# On groups and fields definable in $1$-h-minimal fields

Juan Pablo Acosta López  Department of Mathematics, Ben Gurion University of
the Negev, Be’er-Sheva 84105, Israel<EMAIL_ADDRESS>and  Assaf
Hasson  Department of Mathematics, Ben Gurion University of the Negev, Be’er-
Sheva 84105, Israel<EMAIL_ADDRESS>
###### Abstract.

We show that an infinite group $G$ definable in a $1$-h-minimal field admits a
strictly $K$-differentiable structure with respect to which $G$ is a (weak)
Lie group, and show that definable local subgroups sharing the same Lie
algebra have the same germ at the identity. We conclude that infinite fields
definable in $K$ are definably isomorphic to finite extensions of $K$ and that
$1$-dimensional groups definable in $K$ are finite-by-abelian-by-finite. Along
the way we develop the basic theory of definable weak $K$-manifolds and
definable morphisms between them.

The authors were supported by ISF grants No. 555/21.

##  1\. Introduction

Various Henselian valued fields are amenable to model theoretic study. Those
include the $p$-adic numbers (more generally, $p$-adically closed fields), and
(non-trivially) valued real closed and algebraically closed fields, as well as
various expansions thereof (e.g. by restricted analytic functions). Recently,
a new axiomatic framework for tame valued fields (of characteristic $0$) was
introduced. This framework, known as Hensel-minimality111In [4] and [5]
various notions of Hensel-minimality – $n$-h-minimality – for
$n\in{\mathbb{N}}\cup\\{\omega\\}$, were introduced. For the sake of clarity
of exposition, we will only discuss $1$-h-minimality., was suggested in [4]
and [5] as a valued field analogue of o-minimality. The notion of
$1$-h-minimality is both broad and powerful. Known examples include, among
others, all pure Henselian valued fields of characteristic $0$ as well as
their expansions by restricted analytic functions. Known tameness consequences
of $1$-h-minimality include a well-behaved dimension theory, and strong
regularity of definable functions (e.g., a generic Taylor approximation
theorem for definable functions).

In the present paper, we initiate a study of groups definable in $1$-h-minimal
fields. Using the above mentioned tameness and regularity conditions provided
by $1$-h-minimality and inspired by similar studies in the o-minimal setting
(initiated in [11]) and in $p$-adically closed fields ([12]) our first theorem
(Proposition 6.4, stated here in a slightly weaker form) is:

######  Theorem 1.

Let $K$ be a $1$-h-minimal field, $G$ an infinite group definable in $K$. Then
$G$ admits a definable weak ${\mathcal{C}}^{k}$ (any $k$) manifold structure
with respect to which $G$ has the structure of a strictly differentiable weak
${\mathcal{C}}^{k}$-Lie group. I.e., the forgetful functor from definable
strictly differentiable weak Lie groups to definable groups is an equivalence
of categories. If algebraic closure coincides with definable closure in $K$,
then a definable weak Lie group is a definable Lie group.

Above by a definable weak Lie group (over $K$) we mean a Lie group whose
underlying $K$-manifold structure may not have a definable (so, in particular,
finite) atlas but can be covered by (the domains of) finitely many compatible
étale maps. We do not know whether this is a necessary requirement for the
correctness of the statement, or an artifact of the proof: we follow Pillay’s
argument in the o-minimal and $p$-adic contexts ([11], [12]), but the fact
that in the present setting finite covers are not generically trivial,
requires that we work with weakly definable manifolds, in the above sense. To
pursue this argument, we have to extend the study of definable functions
beyond what was done in [4] (and its sequel). Specifically, instead of working
with continuously differentiable functions (as is the case in the o-minimal
setting) we are working with strictly differentiable functions, and for those
we prove an inverse function theorem, allowing us to deduce an implicit
function theorem for definable functions as well as other standard
consequences of these theorems.

We do not know whether strict differentiability follows in the $1$-h-minimal
context from continuous differentiability (as is the case in real analysis),
but it can be easily inferred from a multi-variable Taylor approximation
theorem for definable functions available in this context.

Having established that definable groups are Lie, our next theorem establishes
the natural Lie correspondence (asserting that the germ of a definable group
morphism at the identity is determined by its derivative at that point). For
applications it is convenient to state the result for local groups (Corollary
6.11):

######  Theorem 2.

Let $K$ be a $1$-h-minimal field, $U$ and $V$ definable strictly
differentiable local Lie groups and $g,f:U\to V$ definable strictly
differentiable local Lie group morphisms. If we denote $Z=\\{x\in
U:g(x)=f(x)\\}$, then $\dim_{e}Z=\dim(\ker(f^{\prime}(e)-g^{\prime}(e)))$.

We then prove two applications. First, we show – adapting techniques from the
o-minimal context – that every infinite field definable in a $1$-h-minimal
field, $K$, is definably isomorphic to a finite extension of $K$, Proposition
7.3. This generalizes an analogous result for real closed valued fields ([2])
and $p$-adically closed fields ([12]). It will be interesting to know whether
these results can be extended to _interpretable_ fields (in the spirit of [6]
or [9, §6] under suitable additional assumptions on the RV-sort.

Our next application is a proof that definable $1$-dimensional groups are
finite-by-abelian-by-finite, Corollary 8.10. This generalizes analogous
results in the o-minimal context ([11]), in $p$-adically closed fields ([12])
and combines with [1] to give a complete classification of $1$-dimensional
groups definable in ACVF0.

The present paper is a first step toward the study of groups definable in
$1$-h-minimal fields. It seems that more standard results on Lie groups over
complete local fields can be extended to this context. Thus, for example, it
can be shown that any definable local group contains a definable open
subgroups. As the proof is long and involves new techniques we postpone it to
a subsequent paper.

###  1.1. Structure of the paper

In Section 2 we review the basics of $1$-h-minimality and dimension theory in
geometric structures. In Section 3 we prove a multi-variable Taylor
approximation theorem for $1$-h-minimal fields, and formulate some strong
regularity conditions (implied, generically, by Taylor’s theorem) that will be
needed in later parts of the paper. These results are, probably, known to the
experts, and we include them mostly for the sake of completeness and clarity
of exposition (as some of them do not seem to exist in writing).

In Section 4 we prove the inverse function theorem and related theorems on the
local structure of immersions, submersions and constant rank functions. Though
some of the proofs are similar to those of analogous statements in real
analysis (and, more generally, in the o-minimal context) this is not true
throughout. Specifically, $1$-h-minimality is invoked in a crucial way in the
proof that a function with vanishing derivative is locally constant, which –
in turn – is used in our proof of the Lie correspondence for definable groups.

Using the results of the first sections, our study of definable groups starts
in Section 6. We first show that definable groups can be endowed with an,
essentially unique, strictly differentiable weak Lie group structure, and that
the germ of definable group morphisms are determined by their derivative at
the identity. We then define the (definable) Lie algebra associated with a
definable Lie group, and show that it satisfies the familiar properties of Lie
algebras. This is done using a local computation, after characterizing the Lie
bracket as the second order part of the commutator function near the identity.

Section 7 is dedicated to the classification of fields definable in
$1$-h-minimal fields, and in Section 8 we prove our results on definable one
dimensional groups.

##  2\. Preliminaries

In this section we set some background definitions, notation and describe
basic relevant results, used in later sections. Most of the terminology below
is either standard or taken from [4]. Throughout, $K$ will denote a non-
trivially valued field. We will not distinguish, notationally, between the
structure and its universe. Formally, we allow $K$ to be a multi-sorted
structure (with all sorts coming from from $K^{eq}$), but by a definable set
we mean (unless explicitly stated otherwise) a subset of $K^{n}$ definable
with parameters. All tuples are finite, and we write (as is common in model
theory) $a\in K$ for $a\in K^{n}$ for $n=\mathrm{length}(a)$. We apply the
same convention to variables.

To stress the analogy of the current setting with the Real numbers we use
multiplicative notation for the valuation. Thus, the valued group is denoted
$(\Gamma,\cdot)$ and the valuation $|\cdot|:K\to\Gamma_{0}=\Gamma\cup\\{0\\}$,
and if $x\in K^{n}$ we set $|x|:=\max_{1\leq k\leq n}|x_{k}|$.

An open ball of (valuative) radius $r\in\Gamma$ in $K^{n}$ is a set of the
form $B=\\{x\in K^{n}:|x-a|<r\\}$ for $a\in K^{n}$. The balls endow $K$ with a
field topology (the valuation topology). Up until Section 6 all topological
notions mentioned in the text will refer solely to this topology.

We denote $\mathcal{O}:=\\{x:|x|\leq 1\\}$, the valuation ring,
$\mathcal{M}:=\\{x\in{\mathcal{O}}:|x|<1\\}$, the valuation ideal, and
$k:={\mathcal{O}}/{\mathcal{M}}$, the residue field. We also denote
$RV=K^{\times}/1+\mathcal{M}$. More generally, whenever $s\in\Gamma$ and
$s\leq 1$, we denote $\mathcal{M}_{s}=\\{x\in K:|x|<s\\}$, and
$RV_{s}=K^{\times}/1+\mathcal{M}_{s}$. If $K$ has mixed characteristic
$(0,p)$, we denote $RV_{p,n}=RV_{|p|^{n}}$ and
$RV_{p,\bullet}=\bigcup_{n}RV_{p,n}$.

It is convenient, when discussing approximation theorems, to adopt the big-O
notation from real analysis. For the sake of clarity we recall this notation
in the valued field setting:

######  Definition 2.1.

  1. (1)

If $f:U\to K^{m}$ and $g:U\to\Gamma_{0}$ are functions defined in an open
neighborhood of $0$ in $K^{n}$, then $f(x)=O(g(x))$ means that there are
$r,M>0$ in $\Gamma$, such that if $|x|<r$ then $|f(x)|\leq Mg(x)$. We also
denote $f_{1}(x)=f_{2}(x)+O(g(x))$ if $f_{1}(x)-f_{2}(x)=O(g(x))$.

  2. (2)

If $g:U\to K^{r}$, and $s\in\mathbb{N}$, then $O(g(x)^{s})=O(|g(x)|^{s})$.

  3. (3)

If $f:Y\times U\to K^{m}$, is a function where $U$ is an open neighborhood of
$0$ in $K^{n}$, and if $g:U\to\Gamma_{0}$, then $f(y,x)=O_{y}(g(x))$ means
that for every $y\in Y$, there are $r_{y},M_{y}>0$, such that if $|x|<r_{y}$
then $|f(y,x)|\leq M_{y}g(x)$.

As mentioned in the introduction, in the present paper we are working with the
notion of strict differentiability, which we now recall:

######  Definition 2.2.

Let $U\subset K^{n}$ be an open subset and $f:U\to K^{m}$ be a map. Then $f$
is strictly differentiable at $a\in U$ if there is a linear map $A:K^{n}\to
K^{m}$ such that for every $\epsilon>0$, there exists $\delta>0$ satisfying
$|f(x)-f(y)-A(x-y)|\leq\epsilon|x-y|$ for every $x,y$ such that $|x-a|<\delta$
and $|y-a|<\delta$.

$f$ is strictly differentiable in $U$ if it is strictly differentiable at
every point of $U$.

In the situation of the definition the linear map $A$ is uniquely determined
and denoted $f^{\prime}(a)$. If $f$ is strictly differentiable in an open $U$,
then it is continuously differentiable.

######  Definition 2.3.

Let $U\subset K^{n}$ and $V\subset K^{n}$ be open subsets. Then $f:U\to V$ is
a strict diffeomorphism if it is strictly differentiable, bijective and its
inverse is strictly differentiable.

As we will see, a strict diffeomorphism is just a strictly differentiable
diffeomorphism.

Given an open ball $B\subseteq K^{n}$ of radius $r$, a subset $Y$ of $K^{n}$,
and an element $s\in\Gamma$ with $s\leq 1$, we say that $B$ is $s$-next to $Y$
if $B^{\prime}\cap Y=\emptyset$ for $B^{\prime}$ the open ball of radius
$s^{-1}r$ containing $B$.

Note that every point not in the closure of $Y$ is contained in a ball
$s$-next to $Y$. This is because if $B$ is an open ball of radius $r$ disjoint
from $Y$, then every open ball of radius $sr$ contained in $B$ is $s$-next to
$Y$.

Following [4] we say that a finite set $Y\subset K$ prepares the set $X\subset
K$, if for every ball, $B$, disjoint from $Y$ is either disjoint from $X$ or
contained in $X$. More generally, if $s\in\Gamma$ is such that $s\leq 1$, then
$Y$ $s$-prepares $X$ if every open ball $B$ $s$-next to $Y$ is either
contained in $X$ or disjoint from $X$.

If $K$ is a valued field of mixed characteristic $(0,p)$, given an integer
$m\in\mathbb{N}$, an open ball, $B\subseteq K^{n}$, and a set $Y\subseteq
K^{n}$, we say that $B$ is $m$-next to $Y$ if it is $|p|^{m}$-next to $Y$.
Similarly, if $s\in\Gamma$ and $s\leq 1$ then $B$ is $m$-$s$-next to $Y$ if it
is $|p|^{m}s$-next to $Y$. Given a finite $Y\subset K$ and $X\subset K$, we
say that $Y$ $m$-prepares (resp. $m$-$s$-prepares) the set $X$ if $Y$
$|p|^{m}$-prepares $X$ (resp. $Y$ $|p|^{m}s$-prepares $X$).

Next, we recall the definitions of 1-h-minimality defined in the equi-
characteristic $0$ ([4]) and in mixed characteristic ([5]) settings:

######  Definition 2.4.

Let $K$ be an $\aleph_{0}$-saturated non-trivially valued field of
characteristic $0$, which is a structure in a language extending the language
of valued fields.

  1. (1)

If $K$ has residue characteristic $0$ then $K$ is $1$-h-minimal, if for any
$s\leq 1$ in $\Gamma$ any $A\subseteq K$, $A^{\prime}\in RV_{s}$ (a singleton)
and every $(A\cup RV\cup A^{\prime})$-definable set $X\subset K$, there is an
$A$-definable finite set $Y\subset K$ $s$-preparing $X$.

  2. (2)

If $K$ has mixed characteristic $(0,p)$ then $K$ is $1$-h-minimal, if for any
$s\leq 1$ in $\Gamma$ any $A\subseteq K$, $A^{\prime}\in RV_{s}$ (a singleton)
and every $(A\cup RV_{p,\bullet}\cup A^{\prime})$-definable set $X\subset K$,
there is $m\in{\mathbb{N}}$ and an $A$-definable finite set $Y\subset K$ which
$m$-$s$-prepares $X$.

In the sequel, when appealing directly to the definition, we will only need
the case $s=1$ (so $A^{\prime}$ does not appear). The parameter $s$ does
appear implicitly, though, when applying properties of $1$-h-minimality such
as generic continuity of definable functions (see [4, Proposition 5.1.1]).

Below we will need to study properties of “one-to-finite definable functions”
(definable correspondences, in the terminology of [15]). It turns out that
statements regarding such objects can sometimes be reduced to statements on
definable functions in expansions of the language by algebraic Skolem
functions (i.e., Skolem functions for definable finite sets). For this, the
following will be convenient (see [4, Proposition 4.3.3], and [5, Proposition
3.2.2]):

######  Fact 2.5.

Suppose $K$ is a 1-h-minimal valued field. Then there exists a language
$\mathcal{L}^{\prime}\supseteq\mathcal{L}$, an elementary extension
$K^{\prime}$ of $K$, and an $\aleph_{0}$-saturated
$\mathcal{L}^{\prime}$-structure on $K^{\prime}$ extending the
$\mathcal{L}$-structure of $K^{\prime}$, such that $K^{\prime}$ is 1-h-minimal
as an $\mathcal{L}^{\prime}$-structure, and such that
$\operatorname{acl}_{\mathcal{L}^{\prime}}(A)=\operatorname{dcl}_{\mathcal{L}^{\prime}}(A)$
for all $A\subseteq K^{\prime}$.

Above and throughout, algebraic and definable closures are always assumed to
be taken in the $K$ sort. In the sequel we will refer to the property
appearing in the conclusion of Fact 2.5 simple as
"$\operatorname{acl}=\operatorname{dcl}$".

######  Remark 2.6.

Given an ${\mathcal{L}}$-definable set, $S$, statements concerning topological
or geometric properties of $S$ are often expressible by first-order
${\mathcal{L}}$-formulas. As the topology on $K$ is definable in the valued
field language, and the dimension of definable sets in $1$-minimal minimal
structure is determined by the topology (see Proposition 2.11), the truth
values of the hypothesis and conclusion of such statements (for our fixed
${\mathcal{L}}$-definable set $S$) are the same in $K$ and in any elementary
extension $K\prec K^{\prime}$, as well as in any $1$-h-minimal expansion of
the latter. Therefore, by Fact 2.5, in the proof of such statements (for a
fixed definable $S$) there is no harm assuming
$\operatorname{acl}=\operatorname{dcl}$.

###  2.1. Geometric structures

Geometric structures were introduced in [8, §2]. Let us recall the definition:
An $\aleph_{0}$-saturated structure, $M$ is _pregeometric_ if
$\operatorname{acl}(\cdot)$ is a matroid, that is, it satisfies the exchange
property:

$\text{if }a\in\operatorname{acl}(Ab)\setminus\operatorname{acl}(A),\text{
then }b\in\operatorname{acl}(Aa)\text{ for singletons }a,b\in M.$

In this situation the matroid gives a notion of dimension, $\dim(a/b)$, the
dimension of a tuple $a$ over a tuple $b$ as the smallest length of a sub-
tuple $a^{\prime}$ of $a$ such that $a\in\operatorname{acl}(a^{\prime}b)$, and
the dimension of a $b$-definable set $X$ as the maximum of the dimensions
$\dim(a/b)$ with $a\in X$ (this does not depend on $b$). As is customary, we
set $\dim(\emptyset)=-\infty$. We recall the basic properties of dimension
(see [8, §2] for all references). This dimension satisfies the additivity
property

$\dim(ab/c)=\dim(a/bc)+\dim(b/c),$

that we will invoke without further reference. We call $a$ and $b$
algebraically independent over $c$ if $\dim(a/bc)=\dim(a/c)$. Note that by
additivity of dimension this is a symmetric relation. Note also that
additivity implies that if $b,c$ are inter-algebraic over $a$, meaning
$b\in\operatorname{acl}(ac)$ and $c\in\operatorname{acl}(ab)$, then
$\dim(b/a)=\dim(c/a)$ (in particular, this holds when $c$ is the image of $b$
under an $a$-definable bijection). If $M$ is a pregeometric structure and
$f:X\to Y$ is a surjective definable function with fibers of constant
dimension $k$, then $\dim(X)=\dim(Y)+k$. This is a consequence of the
additivity formula.

Given an $a$-definable set $X$, a generic element of $X$ over $a$ is an
element $b\in X$ such that $\dim(b/a)=\dim(X)$. By compactness generic
elements can always be found in saturated enough models. We call $Y\subset X$
large if $\dim(X\setminus Y)<\dim(X)$. This is equivalent to $Y$ containing
every generic point of $X$.

A pregeometric structure, $M$, eliminating the quantifier $\exists^{\infty}$
is called geometric. If $M$ is geometric dimension is definable in definable
families. Namely, for $\\{X_{a}\\}_{a\in S}$, a definable family, the set
$\\{a\in S:\dim(X_{a})=k\\}$ is definable.

The following simple fact is a translation of the definition of a pregeometry
to a property of definable sets. Note as an aside that this reformulation
implies that the property of being a pregeometry is preserved under reducts.
I.e., if $M$ is an $\aleph_{0}$-saturated pregeometric
$\mathcal{L}^{\prime}$-structure, and
$\mathcal{L}\subset\mathcal{L}^{\prime}$, then $M$ is also a pregeometric
$\mathcal{L}$-structure. For the sake of completeness, we give the proof:

######  Fact 2.7.

Suppose $M$ is an $\aleph_{0}$-saturated structure. Then $M$ is pregeometric
if and only if for every definable $X\subset M\times M$ if the projection,
$\pi_{1}:X\to M$, into the first factor is finite-to-one, and $\pi_{2}:X\to M$
is the projection into the second factor, then the set $Y=\\{c\in
M:\pi_{2}^{-1}(c)\cap X\text{ is infinite}\\}$ is finite.

###### Proof.

Suppose $M$ is pregeometric and suppose $X\subset M\times M$ is $A$-definable
such that $\pi_{1}^{-1}(x)\cap X$ is finite for all $x\in M$. Suppose also
that $Y=\\{y\in M:\pi_{2}^{-1}(y)\cap X\text{ is infinite}\\}$ is infinite. By
compactness and saturation, we can choose $b\in Y$ such that $\dim(b/A)=1$.
Similarly, we can find $a\in\pi_{2}^{-1}(b)\cap X$ such that $\dim(a/Ab)=1$.
We conclude that $\dim(ab/A)=2$, and so $\dim(X)\geq 2$. This contradicts the
fact that $\pi_{1}^{-1}(x)\cap X$ is finite for all $x\in M$.

For the converse, suppose $A$ is a finite subset of $M$ and $a,b\in M$ are
singletons such that
$a\in\operatorname{acl}(Ab)\setminus\operatorname{acl}(A)$. Then there is an
$A$-definable set $X\subset M\times M$ such that $(b,a)\in X$ and
$\pi_{1}^{-1}(b)\cap X$ is finite, say of cardinality $k$. If we take
$Z=\\{c\in M:\pi_{1}^{-1}(c)\cap X\text{ has cardinality }k\\}$, then we may
replace $X$ by $X\cap Z\times M$, and we may assume that $\pi_{1}^{-1}(c)\cap
X$ is either empty or of constant finite cardinality for all $c\in M$. In this
case, by the hypothesis we conclude that $Y=\\{y\in M:\pi_{2}^{-1}(y)\cap
M\text{ is infinite}\\}$ is finite. Note that $Y$ is $A$-invariant and
definable, so it is $A$-definable. We conclude that $a\notin Y$, because
$a\notin\operatorname{acl}(A)$, and so $b\in\operatorname{acl}(Aa)$ as
required. ∎

The next characterization of the $\operatorname{acl}$-dimension should be well
known:

######  Fact 2.8.

Suppose $M$ is an $\aleph_{0}$-saturated structure, which eliminates the
$\exists^{\infty}$ quantifier. Suppose there is a function, $X\mapsto d(X)$,
from the non-empty definable subsets of (cartesian powers of) $M$ into
$\mathbb{N}$ satisfying:

  1. (1)

If $X\subset M^{n}\times M$ is such that the first coordinate projection
$\pi_{1}:X\to M^{n}$ is finite to one, then $d(X)=d(\pi_{1}(X))$.

  2. (2)

If $X\subset M^{n}\times M$ is such that the first coordinate projection
$\pi_{1}:X\to M^{n}$ has infinite fibers, then $d(X)=d(\pi_{1}(X))+1$.

  3. (3)

If $\pi:M^{n}\to M^{n}$ is a coordinate permutation, then $d(X)=d(\pi(X))$.

  4. (4)

$d(X\cup Y)=\max\\{d(X),d(Y)\\}$.

  5. (5)

$d(M)=1$

  6. (6)

$d(X)=0$ if and only if $X$ is finite.

Then $M$ is a geometric structure and $d$ coincides with its
$\operatorname{acl}$-dimension.

###### Proof.

It suffices to show that $M$ is pregeometric. We use Fact 2.7. Let $X\subset
M\times M$ be such that $\pi_{1}^{-1}(x)\cap X$ is finite for all $x\in M$.
Take $Y=\\{y\in M:\pi_{2}^{-1}(y)\cap X\text{ is infinite}\\}$. Because $M$
eliminates the $\exists^{\infty}$ quantifier we have that $Y$ is definable. If
$Y$ is infinite we conclude that $d(X)\geq d(X\cap\pi_{2}^{-1}(Y))=d(Y)+1=2$.
The first inequality by item (4), the second equality by items (3) and (2),
and the third by item (6). On the other hand $d(X)=d(\pi_{1}(X))\leq d(M)=1$,
the first equality by item (1), the second inequality by item (4) and the
third by item (5). This is a contradiction and finishes the proof.

In order to see that $d(X)=\dim(X)$ for $X\subset M^{n}$ we may proceed by
induction on $n$. The base case $n=1$ follows from item (4), (5) and (6). So
suppose that $X\subset M^{n}\times M$. Denote $Y=\\{x\in
M^{n}:\pi_{1}^{-1}(x)\cap X\text{ is infinite}\\}$. By hypothesis $Y$ is a
definable set. Denote $X_{1}=\pi_{1}^{-1}(Y)\cap X$ and $X_{2}=X\setminus
X_{1}$. Then by items (1),(2) and (4) we conclude that
$d(X)=\max\\{d(X_{1}),d(X_{2})\\}=\max\\{d(Y)+1,d(\pi_{1}(X)\setminus Y)\\}$.
For the same reason we have the formula
$\dim(X)=\max\\{\dim(Y)+1,\dim(\pi_{1}(X)\setminus Y)\\}$, so $d(X)=\dim(X)$
as required. ∎

The next fact is also standard:

######  Fact 2.9.

Suppose $M$ is a geometric structure. Suppose $X\subset M^{n}$ is
$a$-definable. Then there is a partition of $X$ into a finite number of
$a$-definable sets $X=X_{1}\cup\cdots\cup X_{n}$, such that for each member of
the partition $X_{k}$, there is a coordinate projection $\pi:X_{k}\to M^{r}$
which is finite to one and has image of dimension $r$.

######  Remark 2.10.

For this statement we need to allow the identity $\mathrm{id}:M^{n}\to M^{n}$
as a coordinate projection.

Also, recall that $M^{0}$ is a set consisting of one element. For this
statement we also need to allow the constant function $M^{n}\to M^{0}$ as a
coordinate projection.

###### Proof.

By induction on the dimension of the ambient space $n$. Consider the
projection onto the first $n-1$ coordinates $\pi_{1}:M^{n}\to M^{n-1}$. Then
the set $Y\subset M^{n-1}$ of $y$ such that the fibers
$X_{y}=\pi_{1}^{-1}(y)\cap X$ are infinite is definable. So partitioning $X$
we may assume all the nonempty fibers of $X$ over $M^{n-1}$ are finite, or all
are infinite. If all the fibers of $X\to K^{n-1}$ are finite then we finish by
induction.

If all the nonempty fibers are infinite then by the induction there is a
partition $Y=\bigcup_{i}Y_{i}$ and for each $Y_{i}$ there is a coordinate
projection $\tau:Y_{i}\to K^{r}$ with finite fibers and $r=\dim(Y_{i})$.
Denote $\pi_{2}:M^{n}\to M$ the projection onto the last coordinate. Then
setting $X_{i}=X\cap\pi_{1}^{-1}(Y_{i})$, the projection
$\pi(x)=(\tau(\pi_{1}(x)),\pi_{2}(x))$ has the desired properties. ∎

The next proposition is key. It asserts that $1$-h-minimal fields are
geometric, and it connects (combined with the previous fact) topology and
dimension in such structures:

######  Proposition 2.11.

Suppose $K$ is a 1-h-minimal valued field. Then:

  1. (1)

$K$ is a geometric structure.

  2. (2)

Every $X\subset K^{n}$ satisfies $\dim(X)=n$ if and only if $X$ has nonempty
interior. Every $X\subset K^{n}$ satisfies $\dim(X)<n$ if and only if $X$ is
nowhere dense.

  3. (3)

For $X\subset K^{n}$, we have $\dim(X)=\max\limits_{x\in X}\dim_{x}(X)$, where
we denote $\dim_{x}(X)$ the local dimension of $X$ at $x$, defined as
$\dim_{x}(X)=\min\\{\dim(B\cap X):x\in B\text{ is an open ball}\\}$

###### Proof.

This is essentially items (1)-(5) of [4, Proposition 5.3.4] in residue
characteristic $0$ and contained in [5, Proposition 3.1.1] in mixed
characteristic.

For example, assume $K$ has residue characteristic $0$. That $K$ is geometric
is proved in the course of the proof of [4, Proposition 5.3.4]. We can also
derive it from Fact 2.8 and [4, Proposition 5.3.4].

The topological characterization $\dim(X)=n$ if and only if $X$ has nonempty
interior, is a particular case of item (1) in [4, Proposition 5.3.4]. That
$\dim(X)<n$ if and only if $X$ is nowhere dense follows from this. Indeed, if
$\dim(X)=n$, then $X$ has nonempty interior and so it is not nowhere dense. If
$\dim(X)<n$ and $U\subset K^{n}$ is nonempty open, then $\dim(U\setminus
X)=n$, and so $U\setminus X$ has nonempty interior. This implies $X$ is
nowhere dense.

That dimension is the maximum of the local dimensions is item (5) of
Proposition 5.3.4 of [4] ∎

######  Proposition 2.12.

Suppose $K$ is a 1-h-minimal field. Suppose $f:U\to K^{m}$ is a definable
function. Then there is a definable open dense subset $U^{\prime}\subset U$
such that $f:U^{\prime}\to K^{m}$ is continuous.

###### Proof.

This is essentially a particular case of [4, Proposition 5.1.1] in residue
characteristic $0$, and contained in [5, Proposition 3.1.1] in mixed
characteristic.

Indeed, because the intersection of open dense sets is open and dense we
reduce to the case $m=1$. From those propositions one gets that the set $Z$ of
points where $f$ is continuous is dense in $U$. As $Z$ is not nowhere dense we
conclude using item (2) of Proposition 2.11 that $\dim(Z)=n$ and so $Z$ has
nonempty interior. If $V\subset U$ is a nonempty open definable subset then,
as $Z\cap V$ is the set of points at which $f|_{V}$ is continuous, by what we
just proved $Z\cap V$ has nonempty interior. We conclude that the set of
points at which $f$ is continuous has a dense interior in $U$, as desired. ∎

Next we describe a topology for $Y^{[s]}$, the set of subsets of $Y$ of
cardinality $s$, for $Y$ a Hausdorff topological space, and $s$ a positive
integer. We prove a slightly more general statement that will be applied when
$X$ is $Y^{s}\setminus\Delta$, the set of tuples of $Y^{s}$ with distinct
coordinates and the symmetric group, $S_{s}$, on $s$ elements acting on
$Y^{s}$ by coordinate permutation, in which case the orbit space is identified
with $Y^{[s]}$.

######  Fact 2.13.

Suppose $X$ is a Hausdorff topological space, and $G$ a finite group acting on
$X$ by homeomorphisms, and such that every $x\in X$ has a trivial stabilizer
in $G$. Then $X/G$ equipped with the quotient topology is Hausdorff and the
map $p:X\to X/G$ is a closed finite covering map. In fact for every $x\in X$
there is an open set $x\in U\subset X$ such that $\\{gU:g\in G\\}$ are
pairwise disjoint, $p^{-1}p(U)=\bigcup_{g}gU$ and $p|_{gU}$ is a homeomorphism
onto $p(U)$.

###### Proof.

We know that $p$ is open, since $p^{-1}p(U)=\bigcup_{g\in G}gU$ is open for
$U$ open. Consider the orbit $\\{gx\\}_{g\in G}$ of $x$. By assumption, if
$g\neq h$, then $gx\neq hx$. Let $V$ be an open set in $X$ containing
$\\{gx\\}_{g\in G}$. Now, because $X$ is Hausdorff, we conclude that there are
$U_{g}$ open neighborhoods of $gx$, contained in $V$, such that $U_{g}\cap
U_{h}=\emptyset$ for $g\neq h$. If we take $U=\bigcap_{g\in G}g^{-1}U_{g}$,
then $gU\subset U_{g}$ and so $\\{gU:g\in G\\}$ are pairwise disjoint. We
conclude that $p$ is closed and restricted to $gU$ is a homeomorphism. That
$X/G$ is Hausdorff now follows from this. Indeed, if $p(x)\neq p(y)$, then
there are open sets $V_{1}$ and $V_{2}$ of $X$, which are disjoint and such
that $p^{-1}p(x)\subset V_{1}$ and $p^{-1}p(y)\subset V_{2}$. Because $p$ is
closed there are open sets $p(x)\in U_{1}$ and $p(y)\in U_{2}$ in $X/G$ such
that $p^{-1}(U_{i})\subset V_{i}$. We conclude that $U_{1}$ and $U_{2}$ are
disjoint. ∎

######  Proposition 2.14.

Let $K$ be a 1-h-minimal valued field. Suppose $U\subset K^{n}$ is open and
$f:U\to(K^{r})^{[s]}$ is definable. Then there is an open dense definable set
$U^{\prime}\subset U$ such that $f$ is continuous in $U^{\prime}$.

###### Proof.

This statement is equivalent to saying that the interior of the set of points
on which $f$ is continuous is dense. As this property is expressible by a
first order formula we may assume $\operatorname{acl}=\operatorname{dcl}$, see
Fact 2.5 and the remark following it.

In that case we have a definable section $s:(K^{r})^{[s]}\to K^{rs}$, and if
$V\subseteq U$ is open dense such that $sf$ is continuous, as provided by
Proposition 2.12, then $f$ is continuous in $V$. ∎

######  Proposition 2.15.

Suppose $X\subset K^{n}$ is $b$-definable. Then there is finite partition of
$X$ into $b$-definable sets, such that for each element $Y$ of the partition
there is a coordinate projection $\pi:Y\to U$ onto an open set $U\subset
K^{m}$, such that the fibers of $\pi$ all have the same cardinality equal to
$s$, and the associated map $f:U\to(K^{n-m})^{[s]}$ is continuous.

######  Remark 2.16.

As in Remark 2.10, we need to allow the two cases $m=0$ and $m=n$. The set
$K^{0}$ consists of a single point, and has a unique topology.

###### Proof.

This is a consequence of dimension theory and the previous observation. In
more detail, we proceed by induction on the dimension of $X$.

First, recall that $X$ has a finite partition into $b$-definable sets such
that for each set $X^{\prime}$ in the partition there is a coordinate
projection $\pi:X^{\prime}\to K^{r}$ with finite fibers and
$r=\dim(X^{\prime})$, see Fact 2.9.

So now assume $\pi:X\to K^{r}$ is a coordinate projection with finite fibers
and $r=\dim(X)$, and denote $\pi^{\prime}:X\to K^{n-r}$ the projection into
the other coordinates. There is an integer $s$ which bounds the cardinality of
the fibers of $\pi$. If we denote $Y_{k}$ the set of elements $a\in K^{r}$
such that $X_{a}=\pi^{\prime}(\pi^{-1}(a))$ has cardinality $k$ then we get
$Y_{0}\cup\cdots\cup Y_{s}=K^{r}$. Now let $V_{j}\subset Y_{j}$ be open dense
in the interior of $Y_{j}$ and such that the map $V_{j}\to(K^{n-r})^{[j]}$
given by $a\mapsto X_{a}$ is continuous, see Proposition 2.14. Then the set
$\\{x\in X:\pi(x)\in Y_{j}\setminus V_{j},1\leq j\leq s\\}$ is of lower
dimension than $X$, by item 2 of Proposition 2.11, and so we may apply the
induction hypothesis on it. ∎

Recall that a subset $Y\subset X$ of a topological space $X$ is locally closed
if it is the intersection of an open set and a closed set. This is equivalent
to $Y$ being relatively open in its closure. It is also equivalent to, for
every point $y\in Y$, the existence of a neighborhood $V$ of $y$, such that
$Y\cap V$ is relatively closed in $V$.

######  Proposition 2.17.

Suppose $K$ is 1-h-minimal and $X\subset K^{n}$ an $a$-definable set. Then $X$
is a finite union of $a$-definable locally closed subsets of $K^{n}$.

###### Proof.

This is a consequence of Proposition 2.15. Namely, there is a partition of $X$
into a finite union of definable subsets for each of which there is a
coordinate projection with finite fibers onto an open set $U$, so we may
assume $X$ is of this form. We may further assume that the fibers have
constant cardinality $k$ and the associated mapping $U\to(K^{r})^{[k]}$ is
continuous. Then $X$ is closed in $U\times K^{r}$ and so locally closed. ∎

We finish by reviewing a more difficult property of dimension. We will only
use this in Proposition 5.20, Proposition 6.9 and Corollary 6.14, which are
not used in the main theorems.

######  Proposition 2.18.

Suppose $K$ is a 1-h-minimal field and $X\subset K^{n}$. Then
$\dim(\operatorname{cl}(X)\setminus X)<\dim(X)$.

This is item 6 of [4, Proposition 5.3.4] for the residue characteristic $0$
and it is contained in Proposition 3.1.1 of [5] in the mixed characteristic
case.

##  3\. Taylor approximations

In this section we show that, in the $1$-h-minimal setting, the generic one
variable Taylor approximation theorem ([5, Theorem 3.1.2]) implies a multi-
variable version of the theorem. In equicharacteristic $(0,0))$ this is [4,
Theorem 5.6.1]. Though the proof in mixed characteristic is, essentially,
similar; we give the details, for the sake of completeness and in view of the
importance of this result in the sequel.

We then proceed to introducing some regularity conditions for definable
functions (implied, in the present context, by Taylor’s approximation theorem)
necessary for computations related to the Lie algebra of definable groups.

First, we recall the multi-index notation. If
$i=(i_{1},\dots,i_{n})\in\mathbb{N}^{n}$, we denote $|i|=i_{1}+\cdots+i_{n}$
and $i!=i_{1}!\cdots i_{n}!$. For $x=(x_{1},\cdots,x_{n})\in K^{n}$ we denote
$x^{i}=x_{1}^{i_{1}}\cdots x_{n}^{i_{n}}$. Also if $f:U\to K$ is a function
defined in an open set of $K^{n}$, we denote
$f^{(i)}(x)=(\frac{\partial^{i_{1}}}{\partial
x_{1}^{i_{1}}}\cdots\frac{\partial^{i_{n}}}{\partial x_{n}^{i_{n}}}f)(x)$
whenever it exists. Note that we are not assuming equality of mixed
derivatives, but see Corollary 3.6.

######  Proposition 3.1.

Let $K$ be a 1-h-minimal field of residue characteristic $0$. Suppose $f:U\to
K$ is an $a$-definable function with $U\subset K^{n}$ open and let
$r\in{\mathbb{N}}$. Then there is an $a$-definable set $C$, of dimension
strictly smaller than $n$, such that for any open ball, $B\subseteq U$
disjoint from $C$ the derivative $f^{(i)}$ exists in $B$ for every $i$ with
$|i|\leq r$, and has constant valuation in $B$. Moreover,

$\left|f(x)-\sum_{\\{i:|i|<r\\}}\frac{1}{i!}f^{(i)}(x_{0})(x-x_{0})^{i}\right|\leq\max_{\\{i:|i|=r\\}}\left|\frac{1}{i!}f^{(i)}(x_{0})(x-x_{0})^{i}\right|$

For every $x,x_{0}\in B$.

This is [4, Theorem 5.6.1]. Our first order of business is to generalize this
result to positive residue characteristic.

The following fact is proved by a standard compactness argument, and is often
applied implicitly. We add this argument for convenience.

######  Fact 3.2.

Let $M$ be an $\aleph_{0}$-saturated structure, and
$\\{\Phi^{l}(\bar{D})\\}_{l\in I}$ be a family of properties of definable sets
$\bar{D}=(D_{1},\dots,D_{n})$ in $M$, indexed by a directed set $I$. Let $b$
be a tuple in $M$, and $S$ be a $b$-definable set. Assume that

  1. (1)

For all $l$ the property $\Phi^{l}$ is definable in definable families. I.e,
if $\\{D_{i,a}\\}_{a\in T}$ are $b$-definable families, then the set $\\{a\in
T:\Phi^{l}({\bar{D}_{a}})\text{ holds}\\}$ is $b$-definable.

  2. (2)

$\Phi^{l}$ implies $\Phi^{l^{\prime}}$ for all $l\leq l^{\prime}$.

  3. (3)

For every $a\in S$, there are $ba$-definable sets $D_{i,a}$, satisfying
$\Phi^{l_{a}}$ for some $l_{a}\in I$.

Then there are $\\{D_{i,a}\\}_{a\in S}$ $b$-definable families of sets, and a
fixed $l\in I$, such that $\Phi^{l}(\bar{D}_{a})$ holds for every $a\in S$.

######  Remark 3.3.

Formally, $\Phi^{l}$ is a subset of

$\\{(D_{1},\dots,D_{n}):D_{i}\text{ is a definable set and}\\}$

and we say $\Phi^{l}(D_{1},\dots,D_{n})$ holds if the tuple
$(D_{1},\cdots,D_{n})$ belongs to $\Phi^{l}$.

Note also that the tuple $(D_{1},\dots,D_{n})$ can be replaced with
$D_{1}\times\cdots\times D_{n}$, so there is no loss of generality in taking
$\Phi^{l}$ of the form $\Phi^{l}(D)$.

###### Proof.

Let $a\in S$. By hypothesis there are $b$-definable families
$\\{D_{i,a^{\prime}}^{a}\\}_{a^{\prime}\in S_{i}^{0,a}}$ and an element
$l^{a}\in I$, such that $(D_{1,a}^{a},\dots,D_{n,a}^{a})$ satisfies
$\Phi^{l^{a}}$. Consider $S^{a}$ to be the set of $a^{\prime}\in S$ such that
$a^{\prime}\in S_{i}^{0,a}$ for $i=1,\dots,n$, and such that
$\Phi^{l^{a}}(\bar{D}_{a^{\prime}}^{a})$ holds. By hypothesis, this is a
$b$-definable set contained in $S$ and containing $a$.

We conclude that $S=\bigcup_{a\in S}S^{a}$ is a cover of $S$ by $b$-definable
sets, and so by compactness and saturation there is a finite sub-cover, say
$S=S^{1}\cup\dots\cup S^{k}$ for $S^{r}=S^{a_{r}}$. Indeed, if there was no
finite sub-cover, then the partial type expressing $x\in S$ and $x\notin
S^{a}$ for all $a\in S$, is a consistent $b$-type, and so a realization in $M$
would contradict $S=\bigcup_{a\in S}S^{a}$.

Then $D_{i,a}$ defined as $D_{i,a}^{a_{r}}$ if $a\in
S^{r}\setminus\bigcup_{r^{\prime}<r}S^{r^{\prime}}$ satisfy that
$\\{D_{i,a}\\}_{a\in S}$ form $b$-definable families. If we take $l$ such that
$l\geq l^{a_{1}},\dots,l^{a_{k}}$, then we get that $\Phi^{l}(\bar{D}_{a})$
holds for every $a\in S$, as required. ∎

######  Notation 3.4.

If $D\subset E\times F$, and $a\in E$ we often denote $D_{a}=\\{b\in
F:(a,b)\in D\\}$. If $b\in F$ we denote, when no ambiguity can occur,
$D_{b}=\\{a\in E:(a,b)\in D\\}$. If $f:D\to C$ is a function, we let
$f_{a}:D_{a}\to C$ denote the function $f_{a}(b)=f(a,b)$, and similarly
$f_{b}$ for $b\in F$.

######  Proposition 3.5.

Let $K$ be a 1-h-minimal field of positive residue characteristic, $f:U\to K$
an $a$-definable function with $U\subset K^{n}$ open, and let
$r\in{\mathbb{N}}$. Then there is an integer $m$, and a set $C$, closed,
$a$-definable with $\dim(C)<n$, such that for every open ball, $B\subseteq U$
$m$-next to $C$, $f^{(i)}$ exists in $B$ for every $i$ with $|i|\leq r$, and
$f^{(i)}$ has constant valuation in $B$. Moreover,

$\left|f(x)-\sum_{\\{i:|i|<r\\}}\frac{1}{i!}f^{(i)}(x_{0})(x-x_{0})^{i}\right|\leq\max_{\\{i:|i|=r\\}}\left|\frac{1}{i!}f^{(i)}(x_{0})(x-x_{0})^{i}\right|$

For every $x,x_{0}\in B$.

###### Proof.

We proceed by induction on $n$, the case $n=1$ being [5, Theorem 3.1.2].
Assume the result for $n$ and let $f:U\to K$ be an $a$-definable function with
$U\subset K^{n}\times K$ open, $i$ a multi-index with $|i|\leq r$. Then for
every $x\in K^{n}$ there is a finite $ax$-definable set $C_{x}\subset K$ and
an integer $m_{x}$ such that

(1)
$|f_{x}(y)-\sum_{s<r}\frac{1}{s!}f_{x}^{(s)}(y_{0})(y-y_{0})^{s}|\leq|\frac{1}{r!}f_{x}^{(r)}(y_{0})(y-y_{0})^{r}|$

for every $y$ and $y_{0}$ in an open ball $m_{x}$-next to $C_{x}$, and such
that $|f_{x}^{(s)}(y)|$ exists and is constant in any such open ball. By a
standard compactness argument (See Fact 3.2), we may assume that the $C_{x}$
are uniformly definable and that there is some $m\in{\mathbb{N}}$ such that
$m_{x}=m$ for all $x$. Define $C=\cup_{x}\\{x\\}\times C_{x}$. By induction,
for each $y\in K$ we can approximate the functions $g_{s,y}(x)=f_{x}^{(s)}(y)$
defined on $V_{y}=\mathrm{Int}(U_{y}\setminus C_{y})$ up to order $r-s$. By
the same compactness argument we obtain a natural number $m^{\prime}$ and an
$a$-definable family $\\{D_{y}\\}_{y\in K^{n}}$ of subsets $D_{y}\subseteq
V_{y}$ with $\dim(D_{y})<n$ such that $g_{s,y}^{(i)}$ exists and has constant
valuation on any ball $m^{\prime}$-next to $D_{y}$ in $V_{y}$, for every
multi-index, $i$, with $|i|\leq r-s$. Moreover,

(2)
$|g_{s,y}(x)-\sum_{\\{i:|i|<r-s\\}}\frac{1}{i!}g_{s,y}^{(i)}(x_{0})(x-x_{0})^{i}|\leq\max_{\\{i:|i|=r-s\\}}|\frac{1}{i!}g_{s,y}^{(i)}(x_{0})(x-x_{0})^{i}|$

Replacing $m$ and $m^{\prime}$ by their maximum, we may assume $m=m^{\prime}$.
Define $D:=\bigcup_{y}D_{y}\times\\{y\\}$. By additivity of dimension
$\dim(C)\leq n$ and $\dim(D)\leq n$. Finally, take $E=C\cup
D\cup\bigcup_{y}(U_{y}\setminus V_{y})\times\\{y\\}$. Similar dimension
considerations show that $\dim(E)<n+1$.

Note that for $(x,y)\in U\setminus E$ we have that, for $i$ and $s$ such that
$|i|+s\leq r$, $f^{(i,s)}(x,y)$ and $g_{s,y}^{(i)}(x)$ exist and are equal.

Now, for $x\in K^{n}$ define $W_{x}=\mathrm{Int}(U_{x}\setminus E_{x})$, and
for the functions $h_{x,s,i}:y\mapsto f^{(i,s)}(x,y)$ with $s+|i|\leq r$
defined on $W_{x}$, we find a finite set $F_{x}\subset W_{x}$ such that
$\\{F_{x}\\}_{x}$ is an $a$-definable family, and there is an integer
$m^{\prime}$, such that in every ball in $W_{x}$ $m^{\prime}$-next to $F_{x}$,
$h_{x,s,i}$ has constant valuation. We may assume that $m^{\prime}=m$ as
before. Let $G$ be the closure of $E\cup\bigcup_{x}\\{x\\}\times
F_{x}\cup\bigcup_{x}\\{x\\}\times(U_{x}\setminus W_{x})$. Note that
$\dim(G)<n+1$.

Take $B_{1}\times B_{2}$ a ball in $U$, $m$-next to $G$. Then for every $x\in
B_{1}$ we get that $B_{2}$ is $m$-next to both $C_{x}$ and $F_{x}$ and
$B_{2}\subseteq W_{x}$. Similarly, for every $y\in B_{2}$, $B_{1}\subseteq
V_{y}$ is $m$-next to $D_{y}$.

We conclude that for every $(x,y)\in B_{1}\times B_{2}$, $f^{(i,s)}(x,y)$
exists and has constant valuation, for every index $(i,s)$ such that
$|(i,s)|\leq r$. Indeed, we have for every $(x,y),(x^{\prime},y^{\prime})\in
B_{1}\times B_{2}$ that

$|f^{(i,s)}(x^{\prime},y^{\prime})|=|g_{s,y^{\prime}}^{(i)}(x^{\prime})|=|g_{s,y^{\prime}}^{(i)}(x)|=|h_{x,s,i}(y^{\prime})|=|h_{x,s,i}(y)|=|f^{(i,s)}(x,y)|,$

as the second equality follows from the condition on $C_{x}$, and the fourth
from those on $F_{x}$ and $W_{x}$.

Now, if $(x,y)$ and $(x_{0},y_{0})$ are in $B_{1}\times B_{2}$, then equations
1 and 2 hold and for the error term of 1 we have
$|f_{x}^{(r)}(y_{0})|=|f^{(0,r)}(x,y_{0})|=|f^{(0,r)}(x_{0},y_{0})|$. Denote

$M=\max\left\\{\frac{1}{i!s!}|f^{(i,s)}(x_{0},y_{0})(x-x_{0})^{i}(y-y_{0})^{s}|:|(i,s)|=r\right\\}.$

Then Equation 1 yields
$|f(x,y)-\sum_{s<r}\frac{1}{s!}f^{(0,s)}(x,y_{0})(y-y_{0})^{s}|\leq M$. Also
from Equation 2 we have that

$\left|\frac{1}{s!}f^{(0,s)}(x,y_{0})(y-y_{0})^{s}-\sum_{\\{i:|i|<r-s\\}}\frac{1}{i!s!}f^{(i,s)}(x_{0},y_{0})(x-x_{0})^{i}(y-y_{0})^{s}\right|\leq
M.$

Taking the sum over $s$ smaller than $r$ and using the ultrametric inequality
we obtain

$\left|\sum_{s}\frac{1}{s!}f^{(0,s)}(x,y_{0})(y-y_{0})^{s}-\sum_{\\{(i,s):|i|+s<r\\}}\frac{1}{i!s!}f^{(i,s)}(x_{0},y_{0})(x-x_{0})^{i}(y-y_{0})^{s}\right|\leq
M.$

Summing this with Equation 1 and using the ultrametric inequality once more we
conclude. ∎

As a consequence of the previous theorem we obtain that partial derivatives of
definable functions commute generically.

######  Corollary 3.6.

Suppose $f:U\to K$ is a definable function for some open $U\subset K\times K$.
Then there exists a open dense $U^{\prime}\subset U$ such that for every
$(x,y)\in U^{\prime}$

$\frac{\partial}{\partial x}\frac{\partial}{\partial
y}f(x,y)=\frac{\partial}{\partial y}\frac{\partial}{\partial x}f(x,y)$

and, in particular, the terms of the above equation exist in $U^{\prime}$.

Moreover, if $f:U\to K$ is such that the partial derivatives
$\frac{\partial}{\partial x}\frac{\partial}{\partial y}f(x,y)$,
$\frac{\partial}{\partial y}\frac{\partial}{\partial x}f(x,y)$ exist and are
continuous in $U$, then they are equal.

###### Proof.

Take a $1$-dimensional closed $C\subset K\times K$ and $m$ an integer as
provided by the Taylor approximation property for errors of order $3$. We may
also assume that $\pi(C)$ and $m$ satisfy the same Taylor approximation
property for the function $f\pi$, where $\pi$ is the coordinate permutation
$(x,y)\mapsto(y,x)$.

Then for $(x,y),(x_{0},y_{0})\in B_{1}\times B_{2}$ in a ball $m$-next to $C$
we obtain (see Definition 2.1 for the big-$O$ notation) that

$\displaystyle f(x,y)=$ $\displaystyle
f(x_{0},y_{0})+(x-x_{0})^{2}\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}f(x_{0},y_{0})+(y-y_{0})^{2}\frac{1}{2}\frac{\partial^{2}}{\partial
y^{2}}f(x_{0},y_{0})+$
$\displaystyle(x-x_{0})(y-y_{0})\frac{\partial}{\partial
x}\frac{\partial}{\partial y}f(x_{0},y_{0})+O((x-x_{0},y-y_{0})^{3}).$

Similarly,

$\displaystyle f\pi(y,x)=f(x,y)=$ $\displaystyle
f(x_{0},y_{0})+(x-x_{0})^{2}\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}f(x_{0},y_{0})+(y-y_{0})^{2}\frac{1}{2}\frac{\partial^{2}}{\partial
y^{2}}f(x_{0},y_{0})+$
$\displaystyle(x-x_{0})(y-y_{0})\frac{\partial}{\partial
y}\frac{\partial}{\partial x}f(x_{0},y_{0})+O((x-x_{0},y-y_{0})^{3})$

Taking the difference we obtain

$(x-x_{0})(y-y_{0})\frac{\partial}{\partial y}\frac{\partial}{\partial
x}f(x_{0},y_{0})-(x-x_{0})(y-y_{0})\frac{\partial}{\partial
x}\frac{\partial}{\partial y}f(x_{0},y_{0})=O((x-x_{0},y-y_{0})^{3})$

Taking $h=(x-x_{0})=(y-y_{0})$ small we get, $h^{2}(\frac{\partial}{\partial
y}\frac{\partial}{\partial x}f(x_{0},y_{0})-\frac{\partial}{\partial
x}\frac{\partial}{\partial y}f(x_{0},y_{0}))=O(h^{3})$, so
$\frac{\partial}{\partial y}\frac{\partial}{\partial
x}f(x_{0},y_{0})-\frac{\partial}{\partial x}\frac{\partial}{\partial
y}f(x_{0},y_{0})=O(h)$. This is only possible when the left-hand side is $0$,
as desired. ∎

The following notation is intended to look similar to the monomial $ax^{n}$
for $a\in K$ and $x\in K$, for the purpose of expressing the Taylor
approximation of a multivariate function.

######  Definition 3.7.

Let $m,n$ be positive integers, and $r\in{\mathbb{N}}$. Let
$J=J(r,n)=\\{j\in\mathbb{N}^{n}:|j|=r\\}$. Let $a=(a_{j})_{j\in J}$ be such
that $a_{j}\in K^{m}$ for all $j\in J$. Then, for $x\in K^{n}$ we define
$ax^{r}=\sum\limits_{j\in J}a_{j}x^{j}$, where
$x^{j}:=\prod_{i=1}^{n}x^{j(i)}$. Note that $x\mapsto ax^{r}$ is a function
$K^{n}\to K^{m}$.

As an example, consider, in the above notation, the case $r=1$. In this case
$J=\\{e_{1},\dots,e_{n}\\}$ and for $j\in J$ we have $x^{j}=x_{j}$ (where
$x=(x_{1},\dots,x_{n})$), so for $a=(a_{j})_{j\in J}$ with $a_{j}\in K^{m}$ we
get that $ax=A\cdot x$ where $A$ is the matrix whose $j$-th column is $a_{j}$.

######  Definition 3.8.

Let $U\subset K^{k}$ be open, $f:U\to K^{m}$ a function and $a\in U$. We say
that $f$ is $P_{n}$ at $a$ if it is approximable by polynomials of degree $n$
near $a$ in the following sense: there are constants $b_{0},\cdots,b_{n}$
$f(a+x)=\sum_{r\leq n}b_{r}x^{r}+O(x^{n+1})$.

In view of the above example, it follows immediately from the definition that
a $P_{1}$ function is differentiable and for the coefficient $b_{1}$ in the
definition we may take $f^{\prime}(a)$ (or, more precisely
$b_{1}^{t}=f^{\prime}(a)$). It follows from Lemma 3.10 below, that – in fact –
$b_{1}=f^{\prime}(a)^{t}$ whenever $f$ is $P_{n}$ for any $n\geq 1$.

######  Definition 3.9.

Let $U\subset K^{k}$ be open, $f:U\to K^{m}$ a function and $a\in U$. We say
$f$ is $T_{n}$ at $a$ if there is $\gamma\in\Gamma$ such that for every
$x,x^{\prime}$ with $|x-a|,|x^{\prime}-a|<\gamma$, we have $f(x)=\sum_{r\leq
n}c_{r}(x^{\prime})(x-x^{\prime})^{r}+O(x-x^{\prime})^{n+1}$ for $c_{r}$ a
$P_{n-r}$ function at $a$.

Note that in the previous definition, the constant implicit in the notation
$O(x-x^{\prime})^{n+1}$ (see Definition 2.1) does not depend on $x^{\prime}$;
so this definition requires some uniformity with respect to the center
$x^{\prime}$ which is not implied by simply assuming $f$ is $P_{n}$ at every
point of a ball around $a$.

Note also that if $f$ is $T_{n}$ at $c=(b,a)$ then, in particular
$f(z,a+x)=f(z)+f_{1}(z)x+\cdots+f_{n}(z)x^{n}+O(x^{n+1})$, for $P_{n-k}$
functions $f_{k}$ at $b$ (and a constant in $O(x^{n+1})$ uniform in $z$). This
follows from the definition by taking $x=(z,a+x)$ and $x^{\prime}=(z,a)$.

Sums and products of $P_{n}$ (resp. $T_{n}$) functions are $P_{n}$ (resp.
$T_{n}$), and a vector function is $P_{n}$ (resp. $T_{n}$) if and only if its
coordinate functions are $P_{n}$ (resp. $T_{n}$).

We could also require the stronger condition, $ST_{n}$, defined similarly
$T_{n}$, but requiring inductively the functions $c_{r}$ be $ST_{n-r}$ for
$r=1,\dots,n$ (the base case $ST_{0}=T_{0}$). This definition may be more
natural, and one can prove the same results for this notion in what follows,
but the stated definition is enough for Lemma 3.15 which is our main
motivation.

######  Lemma 3.10.

If $f$ is $P_{n}$ at $a$, then for every number $i\leq n$, the coefficients
$b_{i}$ in Definition 3.8 are determined by $f$.

###### Proof.

The problem readily reduces to the case of $f$ a polynomial restricted to some
open neighborhood, $U$, of the origin. I.e., We have to show that if
$\sum_{i\leq n}b_{i}x^{i}=O(x^{n+1})$ in any open $U\subseteq K^{r}$ then
$b_{i}=0$ for all $i$. For $x_{0}$ fixed let $x=tx_{0}$ and consider the
single variable polynomial $P(tx_{0})=\sum_{i}\leq
n(b_{i}x_{0}^{i})t^{i}=O(t^{n+1})$. If we knew the result for $r=1$ this would
give that $b_{i}x_{0}^{i}=0$. Since $x_{0}\in U$ was arbitrary and $U$
contains a cartesian product of $r$ infinite sets, this implies $b_{i}=0$ for
all $i$. So we are reduced to proving the result for $r=1$.

In this case, if $i$ is the smallest with $b_{i}\neq 0$ we get
$x^{i}=O(x^{i+1})$ which is a contradiction. ∎

######  Proposition 3.11.

If $g$ is $P_{n}$ at $a$ and $f$ is $P_{n}$ at $g(a)$ then the composition,
$f\circ g$ is $P_{n}$ at $a$.

If $g$ is $T_{n}$ at $a$ and $f$ is $T_{n}$ at $g(a)$ then $f\circ g$ is
$T_{n}$ at $a$.

###### Proof.

For the first statement we write

$\displaystyle f(g(a+x))=$ $\displaystyle
f(g(a)+g_{1}(a)x+...+g_{n}(a)x^{n}+O(x^{n+1}))=$ $\displaystyle
f(g(a))+f_{1}(g(a))h(a,x)+f_{2}(g(a))h(a,x)^{2}+...+O(h(a,x)^{n+1})=$
$\displaystyle
fg(a)+b_{1}(a)x+\cdots+b_{n}(a)x^{n}+O(x^{n+1})+O(h(a,x)^{n+1}),$

where

  1. (1)

$h(a,x)=g(a,x)-g(a)=g_{1}(a)x+\cdots+g_{n}(a)x^{n}+O(x^{n+1})$

  2. (2)

The second inequality is the application of the assumption that $f$ is $P_{n}$
at $g(a)$.

  3. (3)

The coefficients $b_{i}$ arise by expanding the expression

$f_{k}(a)h(a,x)^{k}=f_{k}(a)(g_{1}(a)x+\cdots+g_{n}(a)x^{n}+O(x^{n+1}))^{k}.$

To conclude, we note that, as in the proof of Lemma 3.10 $h(a,x)=O(x)$, and so
$O(h(a,x)^{n+1})=O(x^{n+1})$.

The proof of the second statement is, essentially, similar:

$\displaystyle f(g(x))=$ $\displaystyle
f(g(x^{\prime})+g_{1}(x^{\prime})(x-x^{\prime})+...+g_{n}(x^{\prime})(x-x^{\prime})^{n}+O(x-x^{\prime})^{n+1})=$
$\displaystyle
f(g(x^{\prime}))+f_{1}(g(x^{\prime}))h(x,x^{\prime})+f_{2}(g(x^{\prime}))h(x,x^{\prime})^{2}+...+f_{n}(g(x^{\prime}))(x-x^{\prime})^{n}+O(h(x,x^{\prime})^{n+1})=$
$\displaystyle
f(g(x^{\prime}))+b_{1}(x^{\prime})(x-x^{\prime})+\cdots+b_{n}(x^{\prime})(x-x^{\prime})^{n}+O(x^{n+1})+O(h(x,x^{\prime})^{n+1})$

Where
$h(x,x^{\prime})=g(x)-g(x^{\prime})=g_{1}(x^{\prime})(x-x^{\prime})+\cdots+g_{n}(x^{\prime})(x-x^{\prime})^{n}+O(x-x^{\prime})^{n+1}=O(x-x^{\prime})$,
and the coordinates of the coefficients $b_{k}(x^{\prime})$ are sums and
products of the coordinates of the coefficients of $f_{i}(g(x^{\prime}))$ and
$g_{j}(x^{\prime})$ with $i,j\leq k$. By what we have just proved, those are
$P_{n-i}$ functions. The constant appearing on
$h(x,x^{\prime})=O(x-x^{\prime})$ does not depend on $x^{\prime}$ because the
$g_{i}$ are continuous at $a$. We conclude that the $b_{k}$ are $P_{n-k}$ at
$a$, as claimed. ∎

######  Proposition 3.12.

Let $K$ be 1-h-minimal and $f:U\to K^{m}$ as definable function. Then there
exists $U^{\prime}\subset U$ definable open and dense such that $f$ is $T_{n}$
at every point of $U^{\prime}$.

In particular for every $f:U\to K^{m}$, there is a definable open dense subset
$U^{\prime}\subset U$ such that $f$ is strictly differentiable in
$U^{\prime}$.

This follows from Taylor’s approximation theorem (Proposition 3.1 in residue
characteristic $0$ and Proposition 3.5 in positive residue characteristic).
The second statement follows because a $T_{1}$ function is strictly
differentiable.

In the next section we show that a strictly differentiable map with invertible
derivative, definable in a 1-h-minimal valued field, is a local homeomorphism.
Here we show that the local inverse is strictly differentiable. We then
proceed to showing that the properties $P_{n}$ and $T_{n}$ are also preserved
in this local inverse, though this latter fact is not used for the proof of
our main results.

######  Proposition 3.13.

Suppose $f:U\to V$ is a bijection where $U\subset K^{n}$ and $V\subset K^{n}$
are open. Suppose $f$ satisfies $|f(x)-f(y)-(x-y)|<|x-y|$ for $x,y\in U$
distinct. Assume $f$ is differentiable at $a$. Then $f^{\prime}(a)$ is
invertible, $f^{-1}$ is differentiable at $b=f(a)$ and
$(f^{-1})^{\prime}(b)=f^{\prime}(f^{-1}(b))^{-1}$.

If $f$ is strictly differentiable at $a$, then $f^{-1}$ is strictly
differentiable at $b$.

###### Proof.

Note that the hypothesis implies $|f(x)-f(x^{\prime})|=|x-x^{\prime}|$. This
implies that $f^{\prime}(a)$ is invertible. Indeed, assume otherwise, and take
$x$ close to $a$ such that $f^{\prime}(a)(x-a)=0$, to get $|f(x)-f(a)|<|x-a|$,
a contradiction.

Assume that $f$ is strictly differentiable at $a$. Take $\epsilon>0$ in
$\Gamma$. Then there is $0<r\in\Gamma$ such that if $|x-a|,|x^{\prime}-a|<r$,
then
$|f(x)-f(x^{\prime})-f^{\prime}(a)(x-x^{\prime})|\leq\epsilon|x-x^{\prime}|$.
If we denote $y=f(x)$ and $y^{\prime}=f(x^{\prime})$, then we have
$|y-y^{\prime}|=|x-x^{\prime}|$, so multiplying the above inequality by
$f^{\prime}(a)^{-1}$ we obtain

$\displaystyle|f^{-1}(y)-f^{-1}(y^{\prime})-f^{\prime}(a)^{-1}(y-y^{\prime})|=$
$\displaystyle|f^{\prime}(a)^{-1}(f^{\prime}(a)(x-x^{\prime})-(f(x)-f(x^{\prime}))|\leq$
$\displaystyle|f^{\prime}(a)^{-1}||f(x)-f(x^{\prime})-f^{\prime}(a)(x-x^{\prime})|\leq$
$\displaystyle\epsilon|f^{\prime}(a)^{-1}||x-x^{\prime}|=\epsilon|f^{\prime}(a)^{-1}||y-y^{\prime}|,$

where, for a linear map, $A$, represented by the matrix $(a_{ij})_{i,j}$ we
denote $|A|=\max_{i,j}|a_{ij}|$, and use the ultra-metric inequality to get
$|Ax|\leq|A||x|$, which we apply to obtain the first inequality in the above
computation.

So we conclude that
$|f^{-1}(y)-f^{-1}(y^{\prime})-f^{\prime}(a)^{-1}(y-y^{\prime})|\leq\epsilon|f^{\prime}(a)|^{-1}|y-y^{\prime}|$
for any $y,y^{\prime}$ such that $|y-b|=|x-a|<r$ and
$|y^{\prime}-b|=|x^{\prime}-a|<r$. We have, thus, shown that $f^{-1}$ is
strictly differentiable at $b$ and
$(f^{-1})^{\prime}(b)=f^{\prime}(f^{-1}(b))^{-1}$.

To show that $f^{-1}$ is differentiable if $f$ is substitute $x^{\prime}=a$ in
the above argument. ∎

######  Proposition 3.14.

Suppose $f:U\to V$ is a bijection where $U\subset K^{n}$ and $V\subset K^{n}$
are open. Suppose $f$ satisfies $|f(x)-f(y)-(x-y)|<|x-y|$ for $x,y\in U$
distinct. Then if $f$ is $P_{n}$ (resp. $T_{n}$) at $b$, $f^{-1}$ is $P_{n}$
(resp. $T_{n}$) at $f(b)$.

###### Proof.

Denote $a=f(b)$. Note that the hypothesis implies than $|f(x)-f(y)|=|x-y|$ for
all distinct $x,y\in U$, so the inverse map $f^{-1}$ is continuous, and in
fact satisfies $|f^{-1}(x)-f^{-1}(y)|=|x-y|$ for distinct $x,y\in V$. In
particular, $f^{-1}$ is $T_{0}$ in $V$.

Now, assume that $f$ is $P_{n}$ at $b$, with $n\geq 1$. In particular, by
Proposition 3.13 it is differentiable and $f^{\prime}(b)$ is invertible.

Apply the fact that $f$ is $P_{n}$ (and see also the discussion following the
definition) to get:

$f(y)-f(b)=f^{\prime}(b)(y-b)+f_{2}(b)(y-b)^{2}+\cdots+f_{n}(b)(y-b)^{n}+O(y-b)^{n+1}.$

Rearranging, we get:

$y-b=f^{\prime}(b)^{-1}(f(y)-f(b))-f^{\prime}(b)^{-1}f_{2}(b)(y-b)^{2}-\cdots-f^{\prime}(b)^{-1}f_{n}(b)(y-b)^{n}+O(y-b)^{n+1}.$

Putting $y=f^{-1}(x)$, and remembering $x-a=O(y-b)$ we conclude

$f^{-1}(x)-f^{-1}(a)=f^{\prime}(f^{-1}(a))^{-1}(x-a)+\sum_{2\leq i\leq
n}c_{i}(a)(f^{-1}(x)-f^{-1}(a))^{i}+O(x-a)^{n+1},$

for some constants $c_{i}(a)$.

Next we proceed to showing (by induction on $k\leq n$) that $f^{-1}$ is
$P_{k}$. As $P_{0}$ follows from the equality $|f^{-1}(x)-f^{-1}(y)|=|x-y|$,
we assume that So suppose $f^{-1}$ is $P_{k-1}$. Using this, we can write
$f^{-1}(x)-f^{-1}(a)=\sum_{1\leq j<k}b_{j}(a)(x-a)^{j}+O(x-a)^{k}$, and apply
a direct computation to obtain that

$c_{i}(a)(f^{-1}(x)-f^{-1}(a))^{i}=\sum_{i\leq j\leq
k}d_{ij}(a)(x-a)^{j}+O(x-a)^{k+1}.$

for some constants $d_{ij}$. Note that as $i\geq 2$ we obtain the improved
error $O(x-a)^{k+1}$. Substituting this in $(\diamond)$, we the conclusion
follows.

Now suppose $f$ is $T_{n}$ at $b$. The proof in this case is similar:

$f^{-1}(x)-f^{-1}(x^{\prime})=f^{\prime}(f^{-1}(x^{\prime}))^{-1}(x-x^{\prime})+\sum_{2\leq
i\leq
n}c_{i}(f^{-1}(x^{\prime}))(f^{-1}(x)-f^{-1}(x^{\prime}))^{i}+O(x-x^{\prime})^{n+1},$

so, as above, if $f^{-1}(x)$ is $T_{k-1}$ we obtain

$f^{-1}(x)-f^{-1}(x^{\prime})=f^{\prime}(f^{-1}(x^{\prime}))^{-1}(x-x^{\prime})+\sum_{2\leq
i\leq k}d_{i}(x^{\prime})(x-x^{\prime})^{i}+O(x-x^{\prime})^{k+1}.$

Here note that $f^{\prime}$ is $P_{n-1}$ at $b$ and so
$f^{\prime}(f^{-1}(x^{\prime}))^{-1}$ is $P_{k-1}$ at $a$. Also, following the
above argument we see that the coordinates of $d_{i}(x^{\prime})$ are sum and
products of functions of the form $b_{i^{\prime}}(x^{\prime})$ with $1\leq
i^{\prime}<i$, for $b_{i^{\prime}}(x^{\prime})$ a $P_{k-1-i^{\prime}}$
function at $a$ (by the induction hypothesis that $f^{-1}$ is $T_{k-1}$), and
functions of the form $c_{i^{\prime}}(f^{-1}(x^{\prime}))$ for
$c_{i^{\prime}}(y^{\prime})$ a $P_{k-i^{\prime}}$ function at $b$,
$i^{\prime}\leq i$ (by the assumption that $f$ is $T_{n}$). So
$d_{i}(x^{\prime})$ is a $P_{k-i}$ at $a$. ∎

The next lemma will be important in our study of the differential structure of
definable groups. For the statement, recall that $O_{x}$ means that the
constant implicit in the notation depends on $x$, see Definition 2.1.

######  Lemma 3.15.

Let $f:U\times V\to K^{r}$ be a definable function, where $U\subset K^{n}$ and
$V\subset K^{m}$ are open sets around $0$. Suppose $f(x,y)$ is $T_{2}$ at
$(0,0)$, and $f(x,y)=O(x,y)^{3}$. If $axy+f(x,y)=O_{x}(y^{2})$, then $a=0$.

###### Proof.

By the definition of $T_{2}$ (with $x=x^{\prime}$, $y^{\prime}=0$) we get
$f(x,y)=f_{0}(x)+f_{1}(x)y+O(y^{2})$, for $f_{1}$ a $P_{1}$ function at $0$
and $f_{0}$ a $P_{2}$ function at $0$. Fixing $x$ and expanding the Taylor
polynomial of $f(x,y)$ the uniqueness of Taylor coefficients (Lemma 3.10)
gives, using our assumption, $axy+f_{0}(x)+f_{1}(x)y=0$.

Expanding $f_{0},f_{1}$ around $0$ and keeping in mind $f(x,y)=O(x,y)^{3}$ we
get $f_{0}(x)=O(x^{3})$ and $f_{1}(x)=O(x^{2})$. Indeed, we have
$f_{0}(x)=b_{0}+b_{1}x+b_{2}x^{2}+O(x^{3})$ and
$f_{1}(x)=c_{0}+c_{1}x+O(x^{2})$, so
$f(x,y)=b_{0}+b_{1}x+b_{2}x^{2}+c_{0}y+c_{1}xy+O(x,y)^{3}=O(x,y)^{3}$, so from
the uniqueness of the Taylor coefficients we get
$b_{0}=b_{1}=b_{2}=c_{0}=c_{1}=0$.

Now from $axy=O(x,y)^{3}$, we get $a=0$, by the uniqueness of Taylor
coefficients again. ∎

##  4\. Strictly differentiable definable maps

In this section we prove an inverse function theorem for definable strictly
differentiable maps in a 1-h-minimal valued fields. This is done adapting a
standard argument from real analysis using Banach’s fixed point theorem. In
the present section we use definable spherical completeness to obtain a
definable version of Banach’s fixed point theorem, implying, almost formally,
the desired inverse function theorem. From the inverse function theorem we
deduce results on the local structure of immersions and submersions in the
usual way. We then proceed to proving a generic version of the theorem on the
local structure of functions of constant rank (Proposition 4.11). This last
result is obtained only generically. The reason is that definable functions
whose partial derivative with respect to a variable $x$ is $0$ on an open set,
$U$ need not be locally constant in $x$ in $U$. For that reason, we give a
different argument for a weaker result, see Proposition 4.8, and the
discussion preceding it.

Throughout the rest of this section, we fix an $\aleph_{0}$-saturated
1-h-minimal valued field $K$.

We start with a fixed point theorem, mentioned in [4, Remark 2.7.3]. We first
note that a version of definable spherical completeness of $1$-h-minimal
fields ([4, Lemma 2.7.1]) holds in positive residue characteristic:

######  Lemma 4.1.

Suppose $K$ has positive residue characteristic $p$. Suppose $\\{B_{i}\\}_{i}$
is a definable chain of open balls or a definable chain of closed balls.
Suppose, further, that for every $i$ there is $j$ such that
$\text{rad}(B_{j})\leq|p|\text{rad}(B_{i})$. Then
$\bigcap_{i}B_{i}\neq\emptyset$.

###### Proof.

The proof is similar to spherical completeness in residue characteristic $0$,
see Lemma 2.7.1 of [4]. It is enough consider the $1$-dimensional case, since
the higher dimensional case follows by considering the coordinate projections.
Note also that our assumption implies that the chain $\\{B_{i}\\}$ has no
minimal element (as such an element would have valuative radius $0$).

The closed case follows from the open case as follows: given a definable chain
$\\{B_{i}\\}_{i\in I}$ of closed balls. For each $i$ let $r_{i}$ be the
valuative radius of $B_{i}$ and let $B_{i}^{\prime}$ be the unique open ball
$B\subseteq B_{i}$ of valuative radius $r_{i}$ with the additional property
that $B\supseteq B_{j}$ for all $j<i$. Obviously,
$\bigcap_{i}B_{i}=\bigcap_{i}B_{i}^{\prime}$ (unless the chain $B_{i}$ has a
minimal element, in which case there is nothing to prove).

Note that, in the above notation, the map $B_{i}\mapsto r_{i}$ is injective,
so there is no harm assuming that $\\{B_{i}\\}$ is indexed by a subset of
$\Gamma$. Thus, our chain $\\{B_{i}\\}$ has index set interpretable in $RV$,
so by [5, Proposition 2.3.2] there is a finite set $C$ $m$-preparing the chain
$\\{B_{i}\\}$ for some $m\in{\mathbb{N}}$. We claim that $C\cap
B_{i}\neq\emptyset$ for all $i\in I$. This would finish the proof, since $C$
is finite. Assume, therefore, that this is not the case, and let $i_{0}\in I$
be such that $B_{i_{0}}\cap C=\emptyset$. By assumption we can find $i<i_{0}$
such that $r_{i}<|p^{m}|r_{i_{0}}$. Then $B_{i}$ is a ball $m$-next to $C$,
and since our chain has no minimal element, any ball $B\varsubsetneq B_{i}$
that is an element of our chain is not $m$-prepared by $C$, a contradiction. ∎

Note that by [3, Exanple 1.5] infinitely ramified $1$-h-minimal fields of
positive residue characteristic need not be definably spherically model
complete. Thus, the extra condition in the assumption of the above lemma is
not superfluous.

######  Proposition 4.2.

Let $B_{r}=\\{x\in K^{n}:|x|\leq r\\}$. Suppose $f:B_{r}\to B_{r}$ is a
definable function. Assume that for distinct $x,y\in B_{r}$, we have

  1. (1)

$|f(x)-f(y)|<|x-y|$ if the residue characteristic is $0$.

  2. (2)

$|f(x)-f(y)|\leq|p||x-y|$ if the residue characteristic is $p>0$.

Then $f$ has a unique fixed point in $B_{r}$.

###### Proof.

Uniqueness is immediate from the hypothesis. For existence take the family of
balls of the form $B(a)_{|f(a)-a|}$. It is a definable chain of balls indexed
by $a\in B_{r}$. Indeed if $a,b\in B_{r}$ are distinct and the balls are
disjoint then $|f(a)-f(b)|=|a-b|$, as the distance of points in disjoint balls
does not change. Note that in positive residue characteristic one has the
additional hypothesis in Lemma 4.1, because $|f(f(a))-f(a)|\leq|p||f(a)-a|$,
by assumption 2 on $f$.

By the appropriate version of definable spherical completeness of 1-h-minimal
fields (Lemma 2.7.1 of [4] for residue characteristic $0$, Lemma 4.1
otherwise), we obtain a point $x$ in the intersection of all balls. Then $x$
is a fixed point of $f$. Indeed, if we assume otherwise then for $y=f(x)$, we
have $|f(y)-y|<|f(x)-x|$ by the hypothesis. On the other hand if $a$ is
arbitrary then, as $x\in B(a)_{|f(a)-a|}$, one has
$|f(x)-f(a)|\leq|x-a|\leq|f(a)-a|$ and so $|f(x)-x|\leq|f(a)-a|$. This is a
contradiction and finishes the proof. ∎

Just as in real analysis this fixed point theorem implies an inverse function
theorem.

######  Proposition 4.3.

Suppose $f:U\to K^{n}$ is a definable function from an open set $U\subset
K^{n}$ satisfying the following “bilipschitz condition”: for every $x,y\in U$
distinct

  1. (1)

$|f(x)-f(y)-(x-y)|<|x-y|$ if the residue characteristic is $0$.

  2. (2)

$|f(x)-f(y)-(x-y)\leq|p||x-y|$ if the residue characteristic is $p>0$.

Then $f(U)$ is open and $f$ is a homeomorphism from $U$ to $f(U)$. If $f$ is
(strictly) differentiable then $f^{-1}$ is (strictly) differentiable.

###### Proof.

Injectivity of the map follows directly from the hypothesis. The same
assumptions also imply that if $x,y\in U$ are distinct then
$|f(x)-f(y)|=|x-y|$, implying continuity of the inverse.

The main difficulty is showing that $f(U)$ is open. Translating, we may assume
$0\in U$ and $f(0)=0$. We have to find an open neighborhood of $0$ in $f(U)$.
Take $r>0$ such that $0\in B_{r}\subset U$. Then $B_{r}\subset f(U)$. Indeed,
if $|a|\leq r$ then, by the same reasoning as above, the function
$g(x)=x+a-f(x)$ satisfies $g(B_{r})\subset B_{r}$. By the assumptions on $f$
this implies that $g$ satisfies the hypothesis of Proposition 4.2. So
$g(x_{0})=x_{0}$ for some $x_{0}$, namely, $a=f(x_{0})$, as claimed.

Differentiability (and strict differentiability) of $f^{-1}$ now follow from
3.13. ∎

We can finally formulate and prove the inverse function theorem for
$1$-h-minimal fields:

######  Proposition 4.4.

Suppose $f:U\to K^{n}$ is a definable function from an open set $U\subset
K^{n}$. Suppose $f$ is strictly differentiable at $a$ and $f^{\prime}(a)$ is
invertible. Then there is an open set $V\subseteq U$ around $a$ such that
$f(V)$ is open and $f:V\to f(V)$ is a bijection whose inverse is strictly
differentiable at $f(a)$.

###### Proof.

By the definition of strict differentiability the function
$f^{\prime}(a)^{-1}f$ satisfies the hypothesis of the previous proposition in
a small open ball around $a$. The conclusion follows. ∎

We do not know whether a definable function, $f:U\to K$, with continuous
partial derivatives, but such that $f$ is not strictly differentiable in $U$
could exist222In real analysis it is well known that a function
$f:U\to{\mathbb{R}}$ is ${\mathcal{C}}^{1}$ in $U$ if and only if it is
strictly differentiable there.. Clearly, sums, products and compositions of
strictly differentiable functions are strictly differentiable, and so are
locally analytic functions. Moreover, strict differentiability is first order
definable, and therefore extends to elementary extensions. Also, by the
generic Taylor approximation theorem, in the $1$-h-minimal context, any
definable function in an open subset of $K^{n}$ is strictly differentiable in
a dense open subset. See Proposition 3.12.

Our next goal is to study definable functions of constant rank. We first note,
that without the assumption of definability, a strictly differentiable
function whose derivative vanishes identically need not be locally constant:

######  Example 4.5.

Consider a function $f:\mathcal{O}\to\mathcal{O}$ that is locally constant in
$B\setminus\\{0\\}$ but near $0$ it grows like $x^{2}$. Such a function $f$
will be strictly differentiable, with $f^{\prime}\equiv 0$, but $f$ is not
locally constant at $0$.

Roughly, a function as in the above example involves an infinite number of
choices, so it is not definable. In contrast we have:

######  Proposition 4.6.

Let $f:U\to K^{m}$ be a function definable in an open set $U\subset K^{n}$.
Assume $f$ is continuous. Assume $f$ is differentiable with derivative $0$ on
an open dense subset of $U$. Then $f$ is locally constant with finite image.

###### Proof.

We proceed by induction on $n$, the dimension of the domain. We may assume
$m=1$.

First assume $n=1$. By the valuative Jacobian property in [4, Corollary 3.1.6]
and [5, Corollary 3.1.3], there is a finite set $C\subset U$ such that in
$U\setminus C$ the function $f$ is locally constant. This implies that the
fibers of $f|_{U\setminus C}$ are of dimension 1, and so the image of $f$ is
finite. As $f$ is continuous it is locally constant, the fibers form a finite
partition of closed and so open sets on which $f$ is constant.

Now assume the proposition is valid for $n$, and suppose $U\subset K^{n}\times
K$. We denote $\pi_{1}:K^{n}\times K\to K^{n}$ the projection onto the first
factor and $\pi_{2}:K^{n}\times K\to K$ the projection onto the second factor.

Let $V\subset U$ be an open dense subset such that $f$ is differentiable at
$V$ with derivative $0$. Denote $C=U\setminus V$. Let $T=\\{x\in
K^{n}\mid\dim(\pi_{1}^{-1}(x)\cap C)=1\\}$ then $\dim(T)<n$. We conclude that
there is an open dense set $W\subset K^{n}$ such that
$\dim(\pi_{1}^{-1}(x)\cap C)=0$ for all $x\in W$. Similarly, there is an open
dense set $P\subset K$ such that $\dim(\pi_{2}^{-1}(x)\cap C)<n$ for every
$x\in P$. Shrinking $V$ to $V\cap\pi_{1}^{-1}(W)\cap\pi_{2}^{-1}(P)$ we may
assume that if $(x,y)\in V$ then $V_{x}$ is an open dense subset of $U_{x}$
and $V_{y}$ is an open dense subset of $U_{y}$. By the induction hypothesis
and the $n=1$ case, we conclude that $f_{x}$ and $f_{y}$ are locally constant.
This implies that the fiber $f^{-1}f(x,y)$ has dimension $n+1$. Indeed, if $B$
is an open neighborhood of $y$ on which $f_{x}$ is constant and for each
$y^{\prime}\in B$ we take $R_{y^{\prime}}\subset U_{x}$ an open neighborhood
of $x$ on which $f_{y^{\prime}}$ is constant, then $f$ is constant on
$\bigcup_{y^{\prime}\in B}R_{y^{\prime}}\times\\{y^{\prime}\\}$.By dimension
considerations, we conclude that the image $f(V)$ is finite. As $f$ is
continuous, $f^{-1}f(V)$ is closed in $U$, and as $V$ is dense in $U$, we
conclude $f(U)=f(V)$ is finite. As $f$ is continuous, we conclude $f$ is
locally constant as before. ∎

Given the previous proposition we may expect that a definable strictly
differentiable function $f:U\to K^{m}$ with open domain $U\subset K^{r}\times
K^{s}$ and satisfying that $D_{y}f(x,y)=0$, is locally of the form
$f(x,y)=g(x)$. Unfortunately this is not true.

######  Example 4.7.

Take $f:\mathcal{O}\times\mathcal{O}\to\mathcal{O}$ defined by $f(x,y)=0$ if
$|y|>|x|$ and $f(x,y)=x^{2}$ if $|y|\leq|x|$. Then $f$ is strictly
differentiable, $f(x,\cdot)$ is locally constant, but $f$ is not of the form
$g(x)$ near $(0,0)$.

It is due to this pathology that the conclusion of Proposition 4.11 below only
holds generically.

Below, we let $D_{y}f(x,y)$ be the differential of the function $f_{x}$, given
by $f_{x}(y)=f(x,y)$; we call this the derivative of $f$ with respect to $y$
(where $y$ can be a tuple of variables).

######  Proposition 4.8.

Suppose $U\subset K^{n}$, and $V\subset K^{r}$ are open and $f:U\times V\to
K^{m}$ is a definable function such that $f$ is continuous and $D_{y}f=0$ on a
dense open subset of $\operatorname{dom}(f)$. Then there exists an open dense
set $U^{\prime}\subset U$ such that $f|_{U^{\prime}}$ is locally of the form
$g(x)$.

###### Proof.

The set, $D$ of points $x\in U$ such that for every point of $\\{x\\}\times V$
$f$ is locally of the form $g(x)$ is definable. More precisely, $x\in D$
exactly when for all $y\in V$, there exists an open ball $B\ni(x,y)$, such
that for all $(x^{\prime},y^{\prime}),(x^{\prime},y^{\prime\prime})\in B$ we
have $f(x^{\prime},y^{\prime})=f(x^{\prime},y^{\prime\prime})$. Thus, the
statement that $D$ has dense interior in $U$ is a first order expressible
property, so we may assume that $\operatorname{acl}=\operatorname{dcl}$, see
Fact 2.5 and the subsequent remark.

In the course of the proof we may replace $U$ by a dense open subset a finite
number of times. Fix, $W\subset U\times V$, a dense open set where $f$ is
differentiable and its derivative with respect to $y$ is $0$. Shrinking $U$ we
may assume $W_{x}\subset V$ is dense open for all $x\in U$. By Proposition 4.6
we know that $f_{x}$ is locally constant with finite image for every $x\in U$
(recall that $f_{x}(y):=f(x,y)$). The sets $\mathrm{Im}(f_{x})$ form a
definable family of finite sets indexed by $x\in U$, so there is uniform
bound, $n$, on their cardinalities. Denoting $A_{k}=\\{x\in
U:|\mathrm{Im}(f_{x})|=k\\}$ we have $U=A_{1}\cup\cdots\cup A_{n}$, so the
union of the interiors of the $A_{k}$ form a dense open subset of $U$. Since
the closures of the $\mathrm{Int}(A_{k})$ are pairwise disjoint, we may assume
that $|\mathrm{Im}(f_{x})|=k$ for all $x$ and some fixed $k$. Since we assumed
that $\operatorname{acl}=\operatorname{dcl}$, there are definable functions
$r_{1},\dots,r_{k}:U\to K^{m}$ such that
$\\{r_{1}(x),\dots,r_{k}(x)\\}=\mathrm{Im}(f_{x})$. By generic continuity of
definable functions (Proposition 2.12) we may assume that $r_{i}$ are all
continuous. Then the sets $B_{i}=\\{(x,y):f(x,y)=r_{i}(x)\\}$ form a finite
partition of $U\times V$ into closed, and so open subsets. ∎

The next two results, describing the local structure of definable maps of full
rank, are standard applications of the inverse function theorem:

######  Proposition 4.9.

Suppose $U\subset K^{k}$ is a definable open set and $f:U\to K^{k}\times
K^{r}$ is a definable, strictly differentiable map. Suppose that for some
$a\in U$ the derivative $f^{\prime}(a)$ has full rank. Then there is a ball
$a\in B\subset U$, a ball $B_{2}\ni 0$, a definable open set $V\subset
K^{k}\times K^{r}$, and a definable strict diffeomorphism $\varphi:V\to
B\times B_{2}$ such that $f(B)\subset V$ and the composition $\varphi f:B\to
B\times B_{2}$ is the inclusion $b\mapsto(b,0)$.

###### Proof.

After a coordinate permutation in the target we may assume the principal
$k\times k$ minor of $f^{\prime}(a)$ is invertible. Consider the function
$g:U\times K^{r}\to K^{k}\times K^{r}$ defined as $g(x,y)=f(x)+(0,y)$. Then
$g$ is strictly differentiable and has invertible derivative at $(a,0)$ so by
the inverse function theorem, Proposition 4.4, we can find a ball $B$ around
$a$ and a ball $B_{2}$ around $0$, and open set $f(a)\in V$ such that $g$
restrict to a strict diffeomorphism $g:B\times B_{2}\to V$. If $i:B\to B\times
B_{2}$ is the inclusion $i(b)=(b,0)$ then we get that $gi=f$, so we conclude
the statement is valid with $\varphi=g^{-1}$. ∎

######  Proposition 4.10.

Suppose $U\subset K^{k}\times K^{r}$ is a definable open set, and $f:U\to
K^{k}$ is a definable strictly differentiable map. Let $a\in U$. Suppose
$f^{\prime}(a)$ has full rank. Then there exists a definable open set $a\in
U^{\prime}\subset U$, a ball $f(a)\in B$, a ball $B_{2}\subseteq K^{r}$, and a
definable strict diffeomorphism $\varphi:B\times B_{2}\to U^{\prime}$, such
that $f(U^{\prime})\subset B$ and the composition $f\varphi:B\times B_{2}\to
B$ is the projection $(b,c)\mapsto b$.

###### Proof.

After applying a coordinate permutation to $U$ we may assume that the
principal $k\times k$ minor of $f^{\prime}(a)$ is invertible.

Consider the function $g:U\to K^{k}\times K^{r}$ defined as
$g(x,y)=(f(x,y),y)$. Then $g$ is strictly differentiable with invertible
differential, so by the inverse function theorem, Proposition 4.4, there is an
open set $a\in U^{\prime}\subset U$ such that $g(U^{\prime})$ is open and
$g:U^{\prime}\to g(U^{\prime})$ is a strict diffeomorphism.

Making $U^{\prime}$ smaller we may assume $g(U^{\prime})=B\times B_{2}$ is a
product of two balls. Then if $p:B\times B_{2}\to B$ is the projection
$p(b,c)=b$, we get that $pg=f$ and so the statement is valid with
$\varphi=g^{-1}$. ∎

We can finally prove our result on the local structure of definable functions
of constant rank:

######  Proposition 4.11.

Let $U\subset K^{k}\times K^{r}$ and $V\subset K^{k}\times K^{s}$ be open
definable sets and let $f:U\to V$ be a definable strictly differentiable map
such that for all $a\in U$ the rank of $f^{\prime}(a)$ is constant equal to
$k$. Then there exist $U^{\prime}\subset U$ and $V^{\prime}\subset V$
definable open sets, such that $f(U^{\prime})\subset V^{\prime}$ and there are
definable strict diffeomorphisms $\varphi_{1}:B_{1}\times B_{2}\to U^{\prime}$
and $\varphi_{2}:V^{\prime}\to B_{1}\times B_{3}$, such that the composition
$\varphi_{2}f\varphi_{1}:B_{1}\times B_{2}\to B_{1}\times B_{3}$ is the map
$(a,b)\mapsto(a,0)$.

###### Proof.

Take a point $(b,c)\in U$. After a coordinate permutation in $U$ and $V$ we
may assume $f^{\prime}(b,c)$ has its first $k\times k$ minor invertible. Then
by the theorem on submersions, Proposition 4.10, applied to the composition of
$f:U\to K^{k}\times K^{s}$ with the projection $K^{k}\times K^{s}\to K^{k}$
onto the first factor, we may assume that $U$ is of the form $B_{1}\times
B_{2}$ and $f$ is of the form $f(x,y)=(x,g(x,y))$. As $f^{\prime}$ has
constant rank equal to $k$ we conclude that $D_{y}g=0$. By the Proposition 4.8
we may assume $g(x,y)$ is of the form $g(x,y)=g(x)$ (after passing to smaller
open balls of $B_{1}$ and $B_{2}$ not necessarily containing $(b,c)$). Now the
function $h:B_{1}\to K^{k}\times K^{s}$ defined by $h(x)=(x,g(x))$ is a
definable strictly differentiable immersion so by the theorem on immersions
4.9 we may after shrinking $B_{1}$ and composing with a definable
diffeomorphism in the target assume that $h$ is of the form $h(x)=(x,0)$. This
finishes the proof. ∎

##  5\. Strictly differentiable definable manifolds

In this section we define definable manifolds in a 1-h-minimal field, and
variants. These are manifolds which are covered by a finite number of
definable charts, with compatibility functions of various kinds.

Throughout, we keep the convention that $K$ is an $\aleph_{0}$-saturated
$1$-h-minimal field. In case $\operatorname{acl}_{K}$ is not the same as
$\operatorname{dcl}_{K}$ it is better to take “étale domains” instead of open
subsets of $K^{n}$ as the local model of the manifold. This is because the
cell decomposition, as provided by Proposition 2.15, decomposes a definable
set into a finite number of pieces, each of which is only a finite cover of an
open set, instead of an open set. We describe this notion formally below:

######  Definition 5.1.

Let $S\subset K^{m}$. A definable function $f:S\to K^{n}$ is (topologically)
étale if it is a local homeomorphism. In other words, for every $x\in T$ there
is a ball $x\in B$ such that $f(B\cap S)$ is open and the inverse map $f(B\cap
S)\to B\cap S\to K^{m}$ is continuous.

Informally, we think of étale maps as similar to open immersions, and will
denote such maps accordingly e.g., $i:U\to K^{n}$. We now proceed to
describing the differential structure of étale maps (or, rather, étale
domains):

######  Definition 5.2.

Suppose $i:U\to K^{n}$ and $j:V\to K^{m}$ are étale maps. A definable function
$f:U\to V$ is strictly differentiable at $x\in U$ if there are balls $x\in B$
and $f(x)\in B^{\prime}$ such that $i:B\cap U\to i(B\cap U)$,
$j:B^{\prime}\cap V\to j(B^{\prime}\cap V)$ are homeomorphisms onto open sets,
such that $f(B\cap U)\subset B^{\prime}\cap V$, and the map $i(B\cap
U)\xrightarrow{i^{-1}}B\cap U\xrightarrow{f}B^{\prime}\cap
V\xrightarrow{j}j(B^{\prime}\cap V)$ is strictly differentiable at $i(x)$. In
this case the derivative $f^{\prime}(x)$ is defined as the derivative of
$i(B\cap U)\to j(B^{\prime}\cap V)$.

The function $f:U\to V$ is called $T_{k}$ at $x$ if the composition $i(B\cap
U)\to j(B^{\prime}\cap V)$ is $T_{k}$ at $i(x)$.

Note that with this definition the given inclusion $U\subset K^{r}$ is not
necessarily strictly differentiable, because the local inverses $i(U\cap B)\to
K^{r}$ of the map $i:U\to K^{n}$ are only topological embeddings, so not
necessarily strictly differentiable.

For the rest of this section let $\mathcal{P}$ stand for any one of the
following adjectives: topological, strictly differentiable, or $T_{n}$.

######  Definition 5.3.

A definable weak $\mathcal{P}$-$n$-manifold is a definable set, $M$, equipped
with a finite number of definable injections, $\varphi_{i}:U_{i}\to M$, and
each $U_{i}$ comes equipped with an étale map $r_{i}:U_{i}\to K^{n}$. We
require further that the sets $U_{ij}:=\varphi_{i}^{-1}(\varphi_{j}(U_{j}))$
are open in $U_{i}$, and that the transition maps $U_{ij}\to U_{ji}$,
$\varphi_{j}^{-1}\varphi_{i}$ are $\mathcal{P}$-maps. We further define:

  1. (1)

A definable weak $\mathcal{P}$-manifolds is a weak $\mathcal{P}$-$n$-manifolds
for some $n$.

  2. (2)

A weak $\mathcal{P}$-manifold is equipped with a topology making the structure
maps, $\varphi_{i}$, open immersions.

  3. (3)

A morphism of definable weak $\mathcal{P}$-manifolds is a definable function
$f:M\to N$, such that for any charts $\varphi_{i}:U_{i}\to M$ and
$\tau_{j}:V_{j}\to N$ the set $W_{ij}=\varphi_{i}f^{-1}\tau_{j}(V_{j})$ is
open in $U_{i}$ and the map $W_{ij}\to V_{j}$ given by
$x\mapsto\tau_{j}^{-1}f\varphi_{i}(x)$ is a $\mathcal{P}$-map.

  4. (4)

A definable $\mathcal{P}$-$n$-manifold is a definable weak
$\mathcal{P}$-$n$-manifold, where the $U_{i}$ are open subsets of $K^{n}$ (and
the maps $U_{i}\to K^{n}$ are inclusions).

  5. (5)

A morphism of definable $\mathcal{P}$-manifolds is a morphism of weak
definable $\mathcal{P}$-manifolds.

Definable weak $K$-manifolds are, immediately from the definition, (abstract)
manifolds over $K$. As such, definable differentiable weak manifold inherit
the classical differential structure. For the sake of completeness we remind
the relevant definitions:

######  Definition 5.4.

If $M$ is a definable strictly differentiable weak manifold and $x\in M$, then
the tangent space of $M$ at $x$, $T_{x}(M)$, is the disjoint union of
$T_{i}=T_{\varphi_{i}^{-1}(x)}(U_{i})=K^{n}$ for $(U_{i},\varphi_{i})$ a chart
around $x$, under the identification of the spaces $T_{i}$ and $T_{j}$
associated with the charts $U_{i},U_{j}$ via the map
$(\varphi_{j}^{-1}\varphi_{i})^{\prime}(\varphi_{i}^{-1}(x))$.

For a strictly differentiable definable morphism $f:M\to N$ of definable
strictly differentiable weak manifolds, we have a map of $K$-vector spaces
$f^{\prime}(x):T_{x}(M)\to T_{f(x)}(N)$ given by the differential of the map
appearing in Definition 5.3 above.

As usual, once we have a chart around a point in a weak strictly
differentiable manifold, we get an identification of $T_{x}(M)$ with $K^{n}$,
but distinct charts may give distinct isomorphisms.

######  Definition 5.5.

A definable (weak) $\mathcal{P}$-Lie group is a group object in the category
of definable (weak) $\mathcal{P}$-manifolds.

######  Lemma 5.6.

Suppose $i:U\to K^{n}$ and $j:V\to K^{m}$ are étale and $f:U\to V$ is a
definable map. Then $f$ is continuous in an open dense subset of $U$.

Also $f$ is strictly differentiable and $T_{k}$ in an open dense subset of
$U$.

###### Proof.

For the statement about continuity, note that $V$ has the subspace topology
(of $V\subset K^{r}$) so we may assume $V=K^{r}$. If we denote $U^{\prime}$
the interior (relative to $U$) of the set of points of $U$ where $f$ is
continuous, then in every ball $B$ where $i$ is a homeomorphism $i:B\cap U\to
i(B\cap U)$ we get that $B\cap U\cap U^{\prime}$ is dense in $B\cap U$, by
generic continuity of definable functions. We conclude that $U^{\prime}$ is
dense as required.

For strict differentiability and $T_{k}$, by the above we may assume that $f$
is continuous. Let $U^{\prime}$ be the interior of the set of all points where
$f:U\to V$ is strictly differentiable and $T_{k}$. This is a definable open
set. By generic differentiability and generic $T_{k}$ property for functions
defined on open sets, for every point $x\in U$ there is an open ball $B\ni x$,
such that $U\cap B\cap U^{\prime}$ is dense in $U\cap B$. Thus, we conclude
that $U^{\prime}$ is dense in $U$. ∎

Note that the previous lemma implies that a (weak) definable topological
manifold $M$ contains an open dense subset $U\subset M$, which admits a
structure of a (weak) definable $T_{n}$ manifold extending the given (weak)
definable topological manifold structure. As a consequence of Proposition 5.7
below we have that this structure on $U$ is unique up to isomorphism and
restriction to a definable open dense subset. For that reason, several of the
statements below hold (essentially unaltered) for definable weak manifolds
(without further assumptions on differentiability or $T_{n}$). For the sake of
clarity of the exposition, we keep these assumptions.

######  Proposition 5.7.

If $f:M\to N$ is a definable function and $M$, $N$ are definable weak
$\mathcal{P}$-manifolds, then $f$ is a $\mathcal{P}$-map in an open dense set
of $M$.

###### Proof.

Considering the charts in $M$ we may assume $M=U\to K^{n}$ is étale. Now if
$(V_{i},\tau_{i})$ are charts for $N$, then $f^{-1}\tau_{i}(V_{i})$ cover $U$,
and so the union of their interiors is open dense in $U$. So we may assume
$N=V\to K^{m}$ is étale. This case is Lemma 5.6. ∎

Recall that the local dimension of a definable set $X$ is defined as

$\dim_{x}X=\min\\{\dim(B\cap X):x\in B\text{ is a definable open neighborhood
of }x\text{ in }M\\}.$

The next lemma is standard:

######  Lemma 5.8.

Suppose $M$ is a definable topological weak manifold. Let $X\subset M$ be a
definable subset. Then $\dim(X)=\max_{x\in X}\dim_{x}(X)$.

If $G$ is a definable weak topological group and $H$ is a subgroup, then the
dimension of $H$ is the local dimension of $H$ at any point.

###### Proof.

If $M=U_{1}\cup\cdots\cup U_{n}$ is a covering by open sets and
$\varphi_{i}:U_{i}\to V_{i}$ is a homeomorphism onto a set $V_{i}$, with an
étale map $V_{i}\to K^{n}$, then ${\rm dim}(X)=\max_{i}({\rm
dim}(\varphi_{i}(X\cap U_{i})))$, and the local dimension of $X$ at $x\in
X\cap U_{i}$ is the local dimension of $\varphi_{i}(X\cap U_{i})$ at
$\varphi_{i}(x)$, so we reduce to the case $M=V$ is étale over $K^{n}$. In
fact, the result is true whenever $M\subset K^{m}$ with the subspace topology,
as then the local dimension of $X\subset M$ at a point $x$ equals the local
dimension of $X$ at $x$ in $K^{m}$, and so the result follows from Proposition
2.11(3).

If $G$ is a definable weak topological group and $H$ is a subgroup, then the
local dimension of $H$ at any point $h\in H$ is constant independent of $h$.
Indeed the left translation $L_{h}:G\to G$ is a definable homeomorphism, that
sends $e$ to $h$ and satisfies $L_{h}(H)=H$, so ${\rm dim}_{e}(H)={\rm
dim}_{h}(H)$. ∎

######  Proposition 5.9.

Suppose $T\subset K^{m}$ is such that there is a coordinate projection
$\pi:T\to U$ onto an open subset $U\subset K^{n}$ and such that the fibres of
$\pi$ are finite of constant cardinality, $s$. Assume the associated map
$f:U\to(K^{m-n})^{[s]}$ is continuous. Then $T\to K^{n}$ is étale.

###### Proof.

Let $x\in T$. Replacing $U$ by a smaller neighborhood around $\pi(x)$ we may
assume, using Fact 2.13, that $f$ lifts to a continuous function
$g:U\to(K^{m-n})^{s}$, $g=(g_{1},\cdots,g_{s})$. In this case one gets that
$T$ is homeomorphic to $\bigsqcup_{i=1}^{s}U$ over $U$, via the map
$(a,i)\mapsto(a,g_{i}(a))$.

∎

######  Lemma 5.10.

Suppose $M=\bigcup_{i=1}^{r}\varphi_{i}(U_{i})$ where $\varphi_{i}:U_{i}\to M$
are definable functions, such that the $U_{i}$ are definable (weak)
$\mathcal{P}$-$n$-manifolds. Suppose further that for all $i,j$ the sets
$U_{ij}:=\varphi_{i}^{-1}(\varphi_{j}(U_{j}))$ are open in $U_{i}$, and the
transition maps $U_{ij}\to U_{ji}$ given by
$x\mapsto\varphi_{j}^{-1}\varphi_{i}(x)$ are $\mathcal{P}$-maps Then $M$ has a
unique structure of a definable (weak) $\mathcal{P}$-$n$-manifold such that
$\varphi_{i}:U_{i}\to M$ is an open immersion.

The proof is straightforward and omitted.

######  Proposition 5.11.

Suppose $M$ is a definable weak topological manifold, then $X\subset M$ is
large if and only if the interior of $X$ in $M$ is dense in $M$.

###### Proof.

Because the dimension of $M\setminus X$ is the maximum of the local dimension
at its points by Lemma 5.8, we conclude that both conditions are local, and so
we may assume $M=U\subset K^{n}$ is open. Here the result follows from
dimension theory. ∎

######  Proposition 5.12.

Suppose $M$ is a weak definable topological manifold. Suppose $X\subset M$ is
definable. Then $X$ is a finite union of locally closed definable subsets of
$M$.

###### Proof.

There is an immediate reduction to the case where $M=U\to K^{n}$ is étale. In
this case $U$ has the subspace topology $U\subset K^{s}$ for some $s$. So it
is enough to prove this for $X\subset K^{s}$. This is a consequence of
Proposition 2.17. ∎

In case $\operatorname{acl}=\operatorname{dcl}$ a weak manifold is generically
a manifold:

######  Proposition 5.13.

Suppose $\operatorname{acl}=\operatorname{dcl}$.

If $M$ is a definable weak $\mathcal{P}$-manifold, then there is a definable
open dense subset $U\subset M$ which is a definable $\mathcal{P}$-manifold.

###### Proof.

There is an immediate reduction to the case in which $i:M=U\to K^{n}$ is
étale. Let $r$ be an uniform bound for the cardinality of the fibers of $U$.
In this case if we denote $X_{k}\subset K^{n}$ the set of points $x$ such that
$i^{-1}(x)$ has cardinality $x$, and $U_{k}\subset X_{k}$ is the interior of
$X_{k}$, then $\bigcup_{k\leq r}U_{k}$ is open dense in in $K^{n}$. Replacing
$U$ with $i^{-1}(U_{k})$ we may assume that the nonempty fibers of $i$ have
constant cardinality. From the assumption
$\operatorname{acl}=\operatorname{dcl}$ we conclude that the map
$i(U)\to(K^{r})^{[s]}$ lifts to a definable map $i(U)\to(K^{r})^{s}$. There is
an open dense subset $V^{\prime}\subset i(U)$ such that $V^{\prime}\to
i(U)\to(K^{r})^{s}$ is a $\mathcal{P}$-map, see Proposition 5.6, and we
conclude that $i^{-1}(V^{\prime})\cong\bigsqcup_{i=1}^{s}V^{\prime}$ over
$V^{\prime}$, which is clearly a $\mathcal{P}$-manifold. ∎

It seems possible that in this situation a weak manifold is already a
manifold, but we do not need this so we do not try to prove it.

The next couple of results are not used in the main theorems, but may be of
independent interest.

######  Definition 5.14.

Suppose $M$ and $N$ are definable strictly differentiable weak manifolds, and
$f:M\to N$ a definable strictly differentiable function.

Then $f$ is called an immersion if the derivative $f^{\prime}(x)$ is injective
at all points $x\in M$.

$f$ is called an embedding if $f$ is an immersion and a homeomorphism onto its
image.

$f$ is called a submersion if the derivative $f^{\prime}(x)$ is surjective for
all $x\in M$.

These notions have the expected properties.

######  Proposition 5.15.

Suppose $f:M\to N$ is a strictly differentiable definable map of strictly
differentiable definable weak manifolds. If $f$ is an immersion then $M$
satisfies the following universal property: For every strictly differentiable
weak definable manifold $P$, and $g:P\to M$, the function $g$ is strictly
differentiable and definable if and only is $fg$ is strictly differentiable
and $g$ is definable and continuous.

If $f$ is an embedding and $g:P\to M$ is as above, then $g$ is a strictly
differentiable definable map if and only if $fg$ is a strictly differentiable
definable map.

######  Proposition 5.16.

If $f:M\to N$ is a surjective submersion, then a map $g:N\to K$ is a strictly
differentiable definable function if and only if the composition $gf$ is
strictly differentiable and definable.

These two properties are a consequence of the theorems on the local structure
of immersions and submersions, Propositions 4.9 and 4.10. We leave the details
for the interested reader to fill.

Suppose $M$ is a definable strictly differentiable weak manifold. If $M\to N$
is a surjective map of sets, it determines at most one structure of a
definable strictly differentiable weak manifold on $N$, in such a way that
$M\to N$ is a submersion. Also, an injective map $N\to M$ determines at most
one structure of a strictly differentiable weak manifold on $N$, in such a way
that $N\to M$ is an embedding. The subsets $N\subset M$ admitting such a
structure are called submanifolds of $M$. We also get that if $N$ is a
definable topological space, and $N\to M$ is a definable and continuous
function, then there is at most one structure of a strictly differentiable
definable manifold on $N$ extending the given topology, and for which $N\to M$
is an immersion. In other words the strictly differentiable weak manifold
structure that makes $N\to M$ an embedding is determined by the set $N$, and
the strictly differentiable weak manifold structure that makes $N\to M$ an
immersion is determined by the topological space $N$.

######  Proposition 5.17.

Suppose $M,N$ are definable strictly differentiable weak manifolds, and let
$f:M\to N$ be an injective definable map. Then there is a definable open dense
subset $U\subset M$ such that $f|_{U}$ is an immersion.

###### Proof.

By Proposition 5.7 we may assume $f$ is strictly differentiable. We have to
show that the interior of the set $\\{x\in M:f^{\prime}(x)\text{ is
injective}\\}$ is dense in $M$. If this is not the case, we can find an open
nonempty subset of $M$ such that $f$ is not an immersion at any point.

So suppose $M$ is an open subset of $K^{n}$, $N$ is an open subset of $K^{m}$
and $f$ is not an immersion at any point. For dimension reasons $n\leq m$. If
we define $X_{k}$ to be the set of points $x$ of $M$ such that $f^{\prime}(x)$
is of rank $k$ then $X_{0}\cup\dots\cup X_{n-1}=M$, and so if $U_{r}$ is the
interior of $X_{r}$ we have that $U_{1}\cup\cdots\cup U_{n-1}$ is open dense
in $M$. So we may assume that $f$ is of constant rank. This contradicts the
result in Proposition 4.11, since the map $(x,y)\mapsto(x,0)$ is not
injective. ∎

The following facts are standard, and are probably known:

######  Fact 5.18.

Suppose $X$ is a Hausdorff space and $X\to Y$ is a surjective continuous map,
which is a local homemorphism with fibers of constant cardinality $s$. Then
the map $t:Y\to X^{[s]}$ given by the fibers of $p$, is continuous.

###### Proof.

Let $\pi:X^{s}\setminus\Delta\to X^{[s]}$ be the canonical projection. Take
$y\in Y$ and $\\{x_{1},\dots,x_{s}\\}=p^{-1}(y)=t(y)$. A basic open
neighborhood of $t(y)$ is of the form $\pi(U_{1}\times\dots\times U_{s})$ for
$x_{k}\in U_{k}$ open and $U_{k}$ pairwise disjoint. Shrinking $U_{k}$ we may
assume $p|_{U_{k}}$ is a homemorphism onto an open set. If
$V=\bigcap_{k}p(U_{k})$, then $t^{-1}(V)\subset\pi(U_{1}\times\dots\times
U_{s})$. ∎

######  Fact 5.19.

Let $X,Y,Z$ be topological spaces, $p:X\to Z$, $q:Y\to Z$ be surjective
continuous functions and $f:X\to Y$ a continuous bijection such that $qf=p$.
Assume that $X$ and $Y$ are Hausdorff spaces, and $p:X\to Z$ has finite fibers
of constant cardinality, $s$. If the map $t:Z\to X^{[s]}$, given by $x\mapsto
p^{-1}(x)$ is continuous then $f$ is a homeomorphism.

###### Proof.

Since $f$ is continuous and bijective, we only need to show that it is open,
which is a local property. Fix some $x\in X$ and $z=p(z)$. By Fact 2.13 the
map $Z\to X^{[s]}$ lifts, locally near $z$, to a continuous map
$(l_{1},\dots,l_{s}):Z\to X^{s}$. Shrinking $Z$ to this neighborhood, and
reducing $X$ and $Y$ accordingly, we may assume that $X$ is homeomorphic to
$\bigsqcup_{i\leq s}Z$ over $Z$, via the homeomorphism $F_{l}:\bigsqcup_{i\leq
s}Z\to X$, given by $(i,z)\mapsto l_{i}(z)$. To see that this is a
homeomorphism, note that the image of the $i$-th-cofactor via $F_{l}$ is the
set $X_{i}=\\{x\in X\mid x=l_{i}p(x)\\}$, which is closed in $X$ (because $X$
is Hausdorff). Since there are only finitely many $X_{i}$ (and they are
pairwise disjoint) they are also open. Finally, the inverse of $F_{l}$
restricted to $X_{i}$ coincides with $p$ which is continuous. So $f$
restricted to $F_{l}(Z)$ is a homeomorphism, so $f$ is open at $x$. Since $x$
was arbitrary, the conclusion follows.

Similarly, we have a homeomorphism $F_{fl}:\bigsqcup_{i\leq s}Z\to Y$, which
is compatible with $f$ in the sense that $fF_{l}=F_{fl}$. ∎

######  Proposition 5.20.

Let $M,N$ be strictly differentiable weak manifolds, and $f:M\to N$ be an
injective definable function. Then there is a dense open $U\subset M$ such
that $f|_{U}$ is an embedding.

###### Proof.

By Proposition 5.17 we may assume that $f$ is an immersion. If
$V_{1},\dots,V_{n}$ is a finite open cover of $N$ and the statement is valid
for $f:f^{-1}V_{i}\to V_{i}$, then it is also valid for $f$. So we may assume
$N=V\to K^{m}$ is étale. From the definition we have that $V\subset K^{d}$ has
the subspace topology. We have already seen that $f:M\to V$ is an immersion,
so if it is a topological embedding into $K^{d}$, then it is an embedding into
$V$. So we may assume $N=K^{m}$.

Now consider $U_{1}\cup\dots\cup U_{n}=M$ a finite open cover of $M$. Assume
$f|_{U_{i}}$ is an embedding. Define
$U_{i}^{\prime}=\text{Int}(U_{i}\setminus\bigcup_{j<i}U_{i})$, and
$U_{i}^{\prime\prime}=U_{i}^{\prime}\setminus\bigcup_{j\neq
i}f^{-1}\operatorname{cl}(f(U_{j}^{\prime}))$. Note that
$\bigcup_{i}U_{i}^{\prime}\subset M$ is an open dense set. Also, note that
$f(U_{i}^{\prime}\setminus U_{i}^{\prime\prime})=\bigcup_{j\neq
i}f(U_{i}^{\prime})\cap\operatorname{cl}(f(U_{j}^{\prime}))\subset\bigcup_{j\neq
i}\operatorname{cl}(f(U_{j}^{\prime}))\setminus f(U_{j}^{\prime})$. So we
conclude that $\dim(U_{i}^{\prime}\setminus U_{i}^{\prime\prime})<\dim(M)$, by
Proposition 2.18. Thus, replacing $U_{i}$ with $U_{i}^{\prime\prime}$ we may
assume $\operatorname{cl}(f(U_{i}))\cap f(U_{j})=\emptyset$ for distinct
$i,j$. In this case one verifies that $f$ is a topological embedding.

We are thus reduced to the case where $M=U\to K^{n}$ is étale.

Consider for each $I\subset\\{1,\dots,m\\}$ of size $n$, the set $A_{I}$ of
$x\in U$ such that the $I$-th-minor of $f^{\prime}(x)$ is invertible. As
$U=\bigcup_{I}A_{I}$ we conclude that
$U^{\prime}=\bigcup_{I}\text{Int}(A_{I})$ is open dense in $U$, so by the
reduction in the previous paragraph we may assume that the composition of
$f:U\to K^{m}$ with the projection onto the first $n$ coordinates $p:K^{m}\to
K^{n}$ is an étale immersion. If $s$ is a uniform bound for the size of the
fibers of $U$ over $K^{n}$, then we can take $A_{k}$ the set of $x\in K^{m}$
such that the fiber $(pf)^{-1}(x)$ has $k$ elements, and consider
$U^{\prime}=\bigcup_{k\leq s}\text{Int}(A_{k})$. So we may assume that if
$V=pf(U)$, the fibers of $U$ over $V$ have the same size $s$.

In this case the function $V\to U^{[s]}$, given by $x\mapsto(pf)^{-1}(x)$, is
continuous, and the function $f$ is topological homeomorphism $U\to f(U)$, see
Facts 5.18 and 5.19. ∎

We can now prove a $1$-h-minimal version of Sard’s Lemma (compare with [16,
Theorem 2.7] for an analogous result in the o-minimal setting). Namely, given
a definable strictly differentiable morphism, $f$, of definable weak
manifolds, call a value $x$ of $f$ regular when $f$ is a submersion at every
point of $f^{-1}(x)$. The statement is, then, that the set of singular values
is small:

######  Proposition 5.21.

Suppose $M$ and $N$ are definable strictly differentiable weak manifolds. If
$f:M\to N$ is a strictly differentiable map, then there exists an open dense
subset $U\subset N$ such that $f:f^{-1}(U)\to U$ is a submersion.

###### Proof.

We have to see that the image via $f$ of the set of points $x\in M$ such that
$f^{\prime}(x)$ is not surjective, is nowhere dense in $N$. This property is
expressible by a first order formula, so we may assume that
$\operatorname{acl}=\operatorname{dcl}$, see Fact 2.5 and the subsequent
remark.

Let $m=\dim(N)$. Let $X\subset M$ a definable set such that for all $x\in X$,
$f^{\prime}(x)$ is not surjective. We have to see that $\dim f(X)<m$. We do
this by induction on the dimension of $X$. The base case, when $X$ is finite,
is trivial.

The dimension of $f(X)$ is the maximum of the local dimensions at points, see
Proposition 5.8. So we may assume $N\subset K^{m}$ is open. Covering $M$ by a
finite number of charts we may assume $M\to K^{n}$ is étale, say $M\subset
K^{r}$.

Then by Proposition 2.15 there exists a finite partition of $X$ into definable
sets such that if $X^{\prime}$ is an element of the partition, there exists a
coordinate projection $p:K^{r}\to K^{l}$, which restricted to $X^{\prime}$ is
a surjection $X^{\prime}\to U$ onto an open subset $U\subset K^{l}$ with
finite fibers of constant cardinality. If we prove that $\dim f(X^{\prime})<m$
for every element $X^{\prime}$ of the partition then also $\dim f(X)<m$. So we
may assume there is a coordinate projection $p:X\to U$ onto an open subset
$U\subset K^{l}$, such that $p^{-1}(u)$ has $t$ elements for all $u\in U$.
From the assumption $\operatorname{acl}=\operatorname{dcl}$, we get that there
are definable sections $s_{1},\dots,s_{t}:U\to X$, such that
$\\{s_{1}(u),\cdots,s_{t}(u)\\}=p^{-1}(u)$, for all $u\in U$. As
$X=\bigcup_{i\leq t}s_{i}(U)$, we may assume $p:X\to U$ is a bijection with
inverse $s:U\to X$. The map $s:U\to M$ becomes strictly differentiable in an
open dense $V\subset U$, see Proposition 5.7. As $s(U\setminus V)$ has smaller
dimension than $X$, we may assume that $s$ is strictly differentiable. Now
note that $s$ has image in $X$ and so the composition $U\to M\to N$ has
derivative which is not surjective at any point of $U$. So we have reduced to
the case in which $M=U\subset K^{n}$ is open and $X=U$.

If we consider $A_{k}\subset U$, the set defined by $A_{k}=\\{x\in
U:f^{\prime}(x)\text{ has rank }k\\}$, then $\bigcup_{k<m}A_{k}=U$ and so
$\bigcup_{k<m}\text{Int}(A_{k})$ is open and dense in $U$. We conclude by the
induction hypothesis that the image of
$U\setminus\bigcup_{k<m}\text{Int}(A_{k})$ is nowhere dense in $N$, and so we
may assume that $f^{\prime}(x)$ has constant rank $k$ in $U$, for a $k<m$.

Consider the set $Y=\\{x\in U:\dim f^{-1}f(x)\geq\dim(U)-k\\}$. Then $Y$ is
definable, because dimension is definable in definable families. Also, $Y$ has
dense interior by the constant rank theorem Proposition 4.11. So once more by
the induction hypothesis we may assume $f^{-1}f(x)$ has dimension at least
$\dim(U)-k$ for all $x\in U$. Then the dimension of $f(U)$ is at most $k$, by
the additivity of dimension. ∎

If $M\to N$ is a map of strictly differentiable weak manifolds, and $y\in N$
is a regular value, then one can show $f^{-1}(y)\subset M$ is a strictly
differentiable weak submanifold.

##  6\. Definable Lie groups

In this section we show that every definable group is a definable weak Lie
group and that the germ of a definable weak Lie group morphism is determined
by its derivative at the identity.

The proof of the following lemma was communicated to us by Martin Hils.

######  Lemma 6.1.

Suppose $G$ is a group $a$-definable in a pregeometric theory, and that
$X,Y\subset G$ are non-empty $a$-definable sets of dimension smaller than $G$.
If $g\in G$ is such that $\dim(gX\cap Y)=\dim(X)$, then
$\dim(g/a)\leq\dim(Y)$.

In particular, there exists $g\in G$ such that $\dim(gX\cap Y)<\dim(X)$

###### Proof.

Denote $d=\dim(X)$ and $d^{\prime}=\dim(Y)$. Suppose $\dim(gX\cap Y)=d$. Note
that $d\leq d^{\prime}$. Let $h^{\prime}\in gX\cap Y$ be such that
$\dim(h^{\prime}/ag)=d$. Let $h=g^{-1}h^{\prime}$.

As $h\in X$ we have $d\geq\dim(h/a)\geq\dim(h/ag)=\dim(h^{\prime}/ag)=d$. The
first inequality is because $h\in X$, the third one because $h$ and
$h^{\prime}$ are inter-definable over $ag$, and the fourth one by choice of
$h^{\prime}$. We conclude that $h$ and $g$ are algebraically independent over
$a$.

Then we obtain that
$d^{\prime}\geq\dim(h^{\prime}/a)\geq\dim(h^{\prime}/ah)=\dim(g/ah)=\dim(g/a)$.
The first inequality because $h^{\prime}\in Y$, the third equality because
$h^{\prime}$ and $g$ are inter-definable over $ah$, and the fourth equality
because $h$ and $g$ are algebraically independent over $a$.

For the second statement, note that if $g\in G$ is such that
$\dim(g/a)=\dim(G)$, or more generally $\dim(g/a)>d^{\prime}$, then
$\dim(gX\cap Y)<\dim(Y)$. ∎

The next lemma generalizes Lemma 2.4 of [11] for o-minimal theories. Pillay’s
proof can be seen to generalize, with some effort, to geometric theories. We
give a different proof:

######  Lemma 6.2.

Suppose, $G$ is a group definable in a pregeometric theory and suppose
$X\subset G$ is such that $\dim(G\setminus X)<\dim(G)$. Then a finite number
of translates of $X$ cover $G$.

###### Proof.

Suppose we have $g_{0},\cdots,g_{n}\in G$ such that
$\dim(G\setminus(\bigcup_{k}g_{k}X))=m$. By the Lemma 6.1 applied to
$G\setminus X$ and $G\setminus(\bigcup_{k}g_{k}X)$ we get that there is
$g_{n+1}\in G$ such that $\dim(G\setminus\bigcup_{k\leq(n+1)}g_{k}X)<m$, which
finishes the proof. ∎

######  Lemma 6.3.

Suppose $G$ is a definable group in a pregeometric theory and $V\subset G$ is
large. Then every $g\in G$ is a product of two elements in $V$.

###### Proof.

The proof of [11, Lemma 2.1] works: if we take $h\in G$ generic over $g$, then
$h^{-1}g$ is also generic over $g$, and so $h,h^{-1}g\in V$ and their product
is $g$. ∎

######  Proposition 6.4.

A definable group can be given the structure of a definable strictly
differentiable weak $T_{k}$-Lie group. The forgetful functor from definable
strictly differentiable weak Lie groups to definable groups is an equivalence
of categories.

If $\operatorname{acl}=\operatorname{dcl}$ the forgetful functor from
definable strictly differentiable $T_{k}$-Lie groups to definable groups is an
equivalence of categories.

###### Proof.

That the forgetful functor is full follows from Proposition 5.7. Indeed,
suppose $G$ and $H$ are strictly differentiable or $T_{k}$-Lie groups, and let
$f:G\to H$ be a definable group morphism. Then by Proposition 5.7 there is an
open dense $U\subset G$ such that $f:U\to H$ is strictly differentiable or
$T_{k}$. If $g_{0}\in U$ is arbitrary, and $g\in G$, consider the formula
$f=L_{f(g)f(g_{0})^{-1}}fL_{g_{0}g^{-1}}$, where we are denoting $L_{h}$ the
left translate by $h$. Now, $L_{g_{0}g^{-1}}$ is a strict diffeomorphism or
$T_{k}$-isomorphism which sends $g$ to $g_{0}$, and $L_{f(g)f(g_{0})^{-1}}$ is
a strict diffeomorphism or $T_{k}$-isomorphism. We conclude that $f$ being
strictly differentiable or $T_{k}$ at $g_{0}$ implies that $f$ is strictly
differentiable of $T_{k}$ at $g$.

To see that the forgetful functor is essentially surjective one follows the
proof of [11, Proposition 2.5]. Namely, let $G$ be of dimension $n$. Decompose
$G$ as in Proposition 2.15, and let $V_{0}\subset G$ be the union of the
$n$-dimensional pieces $U_{0},\cdots,U_{r}$. Give $V_{0}$ the structure of a
weak strictly differentiable manifold with charts the inclusions $U_{i}\to
V_{0}$, see Proposition 5.9.

Note that $V_{0}^{-1}\subset G$ is large in $G$, as the inverse function is a
definable bijection sending the large subset $V_{0}$ onto $V_{0}^{-1}$. As the
intersection of two large sets is large, we conclude that $V_{0}\cap
V_{0}^{-1}$ is large in $G$. A fortiori, $V_{0}\cap V_{0}^{-1}$ is large in
$V_{0}$ and so it contains an open dense subset of $V_{0}$, see Proposition
5.11. Let $V_{1}\subset V_{0}\cap V_{0}^{-1}$ be open dense in $V_{0}$ such
that the inverse function on $V_{1}$ (and into $V_{0}$) is strictly
differentiable and $T_{k}$, see Proposition 5.7.

In a similar way, we have that $V_{0}\times V_{0}\cap m^{-1}(V_{0})$ is large
in $G\times G$ Indeed, $m^{-1}(V_{0})$ is the inverse image of the large
subset $G\times V_{0}$ of $G\times G$ under the definable bijection
$(\operatorname{Id},m):G\times G\to G\times G$. In the same way as before, we
find $Y_{0}\subset V_{0}\times V_{0}\cap m^{-1}(V_{0})$ open and dense in
$V_{0}\times V_{0}$ such that the multiplication map $Y_{0}\to V_{0}$ is
strictly differentiable and $T_{k}$.

Now we take

$V_{1}^{\prime}=\\{g\in V_{1}:(h,g),(h^{-1},hg)\in Y_{0}\text{ for all
}h\text{ generic over }g\\}.$

Note that $V_{1}^{\prime}$ is definable, because $g\in V_{1}^{\prime}$ is
equivalent to $\dim(G\setminus X_{g})<n$ for $X_{g}=\\{h\in
G:(h,g),(h^{-1},hg)\in Y_{0}\\}$, and dimension is definable in definable
families in geometric theories. Note also that $V_{1}^{\prime}$ is large in
$G$, because if $g\in G$ is generic and $h\in G$ is generic over $g$, then
$(h,g)$ is generic in $G\times G$, and $(h^{-1},hg)$, being the image of a
definable bijection at $(h,g)$ is also generic in $G\times G$, so they belong
to $Y_{0}$, because $Y_{0}$ is large in $G\times G$.

Now take $V_{2}$ the interior of $V_{1}^{\prime}$ in $V_{0}$ and $V=V_{2}\cap
V_{2}^{-1}$. Then $V_{2}$ is large in $V_{0}$ by Proposition 5.11, and so it
is also large in $G$. So we conclude that $V$ is an open dense subset of
$V_{0}$.

Define also $Y=\\{(g,h):g,h,gh\in V,(g,h)\in Y_{0}\\}$, then $Y$ is open dense
in $V_{0}\times V_{0}$. This is because $Y$ is large in $G\times G$, with
arguments as above, and it is open in $Y_{0}$, because multiplication is
continuous in $Y_{0}$.

Then we have shown:

  1. (1)

$V$ is large in $G$.

  2. (2)

$Y$ is dense open subset of $V\times V$, and multiplication $Y\to V$ is
strictly differentiable and $T_{k}$.

  3. (3)

Inversion is a strictly differentiable $T_{k}$-map from $V$ onto $V$.

  4. (4)

If $g\in V$ and $h\in G$ is generic in $G$ over $g$ then $(h,g),(h^{-1},hg)\in
Y$

For the last item, note that $h,hg,h^{-1}\in G$ are generic, and so they
belong to $V$. Also, because $g\in V_{1}^{\prime}$, one has that
$(h,g),(h^{-1},hg)\in Y_{0}$.

From this one gets

  1. (a)

For every $g,h\in G$ the set $Z=\\{x\in V:gxh\in V\\}$ is open and $Z\to V$
given by $x\mapsto gxh$ is strictly differentiable and $T_{k}$.

  2. (b)

For every $g,h\in G$, the set $W=\\{(x,y)\in V\times V:gxhy\in V\\}$ is open
in $V\times V$ and the map $W\to V$ given by $(x,y)\mapsto gxhy$ is strictly
differentiable and $T_{k}$.

Indeed, for (a), assume $x_{0}\in Z$, take $h_{1}$ generic over $h$ and $k$
generic over $g,x,h,h_{1}$. Take $h_{2}=h_{1}^{-1}h$. Note that
$h_{1},h_{2}\in V$. Now one writes $f(x)=gxh$ as a composition of strictly
differentiable and $T_{k}$ functions defined on an open neighborhood of
$x_{0}$ in the following way. Consider the set $Z_{1}=\\{x\in V:(kg,x)\in
Y,(kgx,h_{1})\in Y,(kgxh_{1},h_{2})\in Y,(k^{-1},kgxh)\in Y\\}$, then by item
2 we have that $Z_{1}$ is open and the map $x\mapsto
gxh=k^{-1}(((kgx)h_{1})h_{2})$ is a composition of strictly differentiable and
$T_{k}$ functions. Also $x_{0}\in Z_{1}$ by item 4.

Similarly for (b) given $(x_{0},y_{0})\in W$ the set

$W_{1}=\\{(x,y)\in
V:(kg,x),(kgx,h_{1}),(kgxh_{1},h_{2}),(kgxh,y),(k^{-1},kgxhy)\in Y\\}$

is open by item (3), contains $(x_{0},y_{0})$ by item (4) and in $W_{1}$ the
required map is a composition of strictly differentiable and $T_{k}$
functions.

By (1) above and Lemma 6.2 a finite number of translates,
$g_{0}V,\dots,g_{n}V$, cover $G$. Consider the maps $\varphi_{i}:V\to G$ given
by $\varphi_{i}(x)=g_{i}x$. It is straightforward to verify, using (a), (b)
and (3) above that these charts endow $G$ with a (unique) structure of a
strictly differentiable or $T_{k}$ manifold, as in Lemma 5.10, and with this
structure $G$ is a Lie group.

For example, to see that the transition maps are strictly differentiable or
$T_{k}$, we have to see that the sets $V\cap g_{i}^{-1}g_{j}V$ are open, and
the maps $\varphi_{i,j}:V\cap g_{i}^{-1}g_{j}V\to V\cap g_{j}^{-1}g_{i}V$
given by $x\mapsto g_{j}^{-1}g_{i}x$ are strictly differentiable or $T_{k}$.
This is a particular case of (a). Similarly, (b) translates into the
multiplication being strictly differentiable or $T_{k}$ and (a) and (3)
translate into the inversion being strictly differentiable and $T_{k}$.

When $\operatorname{acl}=\operatorname{dcl}$ an appropriate version of cell
decomposition in Proposition 2.15 gives the result by repeating the above
proof. Alternatively, we can see it directly from the result we have just
proved and Proposition 5.13 and Lemma 6.2 (and the appropriate version of the
Lemma 5.10). Indeed, if $G$ is a definable group in a 1-h-minimal field with
$\operatorname{acl}=\operatorname{dcl}$, then $G$ has the structure of a weak
strictly differentiable or $T_{k}$-Lie group. By Proposition 5.13 there is an
open dense $U\subset G$ such that $U$ is a strictly differentiable or
$T_{k}$-manifold. By Proposition 5.11 and Lemma 6.2 a finite number of
translates of $U$ cover $G$, $g_{1}U\cup\cdots\cup g_{n}U=G$. Then the
functions $\varphi_{i}:U\to G$ given by $x\mapsto g_{i}x$ form a gluing data
for $G$ which makes it a strictly differentiable or $T_{k}$-Lie group. ∎

As the previous result implies that every definable group $G$ admits a
structure of a definable weak Lie group which is unique up to a unique
isomorphism, whenever we mention a property of the weak Lie group structure we
understand it with respect to this structure.

######  Definition 6.5.

A definable strictly differentiable local Lie group is given by a definable
open set containing a distinguished point $e\in U\subset K^{n}$, a definable
open subset $e\in U_{1}\subset U$, and definable strictly differentiable maps
$U_{1}\times U_{1}\to U$ denoted as $(a,b)\mapsto a\cdot b$ and $U_{1}\to U$
denoted as $a\mapsto a^{-1}$, such that there exists $e\in U_{2}\subset U_{1}$
definable open such that

  * •

$a\cdot e=e\cdot a=a$ for $a\in U_{2}$.

  * •

If $a,b,c\in U_{2}$ then $a\cdot b\in U_{1},b\cdot c\in U_{1}$ and $(a\cdot
b)\cdot c=a\cdot(b\cdot c)$.

  * •

If $a\in U_{2}$ then $a^{-1}\in U_{1}$ and $a\cdot a^{-1}=a^{-1}\cdot a=e$.

Given two definable strictly differentiable local Lie groups, $U$ and $V$, a
definable strictly differentiable local Lie group morphism is given by a
definable strictly differentiable map $f:U^{\prime}\to V_{1}$ for a $e\in
U^{\prime}\subset U_{1}$ open, with $U_{1}$ and $V_{1}$ as in the above
definition, and such that $f(e)=e$, $f(a\cdot b)=f(a)\cdot f(b)$ and
$f(a^{-1})=f(a)^{-1}$ for $a\in U^{\prime}$. Also two such maps $f_{1}$ and
$f_{2}$ are identified as morphisms if they have the same germ around $0$, in
other words, if there is a definable open neighborhood of the identity
$W\subset\operatorname{dom}(f_{1})\cap\operatorname{dom}(f_{2})$ such that
$f_{1}|_{W}=f_{2}|_{W}$.

It is common to only consider local groups where $e=0$, and translating we see
that every local group is isomorphic to one with this condition. In this case
we denote the distinguished element by $e$ whenever we emphasize its role as a
local group identity.

We will usually identify a local group with its germ at $e$. In those terms,
the prototypical example of a local Lie group is the germ around the identity
of a Lie group.

The following fact is a well known application of the chain rule. We give the
short proof for completeness:

######  Fact 6.6.

Suppose $U$ is a local definable strictly differentiable Lie group. Then the
multiplication map $m:U_{1}\times U_{1}\to U_{0}$ has derivative the
$m^{\prime}(0)(u,v)=u+v$. The inverse $i:U_{1}\to U_{0}$ has derivative
$i^{\prime}(0)(x)=-x$. The $n$-power $p_{n}:U_{n}\to U_{0}$ has derivative
$p_{n}^{\prime}(0)(x)=nx$

###### Proof.

The formula for $m^{\prime}(0)$ follows formally from the equations
$m(x,0)=x,m(0,y)=y$. Indeed if $m(x,y)=ax+by+o(x,y)$, then plugging $y=0$ we
obtain $a=1$ and plugging $x=0$ we obtain $b=1$. Here we are using the small
$o$ notation, $f=o(x,y)$, meaning that for all $\epsilon>0$ there is an $r$
such that if $|(x,y)|<r$ then $|f(x,y)|\leq\epsilon|(x,y)|$, and we are using
the uniqueness of derivatives for the strictly differentiable functions
$m(x,0)$ and $m(0,y)$.

From this the formula for $i^{\prime}(0)$ follows from $m(x,i(x))=0$ and the
chain rule.

The formula for $p_{n}$ follows inductively from the chain rule and
$p_{n}(x)=m(p_{n-1}(x),x)$. ∎

We give some results on subgroups and quotient groups These are not needed for
the main applications.

######  Proposition 6.7.

Suppose $f:G\to H$ is a surjective definable group morphism. Then $f$ is a
submersion.

###### Proof.

This is a consequence of Sard’s Lemma, Proposition 5.21. ∎

######  Fact 6.8.

Suppose $X$ is a topological space and $Y\subset X$ is a finite union of
locally closed subsets of $X$. Then every open nonempty subset of $X$ contains
an open nonempty subset which is disjoint from $Y$ or contained in $Y$.

###### Proof.

The property mentioned is closed under Boolean combinations and is true for
open subsets. ∎

######  Proposition 6.9.

Suppose $G$ is a definable group and $H\subset G$ is a definable subgroup.
Then $H$ is closed in $G$.

###### Proof.

Recall that $H$ is a finite union of locally closed subsets of $G$, see for
instance Proposition 5.12. So by applying Fact 6.8 to $H\subset\bar{H}$, we
conclude that $H$ has nonempty relative interior in $\bar{H}$. As $H$ is a
subgroup we conclude by translation that $H$ is open in $\bar{H}$. An open
subgroup is the complement of some of its translates, so it is also closed. We
conclude that $H=\bar{H}$ is closed. ∎

######  Proposition 6.10.

Suppose $H\subset G$ is a subgroup of $G$. Then with the structure of weak
definable strictly differentiable manifolds on $G$ and $H$, the inclusion
$i:H\to G$ is a closed embedding.

###### Proof.

By Proposition 5.20, there is an open dense set $U\subset H$ such that
$i|_{U}$ is an embedding. Replacing $U$ by
$U^{\prime}=U\setminus\operatorname{cl}(i(H)\setminus i(U))$ if necessary, and
keeping in mind Proposition 2.18 to show $U^{\prime}$ is large in $H$, we may
assume $i(U)$ is open in $i(H)$. By translation we conclude that $i$ is an
immersion. Also for an open set $V\subset H$ we have that
$i(V)=i(\bigcup_{h\in H}hU\cap V)=\bigcup_{h}hi(U\cap h^{-1}V)$ is open in
$i(H)$. Since $i$ is injective, the conclusion follows. ∎

As a consequence of the theorem on constant rank functions, Proposition 4.11,
we have the following result:

######  Corollary 6.11.

Suppose $U$ and $V$ are definable strictly differentiable local Lie groups and
let $g,f:U\to V$ be definable strictly differentiable local Lie group
morphisms. If we denote $Z=\\{x\in U:g(x)=f(x)\\}$, then
$\dim_{e}Z=\dim(\ker(f^{\prime}(e)-g^{\prime}(e)))$.

In particular if $G$ and $H$ are definable strictly differentiable weak Lie
groups and $g,f$ are definable strictly differentiable Lie group morphisms
then $\dim\\{x:f(x)=g(x)\\}=\dim(\ker(f^{\prime}(e)-g^{\prime}(e)))$.

###### Proof.

The second result follows from the first because of Lemma 5.8.

In order to keep the proof readable we only verify the first statement in the
case of weak Lie groups. The proof for local Lie groups is similar. By
translating in $G$ we see that the map $f\cdot g^{-1}:G\to G$ has, at any
point of $G$, derivatives of constant rank equal to ${\rm dim}(G)-k$, for
$k=\dim(\text{ker}(f^{\prime}(e)-g^{\prime}(e)))$. Indeed, if $u\in G$, then
$(f\cdot g^{-1})L_{u}=L_{f(u)}R_{g(u)^{-1}}(f\cdot g^{-1})$, where $L_{u}$,
$R_{u}$ denote the left and right translates by $u$, respectively. By the
chain rule we get $(f\cdot
g^{-1})^{\prime}(u)L_{u}^{\prime}(e)=(L_{f(u)}R_{g(u)^{-1}})^{\prime}(f(u)\cdot
g(u)^{-1})(f\cdot g^{-1})^{\prime}(e)$. As $L_{u}$ and $L_{f(u)}R_{g(u)}$ are
definable strict diffeomorphisms, we have that their derivatives at any point
are vector space isomorphisms, so we conclude that the rank of $(f\cdot
g^{-1})^{\prime}(u)$ equals the rank of $(f\cdot
g^{-1})^{\prime}(e)=f^{\prime}(e)-g^{\prime}(e)$ (see Fact 6.6), as desired.

By the theorem on constant rank functions, Proposition 4.11, we conclude that
there are nonempty open sets $U\subset G$ and $V\subset H$, balls
$B_{1},B_{2}$ and $B_{3}$ around the origin and definable strictly
differentiable isomorphisms $\varphi_{1}:U\to B_{1}\times B_{2}$ and
$\varphi_{2}:V\to B_{1}\times B_{3}$, such that $f(U)\subset V$ and
$\varphi_{2}f=\varphi_{1}\alpha$ for $\alpha:B_{1}\times B_{3}\to B_{1}\times
B_{3}$ the function $(x,y)\mapsto(x,0)$. Translating in $G$ we may assume
$e\in U$. More precisely, from the formula $(f\cdot
g^{-1})L_{u}=(L_{f(u)}R_{g(u)^{-1}})(f\cdot g^{-1})$ discussed before, if
$u\in U$ maps to $(0,0)$ under $\varphi_{1}$, then $e\in u^{-1}U$, so we may
replace $(U,V,\varphi_{1},\varphi_{2})$ by
$(u^{-1}U,f(u)^{-1}Vg(u),\varphi_{1}L_{u},\varphi_{2}L_{f(u)}R_{g(u)}^{-1})$.
Note also that $\varphi_{1}(e)=(0,0)$.

In this case we obtain $\\{0\\}\times B_{2}=\varphi_{1}(Z\cap U)$ so the local
dimension of $e$ at $Z$ is the local dimension of $\\{0\\}\times B_{2}\subset
B_{1}\times B_{2}$ at $(0,0)$, which is the dimension of $B_{2}$ and is as in
the statement. ∎

In the particular case the dimension
$\dim(\text{ker}(f^{\prime}(e)-g^{\prime}(e)))$ of the previous statement
equals the dimension of $G$ we get:

######  Corollary 6.12.

Suppose $U$ and $V$ are definable strictly differentiable local Lie groups and
let $g,f:U\to V$ be definable strictly differentiable local Lie group
morphisms. Then $f$ and $g$ are equal (as local Lie group morphisms) if and
only if $f^{\prime}(0)=g^{\prime}(0)$.

In particular if $G$ and $H$ are definable strictly differentiable weak Lie
groups and $g,f$ are definable strictly differentiable Lie group morphisms
then $f$ and $g$ coincide in an open neighborhood of the identity $e$ if and
only if $f^{\prime}(e)=g^{\prime}(e)$.

The following two corollaries are not needed for the sequel, but may be
interesting on their own right.

######  Corollary 6.13.

Suppose $H_{1}$ and $H_{2}$ are subgroups of the strictly differentiable
definable weak Lie group $G$. Then $T_{e}(H_{1}\cap H_{2})=T_{e}(H_{1})\cap
T_{e}(H_{2})$ as subspaces of $T_{e}(G)$.

###### Proof.

This is a consequence of Corollary 6.11. Indeed, we know $H_{1}$, $H_{2}$ and
$H_{1}\cap H_{2}$ are strictly differentiable definable weak Lie groups and
the inclusion maps $H_{1}\cap H_{2}\to H_{i}$ and $H_{i}\to G$ are strictly
differentiable immersions, for example by Proposition 6.10, so the statement
makes sense. We also have the diagonal map $\Delta:H_{1}\cap H_{2}\to
H_{1}\times H_{2}$ is the equalizer of the two projections $p_{1}:H_{1}\times
H_{2}\to G$ and $p_{2}:H_{1}\times H_{2}\to G$. The kernel of the
$p_{1}^{\prime}(e)-p_{2}^{\prime}(e)$ is the image under the diagonal map of
$T_{e}(H_{1})\cap T_{e}(H_{2})$. So by the equality of the dimensions in
Corollary 6.11 we conclude $T_{e}(H_{1}\cap H_{2})=T_{e}(H_{1})\cap
T_{e}(H_{2})$. ∎

######  Corollary 6.14.

If $G$ is a definable strictly differentiable weak Lie group and $H_{1},H_{2}$
are subgroups, then there is $U\subset G$ an open neighborhood of $e$ such
that $U\cap H_{1}=U\cap H_{2}$ if and only if $T_{e}(H_{1})=T_{e}(H_{2})$.

###### Proof.

By Corollary 6.13 we get $T_{e}(H_{3})=T_{e}(H_{1})=T_{e}(H_{2})$ for
$H_{3}=H_{1}\cap H_{2}$. Then as the inclusion $H_{3}\to H_{1}$ produces an
isomorphism of tangent spaces at the identity we conclude by the inverse
function theorem 4.4 that there is $U\subset G$ an open neighborhood of the
identity, such that $U\cap H_{1}=U\cap H_{3}$. Note that this also uses that
the topology of $H_{1}$ and $H_{3}$ which makes them strictly differentiable
definable Lie groups coincides with the subgroup topology coming from $G$, see
Proposition 6.10.

Symmetrically we have $U^{\prime}\cap H_{2}=U^{\prime}\cap H_{3}$ for some
open $U^{\prime}$. ∎

Next we give the familiar definition of the Lie bracket in $T_{e}(G)$ for the
definable Lie group $G$, and show it forms a Lie algebra.

######  Definition 6.15.

Suppose $G$ is a definable strictly differentiable weak Lie group. For $g\in
G$ we consider the map $c_{g}:G\to G$ defined by $c_{g}(h)=ghg^{-1}$. Then
$c_{g}$ is a definable group morphism and so it is strictly differentiable,
see Proposition 5.7. Its derivative produces a map
$\mathrm{Ad}:G\to\mathrm{Aut}_{K}(T_{e}(G))$, $g\mapsto c_{g}^{\prime}(e)$
which is a definable map and a group morphism by the chain rule and the
equation $c_{g}c_{h}=c_{gh}$. Then $\mathrm{Ad}$ is strictly differentiable
and so its derivative at $e$ gives a linear map
$\mathrm{ad}:T_{e}(G)\to\mathrm{End}_{K}(T_{e}(G))$. In other words this gives
a bilinear map $(x,y)\mapsto\mathrm{ad}(x)(y)$, $T_{e}(G)\times T_{e}(G)\to
T_{e}(G)$ denoted $(x,y)\mapsto[x,y]$.

This map is called the Lie bracket.

######  Proposition 6.16.

Let $G$ be a definable weak $T_{2}$-Lie group. Let $0\in U\subset K^{n}$ be an
open set and $i:U\to G$ a $T_{2}$-diffeomorphism of $U$ onto an open subset of
$G$, that sends $0$ to $e$. Make $U$ into a local definable group via $i$.
Then under the identification $i^{\prime}(0):K^{n}\to T_{e}(G)$ we have that
the Lie bracket is characterized by the property $x\cdot y\cdot x^{-1}\cdot
y^{-1}=[x,y]+O(x,y)^{3}$, for $x,y\in U$.

###### Proof.

We have that the function $f(x,y)=x\cdot y\cdot x^{-1}$ satisfies $f(0,y)=y$
and $f(x,0)=0$, so its Taylor approximation of order 2 is of the form
$f(x,y)=y+axy+O(x,y)^{3}$. Indeed it is of the form
$a_{0}+a_{1}x+a_{2}y+a_{3}x^{2}+a_{4}xy+a_{5}y^{2}+O(x,y)^{3}$ and plugging
$x=0$ and using the uniqueness of the Taylor approximation we get
$a_{0}=a_{5}=0$ and $a_{2}=1$, and a similar argument with $y=0$ gives
$a_{1}=a_{3}=0$, so $f(x,y)=y+axy+O(x,y)^{3}$ as claimed.

From the definition of $\mathrm{Ad}(x)$ we get
$f(x,y)=\mathrm{Ad}(x)y+O_{x}(y^{2})$ where $O_{x}$ means the coefficient may
depend on $x$.

Note that the definition of $\mathrm{ad}(x)$ gives
$\mathrm{Ad}(x)(y)=y+[x,y]+O(x^{2}y)$. Indeed, we have
$\mathrm{Ad}(x)=\mathrm{Ad}(0)+\mathrm{Ad}^{\prime}(0)(x)+O(x^{2})=I+\mathrm{ad}(x)+O(x^{2})$,
where $I$ is the identity matrix, and evaluating at $y$ we conclude
$\mathrm{Ad}(x)y=y+[x,y]+O(x^{2}y)$.

We conclude that $y+axy+O(x,y)^{3}=y+[x,y]+O_{x}(y^{2})$. This implies
$[x,y]=axy$. See Lemma 3.15. Now from $x\cdot y\cdot x^{-1}=y+axy+O(y,x)^{3}$,
and the formula $x\cdot y^{-1}=x-y+b_{0}x^{2}+b_{1}xy+b_{2}y^{2}+O(x,y)^{3}$
(see Fact 6.6), we get $x\cdot y\cdot x^{-1}\cdot y^{-1}=(x\cdot y\cdot
x^{-1})\cdot y^{-1}=axy+b_{3}y^{2}+O(x,y)^{3}$. On the other hand if
$c(x,y)=x\cdot y\cdot x^{-1}\cdot y^{-1}$ then $c(0,y)=0$ implies that
$b_{3}=0$, as required. ∎

######  Proposition 6.17.

Let $G$ be a definable strictly differentiable weak Lie group. Then
$(T_{e}(G),[,])$ is a Lie algebra.

###### Proof.

We have to prove $[x,x]=0$ and the Jacobi identity. We will use the
characterization of Proposition 6.16 (we may assume $G$ is $T_{2}$ by
Proposition 6.4). $[x,x]=0$ now follows immediately.

The idea of proof of the Jacobi identity is to express $xyz$ as $f(x,y,z)zyx$
in two different ways using associativity, the first one permutes from left to
right, the second permutes $yz$ and then permutes from left to right. The
details follow.

Writing $c(x,y)=xyx^{-1}y^{-1}$ one has

$xyz=c(x,y)yxz=c(x,y)yc(x,z)zx=c(x,y)([y,c(x,z)]+O(y,c(x,z))^{3})c(x,z)yzx=c(x,y)([y,[x,z]]+O(x,y,z)^{4})c(x,z)c(y,z)zyx=(c(x,y)+[y,[x,z]]+c(x,z)+c(y,z)+O(x,y,z)^{4})zyx$.

At the last step we use the formula $xy=x+y+O(x,y)^{2}$ And on the other hand

$xyz=xc(y,z)zy=([x,c(y,z)]+O(x,c(y,z))^{3})c(y,z)xzy=([x,[y,z]]+O(x,y,z)^{4})c(y,z)c(x,z)zxy=([x,[y,z]]+O(x,y,z)^{4})c(y,z)c(x,z)zc(x,y)yx=([x,[y,z]]+O(x,y,z)^{4})c(y,z)c(x,z)([z,[x,y]]+O(x,y,z)^{4})c(x,y)zyx=([x,[y,z]]+c(y,z)+c(x,z)+c(x,y)+[z,[x,y]]+O(x,y,z)^{4})zyx$.

From this we get $[y,[x,z]]=[x,[y,z]]+[z,[x,y]]+O(x,y,z)^{4}$ and from the
uniqueness of Taylor expansions we obtain $[y,[x,z]]=[x,[y,z]]+[z,[x,y]]$
which is the Jacobi identity. ∎

Given a strictly differentiable definable weak Lie group $G$, we denote