# De Rham algebras of closed quasiregularly elliptic manifolds are Euclidean

Susanna Heikkilä  Department of Mathematics and Statistics, P.O. Box 68
(Pietari Kalmin katu 5), FI-00014 University of Helsinki, Finland
<EMAIL_ADDRESS>and  Pekka Pankka  Department of Mathematics
and Statistics, P.O. Box 68 (Pietari Kalmin katu 5), FI-00014 University of
Helsinki, Finland<EMAIL_ADDRESS>
###### Abstract.

We show that, if a closed, connected, and oriented Riemannian $n$-manifold $N$
admits a non-constant quasiregular mapping from the Euclidean $n$-space
$\mathbb{R}^{n}$, then the de Rham cohomology algebra $H_{\mathrm{dR}}^{*}(N)$
of $N$ embeds into the exterior algebra ${\bigwedge}^{*}\mathbb{R}^{n}$. As a
consequence, we obtain a homeomorphic classification of closed simply
connected quasiregularly elliptic $4$-manifolds.

###### 2010 Mathematics Subject Classification:

Primary 30C65; Secondary 57M12, 30L10

This work was supported in part by the Academy of Finland project #332671.
This work is also supported in part by the National Science Foundation under
Grant No. DMS-1928930 while P.P. participated in a program hosted by the
Mathematical Sciences Research Institute (MSRI) in Berkeley, California,
during the Spring 2022 semester.

##  1\. Introduction

In this article we prove the following result on the de Rham cohomology of
quasiregularly elliptic manifolds.

######  Theorem 1.1.

Let $N$ be a closed quasiregularly elliptic $n$-manifold, $n\geqslant 2$. Then
there exists an embedding of graded algebras
$H^{*}_{\mathrm{dR}}(N)\to\bigwedge^{*}\mathbb{R}^{n}$.

A connected and oriented Riemannian $n$-manifold $N$ is called _quasiregularly
elliptic_ if there exists a non-constant quasiregular mapping
$\mathbb{R}^{n}\to N$. A continuous map $f\colon M\to N$ between oriented
Riemannian $n$-manifolds, $n\geqslant 2$, is _quasiregular_ if $f$ belongs to
the Sobolev space $W_{{\operatorname{loc}}}^{1,n}(M,N)$ and there exists a
constant $K\geqslant 1$ for which

(1.1) $\lVert Df\rVert^{n}\leqslant KJ_{f}\quad\text{ a.e. in }M;$

here $\lVert Df\rVert$ is the operator norm of the weak differential $Df$ and
$J_{f}$ is the Jacobian determinant of $f$.

Theorem 1.1 may be viewed as a quasiconformal analog of a classical result in
degree theory: _For a non-zero degree mapping $f\colon M\to N$ between
orientable closed manifolds, the induced map $f^{*}\colon H^{*}(N)\to
H^{*}(M)$ in cohomology is an embedding of graded algebras._ This analogy can
be taken further by observing that, if there exists a piecewise linear
branched covering $\mathbb{T}^{n}\to N$, which is quasiregular with respect to
smooth structures of $\mathbb{T}^{n}$ and $N$, then $N$ is a quasiregularly
elliptic.

Theorem 1.1 has an antecedent in Kangasniemi’s theorem [12]: _If a closed,
connected, and oriented Riemannian $n$-manifold $N$ admits a non-constant and
non-injective uniformly quasiregular mapping $N\to N$, then there exists an
embedding of graded algebras
$H^{*}_{\mathrm{dR}}(N)\to\bigwedge^{*}\mathbb{R}^{n}$_. Regarding
Kangasniemi’s theorem, we recall that a quasiregular self-map is _uniformly
quasiregular_ if the distortion inequality (1.1) holds for all the iterates of
the map with the same constant. We also note that such closed manifolds are
quasiregularly elliptic; see Martin, Mayer, and Peltonen [17].

In terms of known results on the cohomology of closed quasiregularly elliptic
manifolds, Theorem 1.1 extends a theorem of Prywes [24]: _Let $N$ be a closed
quasiregularly elliptic $n$-manifold. Then $\dim
H^{k}_{\mathrm{dR}}(N)\leqslant\dim\bigwedge^{k}\mathbb{R}^{n}=\binom{n}{k}$_.
This dimension bound for the de Rham cohomology theorem is sharp as the
$n$-torus $\mathbb{T}^{n}$ is quasiregularly elliptic. Prywes’s theorem
answers a question of Gromov [7, p. 200] whether there exists closed simply
connected manifolds, which are not quasiregularly elliptic. This is indeed the
case as, for example, $\\#^{4}(\mathbb{S}^{2}\times\mathbb{S}^{2})$ is not
quasiregularly elliptic by Prywes’s theorem. We refer to Bonk and Heinonen [1]
for a previous result on distortion dependent bounds on the de Rham cohomology
of quasiregularly elliptic manifolds. We also recall a theorem of Peltonen
[22] which shows that connected sums $\mathbb{T}^{n}\\#N$, where $N$ is not a
rational homology sphere, are not quasiregularly elliptic.

We also note in passing that dimension bounds on the de Rham cohomology in
Prywes’s theorem do not hold for open quasiregularly elliptic manifolds.
Indeed, for any finite set $P\subset\mathbb{S}^{n}$ there exists a
quasiregular mapping $f\colon\mathbb{R}^{n}\to\mathbb{S}^{n}$ for which
$f\mathbb{R}^{n}=\mathbb{S}^{n}\setminus P$; see [5]. Thus, it is known that
the $(n-1)$-cohomology of an open quasiregularly elliptic manifold $N$ may
have arbitrarily large finite dimension. The dependence of the dimension of
$H_{\mathrm{dR}}^{k}(N)$ on the distortion constant is not known in general.

###  1.1. Classification of closed simply connected quasiregularly elliptic
$4$-manifolds

Our main application of Theorem 1.1 is a classification of closed simply
connected quasiregularly elliptic $4$-manifolds.

In dimensions $n=4m$, Theorem 1.1 imposes upper bounds for the middle-
dimensional Betti numbers $\beta^{+}_{2m}(N)$ and $\beta^{-}_{2m}(N)$ given by
positive and negative definite subspaces of the intersection form
$H^{2m}(N;\mathbb{R})\times H^{2m}(N;\mathbb{R})\to\mathbb{R}$ of a closed
quasiregularly elliptic manifold $N$. In particular, in dimension $n=4$, both
Betti numbers $\beta^{+}_{2}(N)$ and $\beta^{-}_{2}(N)$ are at most three.
Therefore the classification of smooth, closed, simply connected, and oriented
$4$-manifolds yields that $N$ is homeomorphic to one of the manifolds

(1.2)
$\\#^{k}(\mathbb{S}^{2}\times\mathbb{S}^{2})\quad\text{or}\quad\\#^{j}\mathbb{C}P^{2}\\#^{i}\overline{\mathbb{C}P^{2}},$

where $k,j,i\in\\{0,1,2,3\\}$ and
$\\#^{0}(\mathbb{S}^{2}\times\mathbb{S}^{2})=\\#^{0}\mathbb{C}P^{2}\\#^{0}\overline{\mathbb{C}P^{2}}=\mathbb{S}^{4}$.

Piergallini and Zuddas show in [23] that these manifolds are quasiregularly
elliptic. Indeed, by a result Piergallini and Zuddas [23], for the manifolds
$N$ in (1.2) there exists a piecewise linear branched covering
$\mathbb{T}^{4}\to N$, where $\mathbb{T}^{4}$ and $N$ have the standard PL
structure compatible with the usual smooth structure. Thus composing a such
branched covering $\mathbb{T}^{4}\to N$ with a locally isometric Riemannian
covering map $\mathbb{R}^{4}\to\mathbb{T}^{4}$, we obtain the required
quasiregular mapping $\mathbb{R}^{4}\to N$.

Combining Theorem 1.1 with this result of Piergallini and Zuddas we obtain the
following classification.

######  Corollary 1.2.

A closed simply connected $4$-manifold is quasiregularly elliptic if and only
if it is homeomorphic to either $\\#^{k}(\mathbb{S}^{2}\times\mathbb{S}^{2})$
or $\\#^{j}\mathbb{C}P^{2}\\#^{i}\overline{\mathbb{C}P^{2}}$ for some
$k,j,i\in\\{0,1,2,3\\}$.

Similar classifications of quasiregularly elliptic manifolds, based on general
classification theorems for manifolds, are known in dimensions two and three.
In dimension two, the result is classically deduced from uniformization of
Riemann surfaces and the Stoïlow theorem for quasiregular mappings. In
dimension three, the classification follows from Perelman’s solution to the
Geometrization conjecture and a growth bound for the fundamental group. This
reduction of the classification to the Geometrization conjecture is due to
Jormakka [11]. We recall briefly these results from the point of view of
Prywes’s theorem and Theorem 1.1.

In dimension $n=2$, a closed quasiregularly elliptic $2$-manifold $N$
satisfies $\dim H^{1}_{\mathrm{dR}}(N)\leqslant 2$. Since $N$ is oriented, we
immediately observe that $N$ is homeomorphic to either $\mathbb{S}^{2}$ or
$\mathbb{T}^{2}$ by classification of orientable closed surfaces.

For a closed quasiregularly elliptic $3$-manifold $N$, we deduce from Prywes’s
theorem that $\dim H^{1}_{\mathrm{dR}}(N)=\dim H^{2}_{\mathrm{dR}}(N)\leqslant
3$. By elementary algebra, we obtain from Theorem 1.1 that $\dim
H^{1}_{\mathrm{dR}}(N)\neq 2$. Thus $N$ has the de Rham cohomology of one of
the manifolds $\mathbb{S}^{3}$, $\mathbb{S}^{2}\times\mathbb{S}^{1}$, or
$\mathbb{T}^{3}$. This homological result is sharp in the sense that, by
Jormakka’s classification theorem [11], the only closed quasiregularly
elliptic $3$-manifolds are $\mathbb{S}^{3}$,
$\mathbb{S}^{2}\times\mathbb{S}^{1}$, $\mathbb{T}^{3}$, and their quotients.
We find it interesting that, in fact, all closed quasiregularly elliptic
manifold in dimensions two and three admit a branched covering from a torus.

In dimension four, to our knowledge, the only known examples of closed simply
connected quasiregularly elliptic $4$-manifolds, before the result of
Piergallini and Zuddas, were $\mathbb{S}^{4}$,
$\mathbb{S}^{2}\times\mathbb{S}^{2}$,
$\\#^{2}(\mathbb{S}^{2}\times\mathbb{S}^{2})$, and $\mathbb{C}P^{2}$; here the
first two are given by elementary maps and the latter two are given by
constructions of Rickman [28] and Luisto–Prywes [16], respectively.

###  1.2. Quasiregular mappings from a ball

Theorem 1.1 admits a reformulation for quasiregular mappings from a unit ball
into closed manifolds. This reformulation in turn yields a cohomological
Montel theorem for quasiregular mappings satisfying a doubling condition. For
the statements, let $N$ be an oriented Riemannian $n$-manifold, let
$K\geqslant 1$ and $D>1$ be constants, and let $\mathcal{F}_{K,D}(N)$ be a
family of all non-constant quasiregular mappings $f\colon B_{2}^{n}\to N$
satisfying the distortion inequality (1.1) with constant $K$ and an additional
doubling condition

$\int_{B_{2}^{n}}f^{*}\mathrm{vol}_{N}\leqslant
D\int_{B^{n}}f^{*}\mathrm{vol}_{N}.$

Here and in what follows, $B^{n}$ and $B_{2}^{n}$ denote the Euclidean balls
of radius $1$ and $2$ centered at the origin, respectively. In what follows,
we also denote

$A(f)=\int_{B^{n}}f^{*}\mathrm{vol}_{N}.$

######  Theorem 1.3.

Let $N$ be a closed, connected, and oriented Riemannian $n$-manifold,
$n\geqslant 2$, for which

$\sup_{f\in\mathcal{F}_{K,D}(N)}A(f)=\infty.$

Then there exists an embedding of graded algebras
$H^{*}_{\mathrm{dR}}(N)\to\bigwedge^{*}\mathbb{R}^{n}$.

We conclude that, if $H^{*}_{\mathrm{dR}}(N)$ is not a subalgebra of
$\bigwedge^{*}\mathbb{R}^{n}$, there exists a uniform upper bound for the area
$A(f)$, and hence also for the energy, of mappings $f$ in
$\mathcal{F}_{K,D}(N)$. This uniform upper bound replaces the degree bound in
the Gromov’s compactness theorem for quasiregular maps [21] and, as a
consequence of these two results, we obtain the following cohomological
version of Montel’s theorem for quasiregular maps in $\mathcal{F}_{K,D}(N)$;
see Rickman [26, Corollary IV.3.14] for Rickman’s Montel theorem for
quasiregular mappings without an additional doubling assumption.

######  Corollary 1.4.

Let $N$ be a closed, connected, and oriented Riemannian $n$-manifold,
$n\geqslant 2$. If $H^{*}_{\mathrm{dR}}(N)$ is not a subalgebra of
$\bigwedge^{*}\mathbb{R}^{n}$, then $\overline{\mathcal{F}_{K,D}(N)}$ is
compact with respect to the topology of local uniform convergence. In
particular, in this case, $\mathcal{F}_{K,D}(N)$ is a normal family.

We finish this introduction by stating a more general result on weak limits of
normalized Jacobians of mappings in $\mathcal{F}_{K,D}(N)$. This statement may
be viewed as non-dynamical version of Kangasniemi’s absolute continuity
theorem for invariant measures for uniformly quasiregular mappings [13]. For
the statement, for each $f\in\mathcal{F}_{K,D}(N)$, let $\nu_{f}$ be the
measure

$\nu_{f}=\frac{1}{A(f)}f^{*}\mathrm{vol}_{N}.$

We call $\nu_{f}$ the _normalized measure of $f$_. Note that, clearly,
$\nu_{f}\ll\mathrm{vol}_{\mathbb{R}^{n}}$.

For a closed manifold $N$, which is not a rational homology sphere, we have
the following dichotomy for sequences in $\mathcal{F}_{K,D}(N)$; for the
statement, note that a sequence in $\mathcal{F}_{K,D}(N)$ need not have a
locally uniformly converging subsequence.

######  Theorem 1.5.

Let $N$ be a closed, connected, and oriented Riemannian $n$-manifold,
$n\geqslant 2$, which is not a rational homology sphere. Then a sequence
$(f_{j})$ in $\mathcal{F}_{K,D}(N)$ has a subsequence $(f_{j_{i}})$ for which
either $(f_{j_{i}})$ converges to a constant map or the normalized measures
$\nu_{f_{j_{i}}}$ converge vaguely to a measure
$\nu\ll\mathrm{vol}_{\mathbb{R}^{n}}$ in $B_{2}^{n}$.

Theorem 1.5 is sharp in the sense that, for $N=\mathbb{S}^{n}$, there exists a
sequence of mappings $(f_{j}\colon B_{2}^{n}\to\mathbb{S}^{n})$ for which the
sequence $(\nu_{f_{j}})$ of normalized measures converges vaguely to a measure
$\nu$, which is singular with respect to the Lebesgue measure. Recall that
measures $(\nu_{f_{j}})$ _converge vaguely to $\nu$_ if
$\int\varphi\nu_{f_{j}}\to\int\varphi\nu$ for each $\varphi\in
C_{0}(B_{2}^{n})$.

######  Remark 1.6.

Theorem 1.5 holds also for covering spaces of closed manifolds which are not
rational homology spheres. In particular, Theorem 1.5 holds for quasiregular
mappings in $\mathcal{F}_{K,D}(\mathbb{R}^{n})$.

###  1.3. Idea of the proof of Theorem 1.3

Since Theorem 1.1 reduces to Theorem 1.3, we discuss now the ideas in terms of
Theorem 1.3. To that end, let $N$ be a closed, connected, and oriented
Riemannian $n$-manifold and let

$h\colon H^{*}_{\mathrm{dR}}(N)\to\Omega^{*}(N),\quad c\mapsto h_{c},$

be the mapping which associates to each de Rham class $c\in
H^{*}_{\mathrm{dR}}(N)$ its unique harmonic representative $h_{c}\in c$. The
mapping $h$ is typically only a graded linear mapping and not an algebra
homomorphism, since harmonic forms typically do not form an algebra. We refer
to Kotschick [14] for a discussion on so-called _formal Riemannian manifolds_
, whose harmonic forms form an algebra, and to Kangasniemi [12] for the role
of formal measurable Riemannian metrics in the uniformly quasiregular
dynamics.

We define, for each $f\in\mathcal{F}_{K,D}(N)$, a normalized pull-back

$f^{\\#}\colon H^{*}_{\mathrm{dR}}(N)\to\mathscr{W}^{*}(B_{2}^{n}),\quad
H^{k}_{\mathrm{dR}}(N)\ni c\mapsto A(f)^{-\frac{k}{n}}f^{*}(h_{c}),$

where $A(f)^{-\frac{k}{n}}=1$ for $k=0$ and $\mathscr{W}^{*}(B_{2}^{n})$ is
the graded algebra

$\mathscr{W}^{k}(B_{2}^{n})=\left\\{\begin{array}[]{ll}W^{d,\frac{n}{k}}(B_{2}^{n};\bigwedge^{k}\mathbb{R}^{n}),&\text{for
}0<k\leqslant n\\\
\ker(d\colon\Omega^{0}(B_{2}^{n})\to\Omega^{1}(B_{2}^{n})),&\text{for
}k=0\end{array}\right.$

of Sobolev forms on $B_{2}^{n}$. In particular, $\mathscr{W}^{0}(B_{2}^{n})$
consists merely of constant functions on $B_{2}^{n}$. Here
$W^{d,\frac{n}{k}}(B_{2}^{n};\bigwedge^{k}\mathbb{R}^{n})$ is the partial
Sobolev space of $\frac{n}{k}$-integrable $k$-forms on $B_{2}^{n}$; see
Section 3 for details.

It is easy to observe that $f^{\\#}$ is an injective graded linear mapping and
not an algebra homomorphism unless $h$ is an algebra homomorphism.
Nevertheless, if $(f_{j})$ is such a sequence in $\mathcal{F}_{K,D}(N)$ that
$A(f_{j})\to\infty$ and that the sequence
$(f_{j}^{\\#}|_{H_{\mathrm{dR}}^{k}(N)}\colon
H_{\mathrm{dR}}^{k}(N)\to\mathscr{W}^{k}(B_{2}^{n}))$ converges weakly for
each $k=1,\ldots,n-1$, then the limit operators yield an embedding of graded
algebras.

######  Theorem 1.7.

Let $N$ be a closed, connected, and oriented Riemannian $n$-manifold,
$n\geqslant 2$. Let $(f_{j})$ be a sequence in $\mathcal{F}_{K,D}(N)$ for
which $A(f_{j})\to\infty$ and $f_{j}^{\\#}\rightharpoonup L$, where

$f_{j}^{\\#},L\colon\bigoplus_{k=1}^{n-1}H^{k}_{\mathrm{dR}}(N)\to\bigoplus_{k=1}^{n-1}L^{\frac{n}{k}}(B_{2}^{n};{\bigwedge}^{k}\mathbb{R}^{n}).$

Then $L$ extends to an embedding of graded algebras

$L\colon H^{*}_{\mathrm{dR}}(N)\to\mathscr{W}^{*}(B_{2}^{n})$

satisfying

$\int_{B^{n}}L([\mathrm{vol}_{N}])=1.$

Having Theorem 1.7 at our disposal, Theorem 1.3 follows by showing that, for
each sequence $(f_{j})$ satisfying $A(f_{j})\to\infty$, there exists a
subsequence whose normalized pull-backs converge weakly for each
$H_{\mathrm{dR}}^{k}(N)$, $k=1,\ldots,n-1$, to a linear map $L\colon
H_{\mathrm{dR}}^{k}(N)\to
L^{\frac{n}{k}}(B_{2}^{n};{\bigwedge}^{k}\mathbb{R}^{n})$. Since these linear
maps yield an embedding of algebras $L\colon
H_{\mathrm{dR}}^{*}(N)\to\mathscr{W}^{*}(B_{2}^{n})$, it suffices then to
observe that there exists a set of positive measure $E\subset B_{2}^{n}$ for
which the map

$H^{*}_{\mathrm{dR}}(N)\to{\bigwedge}^{*}\mathbb{R}^{n},\quad
c\mapsto(L(c))(x_{0}),$

is a well-defined embedding of graded algebras for $x_{0}\in E$.

###  1.4. Organization of the article

This article is organized as follows.

In Section 2, we give the reduction of Theorem 1.1 to Theorem 1.3. After this,
in Section 3, we discuss the normalized pull-backs and the Sobolev algebra
$\mathscr{W}^{*}$. In Sections 4 and 5, we prove a preliminary version of
Theorem 1.7 which shows that the limit extends to an algebra homomorphism. In
Section 6, we prove the embedding part of Theorem 1.7. In Section 7, we
finalize the proof of our main theorem, Theorem 1.3.

In Section 8, we discuss a sequence of quasiregular mappings
$B_{2}^{n}\to\mathbb{S}^{n}$ showing the sharpness of Theorem 1.5. In Section
9, we prove Theorem 1.5. In Appendix A, we briefly recall the classification
of smooth, closed, simply connected, and oriented $4$-manifolds. Finally, in
Appendix B, we discuss briefly the quasiregularity of the piecewise linear
branched covers $\mathbb{T}^{4}\to N$ of Piergallini and Zuddas.

###  1.5. Acknowledgements

We thank Ilmari Kangasniemi, Jani Onninen, Eden Prywes, and Juan Souto for
discussions on these topics over the years. We also thank Eero Hakavuori and
Toni Ikonen for a helpful remark at the right time.

##  2\. Reduction of Theorem 1.1 to Theorem 1.3

In this section, we discuss the reduction of Theorem 1.1 to Theorem 1.3. Since
both Theorems 1.1 and 1.3 hold trivially if the target manifold is a rational
homology sphere, the reduction is based on the following result; see also [24,
Section 4].

For the statement, we say that a quasiregularly elliptic manifold $N$ is _$K$
-quasiregularly elliptic_ if there exists a non-constant quasiregular mapping
$\mathbb{R}^{n}\to N$ satisfying the distortion inequality (1.1) with constant
$K$.

######  Proposition 2.1.

Let $N$ be a closed $K$-quasiregularly elliptic $n$-manifold, $n\geqslant 2$,
which is not a rational homology sphere. Then there exists a constant
$D=D(n)>1$ for which

$\sup_{f\in\mathcal{F}_{K,D}(N)}A(f)=\infty.$

We deduce Proposition 2.1 from the following two results. The first result is
a lower growth bound for quasiregular mappings into manifolds with non-trivial
homology due to Bonk and Heinonen [1]; see also [8] for the corresponding
result in the setting of quasiregular curves.

######  Theorem 2.2.

[1, Theorem 1.11] Let $N$ be a closed, connected, and oriented Riemannian
$n$-manifold, $n\geqslant 2$, and let $f\colon\mathbb{R}^{n}\to N$ be a non-
constant quasiregular map satisfying the distortion inequality (1.1) with
constant $K$. If $N$ is not a rational homology sphere, then there exists a
constant $\varepsilon=\varepsilon(n,K)>0$ satisfying

$\liminf_{r\to\infty}r^{-\varepsilon}\int_{B^{n}(r)}f^{*}\mathrm{vol}_{N}>0.$

The second result is a version of Rickman’s Hunting Lemma [25, Lemma 5.1]. The
following formulation is due to Bonk and Poggi-Corradini [2].

######  Lemma 2.3.

[2, Lemma 2.1] Let $\mu$ be an atomless Borel measure on $\mathbb{R}^{n}$
satisfying $\mu(\mathbb{R}^{n})=\infty$ and $\mu(B)<\infty$ for every ball
$B\subset\mathbb{R}^{n}$. Then there exists a constant $D=D(n)>1$ with the
property that, for every $m\in\mathbb{N}$, there exists a ball
$B\subset\mathbb{R}^{n}$ for which

$m\leqslant\mu(2B)\leqslant D\mu(B).$

Proposition 2.1 is an immediate consequence of these results.

###### Proof of Proposition 2.1.

Since $N$ is $K$-quasiregularly elliptic, there exists a non-constant
quasiregular map $f\colon\mathbb{R}^{n}\to N$ satisfying the distortion
inequality (1.1) with constant $K$. By Theorem 2.2, the Borel measure $\mu$ on
$\mathbb{R}^{n}$ defined by $\mu(E)=\int_{E}f^{*}\mathrm{vol}_{N}$ satisfies
the assumptions of Lemma 2.3. Hence, there exist a constant $D=D(n)>1$ and a
sequence of balls $(B^{n}(a_{m},r_{m}))$ for which

$m\leqslant\int_{B^{n}(a_{m},2r_{m})}f^{*}\mathrm{vol}_{N}\leqslant
D\int_{B^{n}(a_{m},r_{m})}f^{*}\mathrm{vol}_{N}.$

Now the maps $f_{m}\colon B_{2}^{n}\to N$, $x\mapsto f(r_{m}x+a_{m})$, form a
sequence in $\mathcal{F}_{K,D}(N)$ satisfying $A(f_{m})\to\infty$. The claim
follows. ∎

##  3\. Sobolev algebra $\mathscr{W}^{*}$ and normalized pull-backs

We begin by recalling briefly the necessary preliminaries on Sobolev
differential forms. A $(k+1)$-form $d\omega\in
L_{{\operatorname{loc}}}^{1}(B_{2}^{n};\bigwedge^{k+1}\mathbb{R}^{n})$ is the
_weak exterior derivative_ of a $k$-form $\omega\in
L_{{\operatorname{loc}}}^{1}(B_{2}^{n};\bigwedge^{k}\mathbb{R}^{n})$ if

$\int_{B_{2}^{n}}\omega\wedge
d\varphi=(-1)^{k+1}\int_{B_{2}^{n}}d\omega\wedge\varphi$

for every $\varphi\in\Omega_{0}^{n-k-1}(B_{2}^{n})$. By [10, Lemma 3.6], the
exterior derivative commutes with the pullback of a quasiregular map. More
precisely, let $N$ be a closed, connected, and oriented Riemannian
$n$-manifold and let $f\colon B_{2}^{n}\to N$ be a quasiregular map. Then, for
$\alpha\in\Omega^{k}(N)$, the form $f^{*}(d\alpha)\in
L^{\frac{n}{k+1}}_{{\operatorname{loc}}}(B_{2}^{n};\bigwedge^{k+1}\mathbb{R}^{n})$
is the weak exterior derivative of $f^{*}\alpha\in
L^{\frac{n}{k}}_{{\operatorname{loc}}}(B_{2}^{n};\bigwedge^{k}\mathbb{R}^{n})$,
i.e., $d(f^{*}\alpha)=f^{*}(d\alpha)$; here the integrability of $f^{*}\alpha$
and $f^{*}(d\alpha)$ follow from the point-wise inequality

$\left\lvert f^{*}\beta\right\rvert\leqslant(\left\lvert\beta\right\rvert\circ
f)\lVert Df\rVert^{\ell}$

for $\beta\in\Omega^{\ell}(N)$.

Let $W^{d,\frac{n}{k}}(B_{2}^{n};\bigwedge^{k}\mathbb{R}^{n})$ denote the
Sobolev space of forms $\omega$ belonging to
$L^{\frac{n}{k}}(B_{2}^{n};\bigwedge^{k}\mathbb{R}^{n})$ having a weak
exterior derivative $d\omega\in
L^{\frac{n}{k}}(B_{2}^{n};\bigwedge^{k+1}\mathbb{R}^{n})$. Suppose now that
$f\colon B_{2}^{n}\to N$ is a quasiregular map with integrable Jacobian, that
is, $J_{f}\in L^{1}(B_{2}^{n})$. Then, for a closed form
$\alpha\in\Omega^{k}(N)$, the pull-back $f^{*}\alpha$ belongs to
$W^{d,\frac{n}{k}}(B_{2}^{n};\bigwedge^{k}\mathbb{R}^{n})$ since $f^{*}\alpha$
is weakly closed.

Let
$f^{!}\colon\Omega^{*}(N)\to\oplus_{k=0}^{n}L^{\frac{n}{k}}(B_{2}^{n};\bigwedge^{k}\mathbb{R}^{n})$
be the linear map

$\Omega^{k}(N)\ni\alpha\mapsto A(f)^{-\frac{k}{n}}f^{*}\alpha,$

where $A(f)^{-\frac{k}{n}}=1$ for $k=0$; recall that

$A(f)=\int_{B^{n}}f^{*}\mathrm{vol}_{N}.$

It is easily seen that $f^{!}$ commutes with the exterior product and that
$f^{!}\circ h=f^{\\#}$. We also note that $f^{!}(\mathrm{vol}_{N})$ is the
normalized measure $\nu_{f}$ of $f$.

Next, we record an elementary lemma stating that, if
$f\in\mathcal{F}_{K,D}(N)$, then for $k$-forms the operator $f^{!}$ is bounded
from $\Omega^{k}(N)\to L^{\frac{n}{k}}(B_{2}^{n};\bigwedge^{k}\mathbb{R}^{n})$
and the operator norm of $f^{!}$ is bounded by $K$ and $D$.

######  Lemma 3.1.

Let $N$ be a closed, connected, and oriented Riemannian $n$-manifold,
$n\geqslant 2$, and let $f\in\mathcal{F}_{K,D}(N)$. Then

$\lVert f^{!}(\alpha)\rVert_{\frac{n}{k},B_{2}^{n}}\leqslant
DK\lVert\alpha\rVert_{\infty}$

for every $\alpha\in\Omega^{k}(N)$ and for $k=0,\ldots,n$.

###### Proof.

The claim holds trivially for $\alpha\in\Omega^{0}(N)$ as $\left\lvert
f^{!}(\alpha)\right\rvert(x)=\left\lvert\alpha(f(x))\right\rvert$ for each
$x\in B_{2}^{n}$. Hence, let $0<k\leqslant n$ and $\alpha\in\Omega^{k}(N)$.
Then

$\displaystyle\lVert f^{!}(\alpha)\rVert_{\frac{n}{k},B_{2}^{n}}$
$\displaystyle=A(f)^{-\frac{k}{n}}\left(\int_{B_{2}^{n}}\left\lvert
f^{*}\alpha\right\rvert^{\frac{n}{k}}\right)^{\frac{k}{n}}$
$\displaystyle\leqslant\lVert\alpha\rVert_{\infty}A(f)^{-\frac{k}{n}}\left(\int_{B_{2}^{n}}\lVert
Df\rVert^{n}\right)^{\frac{k}{n}}$
$\displaystyle\leqslant\lVert\alpha\rVert_{\infty}K^{\frac{k}{n}}A(f)^{-\frac{k}{n}}\left(\int_{B_{2}^{n}}f^{*}\mathrm{vol}_{N}\right)^{\frac{k}{n}}$
$\displaystyle\leqslant\lVert\alpha\rVert_{\infty}K^{\frac{k}{n}}D^{\frac{k}{n}}\leqslant
DK\lVert\alpha\rVert_{\infty}.$

This concludes the proof. ∎

We finish this section by showing that limits of exact forms are negligible
under sequences $(f_{j}^{!})$, if $(f_{j})$ is a sequence in
$\mathcal{F}_{K,D}(N)$ satisfying $A(f_{j})\to\infty$; see also [24]. For a
similar observation, based on integration by parts, in the context of entire
maps and in quasiregular dynamics see [20], [18], [12], and [8].

######  Lemma 3.2.

Let $N$ be a closed, connected, and oriented Riemannian $n$-manifold,
$n\geqslant 2$, and let $(f_{j})$ be a sequence in $\mathcal{F}_{K,D}(N)$
satisfying $A(f_{j})\to\infty$. Then

$\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi\wedge f_{j}^{!}(d\alpha)=0$

for $\alpha\in\Omega^{k}(N)$, $\varphi\in\Omega_{0}^{n-k-1}(B_{2}^{n})$, and
$k=0,\ldots,n$.

###### Proof.

We may assume that $0\leqslant k<n$. Let $\alpha\in\Omega^{k}(N)$ and let
$\varphi\in\Omega_{0}^{n-k-1}(B_{2}^{n})$. By Hölder’s inequality and Lemma
3.1, we obtain

$\displaystyle\left\lvert\int_{B_{2}^{n}}\varphi\wedge
f_{j}^{!}(d\alpha)\right\rvert$
$\displaystyle\quad=A(f_{j})^{-\frac{k+1}{n}}\left\lvert\int_{B_{2}^{n}}\varphi\wedge
f_{j}^{*}(d\alpha)\right\rvert=A(f_{j})^{-\frac{k+1}{n}}\left\lvert\int_{B_{2}^{n}}d\varphi\wedge
f_{j}^{*}\alpha\right\rvert$
$\displaystyle\quad=A(f_{j})^{-\frac{1}{n}}\left\lvert\int_{B_{2}^{n}}d\varphi\wedge
f_{j}^{!}(\alpha)\right\rvert\leqslant\frac{C(n)}{A(f_{j})^{\frac{1}{n}}}\int_{B_{2}^{n}}\left\lvert
d\varphi\right\rvert\left\lvert f_{j}^{!}(\alpha)\right\rvert$
$\displaystyle\quad\leqslant\frac{C(n)}{A(f_{j})^{\frac{1}{n}}}\lVert
d\varphi\rVert_{\frac{n}{n-k},B_{2}^{n}}\lVert
f_{j}^{!}(\alpha)\rVert_{\frac{n}{k},B_{2}^{n}}$
$\displaystyle\quad\leqslant\frac{C(n)}{A(f_{j})^{\frac{1}{n}}}\lVert
d\varphi\rVert_{\frac{n}{n-k},B_{2}^{n}}DK\lVert\alpha\rVert_{\infty}.$

The claim follows. ∎

##  4\. Limits of normalized pull-backs are Sobolev-Poincaré limits

We begin by recalling the Poincaré operator of Iwaniec and Lutoborski [9,
Section 4]: _There exists a graded compact linear operator_

$T\colon\bigoplus_{k=1}^{n-1}L^{\frac{n}{k}}(B_{2}^{n};{\bigwedge}^{k}\mathbb{R}^{n})\to\bigoplus_{k=1}^{n-1}L^{\frac{n}{k}}(B_{2}^{n};{\bigwedge}^{k-1}\mathbb{R}^{n})$

_satisfying $\mathrm{id}=dT+Td$. In particular, $dT(\omega)=\omega$ for every
weakly closed form
$\omega\in\oplus_{k=1}^{n-1}L^{\frac{n}{k}}(B_{2}^{n};\bigwedge^{k}\mathbb{R}^{n})$_.

The following lemma is a reformulation of [24, Lemma 4.1]. We recall the proof
for the reader’s convenience. Recall also that, for a quasiregular map
$f\colon B_{2}^{n}\to N$,

$f^{\\#}\colon
H_{\mathrm{dR}}^{*}(N)\to\mathscr{W}^{*}(B_{2}^{n}),\;H_{\mathrm{dR}}^{k}(N)\ni
c\mapsto A(f)^{-\frac{k}{n}}f^{*}(h_{c})=(f^{!}\circ h)(c).$

######  Lemma 4.1.

Let $N$ be a closed, connected, and oriented Riemannian $n$-manifold,
$n\geqslant 2$. Let $(f_{j})$ be a sequence in $\mathcal{F}_{K,D}(N)$ for
which $f_{j}^{\\#}\rightharpoonup L$, where

$f_{j}^{\\#},L\colon\bigoplus_{k=1}^{n-1}H^{k}_{\mathrm{dR}}(N)\to\bigoplus_{k=1}^{n-1}L^{\frac{n}{k}}(B_{2}^{n};{\bigwedge}^{k}\mathbb{R}^{n}).$

Then there exist a subsequence $(f_{j_{i}})$ and a graded linear operator

$\widehat{L}\colon\bigoplus_{k=1}^{n-1}H_{\mathrm{dR}}^{k}(N)\to\bigoplus_{k=1}^{n-1}W^{d,\frac{n}{k}}(B_{2}^{n};{\bigwedge}^{k-1}\mathbb{R}^{n})$

satisfying $T(f_{j_{i}}^{\\#})\to\widehat{L}$. In particular,
$d\widehat{L}=L$.

###### Proof.

Let $c_{1},\ldots,c_{m}$ be a basis of
$\oplus_{k=1}^{n-1}H_{\mathrm{dR}}^{k}(N)$. Since

$\lVert f_{j}^{\\#}(c_{\ell})\rVert_{\frac{n}{k_{\ell}},B_{2}^{n}}\leqslant
DK\lVert h(c_{\ell})\rVert_{\infty},$

where $c_{\ell}\in H^{k_{\ell}}(N)$, for each $\ell=1,\ldots,m$ by Lemma 3.1
and $T$ is a compact operator, there exist a subsequence $(f_{j_{i}})$ and
forms
$\tau_{1},\ldots,\tau_{m}\in\oplus_{k=1}^{n-1}L^{\frac{n}{k}}(B_{2}^{n};\bigwedge^{k-1}\mathbb{R}^{n})$
satisfying $T(f_{j_{i}}^{\\#}(c_{\ell}))\to\tau_{\ell}$. Since
$dT(f_{j}^{\\#}(c_{\ell}))=f_{j}^{\\#}(c_{\ell})$, we have

$\displaystyle\int_{B_{2}^{n}}\tau_{\ell}\wedge d\varphi$
$\displaystyle=\lim_{i\to\infty}\int_{B_{2}^{n}}T(f_{j_{i}}^{\\#}(c_{\ell}))\wedge
d\varphi=(-1)^{k_{\ell}}\lim_{i\to\infty}\int_{B_{2}^{n}}f_{j}^{\\#}(c_{\ell})\wedge\varphi$
$\displaystyle=(-1)^{k_{\ell}}\int_{B_{2}^{n}}L(c_{\ell})\wedge\varphi$

for every $\varphi\in\Omega_{0}^{n-k_{\ell}}(B_{2}^{n})$. Hence
$d\tau_{\ell}=L(c_{\ell})$. We define $\widehat{L}$ to be the linear map
defined by $c_{\ell}\mapsto\tau_{\ell}$. The claim follows. ∎

Motivated by Lemma 4.1 we introduce the following notion. Let $N$ be a closed,
connected, and oriented Riemannian $n$-manifold, $n\geqslant 2$, and let
$(f_{j})$ be a sequence in $\mathcal{F}_{K,D}(N)$. We say that a graded linear
operator

$\widehat{L}\colon\bigoplus_{k=1}^{n-1}H_{\mathrm{dR}}^{k}(N)\to\bigoplus_{k=1}^{n-1}W^{d,\frac{n}{k}}(B_{2}^{n};{\bigwedge}^{k-1}\mathbb{R}^{n})$

is a _Sobolev–Poincaré limit of $(f_{j}^{\\#})$_ if $d\widehat{L}$ is the weak
limit of the operators $f_{j}^{\\#}$ and $\widehat{L}$ is the limit of the
operators $T(f_{j}^{\\#})$.

The weak exterior derivative of a Sobolev–Poincaré limit commutes with the
wedge product in a weak sense; see also [24, Lemma 4.2]. For the statement, we
recall that a sequence $(\mu_{j})$ of Radon measures on $B_{2}^{n}$ _converges
vaguely to a Radon measure $\mu$ on $B_{2}^{n}$_ if

$\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi\,\mathrm{d}\mu_{j}=\int_{B_{2}^{n}}\varphi\,\mathrm{d}\mu$

for every $\varphi\in C_{0}(B_{2}^{n})$.

######  Proposition 4.2.

Let $N$ be a closed, connected, and oriented Riemannian $n$-manifold,
$n\geqslant 2$. Let $(f_{j})$ be a sequence in $\mathcal{F}_{K,D}(N)$ and let
the operators $(f_{j}^{\\#})$ have a Sobolev–Poincaré limit $\widehat{L}$. Let
also $k_{1},k_{2}\in\\{1,\ldots,n-1\\}$ satisfy $k_{1}+k_{2}\leqslant n$.
Then, for every $c_{1}\in H_{\mathrm{dR}}^{k_{1}}(N)$ and $c_{2}\in
H_{\mathrm{dR}}^{k_{2}}(N)$, we have $f_{j}^{\\#}(c_{1})\wedge
f_{j}^{\\#}(c_{2})\rightharpoonup d\widehat{L}(c_{1})\wedge
d\widehat{L}(c_{2})$. Here, the weak convergence is the usual weak convergence
in
$L^{\frac{n}{k_{1}+k_{2}}}(B_{2}^{n};\bigwedge^{k_{1}+k_{2}}\mathbb{R}^{n})$
if $k_{1}+k_{2}<n$ and the vague convergence of measures if $k_{1}+k_{2}=n$.

###### Proof.

Let $c_{i}\in H_{\mathrm{dR}}^{k_{i}}(N)$ for $i=1,2$. By Lemma 3.1, the
sequence $(f_{j}^{\\#}(c_{1})\wedge
f_{j}^{\\#}(c_{2}))=(f_{j}^{!}(h(c_{1})\wedge h(c_{2}))$ is bounded in
$L^{\frac{n}{k_{1}+k_{2}}}(B_{2}^{n};\bigwedge^{k_{1}+k_{2}}\mathbb{R}^{n})$.
Moreover, $\Omega_{0}^{n-k_{1}-k_{2}}(B_{2}^{n})$ is dense in
$L^{\frac{n}{n-k_{1}-k_{2}}}(B_{2}^{n};\bigwedge^{n-k_{1}-k_{2}}\mathbb{R}^{n})$
with respect to the $L^{\frac{n}{n-k_{1}-k_{2}}}$-norm if $k_{1}+k_{2}<n$ and
$C_{0}^{\infty}(B_{2}^{n})$ is dense in $C_{0}(B_{2}^{n})$ with respect to the
$L^{\infty}$-norm. Thus, it suffices to prove that

$\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi\wedge f_{j}^{\\#}(c_{1})\wedge
f_{j}^{\\#}(c_{2})=\int_{B_{2}^{n}}\varphi\wedge d\widehat{L}(c_{1})\wedge
d\widehat{L}(c_{2})$

for every $\varphi\in\Omega_{0}^{n-k_{1}-k_{2}}(B_{2}^{n})$.

Let $\varphi\in\Omega_{0}^{n-k_{1}-k_{2}}(B_{2}^{n})$. By the triangle
inequality, it suffices to estimate the terms

$\left\lvert\int_{B_{2}^{n}}\varphi\wedge(d\widehat{L}(c_{1})-f_{j}^{\\#}(c_{1}))\wedge
d\widehat{L}(c_{2})\right\rvert$

and

$\left\lvert\int_{B_{2}^{n}}\varphi\wedge
f_{j}^{\\#}(c_{1})\wedge(d\widehat{L}(c_{2})-f_{j}^{\\#}(c_{2}))\right\rvert.$

Since

$\varphi\wedge d\widehat{L}(c_{2})\in
L^{\frac{n}{k_{2}}}(B_{2}^{n};{\bigwedge}^{n-k_{1}}\mathbb{R}^{n})\subset
L^{\frac{n}{n-k_{1}}}(B_{2}^{n};{\bigwedge}^{n-k_{1}}\mathbb{R}^{n})$

and $f_{j}^{\\#}(c_{1})\rightharpoonup d\widehat{L}(c_{1})$ in
$L^{\frac{n}{k_{1}}}(B_{2}^{n};\bigwedge^{k_{1}}\mathbb{R}^{n})$, the first
term converges to zero. For the second term, since
$dT(f_{j}^{\\#}(c_{2}))=f_{j}^{\\#}(c_{2})$, we have the estimate

$\displaystyle\left\lvert\int_{B_{2}^{n}}\varphi\wedge
f_{j}^{\\#}(c_{1})\wedge(d\widehat{L}(c_{2})-f_{j}^{\\#}(c_{2}))\right\rvert$
$\displaystyle\quad=\left\lvert\int_{B_{2}^{n}}\varphi\wedge
f_{j}^{\\#}(c_{1})\wedge
d(\widehat{L}(c_{2})-T(f_{j}^{\\#}(c_{2})))\right\rvert$
$\displaystyle\quad=\left\lvert\int_{B_{2}^{n}}d\varphi\wedge
f_{j}^{\\#}(c_{1})\wedge(\widehat{L}(c_{2})-T(f_{j}^{\\#}(c_{2}))\right\rvert$
$\displaystyle\quad\leqslant C(n,\varphi)\int_{B_{2}^{n}}\left\lvert
f_{j}^{\\#}(c_{1})\right\rvert\left\lvert\widehat{L}(c_{2})-T(f_{j}^{\\#}(c_{2}))\right\rvert.$

By Hölder’s inequality and Lemma 3.1, we further have that

$\displaystyle\int_{B_{2}^{n}}\left\lvert
f_{j}^{\\#}(c_{1})\right\rvert\left\lvert\widehat{L}(c_{2})-T(f_{j}^{\\#}(c_{2})\right\rvert$
$\displaystyle\quad\leqslant\lVert
f_{j}^{\\#}(c_{1})\rVert_{\frac{n}{n-k_{2}},B_{2}^{n}}\lVert\widehat{L}(c_{2})-T(f_{j}^{\\#}(c_{2}))\rVert_{\frac{n}{k_{2}},B_{2}^{n}}$
$\displaystyle\quad\leqslant C(n,k_{1},k_{2})\lVert
f_{j}^{\\#}(c_{1})\rVert_{\frac{n}{k_{1}},B_{2}^{n}}\lVert\widehat{L}(c_{2})-T(f_{j}^{\\#}(c_{2}))\rVert_{\frac{n}{k_{2}},B_{2}^{n}}$
$\displaystyle\quad\leqslant
C(n,k_{1},k_{2},K,D,c_{1})\lVert\widehat{L}(c_{2})-T(f_{j}^{\\#}(c_{2}))\rVert_{\frac{n}{k_{2}},B_{2}^{n}}.$

Since $T(f_{j}^{\\#}(c_{2}))\to\widehat{L}(c_{2})$ in
$L^{\frac{n}{k_{2}}}(B_{2}^{n};\bigwedge^{k_{2}}\mathbb{R}^{n})$, the claim
follows. ∎

Next, we prove that the exterior derivative of a Sobolev–Poincaré limit
commutes with the wedge product provided that the normalizations grow without
bound.

######  Lemma 4.3.

Let $N$ be a closed, connected, and oriented Riemannian $n$-manifold,
$n\geqslant 2$. Let $(f_{j})$ be a sequence in $\mathcal{F}_{K,D}(N)$ for
which $A(f_{j})\to\infty$ and let the operators $(f_{j}^{\\#})$ have a
Sobolev–Poincaré limit $\widehat{L}$. Then

$d\widehat{L}(c_{1}\wedge c_{2})=d\widehat{L}(c_{1})\wedge
d\widehat{L}(c_{2})$

for every $c_{1}\in H_{\mathrm{dR}}^{k_{1}}(N)$ and $c_{2}\in
H_{\mathrm{dR}}^{k_{2}}(N)$ with $k_{1},k_{2}\in\\{1,\ldots,n-1\\}$ satisfying
$k_{1}+k_{2}<n$.

###### Proof.

Let $k_{1},k_{2}\in\\{1,\ldots,n-1\\}$ satisfy $k_{1}+k_{2}<n$. Let $c_{1}\in
H_{\mathrm{dR}}^{k_{1}}(N)$ and $c_{2}\in H_{\mathrm{dR}}^{k_{2}}(N)$. Then
there exists a form $\tau\in\Omega^{k_{1}+k_{2}-1}(N)$ for which
$h(c_{1}\wedge c_{2})=h(c_{1})\wedge h(c_{2})+d\tau$. Since
$f_{j}^{\\#}(c_{1}\wedge c_{2})\rightharpoonup d\widehat{L}(c_{1}\wedge
c_{2})$ in
$L^{\frac{n}{k_{1}+k_{2}}}(B_{2}^{n};\bigwedge^{k_{1}+k_{2}}\mathbb{R}^{n})$,
we have

$\displaystyle\int_{B_{2}^{n}}\varphi\wedge d\widehat{L}(c_{1}\wedge c_{2})$
$\displaystyle=\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi\wedge
f_{j}^{\\#}(c_{1}\wedge c_{2})$
$\displaystyle=\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi\wedge(f_{j}^{\\#}(c_{1})\wedge
f_{j}^{\\#}(c_{2})+f_{j}^{!}(d\tau))$
$\displaystyle=\int_{B_{2}^{n}}\varphi\wedge d\widehat{L}(c_{1})\wedge
d\widehat{L}(c_{2})$

for every $\varphi\in\Omega_{0}^{n-k_{1}-k_{2}}(B_{2}^{n})$ by Lemma 3.2 and
Proposition 4.2. The claim follows. ∎

The following result shows that, if the target manifold $N$ is not rational
homology sphere and $(f_{j})$ is a sequence in $\mathcal{F}_{K,D}(N)$
satisfying $A(f_{j})\to\infty$, then a Sobolev–Poincaré limit $\widehat{L}$ of
the operators $(f_{j}^{\\#})$ yields a vague limit for the normalized measures
$\nu_{f_{j}}$. Furthermore, the limiting measure is an absolutely continuous
measure; see Sections 8 and 9 for further discussion on the vague convergence
of normalized measures.

######  Theorem 4.4.

Let $N$ be a closed, connected, and oriented Riemannian $n$-manifold,
$n\geqslant 2$, which is not a rational homology sphere. Let $(f_{j})$ be a
sequence in $\mathcal{F}_{K,D}(N)$ for which $A(f_{j})\to\infty$ and let the
operators $(f_{j}^{\\#})$ have a Sobolev–Poincaré limit $\widehat{L}$. Then
there exists a non-negative form $\nu_{d\widehat{L}}\in
L^{1}(B_{2}^{n};\bigwedge^{n}\mathbb{R}^{n})$ for which
$\nu_{f_{j}}\rightharpoonup\nu_{d\widehat{L}}$ vaguely as measures.

###### Proof.

Since $N$ is not a rational homology sphere, there exist, by the Poincaré
duality, an index $k\in\\{1,\ldots,n-1\\}$ and forms $c\in
H_{\mathrm{dR}}^{k}(N)$ and $c^{\prime}\in H_{\mathrm{dR}}^{n-k}(N)$ for which
$\int_{N}c\wedge c^{\prime}\neq 0$. Then

$\nu_{d\widehat{L}}=\frac{\mathrm{vol}(N)}{\int_{N}c\wedge
c^{\prime}}d\widehat{L}(c)\wedge d\widehat{L}(c^{\prime})$

is a well-defined $n$-form in $L^{1}(B_{2}^{n};\bigwedge^{n}\mathbb{R}^{n})$,
which we consider also as a measure.

Let $\tau\in\Omega^{n-1}(N)$ be a form satisfying

$\mathrm{vol}(N)h(c)\wedge h(c^{\prime})=\left(\int_{N}c\wedge
c^{\prime}\right)(\mathrm{vol}_{N}+d\tau).$

Since $\lVert f_{j}^{!}(\mathrm{vol}_{N})\rVert_{1,B_{2}^{n}}\leqslant D$ and
$C_{0}^{\infty}(B_{2}^{n})$ is dense in $C_{0}(B_{2}^{n})$ with respect to the
$L^{\infty}$-norm, it suffices to prove that

$\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi\nu_{f_{j}}=\int_{B_{2}^{n}}\varphi\nu_{d\widehat{L}}$

for every $\varphi\in C_{0}^{\infty}(B_{2}^{n})$. By Proposition 4.2 and Lemma
3.2, we obtain

$\displaystyle\int_{B_{2}^{n}}\varphi\nu_{d\widehat{L}}$
$\displaystyle=\frac{\mathrm{vol}(N)}{\int_{N}c\wedge
c^{\prime}}\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi f_{j}^{\\#}(c)\wedge
f_{j}^{\\#}(c^{\prime})$ $\displaystyle=\frac{\mathrm{vol}(N)}{\int_{N}c\wedge
c^{\prime}}\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi f_{j}^{!}(h(c))\wedge
f_{j}^{!}(h(c^{\prime}))$
$\displaystyle=\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi
f_{j}^{!}(\mathrm{vol}_{N}+d\tau)$
$\displaystyle=\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi\nu_{f_{j}}$

for every $\varphi\in C_{0}^{\infty}(B_{2}^{n})$. Since the forms
$\nu_{f_{j}}$ are non-negative, so is $\nu_{d\widehat{L}}$. ∎

We finish this section by showing that, when restricted to the closed unit
ball $\bar{B}^{n}$, the vague convergence of measures in Theorem 4.4 improves
into weak convergence of probability measures.

######  Corollary 4.5.

Let $N$ be a closed, connected, and oriented Riemannian $n$-manifold,
$n\geqslant 2$, which is not a rational homology sphere. Let $(f_{j})$ be a
sequence in $\mathcal{F}_{K,D}(N)$ for which $A(f_{j})\to\infty$ and let the
operators $(f_{j}^{\\#})$ have a Sobolev–Poincaré limit $\widehat{L}$. Let
$\nu_{d\widehat{L}}\in L^{1}(B_{2}^{n};\bigwedge^{n}\mathbb{R}^{n})$ be the
limit measure in Theorem 4.4. Then
$\nu_{f_{j}}\llcorner\bar{B}^{n}\rightharpoonup\nu_{d\widehat{L}}\llcorner\bar{B}^{n}$.
In particular, $\nu_{d\widehat{L}}(B^{n})=1$.

###### Proof.

Let $\zeta\in C(\bar{B}^{n})$ be a non-negative function. Let $\eta\in
C_{b}(B_{2}^{n})$ be a non-negative bounded function satisfying
$\eta|_{\bar{B}^{n}}=\zeta$. Then
$\eta\nu_{f_{j}}\rightharpoonup\eta\nu_{d\widehat{L}}$ vaguely as measures.

Let $0<r<1<R<2$. Let $\psi_{r}\in C_{0}(B^{n})$ and $\psi_{R}\in
C_{0}(B^{n}(0,R))$ satisfy $\chi_{B^{n}(0,r)}\leqslant\psi_{r}\leqslant 1$ and
$\chi_{B^{n}}\leqslant\psi_{R}\leqslant 1$. Since $\eta\nu_{d\widehat{L}}$ is
non-negative, we obtain

$\int_{\bar{B}^{n}(0,r)}\eta\nu_{d\widehat{L}}\leqslant\int_{B_{2}^{n}}\psi_{r}\eta\nu_{d\widehat{L}}=\lim_{j\to\infty}\int_{B_{2}^{n}}\psi_{r}\eta\nu_{f_{j}}\leqslant\liminf_{j\to\infty}\int_{B^{n}}\eta\nu_{f_{j}}$

and

$\int_{B^{n}(0,R)}\eta\nu_{d\widehat{L}}\geqslant\int_{B_{2}^{n}}\psi_{R}\eta\nu_{d\widehat{L}}=\lim_{j\to\infty}\int_{B_{2}^{n}}\psi_{R}\eta\nu_{f_{j}}\geqslant\limsup_{j\to\infty}\int_{B^{n}}\eta\nu_{f_{j}}.$

Since $\nu_{d\widehat{L}}\in L^{1}(B_{2}^{n};\bigwedge^{n}\mathbb{R}^{n})$,

$\liminf_{j\to\infty}\eta\nu_{f_{j}}(B^{n})\geqslant\eta\nu_{d\widehat{L}}(B^{n})=\eta\nu_{d\widehat{L}}(\bar{B}^{n})\geqslant\limsup_{j\to\infty}\eta\nu_{f_{j}}(B^{n}).$

Thus

(4.1)
$\lim_{j\to\infty}\int_{\bar{B}^{n}}\zeta\nu_{f_{j}}=\int_{\bar{B}^{n}}\zeta\nu_{d\widehat{L}}.$

Equality (4.1) is then obtained for an arbitrary function $\zeta\in
C(\bar{B}^{n})$ by writing $\zeta=\max(\zeta,0)-\max(-\zeta,0)$.

Especially, since $\nu_{d\widehat{L}}\in
L^{1}(B_{2}^{n};\bigwedge^{n}\mathbb{R}^{n})$, we have, for $\zeta=1$, that

$\int_{B^{n}}\nu_{d\widehat{L}}=\int_{\bar{B}^{n}}\nu_{d\widehat{L}}=\lim_{j\to\infty}\int_{\bar{B}^{n}}\nu_{f_{j}}=\lim_{j\to\infty}\int_{B^{n}}\nu_{f_{j}}=1.$

∎

##  5\. Weak exterior derivatives of Sobolev–Poincaré limits extend to algebra
homomorphisms

In this section, we prove that, if $N$ is not a rational homology sphere and
$\widehat{L}$ is a Sobolev–Poincaré limit induced by a sequence $(f_{j})$ in
$\mathcal{F}_{K,D}(N)$ satisfying $A(f_{j})\to\infty$, then the map
$\Phi_{d\widehat{L}}\colon
H_{\mathrm{dR}}^{*}(N)\to\mathscr{W}^{*}(B_{2}^{n})$ defined by

$H_{\mathrm{dR}}^{k}(N)\ni c\mapsto\left\\{\begin{array}[]{ll}\text{constant
function }h_{c},&\text{for }k=0\\\ d\widehat{L}(c),&\text{for }1\leqslant
k\leqslant n-1\\\ \mathrm{vol}(N)^{-1}(\int_{N}c)\nu_{d\widehat{L}},&\text{for
}k=n\end{array}\right.$

is a graded algebra homomorphism; here $\nu_{d\widehat{L}}$ is the limit
$n$-form in Theorem 4.4.

######  Proposition 5.1.

Let $N$ be a closed, connected, and oriented Riemannian $n$-manifold,
$n\geqslant 2$, which is not a rational homology sphere. Let $(f_{j})$ be a
sequence in $\mathcal{F}_{K,D}(N)$ for which $A(f_{j})\to\infty$ and let the
operators $(f_{j}^{\\#})$ have a Sobolev–Poincaré limit $\widehat{L}$. Then
$\Phi_{d\widehat{L}}$ is a graded algebra homomorphism.

###### Proof.

Clearly $\Phi_{d\widehat{L}}$ is a graded linear map. It remains to prove that
$\Phi_{d\widehat{L}}$ is an algebra homomorphism. Let $c_{1}\in
H_{\mathrm{dR}}^{k_{1}}(N)$ and $c_{2}\in H_{\mathrm{dR}}^{k_{2}}(N)$,
$k_{1},k_{2}\in\\{0,\ldots,n\\}$. Without loss of generality, we may assume
that $k_{1},k_{2}>0$ and that $k_{1}+k_{2}\leqslant n$. If $k_{1}+k_{2}<n$,
then, by Lemma 4.3, we have that

$\Phi_{d\widehat{L}}(c_{1}\wedge c_{2})=d\widehat{L}(c_{1}\wedge
c_{2})=d\widehat{L}(c_{1})\wedge
d\widehat{L}(c_{2})=\Phi_{d\widehat{L}}(c_{1})\wedge\Phi_{d\widehat{L}}(c_{2}).$

Suppose now that $k_{1}+k_{2}=n$ and let $\tau\in\Omega^{n-1}(N)$ be a form
satisfying

$\mathrm{vol}(N)h(c_{1})\wedge h(c_{2})=\left(\int_{N}c_{1}\wedge
c_{2}\right)(\mathrm{vol}_{N}+d\tau).$

By Proposition 4.2, Lemma 3.2, and Theorem 4.4, we have

$\displaystyle\int_{B_{2}^{n}}\varphi\Phi_{d\widehat{L}}(c_{1})\wedge\Phi_{d\widehat{L}}(c_{2})$
$\displaystyle=\int_{B_{2}^{n}}\varphi d\widehat{L}(c_{1})\wedge
d\widehat{L}(c_{2})$ $\displaystyle=\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi
f_{j}^{\\#}(c_{1})\wedge f_{j}^{\\#}(c_{2})$
$\displaystyle=\frac{\int_{N}c_{1}\wedge
c_{2}}{\mathrm{vol}(N)}\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi
f_{j}^{!}(\mathrm{vol}_{N}+d\tau)$ $\displaystyle=\frac{\int_{N}c_{1}\wedge
c_{2}}{\mathrm{vol}(N)}\lim_{j\to\infty}\int_{B_{2}^{n}}\varphi\nu_{f_{j}}$
$\displaystyle=\frac{\int_{N}c_{1}\wedge
c_{2}}{\mathrm{vol}(N)}\int_{B_{2}^{n}}\varphi\nu_{d\widehat{L}}=\int_{B_{2}^{n}}\varphi\Phi_{d\widehat{L}}(c_{1}\wedge
c_{2})$

for every $\varphi\in C_{0}^{\infty}(B_{2}^{n})$. Thus
$\Phi_{d\widehat{L}}(c_{1})\wedge\Phi_{d\widehat{L}}(c_{2})=\Phi_{d\widehat{L}}(c_{1}\wedge
c_{2})$. This concludes the proof. ∎

##  6\. Limits of normalized pull-backs are algebra monomorphisms

In this section, we recall the statement of Theorem 1.7 and give its proof.

######  Theorem 1.7.

Let $N$ be a closed, connected, and oriented Riemannian $n$-manifold,
$n\geqslant 2$. Let $(f_{j})$ be a sequence in $\mathcal{F}_{K,D}(N)$ for
which $A(f_{j})\to\infty$ and $f_{j}^{\\#}\rightharpoonup L$, where

$f_{j}^{\\#},L\colon\bigoplus_{k=1}^{n-1}H^{k}_{\mathrm{dR}}(N)\to\bigoplus_{k=1}^{n-1}L^{\frac{n}{k}}(B_{2}^{n};{\bigwedge}^{k}\mathbb{R}^{n}).$

Then $L$ extends to an embedding of graded algebras

$L\colon H^{*}_{\mathrm{dR}}(N)\to\mathscr{W}^{*}(B_{2}^{n})$

satisfying

$\int_{B^{n}}L([\mathrm{vol}_{N}])=1.$

###### Proof.

We may assume that $N$ is not a rational homology sphere. By Lemma 4.1, the
operator $L$ is the weak exterior derivative of a Sobolev–Poincaré limit
$\widehat{L}$ of the operators $(f_{j_{i}}^{\\#})$ for some subsequence
$(f_{j_{i}})$. By Proposition 5.1, the map $\Phi_{L}=\Phi_{d\widehat{L}}\colon
H_{\mathrm{dR}}^{*}(N)\to\mathscr{W}^{*}(B_{2}^{n})$ is an algebra
homomorphism and $\Phi_{L}$ extends $L$ by definition. Moreover, by definition
and Corollary 4.5, we have

$\int_{B^{n}}\Phi_{L}([\mathrm{vol}_{N}])=\nu_{d\widehat{L}}(B^{n})=1.$

It remains to prove that $\Phi_{L}$ is injective. Since
$\Phi_{L}([\mathrm{vol}_{N}])\neq 0$, it suffices to show that $\Phi_{L}$ is
injective on $H_{\mathrm{dR}}^{k}(N)$ for $1\leqslant k\leqslant n-1$. Let
$c\in H_{\mathrm{dR}}^{k}(N)$ be non-zero. There exists $c^{\prime}\in
H_{\mathrm{dR}}^{n-k}(N)$ for which $\int_{N}c\wedge c^{\prime}\neq 0$. Since

$\Phi_{L}(c)\wedge\Phi_{L}(c^{\prime})=\Phi_{L}(c\wedge
c^{\prime})=\frac{\int_{N}c\wedge
c^{\prime}}{\mathrm{vol}(N)}\Phi_{L}([\mathrm{vol}_{N}])\neq 0,$

we have that $\Phi_{L}(c)\neq 0$. This concludes the proof. ∎

##  7\. Proof of Theorem 1.3

In this section, we recall the statement of Theorem 1.3 and finish its proof.

######  Theorem 1.3.

Let $N$ be a closed, connected, and oriented Riemannian $n$-manifold,
$n\geqslant 2$, for which

$\sup_{f\in\mathcal{F}_{K,D}(N)}A(f)=\infty.$

Then there exists an embedding of graded algebras
$H^{*}_{\mathrm{dR}}(N)\to\bigwedge^{*}\mathbb{R}^{n}$.

###### Proof.

Since $\sup_{f\in\mathcal{F}_{K,D}(N)}A(f)=\infty$ and
$\oplus_{k=1}^{n-1}H_{\mathrm{dR}}^{k}(N)$ is finite-dimensional, by Lemma 3.1
and the Banach–Alaoglu theorem, there exist a sequence $(f_{j})$ in
$\mathcal{F}_{K,D}(N)$ for which $A(f_{j})\to\infty$ and an operator

$L\colon\oplus_{k=1}^{n-1}H_{\mathrm{dR}}^{k}(N)\to\oplus_{k=1}^{n-1}L^{\frac{n}{k}}(B_{2}^{n};{\bigwedge}^{k}\mathbb{R}^{n}),$

which is the weak limit of the sequence $(f_{j}^{\\#})$. By Theorem 1.7, the
operator $L$ extends to an embedding of graded algebras $L\colon
H_{\mathrm{dR}}^{*}(N)\to\mathscr{W}^{*}(B_{2}^{n})$ satisfying

$\int_{B^{n}}L([\mathrm{vol}_{N}])=1.$

Let $c_{1},\ldots,c_{m}$ be a basis of $H_{\mathrm{dR}}^{*}(N)$. We fix point-
wise Borel representatives for $L([\mathrm{vol}_{N}])$, $L(c_{\ell_{1}}\wedge
c_{\ell_{2}})$, and $L(c_{\ell_{1}})\wedge L(c_{\ell_{2}})$, where
$\ell_{1},\ell_{2}=1,\ldots,m$. Let

$E=\\{x\in B^{n}\colon(L([\mathrm{vol}_{N}]))(x)\neq 0\\}$

and

$E_{\ell_{1},\ell_{2}}=\\{x\in B^{n}\colon(L(c_{\ell_{1}}\wedge
c_{\ell_{2}}))(x)=(L(c_{\ell_{1}}))(x)\wedge(L(c_{\ell_{2}}))(x)\\}$

for $\ell_{1},\ell_{2}=1,\ldots,m$. Since the set $E$ has positive
$n$-dimensional Lebesgue measure and the sets $E_{\ell_{1},\ell_{2}}$ have
full $n$-dimensional Lebesgue measure in $B^{n}$, the set
$E^{\prime}=\cap_{\ell_{1},\ell_{2}=1}^{m}E\cap E_{\ell_{1},\ell_{2}}$ has
positive $n$-dimensional Lebesgue measure.

Let $x_{0}\in E^{\prime}$. Now the map

$\mathscr{L}\colon
H_{\mathrm{dR}}^{*}(N)\to{\bigwedge}^{*}\mathbb{R}^{n},\quad
c\mapsto(L(c))(x_{0}),$

is a graded algebra homomorphism, which is injective on
$H_{\mathrm{dR}}^{n}(N)$. If $N$ is a rational homology sphere, then
$\mathscr{L}$ is trivially a monomorphism. If $N$ is not a rational homology
sphere, the injectivity of $\mathscr{L}$ follows from the non-degeneracy of
the pairing $H_{\mathrm{dR}}^{*}(N)\times
H_{\mathrm{dR}}^{n-*}(N)\to\mathbb{R}$, $(c,c^{\prime})\mapsto\int_{N}c\wedge
c^{\prime}$. ∎

###  7.1. Remark on the use of harmonic representatives

We defined operators $f^{\\#}\colon
H_{\mathrm{dR}}^{*}(N)\to\mathscr{W}^{*}(B_{2}^{n})$ as composition
$f^{\\#}=f^{!}\circ h$, where $h\colon H_{\mathrm{dR}}^{*}(N)\to\Omega^{*}(N)$
is the section of $\ker d\to H_{\mathrm{dR}}^{*}(N)$, $\alpha\mapsto[\alpha]$,
assigning to each cohomology class its harmonic representative. In fact, we
could have taken any section $\xi$ of $\ker d\to H_{\mathrm{dR}}^{*}(N)$,
$\alpha\mapsto[\alpha]$, and defined a normalized $\xi$-pullback as
composition $f^{\xi}=f^{!}\circ\xi$. The limits of normalized $\xi$-pullbacks
are independent of the section $\xi$ in the following sense.

######  Lemma 7.1.

Let $N$ be a closed, connected, and oriented Riemannian $n$-manifold,
$n\geqslant 2$, and let $(f_{j})$ be a sequence in $\mathcal{F}_{K,D}(N)$ for
which $A(f_{j})\to\infty$. Let $\xi\colon
H_{\mathrm{dR}}^{*}(N)\to\Omega^{*}(N)$ be a section of $\ker d\to
H_{\mathrm{dR}}^{*}(N)$, $\alpha\mapsto[\alpha]$, and let

$L,L^{\prime}\colon\bigoplus_{k=1}^{n-1}H_{\mathrm{dR}}^{k}(N)\to\bigoplus_{k=1}^{n-1}L^{\frac{n}{k}}(B_{2}^{n};{\bigwedge}^{k}\mathbb{R}^{n})$

be weak limits of $f_{j}^{\\#}$ and $f_{j}^{\xi}$, respectively. Then
$L=L^{\prime}$.

The proof is a simple modification of the proof of Lemma 4.1 since, for each
$c\in H_{\mathrm{dR}}^{*}(N)$, the form $\xi(c)-h(c)$ is exact. We leave the
details to the interested reader.

###  7.2. Remark on the non-uniqueness of the embedding
$H^{*}_{\mathrm{dR}}(N)\to\mathscr{W}^{*}(B_{2}^{n})$

Unless the manifold $N$ is a rational homology sphere, the embeddings of
graded algebras $H^{*}_{\mathrm{dR}}(N)\to\mathscr{W}^{*}(B_{2}^{n})$ in
Theorem 1.7 are non-unique. Indeed, given a sequence $(f_{j})$ for which the
sequence $f_{j}^{\\#}\rightharpoonup L$ in the sense of Theorem 1.7 which
admits a linear map $Q\in\mathrm{SO}_{n}(\mathbb{R})$ satisfying $Q^{*}L\neq
L$, then $(f_{j}\circ Q)$ is a sequence in $\mathcal{F}_{K,D}(N)$ for which
$(f_{j}\circ Q)^{\\#}\rightharpoonup Q^{*}L\neq L$ in the sense of Theorem
1.7.

Even if we restrict ourselves to embeddings
$H^{*}_{\mathrm{dR}}(N)\to\mathscr{W}^{*}(B_{2}^{n})$ associated to sequences
$(f_{j}\colon B_{2}^{n}\to N)$, which stem from a single quasiregular map
$\mathbb{R}^{n}\to N$, the uniqueness of the limit is not guaranteed. For
statements of positive and negative results, let $f\colon\mathbb{R}^{n}\to N$
be a non-constant quasiregular map and consider the family of all mappings
$f_{a,r}\colon B_{2}^{n}\to N$, $x\mapsto f\circ T_{a,r}$, where
$a\in\mathbb{R}^{n}$, $r>0$, and
$T_{a,r}\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ is the mapping $x\mapsto rx+a$.

We begin with an observation that, for the Riemannian covering map
$\mathbb{R}^{n}\to\mathbb{T}^{n}$, the induced algebra homomorphism
$H_{\mathrm{dR}}^{*}(\mathbb{T}^{n})\to\mathscr{W}^{*}(B_{2}^{n})$ is unique.

######  Example 7.2.

Let $\pi\colon\mathbb{R}^{n}\to\mathbb{T}^{n}$,
$(x_{1},\ldots,x_{n})\mapsto(e^{ix_{1}},\ldots,e^{ix_{n}})$, be the standard
Riemannian covering map and let $\theta_{1},\ldots,\theta_{n}$ be the
$1$-forms for which $\pi^{*}\theta_{i}=dx_{i}$ for each $i=1,\ldots,n$. Let
also $L_{\pi}\colon
H^{*}_{\mathrm{dR}}(\mathbb{T}^{n})\to\mathscr{W}^{*}(B_{2}^{n})$ the
embedding $[\theta_{i_{1}}\wedge\cdots\wedge\theta_{i_{k}}]\to
m_{n}(B^{n})^{-\frac{k}{n}}dx_{i_{1}}\wedge\cdots dx_{i_{k}}$, where
$m_{n}(B^{n})=\int_{B^{n}}\mathrm{vol}_{\mathbb{R}^{n}}$.

Since $\pi$ is a Riemannian covering map, it is $1$-quasiregular. We also have
that all the mappings $\pi_{a,r}=\pi\circ T_{a,r}|_{B_{2}^{n}}\colon
B_{2}^{n}\to\mathbb{T}^{n}$ are $1$-quasiregular. It is also easy to deduce
that $\pi_{a,r}^{\\#}=L_{\pi}$ for each $a\in\mathbb{R}^{n}$ and $r>0$.
Indeed, it suffices to observe that

$\displaystyle\pi_{a,r}^{!}(\theta_{i_{1}}\wedge\cdots\wedge\theta_{i_{k}})$
$\displaystyle=\frac{1}{A(\pi\circ
T_{a,r})^{k/n}}T_{a,r}^{*}\circ\pi^{*}(\theta_{i_{1}}\wedge\cdots\wedge\theta_{i_{k}})$
$\displaystyle=\frac{1}{r^{k}m_{n}(B^{n})^{\frac{k}{n}}}T_{a,r}^{*}(dx_{i_{1}}\wedge\cdots\wedge
dx_{i_{k}})$
$\displaystyle=\frac{1}{m_{n}(B^{n})^{\frac{k}{n}}}dx_{i_{1}}\wedge\cdots\wedge
dx_{i_{k}}$

for each multi-index $(i_{1},\ldots,i_{k})$.

We modify now the Riemannian covering mapping
$\mathbb{R}^{n}\to\mathbb{T}^{n}$ in a sequence of Euclidean balls to obtain a
quasiregular map $\mathbb{R}^{n}\to\mathbb{T}^{n}$ for which the induced
embedding $H_{\mathrm{dR}}^{*}(\mathbb{T}^{n})\to\mathscr{W}^{*}(B_{2}^{n})$
is non-unique.

######  Example 7.3.

Let $Q\in\mathrm{SO}_{n}(\mathbb{R})$ be a linear map, which is not the
identity. Let also $\alpha\colon[0,1]\to\mathrm{SO}_{n}(\mathbb{R})$ be a
geodesic in $\mathrm{SO}_{n}(\mathbb{R})$ from the identity to $Q$. We denote
$Q_{t}=\alpha(t)$ for each $t\in[0,1]$. Let also
$B_{j}=B^{n}(a_{j},r_{j})=B^{n}(2^{j}e_{1},j)$ be a sequence of mutually
disjoint Euclidean balls for which $r_{j}\to\infty$.

We define now $h_{Q}\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ as follows. In the
complement of the union $\bigcup_{j}B_{j}$, the mapping $h_{Q}$ is the
identity. For $x\in B^{n}(a_{j},r_{j}/2)$ , we set
$h_{Q}(x)=a_{j}+r_{j}Q((x-a_{j})/r_{j})$. Finally, if $x\in
B^{n}(a_{j},r_{j})\setminus B^{n}(a_{j},r_{j}/2)$, we set
$h_{Q}(x)=a_{j}+r_{j}Q_{2-2|x-a_{j}|/r_{j}}((x-a_{j})/r_{j})$. Since $\alpha$
is a geodesic and each $Q_{t}$ is an isometry, the mapping $h_{Q}$ is
quasiconformal.

Let now $f\colon\mathbb{R}^{n}\to\mathbb{T}^{n}$ be the quasiregular map
$f=\pi\circ h_{Q}$. Then, the sequence $(f_{-je_{1},j})$ yields the limiting
embedding $L_{\pi}$ and the sequence $(f_{a_{j},r_{j}/2})$ the limiting
embedding $Q^{*}L_{\pi}\neq L_{\pi}$.

##  8\. Example of a singular limit of normalized measures of quasiregular
mappings $B_{2}^{n}\to\mathbb{S}^{n}$

In this subsection, we show that, for $n\geqslant 2$, there exist a constant
$K=K(n)\geqslant 1$ and maps $f_{j}\in\mathcal{F}_{K,3}(\mathbb{S}^{n})$ for
which the normalized measures $\nu_{f_{j}}$ converge vaguely to the Borel
measure $\nu$,

$E\mapsto\frac{1}{2}\mathcal{H}^{1}(E\cap J),$

where $J=[-1,1]\times\\{0\\}^{n-1}$. Since $\nu$ is singular with respect to
the $n$-dimensional Lebesgue measure, this example shows that Theorem 1.5 does
not hold for quasiregular mappings into $\mathbb{S}^{n}$. For the construction
of the maps $f_{j}$, see also [19, Example 24].

Let $\iota\colon\mathbb{R}^{n}\to\mathbb{S}^{n}$ be the stereographic
projection and let $F\colon\mathbb{S}^{n}\to\mathbb{S}^{n}$ be the restriction
of the map
$\mathbb{R}^{n-1}\times\mathbb{C}\to\mathbb{R}^{n-1}\times\mathbb{C}$,
$(x,re^{i\theta})\mapsto(x,re^{i3\theta})$. We remark that $F(z)=z$ for points
$z$ on the equator $\mathbb{S}^{n-1}\subset\mathbb{S}^{n}$ and that the map
$F$ is $K$-quasiregular for $K=K(n)\geqslant 1$.

For $j\in\mathbb{N}$ and for $i=1,\ldots,j$, let
$a_{ij}=(-1+(2i-1)/j,0,\ldots,0)\in\mathbb{R}^{n}$,
$B_{ij}=B^{n}(a_{ij},1/j)$, and $U_{j}=\bigcup_{i=1}^{j}B_{ij}\subset B^{n}$.
Let also $\rho_{ij}\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ be the mapping
$x\mapsto j(x-a_{ij})$. Finally, let $\sigma_{ij}$ be the Möbius
transformation of $\mathbb{S}^{n}$ satisfying
$\sigma_{ij}\circ\iota=\iota\circ\rho_{ij}$.

We define now $f_{j}\colon B_{2}^{n}\to\mathbb{S}^{n}$ to be the mapping
$f_{j}(x)=\iota(x)$ for $x\notin U_{j}$ and
$f_{j}(x)=\sigma_{ij}^{-1}(F(\sigma_{ij}(\iota(x))))$ for $x\in B_{ij}$ and
each $i=1,\ldots,j$. Each map $f_{j}$ is $K$-quasiregular and, in each ball
$B_{ij}$, we have that

$\int_{B_{ij}}f_{j}^{*}\mathrm{vol}_{N}=\int_{\sigma_{ij}^{-1}(\mathbb{S}_{+}^{n})}2+\int_{\sigma_{ij}^{-1}(\mathbb{S}_{-}^{n})}1=\mathrm{vol}(\sigma_{ij}^{-1}(\mathbb{S}_{+}^{n}))+\mathrm{vol}(\mathbb{S}^{n}),$

where $\mathbb{S}_{+}^{n}$ and $\mathbb{S}_{-}^{n}$ are the upper and lower
hemispheres of $\mathbb{S}^{n}$, respectively. Hence

$\int_{B_{2}^{n}}f_{j}^{*}\mathrm{vol}_{N}\leqslant
3j\mathrm{vol}(\mathbb{S}^{n})\leqslant
3\int_{B^{n}}f_{j}^{*}\mathrm{vol}_{N}.$

Thus $f_{j}\in\mathcal{F}_{K,3}(\mathbb{S}^{n})$.

It remains to show that the normalized measures $\nu_{f_{j}}$ converge vaguely
to $\nu$. Let $\psi\in C_{0}(B_{2}^{n})$. Denote

$\mu_{f_{j}}:=\left(\int_{U_{j}}f_{j}^{*}\mathrm{vol}_{N}\right)^{-1}f_{j}^{*}\mathrm{vol}_{N}\quad\text{and}\quad\psi_{ij}:=\int_{J\cap
B_{ij}}\psi\;\mathrm{d}\nu.$

Since

$\frac{1}{A(f_{j})}\int_{B_{2}^{n}\setminus
U_{j}}f_{j}^{*}\mathrm{vol}_{N}\leqslant\frac{1}{j}$

for each $j$, it suffices to show that

$\lim_{j\to\infty}\int_{U_{j}}\psi\mu_{f_{j}}=\int_{B_{2}^{n}}\psi\;\mathrm{d}\nu.$

Since $\sigma_{ij}^{-1}(\mathbb{S}_{+}^{n})=\iota(B^{n}(a_{ij},1/j))$ and
$\iota$ is $\tilde{L}$-bilipschitz in $B^{n}$, where $\tilde{L}=\tilde{L}(n)$,
we have that

$C^{\prime}(n)\left(\frac{1}{\tilde{L}j}\right)^{n}+\mathrm{vol}(\mathbb{S}^{n})\leqslant\int_{B_{ij}}f_{j}^{*}\mathrm{vol}_{N}\leqslant
C^{\prime}(n)\left(\frac{\tilde{L}}{j}\right)^{n}+\mathrm{vol}(\mathbb{S}^{n})$

and thus

$\left\lvert\int_{B_{ij}}\mu_{f_{j}}-\frac{1}{j}\right\rvert\leqslant
C(n)\left(\frac{1}{j}\right)^{n+1}.$

Hence, we have the estimate

$\displaystyle\left\lvert\int_{U_{j}}\psi\mu_{f_{j}}-\int_{B_{2}^{n}}\psi\;\mathrm{d}\nu\right\rvert$
$\displaystyle=\left\lvert\sum_{i=1}^{j}\int_{B_{ij}}\psi\mu_{f_{j}}-\sum_{i=1}^{j}\int_{J\cap
B_{ij}}\psi\;\mathrm{d}\nu\right\rvert$
$\displaystyle\leqslant\sum_{i=1}^{j}\left\lvert\int_{B_{ij}}\psi\mu_{f_{j}}-\psi_{ij}\right\rvert$
$\displaystyle\leqslant\sum_{i=1}^{j}\left\lvert\int_{B_{ij}}(\psi-j\psi_{ij})\mu_{f_{j}}\right\rvert+\left\lvert
j\psi_{ij}\int_{B_{ij}}\mu_{f_{j}}-\psi_{ij}\right\rvert$
$\displaystyle\leqslant\sum_{i=1}^{j}\sup_{B_{ij}}\left\lvert\psi-j\psi_{ij}\right\rvert\int_{B_{ij}}\mu_{f_{j}}+\lVert\psi\rVert_{\infty}\left\lvert\int_{B_{ij}}\mu_{f_{j}}-\frac{1}{j}\right\rvert$
$\displaystyle\leqslant\sum_{i=1}^{j}\sup_{B_{ij}}\left\lvert\psi-j\psi_{ij}\right\rvert
C(n)\frac{1}{j}+\lVert\psi\rVert_{\infty}C(n)\left(\frac{1}{j}\right)^{n+1}$
$\displaystyle\leqslant\max_{1\leqslant i\leqslant
j}\sup_{B_{ij}}\left\lvert\psi-j\psi_{ij}\right\rvert
C(n)+\lVert\psi\rVert_{\infty}C(n)\left(\frac{1}{j}\right)^{n}.$

Since $\psi\in C_{0}(B_{2}^{n})$, it holds that $\max_{1\leqslant i\leqslant
j}\sup_{B_{ij}}\left\lvert\psi-j\psi_{ij}\right\rvert\to 0$. Thus, we obtain
that that the normalized measures $\nu_{f_{j}}$ converge vaguely to the
singular measure $\nu$.

##  9\. Weak limits of normalized measures

In this section, we prove the following result, which yields Corollary 1.4 and
Theorem 1.5 stated in the introduction.

######  Theorem 9.1.

Let $N$ be a closed, connected, and oriented Riemannian $n$-manifold,
$n\geqslant 2$, which is not a rational homology sphere, i.e. $H^{k}(N)\neq 0$
for some $k\in\\{1,\ldots,n-1\\}$. Then, up to passing to a subsequence, a
sequence $(f_{j})$ in $\mathcal{F}_{K,D}(N)$ satisfying
$\sup_{j}A(f_{j})<\infty$ converges locally uniformly to a $K$-quasiregular
map $f\colon B_{2}^{n}\to N$ in $\overline{\mathcal{F}_{K,D}(N)}$.

Theorem 9.1 implies Corollary 1.4 almost immediately.

######  Corollary 1.4.

Let $N$ be a closed, connected, and oriented Riemannian $n$-manifold,
$n\geqslant 2$. If $H^{*}_{\mathrm{dR}}(N)$ is not a subalgebra of
$\bigwedge^{*}\mathbb{R}^{n}$, then $\overline{\mathcal{F}_{K,D}(N)}$ is
compact with respect to the topology of local uniform convergence. In
particular, in this case, $\mathcal{F}_{K,D}(N)$ is a normal family.

###### Proof.

Since the closure $\overline{\mathcal{F}_{K,D}(N)}$ of quasiregular mappings
in $\mathcal{F}_{K,D}(N)$ in the topology of local uniform convergence
consists of $\mathcal{F}_{K,D}(N)$ and constant mappings $B_{2}^{n}\to N$, it
suffices to show that a sequence in $\mathcal{F}_{K,D}(N)$ has a subsequence
converging either to a mapping in $\mathcal{F}_{K,D}(N)$ or to a constant
mapping.

Since $H_{\mathrm{dR}}^{*}(N)$ is not a subalgebra of
$\bigwedge^{*}\mathbb{R}^{n}$, we have that

$\sup_{f\in\mathcal{F}_{K,D}(N)}A(f)<\infty$

by Theorem 1.3 and that $N$ is not a rational homology sphere. Thus, by
Theorem 9.1, each sequence $(f_{j})$ in $\mathcal{F}_{K,D}(N)$ has a
subsequence $(f_{j_{i}})$ converging locally uniformly to a (possibly
constant) quasiregular map $B_{2}^{n}\to N$ in
$\overline{\mathcal{F}_{K,D}(N)}$. We conclude that the family
$\overline{\mathcal{F}_{K,D}(N)}$ is compact. ∎

As a consequence of Theorem 9.1, we obtain also Theorem 1.5.

######  Theorem 1.5.

Let $N$ be a closed, connected, and oriented Riemannian $n$-manifold,
$n\geqslant 2$, which is not a rational homology sphere. Then a sequence
$(f_{j})$ in $\mathcal{F}_{K,D}(N)$ has a subsequence $(f_{j_{i}})$ for which
either $(f_{j_{i}})$ converges to a constant map or the normalized measures
$\nu_{f_{j_{i}}}$ converge vaguely to a measure
$\nu\ll\mathrm{vol}_{\mathbb{R}^{n}}$ in $B_{2}^{n}$.

###### Proof.

If $\sup_{j}A(f_{j})=\infty$, then there exists a subsequence $(f_{j_{i}})$
for which the operators $(f_{j_{i}}^{\\#})$ have a Sobolev–Poincaré limit
$\widehat{L}$ and, by Theorem 4.4,
$\nu_{f_{j_{i}}}\rightharpoonup\nu_{d\widehat{L}}$ vaguely as measures, where
$\nu_{d\widehat{L}}\ll\mathrm{vol}_{\mathbb{R}^{n}}$.

Suppose now that $\sup_{j}A(f_{j})<\infty$. By Theorem 9.1, there exists a
subsequence $(f_{j_{i}})$ converging locally uniformly to a map
$f\in\overline{\mathcal{F}_{K,D}(N)}$. If $f$ is non-constant, then, by the
weak convergence of the Jacobians (see e.g. [27, Lemma VI.8.8]), we have that
$\nu_{f_{j_{i}}}\rightharpoonup\nu_{f}$ vaguely as measures, where
$\nu_{f}\ll\mathrm{vol}_{\mathbb{R}^{n}}$. ∎

For the rest of this section, we will describe the idea of the proof of
Theorem 9.1. This theorem is a particular case of the following version of the
Gromov compactness theorem for quasiregular mappings; see [21, Theorem 1.1].

######  Theorem 9.2.

Let $N$ be a closed, connected, and oriented Riemannian $n$-manifold,
$n\geqslant 2$. Let $(f_{j})$ be a sequence in $\mathcal{F}_{K,D}(N)$ for
which $\sup_{j}A(f_{j})<\infty$. Then, up to passing to a subsequence,
$(f_{j})$ converges locally uniformly to a $K$-quasiregular map $f\colon X\to
N$, for which $J_{f}\in L^{1}(X)$, on some nodal Riemannian $n$-manifold $X$.
Furthermore, the main stratum of $X$ is diffeomorphic to $B_{2}^{n}$ and all
of the bubbles of $X$ are conformal to $\mathbb{S}^{n}$.

Recall that a (pure) nodal $n$-manifold $X$ is a topological space, which in
the complement of a finite set $\mathrm{Sing}(X)\subset X$ is an $n$-manifold,
each point in $\mathrm{Sing}(X)$ separates $X$ into two components, and the
local structure of $X$ near a point $p\in\mathrm{Sing}(X)$ is modeled by two
$n$-dimensional planes in general position in
$\mathbb{R}^{n}\times\mathbb{R}^{n}$. The closure of each connected component
of $X\setminus\mathrm{Sing}(X)$ is a called a _stratum_. In Theorem 9.2, each
strata is a smooth $n$-manifold and, more precisely, one stratum is
diffeomorphic to $B_{2}^{n}$ and all other strata are diffeomorphic to
$n$-spheres $\mathbb{S}^{n}$. We refer to [21, Section 4] for further
discussion.

In [21] this convergence theorem is stated for $K$-quasiregular mappings
$f_{j}\colon M\to N$ of fixed degree $\deg f_{j}$ between closed manifolds and
it was shown that the limit map $f\colon X\to N$ has the same degree in the
nodal sense; see [21, Theorem 1.1] for a precise statement.

In Theorem 9.2, the condition on the degree is replaced by a uniform bound for
the energy $A(f_{j})$. The main difference is that a sequence $(f_{j})$ in
$\mathcal{F}_{K,D}(N)$ may converge to a constant map and hence we may observe
a loss of energy $A(f_{j})$ at the limit. Indeed, it suffices to replace [21,
Proposition 3.1] by the following statement, whose proof is almost verbatim to
the proof of [21, Proposition 3.1].

######  Proposition 9.3.

[21, Proposition 3.1] Let $N$ be a closed and oriented Riemannian $n$-manifold
and let $(f_{j})$ be a sequence in $\mathcal{F}_{K,D}(N)$ satisfying
$\sup_{j}A(f_{j})<\infty$. Then, up to passing to a subsequence, there exists
a finite set $P\subset B_{2}^{n}$ with the following properties:

  * _(1)_

The measures $f_{j}^{*}\mathrm{vol}_{N}$ converge vaguely to a measure $\mu$
satisfying $\mu(B_{2}^{n})\leqslant D\cdot\sup_{j}A(f_{j})$.

  * _2_

There exists a $K$-quasiregular map $f\colon B_{2}^{n}\to N$ with
$f^{*}\mathrm{vol}_{N}=\mu$ on $B_{2}^{n}\setminus P$.

  * _3_

The sequence $(f_{j})$ converges locally uniformly to $f$ in
$B_{2}^{n}\setminus P$.

  * _4_

For each $p\in P$, we have $\mu(p)=d_{p}\cdot\mathrm{vol}(N)$ for some integer
$d_{p}\geqslant 1$.

We continue now to the proof of Theorem 9.1.

###### Proof of Theorem 9.1.

Let $(f_{j})$ be a sequence in $\mathcal{F}_{K,D}(N)$ with
$\sup_{j}A(f_{j})<\infty$. Then, by Theorem 9.2, the sequence $(f_{j})$ has a
subsequence $(f_{j_{i}})$, which converges to a $K$-quasiregular mapping
$f\colon X\to N$, where $X$ is a nodal $n$-manifold having $B_{2}^{n}$ as its
main stratum and (possibly) $n$-spheres as other strata.

Let $S\subset X$ be a stratum in $X$, which is an $n$-sphere. Then
$f|_{S}\colon S\to N$ is a quasiregular map. Since $N$ is not a rational
homology sphere, we have – by the Poincaré duality – that $\deg(f|_{S}\colon
S\to N)=0$. Thus $f|_{S}$ is constant. Hence $f$ is a constant map outside the
main stratum of $X$. Thus $d_{p}=0$ for all $p\in\mathrm{Sing}(X)$. We
conclude that $(f_{j_{i}})$ converges locally uniformly to a $K$-quasiregular
map $B_{2}^{n}\to N$. Moreover, we have

$\int_{B_{2}^{n}}f^{*}\mathrm{vol}_{N}\leqslant\liminf_{j\to\infty}\int_{B_{2}^{n}}f_{j}^{*}\mathrm{vol}_{N}\leqslant
D\lim_{j\to\infty}\int_{B^{n}}f_{j}^{*}\mathrm{vol}_{N}=D\int_{B^{n}}f^{*}\mathrm{vol}_{N}$

by the weak convergence of the Jacobians. This concludes the proof. ∎

##  Appendix A Donaldson–Freedman–Serre classification of closed $4$-manifolds

In this appendix, we recall the part of the topological classification of
smooth, closed, simply connected, and oriented $4$-manifolds we use for
Corollary 1.2. We also give upper bounds for the dimensions of positive and
negative definite subspaces of the intersection form of a $4m$-manifold whose
de Rham algebra embeds as a subalgebra of $\bigwedge^{*}\mathbb{R}^{4m}$.

We begin by recalling the definition of an intersection form. Let $N$ be a
closed, connected, and oriented $4m$-manifold. The _intersection form of $N$_
is the bilinear form

$Q_{N}\colon H^{2m}(N)/\mathrm{Tor}\,H^{2m}(N)\times
H^{2m}(N)/\mathrm{Tor}\,H^{2m}(N)\to\mathbb{Z},$ $(u,v)\mapsto(u\smile
v)([N]),$

where $[N]\in H_{2m}(N)$ is the fundamental class of $N$ associated to the
orientation of $N$. Let $\beta_{2m}^{+}(N)$ and $\beta_{2m}^{-}(N)$ denote the
maximal dimensions of vector subspaces of $H^{2m}(N;\mathbb{R})$ in which
$Q_{N}$ is positive and negative definite, respectively.

For $m=1$, Freedman [6, Theorem 1.5] implies that smooth, closed, simply
connected, and oriented $4$-manifolds are topologically classified by their
intersection forms, i.e., two manifolds are homeomorphic if and only if they
have the same intersection form. On the other hand, theorems of Donaldson [4]
and Serre [29, Section V.2.2] yield a list of symmetric, unimodular, and
bilinear forms which contains every intersection form of a smooth, closed,
simply connected, and oriented $4$-manifold. To avoid unnecessary
technicalities, we do not recite the Donaldson–Serre list in detail, but
observe that we obtain the following table of smooth, closed, simply
connected, and oriented $4$-manifolds $N$ for which
$\beta_{2m}^{+}(N)\leqslant 3$ and $\beta_{2m}^{-}(N)\leqslant 3$:

$\beta_{2}^{-}$ $\beta_{2}^{+}$ | 0 | 1 | 2 | 3
---|---|---|---|---
0 | $\mathbb{S}^{4}$ | $\mathbb{C}P^{2}$ | $\\#^{2}\mathbb{C}P^{2}$ | $\\#^{3}\mathbb{C}P^{2}$
1 | $\overline{\mathbb{C}P^{2}}$ |  $\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$, | $\\#^{2}\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$ | $\\#^{3}\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$
|  | $\mathbb{S}^{2}\times\mathbb{S}^{2}$ |  |
2 | $\\#^{2}\overline{\mathbb{C}P^{2}}$ | $\mathbb{C}P^{2}\\#^{2}\overline{\mathbb{C}P^{2}}$ |  $\\#^{2}\mathbb{C}P^{2}\\#^{2}\overline{\mathbb{C}P^{2}}$, | $\\#^{3}\mathbb{C}P^{2}\\#^{2}\overline{\mathbb{C}P^{2}}$
|  |  | $\\#^{2}\mathbb{S}^{2}\times\mathbb{S}^{2}$ |
3 | $\\#^{3}\overline{\mathbb{C}P^{2}}$ | $\mathbb{C}P^{2}\\#^{3}\overline{\mathbb{C}P^{2}}$ | $\\#^{2}\mathbb{C}P^{2}\\#^{3}\overline{\mathbb{C}P^{2}}$ |  $\\#^{3}\mathbb{C}P^{2}\\#^{3}\overline{\mathbb{C}P^{2}}$,
|  |  |  | $\\#^{3}\mathbb{S}^{2}\times\mathbb{S}^{2}$

For the classification of closed simply connected quasiregularly elliptic
$4$-manifolds $N$ it therefore suffices to observe that
$\beta_{2m}^{\pm}(N)\leqslant 3$. The following lemma records this simple
algebraic consequence of Theorem 1.1.

######  Lemma A.1.

Let $N$ be a smooth, closed, connected, and oriented $4m$-manifold and let
$\Phi\colon H^{*}(N,\mathbb{R})\to\bigwedge^{*}\mathbb{R}^{4m}$ be an
embedding of graded algebras. Then

$\beta_{2m}^{\pm}(N)\leqslant\frac{1}{2}\binom{4m}{2m}.$

###### Proof.

Let $V$ be a definite vector subspace of $Q_{N}$ in $H^{2m}(N;\mathbb{R})$ and
let $\mathscr{R}\colon H^{*}(N;\mathbb{R})\to H_{\mathrm{dR}}^{*}(N)$ be the
de Rham isomorphism. Note that $\mathscr{R}$ is an isomorphism of graded
algebras; see e.g. [3, Theorem III.3.1].

Let $I\colon\bigwedge^{2m}\mathbb{R}^{4m}\to\mathbb{R}$ be the bilinear form
$(\omega,\tau)\mapsto\int_{\mathbb{R}^{4m}}\omega\wedge\tau$. The bilinear
form $I$ is either positive or negative definite in $\Phi(\mathscr{R}(V))$
since

$I(\Phi(\mathscr{R}(u)),\Phi(\mathscr{R}(u)))=\int_{\mathbb{R}^{4m}}\Phi(\mathscr{R}u\wedge\mathscr{R}u)=\frac{\int_{\mathbb{R}^{4m}}\Phi([\mathrm{vol}_{N}])}{\mathrm{vol}(N)}Q_{N}(u,u),$

where $\int_{\mathbb{R}^{4m}}\Phi([\mathrm{vol}_{N}])\neq 0$. Since a definite
vector subspace of $I$ in $\bigwedge^{2m}\mathbb{R}^{4m}$ has dimension at
most $\frac{1}{2}\binom{4m}{2m}$, we obtain that

$\dim V=\dim\Phi(\mathscr{R}(V))\leqslant\frac{1}{2}\binom{4m}{2m}.$

The claim follows. ∎

##  Appendix B Quasiregularity of piecewise linear maps

In this appendix, we discuss the quasiregularity of piecewise linear maps. We
begin by recalling the definition of a smooth triangulation. Let $N$ be a
smooth manifold with a piecewise linear structure. A piecewise linear
homeomorphism $\kappa\colon P\to N$ from a Euclidean polyhedron $P$ into $N$
is a _piecewise smooth_ if there exists a locally finite simplicial complex
$\Sigma$ having the properties that $P=|\Sigma|$ and, for each
$\sigma\in\Sigma$, there exist a neighbourhood
$U\subset\mathbb{R}^{\dim\sigma}$ of $[0,e_{1},\ldots,e_{\dim\sigma}]$, a
smooth map $\lambda\colon U\to N$ of rank $\dim\sigma$, and an affine map
$A\colon[0,e_{1},\ldots,e_{\dim\sigma}]\to\sigma$ satisfying
$\lambda|_{[0,e_{1},\ldots,e_{\dim\sigma}]}=\kappa\circ A$. We say that the
piecewise linear structure of $N$ is compatible with the smooth structure of
$N$ if there exists a piecewise linear homeomorphism, which is piecewise
smooth, from a Euclidean polyhedron into $N$.

We are now ready to state the main result of this appendix.

######  Proposition B.1.

Let $M$ and $N$ be closed and oriented piecewise linear Riemannian
$n$-manifolds, $n\geqslant 2$. Let $p\colon M\to N$ be a non-degenerate
orientation preserving piecewise linear map. If the piecewise linear structure
is compatible with the smooth structure on both $M$ and $N$, then $p$ is a
quasiregular map.

The following result yields the Sobolev-regularity in Proposition B.1. While
the result is well-known to experts, we present its proof for the reader’s
convenience.

######  Lemma B.2.

Let $M$ be a closed piecewise linear smooth $n$-manifold and let $\kappa\colon
P\to M$ be a piecewise linear homeomorphism, which is piecewise smooth. Let
$\Sigma$ be a locally finite simplicial complex satisfying $P=|\Sigma|$ and
let $\rho\colon M\to\mathbb{R}$ be a continuous function having the property
that, for each $n$-simplex $\sigma\in\Sigma$, the restriction
$\rho|_{\kappa(\sigma)}$ has a smooth extension. Then $\rho\in
W^{1,\infty}(M,\mathbb{R})$.

###### Proof.

Since $M$ is closed and $\kappa$ is a homeomorphism, $\Sigma$ has finitely
many $n$-simplices $\sigma_{1},\ldots,\sigma_{j}$. Without loss of generality,
we may assume that each restriction $\kappa|_{\sigma_{i}}$, $i=1,\ldots,j$, is
a smooth map of rank $n$. For each $i\in\\{1,\ldots,j\\}$, there exist a
neighbourhood $U_{i}\subset M$ of $\kappa(\sigma_{i})$ and a smooth function
$\psi_{i}\colon U_{i}\to\mathbb{R}$ satisfying
$\psi_{i}|_{\kappa(\sigma_{i})}=\rho|_{\kappa(\sigma_{i})}$.

Define a differential $1$-form $\alpha\colon M\to TM^{*}$ by setting
$\alpha_{x}=(d\psi_{i})_{x}$ if $x\in\kappa(\mathrm{int}\,\sigma_{i})$ and
$\alpha_{x}=0$ otherwise. Clearly, $\rho\in L^{\infty}(M)$ and $\alpha\in
L^{\infty}(M,TM^{*})$. Thus, it suffices to prove that $\alpha$ is the weak
differential of $\rho$.

Let $\varphi\in\Omega_{0}^{n-1}(M)$. Then

$\displaystyle\int_{M}\rho\wedge d\varphi+\int_{M}\alpha\wedge\varphi$
$\displaystyle=\sum_{i=1}^{j}\left(\int_{\kappa(\sigma_{i})}\psi_{i}\wedge
d\varphi+\int_{\kappa(\sigma_{i})}d\psi_{i}\wedge\varphi\right)$
$\displaystyle=\sum_{i=1}^{j}\int_{\partial\kappa(\sigma_{i})}\psi_{i}\wedge\varphi=\sum_{i=1}^{j}\int_{\kappa(\partial\sigma_{i})}\rho\wedge\varphi$

by Stokes’ theorem on manifolds with corners; see e.g. [15, Theorem 16.25].

For each $i$, let $\delta_{1}^{i},\ldots,\delta_{n+1}^{i}$ be the
$(n-1)$-faces of $\sigma_{i}$. Then

$\sum_{i=1}^{j}\int_{\kappa(\partial\sigma_{i})}\rho\wedge\varphi=\sum_{i=1}^{j}\sum_{j=1}^{n+1}\int_{\kappa(\delta_{j}^{i})}\rho\wedge\varphi.$

For each face $\delta_{\ell}^{i}$, there exists a unique face
$\delta_{\ell^{\prime}}^{i^{\prime}}$ such that the faces $\delta_{\ell}^{i}$
and $\delta_{\ell^{\prime}}^{i^{\prime}}$ agree as sets but have opposite
orientations. Thus, the terms
$\int_{\kappa(\delta_{\ell}^{i})}\rho\wedge\varphi$ and
$\int_{\kappa(\delta_{\ell^{\prime}}^{i^{\prime}})}\rho\wedge\varphi$ cancel
each other in the sum. Hence, the whole sum vanishes and the claim follows. ∎

###### Proof of Proposition B.1.

Let $\kappa_{1}\colon P_{1}\to M$ and $\kappa_{2}\colon P_{2}\to N$ be
piecewise linear homeomorphisms, which are piecewise smooth. Let $\Sigma_{1}$
and $\Sigma_{2}$ be finite simplicial complexes for which
$P_{i}=|\Sigma_{i}|$, $\kappa_{i}$ is smooth and full rank with respect to
$\Sigma_{i}$, and the map $\kappa_{2}^{-1}\circ p\circ\kappa_{1}\colon
P_{1}\to P_{2}$ is simplicial with respect to $\Sigma_{1}$ and $\Sigma_{2}$.
Let $\sigma_{1},\ldots,\sigma_{j}$ be the $n$-simplices of $\Sigma_{1}$.

Since $p$ is non-degenerate, each image
$\kappa_{2}^{-1}(p(\kappa_{1}(\sigma_{i})))$ is an $n$-simplex in
$\Sigma_{2}$. Then, for $i=1,\ldots,j$, there exist a neighbourhood
$U_{i}\subset M$ of $\kappa_{1}(\sigma_{i})$, a neighbourhood
$U_{i}^{\prime}\subset N$ of $p(\kappa_{1}(\sigma_{i}))$, and a diffeomorphism
$\varphi_{i}\colon U_{i}\to U_{i}^{\prime}$ satisfying
$\varphi_{i}|_{\kappa_{1}(\sigma_{i})}=p|_{\kappa_{1}(\sigma_{i})}$ since the
maps $\kappa_{1}|_{\sigma_{i}}$ and
$\kappa_{2}|_{\kappa_{2}^{-1}(p(\kappa_{1}(\sigma_{i})))}$ are injective
smooth maps of rank $n$ and $\kappa_{2}^{-1}\circ p\circ\kappa_{1}$ maps
$\sigma_{i}$ smoothly to $\kappa_{2}^{-1}(p(\kappa_{1}(\sigma_{i})))$.

Let $\iota\colon N\to\mathbb{R}^{m}$ be a Nash embedding. Then $p\circ\iota\in
W^{1,\infty}(M,\mathbb{R}^{m})$ by Lemma B.2. Thus, $p\in W^{1,\infty}(M,N)$
and it remains to show that $p$ has finite distortion.

For almost every $x\in M$, we have $x\in\mathrm{int}\,\kappa(\sigma_{i})$ for
some $i$ and hence

$\frac{\lVert Dp(x)\rVert^{n}}{\det
Dp(x)}\leqslant\frac{\max_{\kappa(\sigma_{i})}\lVert
D\varphi_{i}\rVert^{n}}{\min_{\kappa(\sigma_{i})}\det D\varphi}.$

Since

$\max_{1\leqslant i\leqslant j}\frac{\max_{\kappa(\sigma_{i})}\lVert
D\varphi_{i}\rVert^{n}}{\min_{\kappa(\sigma_{i})}\det D\varphi}<\infty,$

we conclude that $p$ has finite distortion. This concludes the proof. ∎

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