dense in the space of $\mathcal{C}^{\infty}$ maps
$\mathcal{C}^{\infty}(X,\mathbb{S}^{p})$.

Next we illustrate the behavior of regular maps from the product of spheres
$\mathbb{S}^{p}\times\mathbb{S}^{q}$ into the sphere $\mathbb{S}^{p+q}$.

######  Example 5.9.

The results depend strongly on the specific values of $p$ and $q$. In what
follows by a degree of a map we mean a topological degree.

  1. (i)

According to [44, Theorem 12], there exists a regular map
$\mathbb{S}^{4}\times\mathbb{S}^{2}\to\mathbb{S}^{6}$ of degree $1$. Hence, in
view of [5, Corollary 4.2], for every integer $d$ there exists a regular map
${\mathbb{S}^{4}\times\mathbb{S}^{2}\to\mathbb{S}^{6}}$ of degree $d$. Now, it
follows from Hopf’s theorem that each $\mathcal{C}^{\infty}$ map
$\mathbb{S}^{4}\times\mathbb{S}^{2}\to\mathbb{S}^{6}$ is homotopic to a
regular map. Consequently, by Corollary 1.5, the set of regular maps
$\mathcal{R}(\mathbb{S}^{4}\times\mathbb{S}^{2},\mathbb{S}^{6})$ is dense in
the space of $\mathcal{C}^{\infty}$ maps
$\mathcal{C}^{\infty}(\mathbb{S}^{4}\times\mathbb{S}^{2},\mathbb{S}^{6})$.

  2. (ii)

The same argument shows that
$\mathcal{R}(\mathbb{S}^{4}\times\mathbb{S}^{1},\mathbb{S}^{5})$ is dense in
$\mathcal{C}^{\infty}(\mathbb{S}^{4}\times\mathbb{S}^{1},\mathbb{S}^{5})$.

  3. (iii)

By [44, Theorem 14] and [5, Corollary 4.2], for every even integer $d$ there
exists a regular map $\mathbb{S}^{2}\times\mathbb{S}^{2}\to\mathbb{S}^{4}$ of
degree $d$. It is an open problem whether there exists a regular map
$\mathbb{S}^{2}\times\mathbb{S}^{2}\to\mathbb{S}^{4}$ of odd degree; if it
does, then $\mathcal{R}(\mathbb{S}^{2}\times\mathbb{S}^{2},\mathbb{S}^{4})$ is
dense in
$\mathcal{C}^{\infty}(\mathbb{S}^{2}\times\mathbb{S}^{2},\mathbb{S}^{4})$.

  4. (iv)

Assuming that both $p$ and $q$ are odd positive integers, according to Theorem
1.7, a $\mathcal{C}^{\infty}$ map
$f\colon\mathbb{S}^{p}\times\mathbb{S}^{q}\to\mathbb{S}^{p+q}$ can be
approximated by regular maps if and only if it is null homotopic.

##  6 Regular maps into real algebraic groups

For maps into classical groups or Stiefel manifolds (see Example 1.4(iii) for
the notation), we have the following result supporting Conjecture I.

######  Proposition 6.1.

Let $(n,m)$ be a pair of positive integers and let $Y$ be one of the following
real algebraic varieties:

  1. (i)

$Y=\mathrm{O}(m)$ or $Y=\mathrm{SO}(m)$ with $n\leq m-2$ and $n=8k+l$,
$k\in\mathbb{Z}$, $l=2,4,5$ or $6$;

  2. (ii)

$Y=\mathrm{U}(m)$ or $Y=\mathrm{SU}(m)$ with $n\leq 2m-1$ and $n=2k$,
$k\in\mathbb{Z}$;

  3. (iii)

$Y=\mathrm{Sp}(m)$ with $n\leq 4m+1$ and $n=8k+l$, $k\in\mathbb{Z}$, $l=0,1,2$
or $6$;

  4. (iv)

$Y=\mathbb{V}_{r}(\mathbb{F}^{m})$ with $n\leq(m-r+1)d(\mathbb{F})-2$, where
$d(\mathbb{F})=\dim_{\mathbb{R}}{\mathbb{F}}$.

Then the set of regular maps $\mathcal{R}(\mathbb{S}^{n},Y)$ is dense in the
space of $\mathcal{C}^{\infty}$ maps $\mathcal{C}^{\infty}(\mathbb{S}^{n},Y)$.

###### Proof.

According to Bott’s periodicity theorem [18] (see also [31, Chap. 8, Remark
4.2, and 12.2]) the $n$th homotopy group $\pi_{n}(Y)$ is trivial in cases (i),
(ii) and (iii). Furthermore, by [31, Chap. 8, Theorem 5.1], the homotopy group
$\pi_{n}(Y)$ is trivial in case (iv). Therefore the proof is complete in view
of Corollary 1.2. ∎

Next we consider regular maps from $\mathbb{S}^{n}$ into $\mathrm{U}(m)$ or
$\mathrm{SU}(m)$, for $n$ odd with $n\leq 2m-1$. This is harder to handle than
Proposition 6.1(ii) since the homotopy groups $\pi_{n}(\mathrm{U}(m))$ and
$\pi_{n}(\mathrm{SU}(m))$ are nontrivial if $n$ is odd ($n\neq 1$ for the
latter group). We have the following partial result.

######  Theorem 6.2.

Let $(n,m)$ be a pair of positive integers and let $G(m)$ denote either
$\mathrm{U}(m)$ or $\mathrm{SU}(m)$. Assume that $n=2k-1$ is odd and $m\geq
2^{k-1}$. Then the set of regular maps $\mathcal{R}(\mathbb{S}^{n},G(m))$ is
dense in the space of $\mathcal{C}^{\infty}$ maps
$\mathcal{C}^{\infty}(\mathbb{S}^{n},G(m))$.

As a preparation for the proof of Theorem 6.2, we briefly summarize the
discussion contained in [17, Part III B]. Let $k,p$ be two integers with
$1\leq k\leq p$. For any continuous map
$f\colon\mathbb{S}^{2k-1}\to\mathrm{GL}_{p}(\mathbb{C})$, its degree $\deg(f)$
is an integer defined as follows. The map $f$ is homotopic to the composite of
some continuous map $h\colon\mathbb{S}^{2k-1}\to\mathrm{GL}_{k}(\mathbb{C})$
and the inclusion map
$\mathrm{GL}_{k}(\mathbb{C})\hookrightarrow\mathrm{GL}_{p}(\mathbb{C})$. The
first column $h_{1}$ of $h$ determines a continuous map
$h_{1}\colon\mathbb{S}^{2k-1}\to\mathbb{C}^{k}\setminus\\{0\\}$, and therefore
one gets a continuous map

$\psi=\frac{h_{1}}{\left\lVert
h_{1}\right\rVert}\colon\mathbb{S}^{2k-1}\to\mathbb{S}^{2k-1},$

where $\mathbb{S}^{2k-1}$ is regarded as a subset of
$\mathbb{C}^{k}=\mathbb{R}^{2k}$. The topological degree $\deg(\psi)$ of
$\psi$ is divisible by $(k-1)!$, and one sets

$\deg{f}=(-1)^{k-1}\frac{\deg(\psi)}{(k-1)!}.$

The definition of $\deg(f)$ does not depend on the choice of $h$. Furthermore,
one has the following variant of Bott’s periodicity theorem: If $1\leq k\leq
p$, then

$\deg\colon\pi_{2k-1}(\mathrm{GL}_{p}(\mathbb{C}))\to\mathbb{Z},\quad[f]\mapsto\deg(f)$

is a well-defined group isomorphism.

As usual, for any complex matrix $A$, let $A^{*}$ denote its conjugate
transpose. For any positive integer $r$, denote by $I_{r}$ the identity
$r$-by-$r$ matrix.

Suppose given two continuous maps
$f\colon\mathbb{S}^{2k-1}\to\mathrm{GL}_{p}(\mathbb{C})$ and
$g\colon\mathbb{S}^{2l-1}\to\mathrm{GL}_{q}(\mathbb{C})$, where $1\leq k\leq
p$ and $1\leq l\leq q$. The product

$f\\#g\colon\mathbb{S}^{2(k+l)-1}\to\mathrm{GL}_{2pq}(\mathbb{C})$

is a continuous map defined by

$(x,y)\mapsto\begin{pmatrix}F(x)\otimes I_{q}&-I_{p}\otimes G(y)^{*}\\\
I_{p}\otimes G(y)&F(x)^{*}\otimes I_{q}\end{pmatrix},$

where $x\in\mathbb{C}^{k}$, $y\in\mathbb{C}^{l}$,
$\left\lVert(x,y)\right\rVert=1$, while $F$ and $G$ are the homogeneous
extensions of $f$ and $g$, respectively. To be precise,

$F\colon\mathbb{C}^{k}\to\mathrm{Mat}_{p}(\mathbb{C}),\quad
F(x)=\begin{cases}\left\lVert x\right\rVert f\big{(}\frac{x}{\left\lVert
x\right\rVert}\big{)}&\text{for }x\in\mathbb{C}^{k}\setminus\\{0\\},\\\
0&\text{for }x=0,\end{cases}$

where $\mathrm{Mat}_{p}(\mathbb{C})$ is the space of all complex $p$-by-$p$
matrices, and $G\colon\mathbb{C}^{l}\to\mathrm{Mat}_{q}(\mathbb{C})$ is
defined analogously. One has the formula

$\deg(f\\#g)=\deg(f)\deg(g).$

Now, starting with the map

$a\colon\mathbb{S}^{1}\to\mathrm{GL}_{1}(\mathbb{C})=\mathbb{C}\setminus\\{0\\},\quad
z\mapsto z,$

we define a sequence of continuous maps

$a_{k}\colon\mathbb{S}^{2k-1}\to\mathrm{GL}_{2^{k-1}}(\mathbb{C}),\quad
k=1,2,\ldots$

by a recursive formula: $a_{1}=a$ and $a_{k}=a_{k-1}\\#a$ for $k\geq 2$. Since
$\deg(a)=1$, it follows that $\deg(a_{k})=1$ for all $k\geq 1$. Thus, by the
variant of Bott’s periodicity theorem stated above, for every $k\geq 1$ the
homotopy group $\pi_{2k-1}(\mathrm{GL}_{2^{k-1}}(\mathbb{C}))\cong\mathbb{Z}$
is generated by the homotopy class represented by $a_{k}$.

The maps $a_{k}$ have some other useful properties. Let
$A_{k}\colon\mathbb{C}^{k}\to\mathrm{Mat}_{2^{k-1}}(\mathbb{C})$ be the
homogeneous extension of $a_{k}$, $k\geq 1$. Clearly, $A_{1}(z_{1})=z_{1}$ and

$A_{k}(z_{1},\ldots,z_{k})=\begin{pmatrix}A_{k-1}(z_{1},\ldots,z_{k-1})&-\bar{z}_{k}I_{2^{k-2}}\\\
z_{k}I_{2^{k-2}}&A_{k-1}(z_{1},\ldots,z_{k-1})^{*}\end{pmatrix}\quad\text{for
}k\geq 2,$

where $(z_{1},\ldots,z_{k})\in\mathbb{C}^{k}$ and $\bar{z}_{k}$ is the
conjugate of $z_{k}$. It follows that $A_{k}$ is an $\mathbb{R}$-linear map
for $k\geq 1$. Furthermore,

$\displaystyle
A_{k}(z_{1},\ldots,z_{k})A_{k}(z_{1},\ldots,z_{k})^{*}=\Big{(}\sum_{j=1}^{k}z_{j}\bar{z}_{j}\Big{)}I_{2^{k-1}}\quad\text{for
}k\geq 1,$ $\displaystyle\det A_{1}(z_{1})=z_{1}\bar{z}_{1},\quad\det
A_{k}(z_{1},\ldots,z_{k})=\Big{(}\sum_{j=1}^{k}z_{j}\bar{z}_{j}\Big{)}^{2^{k-2}}\quad\text{for
}k\geq 2.$

In particular, $a_{k}$ can be regarded as a regular map
$a_{k}\colon\mathbb{S}^{2k-1}\to\mathrm{SU}(2^{k-1})$ for $k\geq 2$.

###### Proof of Theorem 6.2.

By Proposition 5.5, it is sufficient to show that
$\pi^{\mathrm{alg}}_{2k-1}(G(m))=\pi_{2k-1}(G(m))$.

######  Case 1.

Suppose that $(k,m)=(1,m)$ with $m\geq 1$. It is well known that
${\pi_{1}(\mathrm{SU}(m))=0}$, and the inclusion map
$\mathrm{U}(1)\hookrightarrow\mathrm{U}(m)$ induces an isomorphism
$\pi_{1}(\mathrm{U}(1))\cong\pi_{1}(U(m))$. Case 1 follows since
$\mathrm{U}(1)=\mathbb{S}^{1}$ and
$\pi^{\mathrm{alg}}_{1}(\mathbb{S}^{1})=\pi_{1}(\mathbb{S}^{1})$.

######  Case 2.

Suppose that $k\geq 2$ and $m\geq 2^{k-1}$. The inclusion map
$G(2^{k-1})\hookrightarrow\mathrm{GL}_{2^{k-1}}(\mathbb{C})$ induces an
isomorphism of the homotopy groups $\pi_{2k-1}(G(2^{k-1}))$ and
$\pi_{2k-1}(\mathrm{GL}_{2^{k-1}}(\mathbb{C}))$. As explained above, the
latter group is generated by the homotopy class represented by the map
$a_{k}$. Since $a_{k}$ is a regular map with values in $G(2^{k-1})$, we get
$\pi^{\mathrm{alg}}_{2k-1}(G(2^{k-1}))=\pi_{2k-1}(G(2^{k-1}))$ by Proposition
5.7. Moreover, the inclusion map $G(2^{k-1})\hookrightarrow G(m)$ induces an
isomorphism of the homotopy groups $\pi_{2k-1}(G(2^{k-1}))$ and
$\pi_{2k-1}(G(m))$ (see [31, Chap. 8, Remark 4.2]), and hence
$\pi^{\mathrm{alg}}_{2k-1}(G(m))=\pi_{2k-1}(G(m))$.∎

We have one more result on regular maps into the groups $\mathrm{U}(m)$ or
$\mathrm{SU}(m)$.

######  Proposition 6.3.

Let $(n,m)$ be a pair of positive integers and let $G(m)$ denote either
$\mathrm{U}(m)$ or $\mathrm{SU}(m)$. Assume that $1\leq n\leq 6$. Then the set
of regular maps $\mathcal{R}(\mathbb{S}^{n},G(m))$ is dense in the space of
$\mathcal{C}^{\infty}$ maps $\mathcal{C}^{\infty}(\mathbb{S}^{n},G(m))$,
possibly with the exception of $(n,m)=(5,3)$ or $(n,m)=(6,3)$.

###### Proof.

Suppose that $(n,m)\neq(5,3)$ and $(n,m)\neq(6,3)$. Then the only cases not
already covered by Proposition 6.1 and Theorem 6.2 are $(n,m)$ equal to
$(4,2)$, $(5,2)$ and $(6,2)$. Hence, by Proposition 5.5, it is sufficient to
show that $\pi^{\mathrm{alg}}_{n}(G(2))=\pi_{n}(G(2))$ for $n=4,5$ and $6$.
This is proved in [50, Corollaries 1.2, 1.3 and 1.4] for
$G(2)=\mathrm{SU}(2)=\mathbb{S}^{3}$. The case $G(2)=\mathrm{U}(2)$ follows
since the inclusion map $\mathrm{SU}(m)\hookrightarrow\mathrm{U}(m)$ induces
for all $i\geq 2$ an isomorphism between the corresponding $i$th homotopy
groups, see [31, Chap. 8, 12.2]. ∎

######  Acknowledgements.

The second named author was partially supported by the National Science Center
(Poland) under grant number 2018/31/B/ST1/01059.

We are grateful to Olivier Benoist, János Kollár and Olivier Wittenberg for
very useful comments.

## Appendix
János Kollár

The following unirationality result was conjectured by Bochnak and Kucharz.

######  Proposition A.1.

Let $G$ be a linear real algebraic group of dimension $n$. Then there is a
real algebraic morphism $g\colon\mathbb{R}^{n}\to G$ such that $g(0)=e$ (the
group identity) and the tangent map

$dg_{0}\colon\mathbb{R}^{n}\cong T_{0}\mathbb{R}^{n}\to
T_{e}G\cong\mathbb{R}^{n}\quad\text{is an isomorphism.}$ (A.1)

The original version of the paper used a theorem of Chevalley to prove a
variant, where $g$ was defined on a dense, open subset of $\mathbb{R}^{n}$.
The above stronger form does not change any of the main theorems of the paper,
but allows to formulate some intermediate steps in a more elegant way and
illustrates more precisely unirationality of linear real algebraic groups. It
is worth mentioning that the question whether all such groups are rational
remains open.

First we show how to obtain Proposition A.1 from Chevalley’s theorem. Next,
looking at Chevalley’s method gives a more direct construction. Then we recall
the Cayley transform which gives a birational morphisms for the orthogonal and
unitary groups.

######  A.2Proof using Chevalley’s theorem.

By [21] there is a rational map $\tilde{g}\colon\mathbb{R}^{n}\dasharrow G$
satisfying (A.1). Thus it is enough to write down a morphism
$h\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ whose image lands in the open set
where $\tilde{g}$ is defined, and set $g\coloneqq\tilde{g}\circ h$. We can
take

$h\colon{\mathbf{x}}\mapsto
c\cdot\tfrac{{\mathbf{x}}}{1+||{\mathbf{x}}||^{2}},$

which sends $\mathbb{R}^{n}$ to a small neighborhood of the origin for $0<c\ll
1$. ∎

######  A.3Proof using Chevalley’s method.

We use the Iwasawa decomposition to write
$G=K\times(\mathbb{R}^{\times})^{r}\times U$ where $K$ is compact and $U$ is
unipotent.

The Lie algebra $L_{U}$ of $U$ is nilpotent, so the exponential map
$\exp\colon L_{U}\to U$ is algebraic and even a homeomorphism. The
$\mathbb{R}^{\times}$ factor is the least natural, but we can use
$g\colon\mathbb{R}\to\mathbb{R}^{\times}$ given by
$t\mapsto(1+t+t^{2})(1+t^{2})^{-1}$.

It remains to deal with $K$. For any root $\alpha$ we have a subgroup
$\mathbb{S}^{1}\cong C_{\alpha}\subset K$. Composing it with the inverse of
the stereographic projection from $-1$ we get
$g_{\alpha}\colon\mathbb{R}^{1}\to\mathbb{S}^{1}\cong C_{\alpha}\subset K$.

Now fix an ordering of the roots and send $v=\sum c_{\alpha}\alpha$ to

$g(v)\coloneqq\textstyle{\prod}_{\alpha}g_{\alpha}(c_{\alpha})\in K.\qed$

Note that $g$ does depend on the ordering of the roots. It would be
interesting to find a more natural construction that generalizes the Cayley
transform.

######  Example A.4 (Cayley transform).

Let $A$ be a skew-symmetric $n\times n$ matrix. Then $I+A$ is invertible, and
the Cayley transform is $Q(A)\coloneqq(I-A)(I+A)^{-1}$. Then $Q(A)$ is an
orthogonal matrix, giving the required map $Q\colon
T_{0}\mathrm{SO}_{n}\to\mathrm{SO}_{n}$.

Similarly, starting with a skew-Hermitian matrix $A$ we get $Q\colon
T_{0}\mathrm{U}_{n}\to\mathrm{U}_{n}$.

János Kollár
Princeton University
Princeton NJ 08544-1000 USA
E-mail address<EMAIL_ADDRESS>
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Jacek Bochnak
Le Pont de l’Étang 8
1323 Romainmôtier
Switzerland

E-mail address<EMAIL_ADDRESS>
Wojciech Kucharz
Institute of Mathematics
Faculty of Mathematics and Computer Science
Jagiellonian University
Łojasiewicza 6
30-348 Kraków
Poland

E-mail address<EMAIL_ADDRESS>