# Closed 1-forms in topology and geometric group theory

Michael Farber, Ross Geoghegan and Dirk Schütz  Department of Mathematics,
University of Durham, Durham DH1 3LE, UK<EMAIL_ADDRESS>Department of Mathematics, SUNY Binghamton, NY 13902-6000, USA
<EMAIL_ADDRESS>Department of Mathematics, University of Durham,
Durham DH1 3LE, UK<EMAIL_ADDRESS>
###### Abstract.

In this article we describe relations of the topology of closed 1-forms to the
group theoretic invariants of Bieri-Neumann-Strebel-Renz. Starting with a
survey, we extend these Sigma invariants to finite CW-complexes and show that
many properties of the group theoretic version have analogous statements. In
particular we show the relation between Sigma invariants and finiteness
properties of certain infinite covering spaces. We also discuss applications
of these invariants to the Lusternik-Schnirelmann category of a closed 1-form
and to the existence of a non-singular closed 1-form in a given cohomology
class on a high-dimensional closed manifold.

To S.P. Novikov on the occasion of his 70-th birthday

## Introduction

The last three decades have seen a growing interest in the topology of closed
1-forms ever since S.P. Novikov [23, 24] introduced Morse theoretic techniques
to study classical problems in mathematical physics. In analogy to the Morse-
Smale complex of an ordinary Morse function on a closed manifold, he
constructed a chain complex, now called the Novikov complex, associated to a
closed 1-form whose singularities are non-degenerate in the sense of Morse.
While Novikov’s interest was to study such problems as Kirchhoff type
equations [24, 25, 26], other applications in different areas of mathematics
would soon become apparent. Gradient vector fields of closed 1-forms, for
example, give rise to fascinating results in dynamical systems, and some of
the topics arising this way have been covered in a recent monograph [13] and
survey article [18].

Another quite surprising link has been made to geometric group theory, when it
became clear that the work of Sikorav [29], who applied Novikov theory to
symplectic topology, was closely related to the work of Bieri, Neumann,
Strebel and Renz [6, 7]. More specifically, the geometric invariants
$\Sigma^{k}(G)$ of a group $G$, which contain important information on the
finiteness properties of certain subgroups of $G$ and whose definition is
recalled in Section 1, can be described in terms of vanishing results of a
generalized Novikov homology.

The feature that combines these areas is that closed 1-forms represent
cohomology classes $\xi\in H^{1}(X;{\mathbb{R}})$. If $X$ is a smooth manifold
this is a special case of de Rham theory, and the first-named author has
developed a theory of closed 1-forms on topological spaces for which this
result holds more generally, see [11]. Now if $X$ is connected,
$H^{1}(X;{\mathbb{R}})$ can be identified with the set of homomorphisms ${\rm
Hom}(\pi_{1}(X),{\mathbb{R}})$, where ${\mathbb{R}}$ is considered as a group
with the usual addition. Indeed, the invariants $\Sigma^{k}(G)$ can be viewed
as subsets of the unit sphere in ${\rm Hom}(G,{\mathbb{R}})$.

One purpose of this article is to describe these relations in a general
setting, and to develop the theory of Bieri-Neumann-Strebel-Renz using the
language of the topology of closed 1-forms. In particular, we extend the
notion of Sigma invariants to finite CW-complexes. These invariants
$\Sigma^{k}(X)$ for $k\geq 1$ are defined as generalizations of the group
theoretic versions, and they have similar properties. This generalization is
motivated by the fact that the group theoretic invariants regularly occur in
the topology of closed 1-forms. For example, the condition
$\xi\in\Sigma^{2}(\pi_{1}(M))$ appears implicitly in the work of Latour [21]
as a necessary condition for the existence of a non-singular closed 1-form in
$\xi\in H^{1}(M;{\mathbb{R}})$, where $M$ is a high-dimensional closed
manifold ($\dim M\geq 6$). Another necessary condition of [21], which in fact
implies $\xi\in\Sigma^{2}(\pi_{1}(M))$, is the contractibility of a certain
function space. It turns out that contractibility of this space is equivalent
to $\xi\in\Sigma^{k}(M)$ for all $k\geq 1$. We remark that for compact
3-manifolds the condition $\xi\in\Sigma^{1}(\pi_{1}(M))$ is sufficient for the
existence of a non-singular closed 1-form representing $\xi$, see [6].

Another property of these new Sigma invariants is that they reflect finiteness
properties of infinite abelian covering spaces $q:\overline{X}\to X$. By an
abelian covering we mean a regular covering with
$\pi_{1}(X)/\pi_{1}(\overline{X})$ an abelian group. Denote

$\displaystyle S(X,\overline{X})$ $\displaystyle=$
$\displaystyle\left\\{0\not=\xi:\pi_{1}(X)\to{\mathbb{R}}\,\left|\,\xi|_{\pi_{1}(\overline{X})}=0\right.\right\\}.$

######  Theorem.

Let $X$ be a finite connected CW-complex and $q:\overline{X}\to X$ a regular
covering space with $\pi_{1}(X)/\pi_{1}(\overline{X})$ abelian. For $k\geq 1$,
the following properties are equivalent.

  1. (1)

$\overline{X}$ is homotopy equivalent to a CW-complex with finite
$k$-skeleton.

  2. (2)

$S(X,\overline{X})\subset\Sigma^{k}(X)$.

Furthermore, if $S(X,\overline{X})\subset\Sigma^{\dim X}(X)$, then
$\overline{X}$ is finitely dominated.

The proof of this theorem is given in Section 7.

A common feature in the definitions and techniques is the notion of movability
of subsets of a space $X$. Here movability is meant with respect to a closed
1-form $\omega$ representing a cohomology class $\xi\in
H^{1}(X;{\mathbb{R}})$. Roughly, movability of a set $A\subset X$ means that
there is a homotopy $H$ of $A$ into $X$ starting with the inclusion and such
that for every point $a\in A$ the integral of $\omega$ along the path
$t\mapsto H_{t}(a)$ is large. While homological versions, using chain
homotopies, of this appeared already in Bieri and Renz [7, Thm.C], a
topological version was formulated in a quite different context in developing
a Lusternik-Schnirelmann theory for closed 1-forms, see [11, 12, 13, 14, 18].
Due to the similar nature one expects a closer relation, which we derive in
Section 4. A movability notion for homology classes is developed in Section 8,
which has applications to cup-length estimates for the Lusternik-Schnirelmann
theory of a closed 1-form.

This article is written as a companion to the recent survey article [18] which
was focussing on applications of closed 1-forms in dynamical systems. The
present paper contains a significant amount of new material, although parts of
it are also meant as a survey.

##  1\. Bieri-Neumann-Strebel-Renz Invariants

Let $G$ be a finitely generated group. We want to recall the definition of the
Bieri-Neumann-Strebel-Renz invariants $\Sigma^{k}(G;{\mathbb{Z}})$, introduced
in [6, 7]. We denote

$\displaystyle S(G)$ $\displaystyle=$ $\displaystyle({\rm
Hom}(G,{\mathbb{R}})-\\{0\\})/{\mathbb{R}}_{+},$

that is, we identify nonzero homomorphisms, if one is a positive multiple of
the other. This is a sphere of dimension $r-1$, where $r$ denotes the rank of
the abelianization of $G$. We will identify $S(G)$ with the unit sphere in
${\rm Hom}(G,{\mathbb{R}})$ (after choosing an inner product on the latter)
and simply write $\xi\in S(G)$.

Given $\xi\in S(G)$, we denote

$\displaystyle{\mathbb{Z}}G_{\xi}$ $\displaystyle=$
$\displaystyle\left\\{\left.\sum_{g\in
G}n_{g}g\in{\mathbb{Z}}G\,\right|\,n_{g}=0\mbox{ for }\xi(g)<0\right\\},$

a subring of ${\mathbb{Z}}G$.

We say that the trivial ${\mathbb{Z}}G$-module ${\mathbb{Z}}$ is of type
$FP_{k}$ over ${\mathbb{Z}}G_{\xi}$, if there exists a resolution

(1) $\ldots\longrightarrow F_{i}\longrightarrow
F_{i-1}\longrightarrow\ldots\longrightarrow
F_{0}\longrightarrow\mathbb{Z}\longrightarrow 0$

of ${\mathbb{Z}}$ by free ${\mathbb{Z}}G_{\xi}$-modules with each $F_{i}$
finitely generated for $i\leq k$.

######  Definition 1.

The Bieri-Neumann-Strebel-Renz invariants are now defined as

$\displaystyle\Sigma^{k}(G;{\mathbb{Z}})$ $\displaystyle=$
$\displaystyle\\{\xi\in S(G)\,|\,{\mathbb{Z}}\mbox{ is of type }FP_{k}\mbox{
over }{\mathbb{Z}}G_{\xi}\\}.$

The power of these invariants lies in the fact that they are closely related
to the finiteness properties of subgroups of $G$. Let us recall the relevant
finiteness properties.

######  Definition 2.

For $k\geq 1$ a group $G$ is of type $FP_{k}$, if there is a resolution (1) of
${\mathbb{Z}}$ by free ${\mathbb{Z}}G$-modules with each $F_{i}$ finitely
generated for $i\leq k$. Also, we say that $G$ is of type $F_{k}$, if there is
an Eilenberg-MacLane space for $G$ with finite $n$-skeleton.

We get that $G$ of type $F_{k}$ implies type $FP_{k}$ by looking at the
cellular chain complex of the universal cover of the Eilenberg-MacLane space,
and type $FP_{1}$ is equivalent to type $F_{1}$ which simply means finitely
generated. But type $FP_{2}$ does not imply finitely presented, as the
examples of Bestvina and Brady show [2]. For more information on these
finiteness properties see Brown [9] and Geoghegan [19].

Notice that ${\mathbb{Z}}G$, when viewed as a ${\mathbb{Z}}G_{\xi}$-module for
any $\xi$, is a direct limit of free ${\mathbb{Z}}G_{\xi}$-modules. It is
therefore a flat ${\mathbb{Z}}G_{\xi}$-module. Furthermore, for every
${\mathbb{Z}}G$-module $A$ we have
${\mathbb{Z}}G\otimes_{{\mathbb{Z}}G_{\xi}}A\cong A$. Thus if
$\Sigma^{k}(G;{\mathbb{Z}})\not=\emptyset$ for some $k\geq 1$, we can apply
${\mathbb{Z}}G\otimes_{{\mathbb{Z}}G_{\xi}}-$ to the resolution (1) for some
$\xi\in\Sigma^{k}(G;{\mathbb{Z}})$, to get that $G$ is itself of type
$FP_{k}$.

The following theorem, which we generalize in Section 7, relates finiteness
properties of certain subgroups to the invariants.

######  Theorem 1 (Bieri-Renz, [7]).

Let $G$ be a group of type $FP_{k}$, $N$ a subgroup of $G$ containing the
commutator subgroup of $G$. Then $N$ is of type $FP_{k}$ if and only if
$\Sigma^{k}(G;{\mathbb{Z}})$ contains the subsphere $S(G,N)=\\{\xi\in
S(G)\,|\,N\leq{\rm Ker}\,\xi\\}$.

There also exists a version of Theorem 1 which gives a criterion for $N$ to be
of type $F_{k}$, involving a homotopical invariant $\Sigma^{k}(G)$. We will
see more about this invariant below. Another result, proven in [7] is that all
$\Sigma^{k}(G;{\mathbb{Z}})$ are open subsets of $S(G)$.

In [6] it was shown that even the particular case $k=1$ has very important
applications to group theory.

######  Theorem 2 (Bieri-Neumann-Strebel, [6]).

Let $G$ be a finitely presented group without non-abelian free subgroups. Then

$\displaystyle\Sigma^{1}(G;{\mathbb{Z}})\cup-\Sigma^{1}(G;{\mathbb{Z}})$
$\displaystyle=$ $\displaystyle S(G).$

Here $-\Sigma^{1}(G)$ denotes the image of $\Sigma^{1}(G)$ under the antipodal
map.

We now want to give a more geometrical interpretation of these invariants, for
the moment we will confine ourselves to the case $k=1$ and for simplicity we
will assume that $G$ is finitely presented. Let $X$ be a finite CW-complex
with $\pi_{1}(X)\cong G$ and let $q:\overline{X}\to X$ be the universal
abelian covering. Given a non-zero homomorphism $\xi:G\to{\mathbb{R}}$ we can
build a map $h:\overline{X}\to{\mathbb{R}}$ with $h(gx)=\xi(g)+h(x)$ for all
$g\in G$ and $x\in\overline{X}$ by induction over the skeleta of
$\overline{X}$. Note that $G$ acts on $\overline{X}$ by covering
transformations, with the commutator subgroup acting trivially. Write
$N=h^{-1}([0,\infty))$, then $N$ need not be connected, but has a unique
component on which $h$ is unbounded, see [6, Lm.5.2]. In the result below we
assume that the basepoint of $N$ is chosen in this component.

######  Theorem 3 (Bieri-Neumann-Strebel, [6]).

We have $\xi\in\Sigma^{1}(G;{\mathbb{Z}})$ if and only if
$i_{\\#}:\pi_{1}(N)\to\pi_{1}(\overline{X})$ is an epimorphism, where
$i:N\to\overline{X}$ is the inclusion.

This geometric criterion is closely related to a concept of movability of a
subset of $X$ with respect to a given $\xi$ and which has recently been
studied in connection with a Lusternik-Schnirelmann theory of such $\xi$, see
[11, 13, 18]. Let us recall the definition of a closed 1-form on a topological
space $X$.

######  Definition 3.

A continuous closed 1-form $\omega$ on a topological space $X$ is defined as a
collection $\\{f_{U}\\}_{U\in\mathcal{U}}$ of continuous real-valued functions
$f_{U}:U\to{\mathbb{R}}$ where $\mathcal{U}=\\{U\\}$ is an open cover of $X$
such that for any pair $U,V\in\mathcal{U}$ the difference

$f_{U}|_{U\cap V}-f_{V}|_{U\cap V}:U\cap V\to{\mathbb{R}}$

is a locally constant function. Another such collection
$\\{g_{V}\\}_{V\in\mathcal{V}}$ (where $\mathcal{V}$ is another open over of
$X$) defines an equivalent closed 1-form if the union collection
$\\{f_{U},g_{V}\\}_{U\in\mathcal{U},V\in\mathcal{V}}$ is a closed 1-form,
i.e., if for any $U\in\mathcal{U}$ and $V\in\mathcal{V}$ the function
$f_{U}-g_{V}$ is locally constant on $U\cap V$.

These closed 1-forms behave in the same way as smooth closed 1-forms on
manifolds; they can be integrated along paths $\gamma:[a,b]\to X$, and
integration along loops defines a homomorphism
$\xi_{\omega}:\pi_{1}(X)\to{\mathbb{R}}$. Furthermore, every such homomorphism
can be realized by a closed 1-form. See [18, §3] for details.

######  Example 1.

A continuous function $f:X\to S^{1}$ determines a closed 1-form in the
following way. Think of $S^{1}$ as ${\mathbb{R}}/{\mathbb{Z}}$ and let
$p:{\mathbb{R}}\to{\mathbb{R}}/{\mathbb{Z}}$ be the projection. If
$I=(a,b)\subset{\mathbb{R}}$ is an open interval with $b-a\leq 1$, then $I$ is
homeomorphic to the open subset $p(I)\subset{\mathbb{R}}/{\mathbb{Z}}$ via
$p$. The collection

$\displaystyle\omega$ $\displaystyle=$
$\displaystyle\left\\{(p|_{I})^{-1}\circ f|_{f^{-1}(p(I))}:f^{-1}(p(I))\to
I\right\\}$

then defines a closed 1-form. Furthermore,
$\xi_{\omega}:\pi_{1}(X)\to{\mathbb{R}}$ can be identified with
$f_{\\#}:\pi_{1}(X)\to\pi_{1}(S^{1})$, if the standard generator of
$\pi_{1}(S^{1})$ is identified with $1\in{\mathbb{R}}$.

######  Definition 4.

Let $X$ be a finite connected CW-complex, $G=\pi_{1}(X)$ and $\omega$ a closed
1-form on $X$. A subset $A\subset X$ is called $n$-movable with respect to
$\omega$ and control $C\geq 0$ (where $n$ is an integer), if there is a
homotopy $H:A\times[0,1]\to X$ such that $H(a,0)=a$ or all $a\in A$, and

$\displaystyle\int_{a}^{H_{1}(a)}\omega$ $\displaystyle\geq$ $\displaystyle n$

and

$\displaystyle\int_{a}^{H_{t}(a)}\omega$ $\displaystyle\leq$ $\displaystyle-C$

for all $a\in A$ and $t\in[0,1]$. Here the integral is taken over the path
$s\mapsto H(a,s)$ for $s\in[0,t]$.

This notion of movability has its roots in the Lusternik-Schnirelmann theory
of a closed 1-form, compare [11, 18]. The next Proposition shows that it also
gives a criterion for $\Sigma^{1}(G;{\mathbb{Z}})$.

######  Proposition 1.

Let $X$ be a finite connected CW-complex, $G=\pi_{1}(X)$ and
$\xi:G\to{\mathbb{R}}$ a homomorphism which is represented by a closed 1-form
$\omega$. Then the following are equivalent.

  1. (1)

$\xi\in\Sigma^{1}(G;{\mathbb{Z}})$.

  2. (2)

There is a $C\geq 0$ such that the 1-skeleton $X^{(1)}\subset X$ is
$n$-movable with respect to $\omega$ and control $C$ for every $n>0$.

###### Proof.

(1) $\Longrightarrow$ (2): Let $\overline{X}$ be the universal abelian cover
of $X$ and $h:\overline{X}\to{\mathbb{R}}$ be obtained from the pullback of
$\omega$ to $\overline{X}$. It is easy to see that we have $h(gx)=\xi(g)+h(x)$
for all $x\in\overline{X}$ and $g\in G$. We choose $N=h^{-1}([0,\infty))$, and
let $N^{\prime}\subset N$ be the component such that $h$ is unbounded on
$N^{\prime}$ by [6, Lemma 5.2]. For every cell $\sigma$ of $X$ pick a lift
$\bar{\sigma}\subset h^{-1}((-\infty,-1])$. If $\sigma$ is a 0-cell, we can
find a cellular map $H_{\sigma}:[0,1]\to\overline{X}$ such that
$H_{\sigma}(0)=\bar{\sigma}$ and $H_{\sigma}(1)\in N^{\prime}$. Here $[0,1]$
has the standard cell structure with two 0-cells and one 1-cell. Using
equivariance, this gives an equivariant cellular homotopy
$H^{0}:\overline{X}^{(0)}\times[0,1]\to\overline{X}$. Note that $G$ acts on
$\overline{X}\times[0,1]$ by $g(x,t)=(gx,t)$, and $H^{0}$ induces a homotopy
on $X^{(0)}$ which gives $1$-movability of the 0-skeleton. As the image of
$H^{0}_{1}$ is in the 0-skeleton, we can iterate this homotopy to obtain
$n$-movability for any $n>0$.

Now pick a cell $\bar{\sigma}\subset h^{-1}((-\infty,-1])$ for every 1-cell
$\sigma$ of $X$ and let $u,v$ be the boundary points of $\bar{\sigma}$. By
possibly iterating $H^{0}$, we can assume that $H^{0}(u,1),H^{0}(v,)\in
N^{\prime}$ and since $N^{\prime}$ is connected, we can find a cellular map
$H_{u,v}:[0,1]\to N^{\prime}$ connecting these points.

Note that $H_{u,v}([0,1])$, $H^{0}(\\{u,v\\}\times[0,1])$ and $\bar{\sigma}$
combine to a closed loop in $\tilde{X}$ when suitably oriented. But by Theorem
3 this loop is representable by a loop in $N^{\prime}$. In other words, by
changing the path $H_{u,v}$ suitably in $N^{\prime}$, we can assume that this
loop bounds. Therefore we can extend $H^{0}$ to
$H_{\sigma}:(\overline{X}^{(0)}\cup\bar{\sigma})\times[0,1]\to\overline{X}$
cellularly such that $H_{\sigma}(\bar{\sigma},1)=H_{u,v}([0,1])$. Doing this
for every 1-cell of $X$ and extending equivariantly gives a cellular and
equivariant homotopy $H^{1}:\overline{X}^{(1)}\times[0,1]\to\overline{X}$ such
that $H^{1}_{0}$ is inclusion and $h(H^{1}(x,1))-h(x)\geq 1$.

Since $H^{1}$ is equivariant we get a homotopy $H:X^{(1)}\times[0,1]\to X$
which shows that $X^{(1)}$ is 1-movable with respect to $\omega$. By
compactness there is a $C>0$ such that $X^{(1)}$ is 1-movable with control
$C$. Again the image of $H_{1}$ is in the 1-skeleton of $X$, so we can iterate
the homotopy. Note that iterating the homotopy does not increase the control,
so we get that $X^{(1)}$ is $n$-movable with respect to $\omega$ with control
$C$ for every $n>0$.

(2) $\Longrightarrow$ (1): Let $h:\overline{X}\to{\mathbb{R}}$ and
$N^{\prime}\subset N=h^{-1}([0,\infty))$ be as above. Pick a basepoint
$x_{0}\in N^{\prime}$ with $f(x_{0})\geq C+1$. Let
$\gamma:(S^{1},1)\to(\tilde{X},x_{0})$ be a loop which can be assumed
cellular. By compactness of $S^{1}$ there is a $K\leq 0$ such that
$\gamma(S^{1})\subset f^{-1}([K,\infty))$.

By assumption there is a homotopy
$\bar{H}:\overline{X}^{(1)}\times[0,1]\to\overline{X}$ with $\bar{H}_{0}$ is
inclusion and

$\displaystyle h(\bar{H}_{1}(x))-h(x)$ $\displaystyle\geq$ $\displaystyle C-K$
$\displaystyle h(\bar{H}_{t}(x))-h(x)$ $\displaystyle\geq$ $\displaystyle-C$

for all $x\in\overline{X}$. Let $\mu:\tilde{X}\to[0,1]$ be a map with
$\mu|f^{-1}([C+1,\infty))\equiv 0$ and $\mu|f^{-1}((-\infty,C])\equiv 1$. Now
define $A:S^{1}\times[0,1]\to\tilde{X}$ by
$A(x,t)=\bar{H}(\gamma(x),\mu(\gamma(x))\cdot t)$. Then $A(x,0)=\gamma(x)$,
$A(x,1)=\bar{H}(\gamma(x),\mu(\gamma(x)))\in N^{\prime}$ and
$A(x_{0},t)=x_{0}$ for all $t\in[0,1]$. Therefore
$\xi\in\Sigma^{1}(G;{\mathbb{Z}})$ by Theorem 3. ∎

A criterion analogous to condition (2) of Proposition 1 leads to the
homotopical version of the Bieri-Neumann-Strebel-Renz invariants
$\Sigma^{k}(G)$, introduced in [7]. For this we assume that $X$ is an
Eilenberg-MacLane space with finite $n$-skeleton for some $n\geq 1$.

######  Definition 5.

Let $X$ be as above, $\xi\in S(G)$, $\omega$ a closed 1-form on $X$
representing $\xi$ and $k\geq 0$. We say that $\xi\in\Sigma^{k}(G)$, if there
is $\varepsilon>0$ and a cellular homotopy $H:X^{(k)}\times[0,1]\to X$ such
that $H(x,0)=x$ for all $x\in X^{(k)}$ and

$\displaystyle\int_{\gamma_{x}}\omega$ $\displaystyle\geq$
$\displaystyle\varepsilon$

for all $x\in X^{(k)}$, where $\gamma_{x}:[0,1]\to X$ is given by
$\gamma_{x}(t)=H(x,t)$. Here $X^{(k)}$ denotes the $k$-skeleton of $X$.

The condition that $H$ is cellular ensures that $H_{1}$ has image in
$X^{(k)}$, so that the homotopy can be iterated. As a result we see that
$\varepsilon$ can be arbitrarily large which shows that the definition does
not depend on the particular $\omega$. These iterations all have the same
control $C\geq 0$. Using cellular approximations it is easy to see that
condition (2) of Proposition 1 is equivalent to $\xi\in\Sigma^{1}(G)$.

The above definition is not the usual definition of $\Sigma^{k}(G)$, but it
follows from Proposition 2 below that it agrees with the definition given in
Bieri and Renz [7, §6].

Even though it is generally quite difficult to describe $\Sigma^{k}(G)$ and
$\Sigma^{k}(G;{\mathbb{Z}})$, there are some important classes of groups for
which the Sigma invariants can be determined, for example right-angled Artin
groups, see [22], and Thompson’s group $F$, see [5]. For more information and
applications of these invariants see, for example, [3, 4, 6, 7, 20].

##  2\. Sigma invariants for CW-complexes

There is no particular reason for $X$ to be aspherical in Definition 5, so we
can extend this definition to more general $X$. For simplicity we will assume
that $X$ is a finite connected CW-complex, but it is possible to consider the
case where $X$ has finite $n$-skeleton for some $n\geq 1$, in which case we
can define $\Sigma^{k}(X)$ for $k\leq n$. Let us first define

$\displaystyle S(X)$ $\displaystyle=$ $\displaystyle({\rm
Hom}(H_{1}(X),{\mathbb{R}})-\\{0\\})/{\mathbb{R}}_{+}$

where we again identify a homomorphism with its positive multiples. Clearly
$S(X)=S(G)$ with $G=\pi_{1}(X)$.

The Sigma invariants for CW-complexes are now defined in analogy to Definition
5.

######  Definition 6.

Let $X$ be a finite connected CW-complex, $\xi\in S(X)$, $\omega$ a closed
1-form on $X$ representing $\xi$ and $k\geq 0$. We say that
$\xi\in\Sigma^{k}(X)$, if there is $\varepsilon>0$ and a cellular homotopy
$H:X^{(k)}\times[0,1]\to X$ such that $H(x,0)=x$ for all $x\in X^{(k)}$ and

$\displaystyle\int_{\gamma_{x}}\omega$ $\displaystyle\geq$
$\displaystyle\varepsilon$

for all $x\in X^{(k)}$, where $\gamma_{x}:[0,1]\to X$ is given by
$\gamma_{x}(t)=H(x,t)$.

Let $p:\tilde{X}\to X$ be the universal covering space. Just as in the smooth
manifold case, a closed 1-form $\omega$ pulls back to an exact form on
$\tilde{X}$, $p^{\ast}\omega=dh$ for some $h:\tilde{X}\to{\mathbb{R}}$ with

(2) $\displaystyle h(gx)$ $\displaystyle=$ $\displaystyle h(x)+\xi(g)$

for all $g\in\pi_{1}(X)$ and $x\in\tilde{X}$. A function
$h:\tilde{X}\to{\mathbb{R}}$ with property (2) is called a height function.
Such a height function defines a closed 1-form representing $\xi$. A subset
$N\subset\tilde{X}$ is called a neighborhood of $\infty$ with respect to
$\xi$, if there exists a height function $h_{\xi}:\tilde{X}\to{\mathbb{R}}$
and $a\in{\mathbb{R}}$ such that

$\displaystyle h_{\xi}^{-1}([a,\infty))$ $\displaystyle\subset$ $\displaystyle
N.$

It is easy to check that if $N$ is a neighborhood of $\infty$ with respect to
$\xi$ for some height function, it is also a neighborhood of $\infty$ with
respect to $\xi$ for every other height function.

If a particular height function $h_{\xi}$ is given, we write
$N_{i}=h_{\xi}^{-1}([i,\infty))$ for every $i\in{\mathbb{R}}$.

We can describe $\Sigma^{k}(X)$ in terms of height functions alone. For this
we need one more definition.

######  Definition 7.

Let $X$ be a finite connected CW-complex, $h_{\xi}:\tilde{X}\to{\mathbb{R}}$ a
height function and $\xi\in H^{1}(X;{\mathbb{R}})$ be nonzero. A path to
$\infty$ with respect to $\xi$ is a map $\gamma:[0,\infty)\to\tilde{X}$ such
that for every neighborhood $N$ of $\infty$ there is a $T\geq 0$ such that
$\gamma(t)\in N$ for all $t\geq T$.

Given a path $\gamma$ to $\infty$ we can pick points $\gamma(T_{N})\in N$ for
every neighborhood $N$ of $\infty$ and get an inverse system
$\\{\pi_{\ast}(\tilde{X},N,\gamma(T_{N}))\\}$ where the basepoint change is
done via $\gamma|[T_{N},T_{N^{\prime}}]$. We will often suppress the
basepoints but we want to point out that there is always a path to $\infty$ in
the background. The next proposition shows that the basepath is not important
for our purposes.

######  Proposition 2.

Let $X$ be a finite connected CW-complex, $\xi\in H^{1}(X;{\mathbb{R}})$ be
nonzero and $h_{\xi}:\tilde{X}\to{\mathbb{R}}$ a height function. The
following are equivalent.

  1. (1)

$\xi\in\Sigma^{k}(X)$.

  2. (2)

There is a $\lambda\geq 0$ such that
$j_{\\#}:\pi_{l}(\tilde{X},N_{i})\to\pi_{l}(\tilde{X},N_{i-\lambda})$ is
trivial for all $l\leq k$ and every $i\in{\mathbb{R}}$.

  3. (3)

For every neighborhood $N$ of $\infty$ with respect to $\xi$, there is another
neighborhood $N^{\prime}\subset N$ such that
$j_{\\#}:\pi_{l}(\tilde{X},N^{\prime})\to\pi_{l}(\tilde{X},N)$ is trivial for
all $l\leq k$.

  4. (4)

There is an $\varepsilon>0$ and an equivariant cellular homotopy
$\tilde{H}:\tilde{X}^{(k)}\times[0,1]\to\tilde{X}$, such that $\tilde{H}_{0}$
is inclusion and
$h_{\xi}(\tilde{H}_{1}(\tilde{x}))-h_{\xi}(\tilde{x})\geq\varepsilon$ for all
$\tilde{x}\in\tilde{X}^{(k)}$.

###### Proof.

(1) $\Longleftrightarrow$ (4) as we can lift $H$ to $\tilde{H}$ and
$\tilde{H}$ determines $H$. (2) $\Longrightarrow$ (3) is obvious. (3)
$\Longrightarrow$ (4): We define $H$ by induction on the skeleta of
$\tilde{X}$. The homotopy can always be defined on $\tilde{X}^{(0)}$ as
$\tilde{X}$ is connected and $\xi$ nonzero. Assume that
$H:\tilde{X}^{(k-1)}\times[0,1]\to\tilde{X}$ satisfies the conclusion of (4).
Let $N=N_{0}=h_{\xi}^{-1}([0,\infty))$ be a neighborhood of $\infty$ with
respect to $\xi$ and $N^{\prime}\subset N$ as in (3), and choose a
$\varepsilon>0$. Choose a lift
$\tilde{\sigma}\subset\tilde{X}-N_{-\varepsilon}$ for every $k$-cell $\sigma$
of $X$. By iterating $H$ if necessary, we can assume that
$H_{1}(\partial\tilde{\sigma})\subset N^{\prime}$. Given a characteristic map
$\chi_{\sigma}:(D^{k},S^{k-1})\to(\tilde{X}^{(k)},\tilde{X}^{(k-1)})$, we can
compose $\chi_{\sigma}|S^{k-1}$ with $H$ to get an element of
$\pi_{k}(\tilde{X},N^{\prime})$ which restricts to the trivial element of
$\pi_{k}(\tilde{X},N)$. This gives a homotopy
$\chi:D^{k}\times[0,1]\to\tilde{X}$ with $\chi_{0}=\chi_{\sigma}$ and
$\chi_{1}(D^{k})\subset N$. We can use this homotopy to extend $H$
equivariantly to the $k$-skeleton such that (4) is satisfied.

To see that (4) $\Longrightarrow$ (2) observe that $H$ can be used to homotop
every map $\varphi:(D^{k},S^{k-1})\to(\tilde{X},N_{i})$ to a map with image in
any neighborhood of $\infty$. As the base point should not be moved during the
homotopy, we have to modify $H$ on a ‘buffer zone’ $N_{i-\lambda}-N_{i}$ so
that points mapped to $N_{i}$ will not be changed. Nevertheless we can find
$N_{i-\lambda}$ such that every $\varphi$ can be homotoped to a map
$(D^{k},S^{k-1})\to(N_{i-\lambda})$ which gives (1). ∎

######  Remark 1.

Notice that in (4) we can choose $\varepsilon>0$ arbitrary large: as the
homotopy is cellular, we can simply iterate it to increase the $\varepsilon$.

Our next result shows that the Sigma invariants are in fact open subsets of
$S(X)$, for the group theoretic version of this statement see [6, 7]. For the
proof we need a version of an Abel-Jacobi map. Let $q:\overline{X}\to X$ be
the universal abelian cover of $X$ and let $r=b_{1}(X)$, the first Betti
number of $X$. Then $H_{1}(X)$ acts on $\overline{X}$ by covering translations
and on ${\mathbb{R}}^{r}=H_{1}(X)\otimes{\mathbb{R}}$ by translation.

There exists an equivariant map $A:\overline{X}\to{\mathbb{R}}^{r}$, canonical
up to homotopy, called an Abel-Jacobi map, see [15, Prop.1]. For a different
construction, note that we have a canonical epimorphism
$\pi_{1}(X)\to{\mathbb{Z}}^{r}$ factoring through $H_{1}(X)$. Then $A$ is a
lift of the resulting classifying map $X\to(S^{1})^{r}$.

######  Theorem 4.

Let $X$ be a finite connected CW-complex. For every $k\geq 0$ the set
$\Sigma^{k}(X)$ is open and $\Sigma^{n}(X)=\Sigma^{\dim X}(X)$ for $n\geq\dim
X$.

###### Proof.

Let $h:\tilde{X}\to{\mathbb{R}}^{r}$ by the composition of the covering map
$\bar{p}:\tilde{X}\to\overline{X}$ with the Abel-Jaobi map
$A:\overline{X}\to{\mathbb{R}}^{r}$. Then, given
$\xi:\pi_{1}(X)\to{\mathbb{R}}$ we get a height function by
$h_{\xi}=l_{\xi}\circ h$ where
$l_{\xi}:\pi_{1}(X)/[\pi_{1}(X),\pi_{1}(X)]\otimes{\mathbb{R}}\to{\mathbb{R}}$
is defined by $l_{\xi}([g]\otimes t)=\xi(g)\cdot t$. Now let $\tilde{H}$ be a
homotopy as in Proposition 2 (4) for a $\xi$. Define
$\tilde{K}:S(X)\times\tilde{X}^{(k)}\to{\mathbb{R}}$ by

$\displaystyle\tilde{K}(\xi^{\prime},\tilde{x})$ $\displaystyle=$
$\displaystyle
h_{\xi^{\prime}}(H_{1}(\tilde{x}))-h_{\xi^{\prime}}(\tilde{x}).$

By the choice of $H$ we get $\tilde{K}(\xi,\tilde{x})\geq\varepsilon$ for all
$\tilde{x}\in\tilde{X}$ and some $\varepsilon>0$. It also induces a map
$K:S(X)\times X^{(k)}\to{\mathbb{R}}$. From the compactness of $X^{(k)}$, we
get

$\displaystyle\tilde{K}(\xi^{\prime},\tilde{x})$ $\displaystyle\geq$
$\displaystyle\frac{\varepsilon}{2}$

for all $\tilde{x}\in\tilde{X}^{(k)}$and all $\xi^{\prime}$ in a neighborhood
of $\xi$ in $S(X)$. Therefore Proposition 2 (4) is satisfied for all such
$\xi^{\prime}$. ∎

We get the following relation between $\Sigma^{k}(X)$ and the group-theoretic
version $\Sigma^{k}(\pi_{1}(X))$.

######  Proposition 3.

Let $X$ be a finite connected CW-complex and $k\geq 0$. If $\tilde{X}$ is
$k$-connected, then $\Sigma^{k}(X)=\Sigma^{k}(\pi_{1}(X))$ and
$\Sigma^{k+1}(X)\subset\Sigma^{k+1}(\pi_{1}(X))$.

The inclusion can be proper, as the example
$X={\mathbf{S}}^{1}\vee{\mathbf{S}}^{k}$ with $k\geq 2$ shows.

###### Proof.

Note that we can build a $K(\pi_{1}(X),1)$ out of $X$ by attaching $n$-cells
for $n\geq k+2$. Denote this Eilenberg-MacLane space by $Y$. If
$\xi\in\Sigma^{k+1}(X)$, the cellular homotopy
$\tilde{H}:\tilde{X}^{(k+1)}\times[0,1]\to\tilde{X}$ from Proposition 2 (4)
induces a homotopy
$\tilde{H}^{\prime}:\tilde{Y}^{(k+1)}\times[0,1]\to\tilde{Y}$, since
$\tilde{Y}^{(k+1)}=\tilde{X}^{(k+1)}$ and $\tilde{X}\subset\tilde{Y}$.
Therefore $\xi\in\Sigma^{k+1}(\pi_{1}(X))$.

If $\xi\in\Sigma^{k}(\pi_{1}(X))$, we get a cellular homotopy
$\tilde{H}:\tilde{Y}^{(k)}\times[0,1]\to\tilde{Y}^{(k+1)}$ as in Proposition 2
(4), and since $\tilde{Y}^{(k+1)}=\tilde{X}^{(k+1)}$, this gives
$\xi\in\Sigma^{k}(X)$. ∎

In analogy to the homological invariants $\Sigma^{k}(G,{\mathbb{Z}})$ of Bieri
and Renz [7] we now want to define homological invariants
$\Sigma^{k}(X,{\mathbb{Z}})$. We will in fact introduce a more general
definition for chain complexes which will also generalize the invariants of
[7]. We assume that all chain complexes satisfy $C_{i}=0$ for $i<0$.

######  Definition 8.

Let $R$ be a ring, $n$ a non-negative integer and $C$ a chain complex over
$R$. Then $C$ is of finite $n$-type, if there is a finitely generated
projective chain complex $C^{\prime}$ and a chain map $f:C^{\prime}\to C$ with
$f_{i}:H_{i}(C^{\prime})\to H_{i}(C)$ an isomorphism for $i<n$ and an
epimorphism for $i=n$. In this situation we call $f$ an $n$-equivalence.

It is clear that this is equivalent to the existence of a free $R$-chain
complex $D$ and a chain map $f:D\to C$ inducing an isomorphism on homology,
and such that $D_{i}$ is finitely generated for $i\leq n$.

######  Definition 9.

Let $C$ be a chain complex over ${\mathbb{Z}}G$ and $k\geq 0$. Then

$\displaystyle\Sigma^{k}(C)$ $\displaystyle=$ $\displaystyle\\{\xi\in
S(G)\,|\,C\mbox{ is of finite }k\mbox{-type over }{\mathbb{Z}}G_{\xi}\\}.$

######  Definition 10.

If $X$ is a finite connected CW-complex, we set

$\displaystyle\Sigma^{k}(X;\mathbb{Z})$ $\displaystyle=$
$\displaystyle\Sigma^{k}(C_{\ast}(\tilde{X})).$

The invariants $\Sigma^{n}(G;A)$ of Bieri and Renz [7] are given by
$\Sigma^{k}(G;A)=\Sigma^{k}(P)$, where $P$ is a projective ${\mathbb{Z}}G$
resolution of the $\mathbb{Z}G$-module $A$.

######  Remark 2.

Notice that ${\mathbb{Z}}G$ is a flat ${\mathbb{Z}}G_{\xi}$-module, as
${\mathbb{Z}}G$ is a direct limit of free ${\mathbb{Z}}G_{\xi}$-modules. As
${\mathbb{Z}}G\otimes_{{\mathbb{Z}}G_{\xi}}A\cong A$ for every
${\mathbb{Z}}G$-module $A$, we see that $\xi\in\Sigma^{n}(C)$ implies that $C$
is of finite $n$-type over ${\mathbb{Z}}G$. If the chain complex $C$ is free,
we then get that $C$ is chain-homotopy equivalent to a free chain complex $D$
such that $D_{i}$ is finitely generated for $i\leq n$.

To get an analogue of Proposition 2 we will define a chain complex version of
a height function.

######  Definition 11.

Let $C$ be a finitely generated free chain complex over ${\mathbb{Z}}G$ and
$\xi:G\to{\mathbb{R}}$ a non-zero homomorphism. A valuation on $C$ extending
$\xi$ is a sequence of maps $v:C_{k}\to{\mathbb{R}}_{\infty}$ satisfying the
following

  1. (1)

$v(a+b)\geq\min\\{v(a),v(b)\\}$ for all $a,b\in C_{k}$.

  2. (2)

$v(ga)=\xi(g)+v(a)$ for all $g\in G$, $a\in C_{k}$.

  3. (3)

$v(-a)=v(a)$ for all $a\in C_{k}$.

  4. (4)

$v(\partial a)\geq v(a)$ for all $a\in C_{k}$.

  5. (5)

$v^{-1}(\\{\infty\\})=\\{0\\}$.

Here ${\mathbb{R}}_{\infty}$ denotes the reals together with an element
$\infty$ with the obvious extension of addition and $\geq$.

To define a valuation on a free ${\mathbb{Z}}G$ complex $C$ which is finitely
generated in every degree, let $X_{i}$ be a basis for $C_{i}$. We begin with
setting $v(x)=0$ for all $x\in X_{0}$. The valuation can then be extended in
the obvious way to $C_{0}$. Inductively we now define for $x\in C_{i}$ the
valuation by $v(x)=0$ if $\partial x=0$, or $v(x)=v(\partial x)$, if $\partial
x\not=0$. Again we can extend $v$ to $C_{i}$ which gives the existence of
valuations on $C$.

######  Proposition 4.

Let $X$ be a finite connected CW-complex, $\xi\in H^{1}(X;{\mathbb{R}})$ be
nonzero and $v:C_{\ast}(\tilde{X})\to{\mathbb{R}}_{\infty}$ a valuation
extending $\xi$. The following are equivalent.

  1. (1)

$\xi\in\Sigma^{k}(X,{\mathbb{Z}})$.

  2. (2)

There is a $\lambda\geq 0$ such that $j_{\ast}:H_{l}(\tilde{X},N_{i})\to
H_{l}(\tilde{X},N_{i-\lambda})$ is trivial for all $l\leq k$ and every
$i\in{\mathbb{R}}$.

  3. (3)

For every neighborhood $N$ of $\infty$ with respect to $\xi$, there is another
neighborhood $N^{\prime}\subset N$ such that
$j_{\ast}:H_{l}(\tilde{X},N^{\prime})\to H_{l}(\tilde{X},N)$ is trivial for
all $l\leq k$.

  4. (4)

Given $\varepsilon>0$ there exists a ${\mathbb{Z}}G$-chain map
$A:C_{\ast}(\tilde{X})\to C_{\ast}(\tilde{X})$ chain homotopic to the identity
with $v(A(x))\geq v(x)+\varepsilon$ for all non-zero $x\in C_{i}(\tilde{X})$
with $i\leq k$.

The equivalences of (2),(3) and (4) are similar to the proof of Proposition 2
and will be omitted. For the equivalence to (1) we refer to Appendix A which
treats a more general version.

######  Corollary 5.

Let $X$ be a finite connected CW-complex. For every $k\geq 0$ the set
$\Sigma^{k}(X,{\mathbb{Z}})$ is open and
$\Sigma^{n}(X,{\mathbb{Z}})=\Sigma^{\dim X}(X,{\mathbb{Z}})$ for $n\geq\dim
X$.∎

######  Corollary 6.

Let $X$ be a finite connected CW-complex. Then

  1. (1)

$\Sigma^{1}(X)=\Sigma^{1}(X,{\mathbb{Z}})$.

  2. (2)

$\Sigma^{k}(X)\subset\Sigma^{k}(X,{\mathbb{Z}})$ for $k\geq 2$.

###### Proof.

It is easy to see that Condition (3) of Proposition 2 implies Condition (3) of
Proposition 4 so that $\Sigma^{k}(X)\subset\Sigma^{k}(X,{\mathbb{Z}})$ for
$k\geq 1$. Also, for $k=1$ the chain homotopy of Proposition 4 (3) can easily
be used to realize a homotopy as in Proposition 2 (3). ∎

It follows from the examples of Bestvina and Brady [2] that in general
$\Sigma^{2}(X)\not=\Sigma^{2}(X,{\mathbb{Z}})$, see [22].

##  3\. Novikov rings and homology

Let $G$ be a finitely generated group and $\xi\in S(G)$. We then let

$\displaystyle\widehat{{\mathbb{Z}}G}_{\xi}$ $\displaystyle=$
$\displaystyle\left\\{\left.\sum_{g\in G}n_{g}g\,\right|\,\mbox{for all
}t\in{\mathbb{R}}\,\,\,\\#\\{g\,|\,n_{g}\not=0\mbox{ and
}\xi(g)>t\\}<\infty\right\\},$

a ring containing the group ring ${\mathbb{Z}}G$. If $\xi:G\to{\mathbb{R}}$ is
injective, this is the Novikov ring first defined in [24]. For general $\xi$
we call it the Novikov-Sikorav ring, first introduced in [29].

The relation to the Bieri-Neumann-Strebel-Renz invariants was immediately
apparent once the work of Sikorav in [29] became known. This relation also
extends to our situation and is explained in Proposition 7 below.

If $X$ is a finite connected CW-complex, $p:\overline{X}\to X$ a regular
covering space with $\pi_{1}(\overline{X})\subset{\rm Ker}\,\xi$, we get an
induced homomorphism, also denoted by $\xi:G\to{\mathbb{R}}$, where
$G=\pi_{1}(X)/\pi_{1}(\overline{X})$. We then obtain a finitely generated free
$\widehat{{\mathbb{Z}}G}_{\xi}$-chain complex by setting

$\displaystyle C_{\ast}(X;\widehat{{\mathbb{Z}}G}_{\xi})$ $\displaystyle=$
$\displaystyle\widehat{{\mathbb{Z}}G}_{\xi}\otimes_{{\mathbb{Z}}G}C_{\ast}(\overline{X})$

and we denote by

$\displaystyle H_{\ast}(X;\widehat{{\mathbb{Z}}G}_{\xi})$ $\displaystyle=$
$\displaystyle H_{\ast}(C_{\ast}(X;\widehat{{\mathbb{Z}}G}_{\xi}))$

the resulting homology, called the Novikov-Sikorav homology (Novikov homology
if $\xi$ is injective).

The case of the universal cover is the one directly related to
$\Sigma^{k}(X)$, but the case of abelian covers plays an important role for
the cup-length estimates of the Lusternik-Schnirelmann categories.

######  Proposition 7.

Let $X$ be a finite connected CW-complex, $\xi\in H^{1}(X;{\mathbb{R}})$ non-
zero, $\tilde{X}$ the universal cover of $X$ and $G=\pi_{1}(X)$. Then the
following are equivalent.

  1. (1)

$\xi\in\Sigma^{k}(X;{\mathbb{Z}})$.

  2. (2)

$H_{i}(X;\widehat{{\mathbb{Z}}G}_{-\xi})=0$ for $i\leq k$.

Note that for testing $\xi\in\Sigma^{k}(X;{\mathbb{Z}})$ we have to use the
Novikov-Sikorav ring with respect to $-\xi$. The reason for this is that we
allow infinitely many non-zero coefficients in elements of
$\widehat{{\mathbb{Z}}G}_{\xi}$ in the “negative direction”, which is in line
with the original definition of the Novikov ring [23]. But to adhere to the
convention of [7] one has to complete in the “positive direction”. In order to
stick to the convention used in [18], we have to introduce the minus-sign
above.

###### Proof of Proposition 7.

(1) $\Longrightarrow$ (2): Let $A:C_{\ast}(\tilde{X})\to C_{\ast}(\tilde{X})$
be the chain map chain homotopic to the identity, given by Proposition 4 (4).
Then ${\rm id}-A:C_{i}(X;\widehat{{\mathbb{Z}}G}_{-\xi})\to
C_{i}(X;\widehat{{\mathbb{Z}}G}_{-\xi})$ is an isomorphism with inverse ${\rm
id}+A+A^{2}+\ldots$, which converges over $\widehat{{\mathbb{Z}}G}_{-\xi}$ by
the valuation property in Proposition 4(4). So the map on homology is both an
isomorphism and zero, which means that the homology vanishes.

(2) $\Longrightarrow$ (1): As $C_{\ast}(X;\widehat{{\mathbb{Z}}G}_{-\xi})$ is
free and bounded below, the vanishing of its homology groups up to degree $k$
guarantees the existence of a chain homotopy
$\delta:C_{\ast}(X;\widehat{{\mathbb{Z}}G}_{-\xi})\to
C_{\ast+1}(X;\widehat{{\mathbb{Z}}G}_{-\xi})$ with
$\partial_{i+1}\delta_{i}+\delta_{i-1}\partial_{i}={\rm id}$ for $i\leq k$.
“Cutting off” gives a chain homotopy $\bar{\delta}:C_{\ast}(\tilde{X})\to
C_{\ast}(\tilde{X})$ with
$\partial_{i+1}\bar{\delta}_{i}+\bar{\delta}_{i-1}\partial_{i}={\rm id}-A$.
Then $A$ is a chain map homotopic to the identity, and by approximating
$\delta$ sufficiently well with $\bar{\delta}$, condition (4) of Proposition 4
is satisfied. ∎

##  4\. Relations to the Lusternik-Schnirelmann category of closed 1-forms

We now want to describe a connection to the Lusternik-Schnirelmann theory of
closed 1-forms, which has been introduced in a series of papers [11, 12, 14]
and for which more information, in particular on applications and
calculations, can be found in [13, 18]. Indeed, there exist various different
notions, and the one related closest to $\Sigma^{k}(X)$ is denoted by
${\rm{Cat}}(X,\xi)$. Here again $X$ is a finite CW-complex and $\xi\in
H^{1}(X;{\mathbb{R}})$. Let us recall the definition from [18].

######  Definition 12.

Let $X$ be a finite CW-complex and $\xi\in H^{1}(X;{\mathbb{R}})$. Fix a
closed 1-form $\omega$ representing $\xi$. Then ${\rm{Cat}}(X,\xi)$ is defined
as the minimal integer $k$ such that there exists an open subset $U\subset X$
satisfying

  1. (1)

${\rm{cat}}_{X}(X-U)\leq k$.

  2. (2)

for some homotopy $h:U\times[0,\infty)\to X$ one has

$h(x,0)=x\quad\mbox{ and
}\quad\lim_{t\to\infty}\int_{x}^{h_{t}(x)}\omega=-\infty$

for any point $x\in U$.

  3. (3)

the limit in (2) is uniform in $x\in U$.

The integral is taken along the path $\gamma:[0,t]\to X$ given by
$\gamma(\tau)=h(x,\tau)$.

Here ${\rm{cat}}_{X}(A)$ for $A\subset X$ is the minimal number $i$ such that
there exist open sets $U_{1},\ldots,U_{i}\subset X$ covering $A$, each of
which is null-homotopic in $X$. The invariant ${\rm{Cat}}(X,\xi)$ originally
appeared in [12]. It is easy to see that it does not depend on positive
multiples of $\xi$, therefore it is well defined for $\xi\in S(X)$. Note that
it is also defined for $\xi=0$, in which case we recover the original
Lusternik-Schnirelmann category of $X$, see [18] for details.

The connection to $\Sigma^{k}(X)$ can now be described as follows. Because of
our sign conventions, we obtain another minus-sign in front of a $\xi$.

######  Theorem 5.

Let $X$ be a finite connected CW-complex and $\xi\in H^{1}(X;{\mathbb{R}})$ be
nonzero. Write $n=\dim X$. If $-\xi\in\Sigma^{k}(X)$ for some $k\leq n$. Then
${\rm{Cat}}(X,\xi)\leq n-k$.

###### Proof.

Let $U$ be a small open neighborhood of $X^{(k)}$ in $X$ which deformation
retracts to $X^{(k)}$. Let $H:X^{(k)}\times[0,1]\to X$ be a homotopy starting
with inclusion which lifts to a homotopy as in Proposition 2 (3). Now define
$h:X^{(k)}\times[0,\infty)\to X$ by $h(x,t)=H(H^{m}(x,1),t-m)$, where $m$ is
an integer with $t\in[m,m+1]$. Notice that $H$ is cellular so $H^{m}(x,1)\in
X^{(k)}$. It is easy to see that $h$ combined with the deformation retraction
of $U$ to $X^{(k)}$ gives a homotopy as required in Definition 12. It is well
known that ${\rm{cat}}_{X}(X-X^{(k)})\leq\dim(X)-k$ so the result follows. ∎

There are other definitions for a Lusternik-Schnirelmann category of $\xi$,
denoted by ${\rm{cat}}(X,\xi)$ and ${\rm{cat}}^{1}(X,\xi)$, see [11, 13, 14,
18], satisfying

${\rm{cat}}(X,\xi)\leq{\rm{cat}}^{1}(X,\xi)\leq{\rm{Cat}}(X,\xi).$

Therefore Theorem 5 provides an upper bound for these as well.

##  5\. A Hurewicz type result

We can get a converse of Corollary 6 if we assume that
$\xi\in\Sigma^{2}(\pi_{1}(X))$. This can be described in the following way.

######  Definition 13.

Let $G$ be a finitely presented group and $\xi:G\to{\mathbb{R}}$ a nonzero
homomorphism. Let $X$ be a finite connected CW-complex with $\pi_{1}(X)=G$. We
say $\xi\in\Sigma^{2}(G)$, if there is a $\lambda\geq 0$ such that
$j_{\\#}:\pi_{l}(N_{i})\to\pi_{l}(N_{i-\lambda})$ is trivial for $l\leq 1$ and
every $i\in{\mathbb{R}}$.

Note that we can think of $\xi\in H^{1}(X;{\mathbb{R}})$ so neighborhoods of
$\infty$ are defined as above.

######  Theorem 6.

Let $X$ be a finite connected CW-complex and $\xi\in H^{1}(X;{\mathbb{R}})$
nonzero. Let $k\geq 2$. If
$\xi\in\Sigma^{2}(\pi_{1}(X))\cap\Sigma^{k}(X,{\mathbb{Z}})$, then
$\xi\in\Sigma^{k}(X)$.

The theorem will follow from two lemmas.

######  Lemma 8.

Let $X$ be a finite connected CW-complex and $\xi\in H^{1}(X;{\mathbb{R}})$
nonzero. If $\xi\in\Sigma^{2}(\pi_{1}(X))$, then $\\{\pi_{2}(\tilde{X},N)\\}$
and $\\{H_{2}(\tilde{X},N)\\}$ are pro-isomorphic.

###### Proof.

Notice that

$\displaystyle{\rm
Im}(\pi_{2}(\tilde{X},N_{i})\to\pi_{2}(\tilde{X},N_{i-\lambda}))$
$\displaystyle=$ $\displaystyle{\rm
Im}(\pi_{2}(\tilde{X})\to\pi_{2}(\tilde{X},N_{i-\lambda}))$

if every 1-sphere in $N_{i}$ bounds in $N_{i-\lambda}$, in particular the
images become abelian. As $\xi\in\Sigma^{2}(\pi_{1}(X))$, we can define a
homomorphism $\pi_{2}(\tilde{X},N_{i})\to H_{2}(\tilde{X},N_{i-\lambda})$ as
in a typical proof of the classical Hurewicz Theorem (after possibly
increasing $\lambda$), see, for example, Spanier [30]. The details will be
left to the reader. ∎

######  Lemma 9.

Let $X$ be a finite connected CW-complex and $\xi\in H^{1}(X;{\mathbb{R}})$
nonzero. Let $k\geq 3$ and $\xi\in\Sigma^{k-1}(X)$. Then
$\\{\pi_{k}(\tilde{X},N)\\}$ and $\\{H_{k}(\tilde{X},N)\\}$ are pro-
isomorphic.

###### Proof.

This is similar to the proof of Lemma 8, but easier, as we can define the
homomorphism directly. ∎

These two Lemmas combine to a proof of Theorem 6.

######  Remark 3.

Theorem 6 also follows from Latour’s Theorem 5.10 and 5.3 [21].

##  6\. Functoriality properties

######  Proposition 10.

Let $X$ and $Y$ be finite CW-complexes, $f:X\to Y$ and $g:Y\to X$ maps with
$fg\simeq{\rm id}_{Y}$. Then for all $k\geq 0$ we have

$\displaystyle(f^{\ast})^{-1}(\Sigma^{k}(X))$ $\displaystyle\subset$
$\displaystyle\Sigma^{k}(Y)$
$\displaystyle(f^{\ast})^{-1}(\Sigma^{k}(X,{\mathbb{Z}}))$
$\displaystyle\subset$ $\displaystyle\Sigma^{k}(Y,{\mathbb{Z}}).$

###### Proof.

We give a proof for the homotopy invariant $\Sigma^{k}(Y)$. Choose liftings
$\tilde{f}:\tilde{X}\to\tilde{Y}$ and $\tilde{g}:\tilde{Y}\to\tilde{X}$ with
$\tilde{f}\tilde{g}\simeq{\rm id}_{\tilde{Y}}$ equivariantly. Let $\xi\in
S(Y)$ satisfy $f^{\ast}(\xi)\in\Sigma^{k}(X)$. Let
$h_{\xi}:\tilde{Y}\to{\mathbb{R}}$ be a height function. Then
$h_{\xi}\circ\tilde{f}:\tilde{X}\to{\mathbb{R}}$ is a height function for
$f^{\ast}\xi$.

By cocompactness, there exists a $C\geq 0$ such that

$\displaystyle|h_{\xi}(\tilde{f}\tilde{g}(\tilde{y}))-h_{\xi}(\tilde{y})|$
$\displaystyle\leq$ $\displaystyle C$

for every $\tilde{y}\in\tilde{Y}$.

By assumption, Proposition 2 and Remark 1, there is an equivariant homotopy
$\tilde{H}:\tilde{X}^{(k)}\times[0,1]\to\tilde{X}$ starting with inclusion,
such that

$\displaystyle h_{\xi}\tilde{f}(H(\tilde{x},1))-h_{\xi}\tilde{f}(\tilde{x})$
$\displaystyle\geq$ $\displaystyle C+1$

for all $\tilde{x}\in\tilde{X}$. Therefore

$\displaystyle h_{\xi}\tilde{f}(H(\tilde{g}(\tilde{y}),1))-h_{\xi}(\tilde{y})$
$\displaystyle\geq$ $\displaystyle 1.$

Now $\tilde{f}H(\tilde{g},\cdot)$ can be combined with the homotopy
$\tilde{f}\tilde{g}\simeq{\rm id}_{\tilde{Y}}$ to show that
$\xi\in\Sigma^{k}(Y)$. ∎

######  Corollary 11.

Let $X$ and $Y$ be finite connected CW-complexes and $h:X\to Y$ a homotopy
equivalence. Then $h^{\ast}(\Sigma^{k}(Y))=\Sigma^{k}(X)$ and
$h^{\ast}(\Sigma^{k}(Y,{\mathbb{Z}}))=\Sigma^{k}(X,{\mathbb{Z}})$ for all
$k\geq 0$.

######  Proposition 12.

Let $X$ and $Y$ be finite CW-complexes and $f:X\to Y$ $m$-connected with
$m\geq 1$. Then

$\displaystyle(f^{\ast})^{-1}(\Sigma^{k}(X))$ $\displaystyle\subset$
$\displaystyle\Sigma^{k}(Y)$
$\displaystyle(f^{\ast})^{-1}(\Sigma^{k}(X,{\mathbb{Z}}))$
$\displaystyle\subset$ $\displaystyle\Sigma^{k}(Y,{\mathbb{Z}})$

for all $k\leq m$.

###### Proof.

Add cells of dimension $\geq m+1$ to $X$ to get a (possibly infinite) CW-
complex $X^{\prime}$ containing $X$ such that $f$ extends to a homotopy
equivalence $f^{\prime}:X^{\prime}\to Y$. Let $g:Y\to X^{\prime}$ be a
cellular homotopy inverse. Then $g(Y^{(m)})\subset X$. If
$f^{\ast}(\xi)\in\Sigma^{k}(X)$, the homotopy
$H:\tilde{X}^{(k)}\times[0,1]\to\tilde{X}$ can now be used as in the proof of
Proposition 10 to show that $\xi\in\Sigma^{k}(Y)$ for $k\leq m$. ∎

######  Example 2.

Let $X$ be a finite connected CW-complex and $f:X\to X$ a map. The mapping
torus $M_{f}$ is the quotient space $M_{f}=X\times[0,1]/\,\sim$, where
$(x,0)\sim(f(x),1)$. There is a natural map $g:M_{f}\to{\mathbf{S}}^{1}$ given
by $g([x,t])=\exp 2\pi it$. Let

$\xi=[g]\in[M_{f},{\mathbf{S}}^{1}]=H^{1}(M_{f};{\mathbb{Z}})\subset
H^{1}(M_{f};{\mathbb{R}}).$

The homotopy $h:M_{f}\times[0,1]\to M_{f}$ given by

$\displaystyle H([x,t],s)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{rc}[x,t-s]&t\geq s\\\
\,[f(x),1+t-s]&t\leq s\end{array}\right.$

shows that $-\xi\in\Sigma^{k}(M_{f})$ for all $k\geq 0$.

######  Proposition 13.

Let $X$ be a finite connected CW-complex and $\xi\in H^{1}(X;{\mathbb{Z}})$ be
nonzero. Let $q:\bar{X}\to X$ be the infinite cyclic covering space
corresponding to ${\rm Ker}\,\xi$. Assume that $\bar{X}$ is homotopy
equivalent to a CW-complex $Y$ with finite $k$-skeleton. Then
$\pm\xi\in\Sigma^{k}(X)$.

###### Proof.

Let $h:\bar{X}\to{\mathbb{R}}$ be induced by a height function
$h_{\xi}:\tilde{X}\to{\mathbb{R}}$ and $\zeta:\bar{X}\to\bar{X}$ the
generating covering transformation with $h\zeta(x)>h(x)$ for all
$x\in\bar{X}$. Let $a:Y\to\bar{X}$ and $b:\bar{X}\to Y$ be mutually inverse
homotopy equivalences. The there is a homotopy equivalence $g:M_{b\zeta a}\to
X$ given by

$M_{b\zeta a}\simeq M_{\zeta ab}\simeq M_{\zeta}\simeq X$

where the last homotopy equivalence is given by $[\bar{x},t]\to q(\bar{x})$
for $\bar{x}\in\bar{X}$.

We can assume that $b\zeta a:Y\to Y$ sends the $k$-skeleton to the
$k$-skeleton. Let $\varphi:Y^{(k)}\to Y^{(k)}$ be the restriction of $b\zeta
a$ to $Y^{(k)}$. The induced map $M_{\varphi}\to M_{b\zeta a}$ is
$k$-connected and so there is a $k$-connected map $f:M_{\varphi}\to X$. By
Example 2 $-f^{\ast}(\xi)\in\Sigma^{k}(M_{\varphi})$. It follows from
Proposition 12 that $-\xi\in\Sigma^{k}(X)$. To get $\xi\in\Sigma^{k}(X)$ as
well, replace $\zeta$ by $\zeta^{-1}$. ∎

In the next section we will show that the converse of Proposition 13 also
holds.

##  7\. Domination results for covering spaces

Let $X$ be a finite connected CW-complex and $q:\overline{X}\to X$ a regular
covering space with $\pi_{1}(X)/\pi_{1}(\overline{X})$ abelian. Then we define

$\displaystyle S(X,\overline{X})$ $\displaystyle=$ $\displaystyle\\{\xi\in
S(X)\,|\,q^{\ast}\xi=0\\}.$

In particular, if $\overline{X}$ is the universal abelian covering of $X$,
then $S(X,\overline{X})=S(X)$. More generally, $S(X,\overline{X})$ is a sphere
of dimension $d-1$, where $d$ is the rank of the finitely generated abelian
group $\pi_{1}(X)/\pi_{1}(\overline{X})$.

######  Theorem 7.

Let $X$ be a finite connected CW-complex and $q:\overline{X}\to X$ a regular
covering space with $\pi_{1}(X)/\pi_{1}(\overline{X})$ abelian. Let $k\geq 1$.
Then the following are equivalent.

  1. (1)

$\overline{X}$ is homotopy equivalent to a CW-complex with finite
$k$-skeleton.

  2. (2)

$S(X,\overline{X})\subset\Sigma^{k}(X)$.

Furthermore, if $S(X,\overline{X})\subset\Sigma^{\dim X}(X)$, then
$\overline{X}$ is finitely dominated.

We derive Theorem 7 from a more general version for chain complexes which is a
generalization of [7, Thm.B]. To get an alternative proof for (1)
$\Longrightarrow$ (2) one can use the techniques of [28, Thm.3.2].

If $N$ is a normal subgroup of $G$ with $G/N$ abelian, we write

$\displaystyle S(G,N)$ $\displaystyle=$ $\displaystyle\\{\xi\in
S(G)\,|\,N\leq{\rm Ker}\,\xi\\}$

######  Theorem 8.

Let $C$ be a free ${\mathbb{Z}}G$-chain complex which is finitely generated in
every degree $i\leq n$, and $N$ a normal sugroup of $G$ such that $G/N$ is
abelian. Then $C$ is of finite $n$-type over ${\mathbb{Z}}N$ if and only if
$S(G,N)\subset\Sigma^{n}(C)$.

Theorem 7 follows from Theorem 8 by the work of Wall [32, 33].

###### Proof.

Assume that $C$ is of finite $n$-type over ${\mathbb{Z}}N$. Let $\xi\in
S(G,N)$ and denote $Q=G/N$. Then $\xi$ induces a homomorphism, also denoted
$\xi$, $\xi:Q\to{\mathbb{R}}$. As ${\mathbb{Z}}G$ is free over
${\mathbb{Z}}N$, there is a chain homotopy equivalence $f:P\to C$ over
${\mathbb{Z}}N$ with $P_{j}$ finitely generated free for all $j\leq n$. Then
$f:{\mathbb{Z}}G_{\xi}\otimes_{{\mathbb{Z}}N}P\to{\mathbb{Z}}G_{\xi}\otimes_{{\mathbb{Z}}N}C$
shows that ${\mathbb{Z}}G_{\xi}\otimes_{{\mathbb{Z}}N}C$ is of finite $n$-type
over ${\mathbb{Z}}G_{\xi}$. Also
${\mathbb{Z}}G_{\xi}\otimes_{{\mathbb{Z}}N}C\cong{\mathbb{Z}}Q_{\xi}\otimes C$
where the chain complex on the right has ${\mathbb{Z}}G_{\xi}$ acting
diagonally. The isomorphism is given by $g\otimes c\mapsto\pi(g)\otimes gc$,
where $\pi:G\to Q$ is projection. By [7, Lm.5.2] there is a free resolution
$E_{\ast}\to{\mathbb{Z}}$ over ${\mathbb{Z}}Q_{\xi}$ which is finitely
generated in every degree. Therefore each $E_{p}\otimes C$ is of finite
$n$-type over ${\mathbb{Z}}G_{\xi}$. Let $f:P_{p\,q}\to E_{p}\otimes C_{q}$ be
the corresponding chain map, notice that $P_{p\,q}$ is just a positive power
of ${\mathbb{Z}}G_{\xi}\otimes_{{\mathbb{Z}}N}P_{q}$ depending on the rank of
$E_{p}$. As $C$ is free over ${\mathbb{Z}}N$, we get that
${\mathbb{Z}}Q_{\xi}\otimes C\cong{\mathbb{Z}}G_{\xi}\otimes_{{\mathbb{Z}}N}C$
is free over ${\mathbb{Z}}G_{\xi}$, and we can assume that $f$ is a chain
homotopy equivalence with inverse $g_{q}:E_{p}\otimes C_{q}\to P_{p\,q}$.
Denote by $L:E_{p}\otimes C_{q}\to E_{p}\otimes C_{q+1}$ the chain homotopy
$L:fg\simeq 1$.

For $k\geq 0$ define

$F^{k}:P_{p\,q}\to E_{p-k}\otimes C_{q+k}\mbox{ by }F^{k}=(Ld)^{k}f,$

where $d:E_{p}\to E_{p-1}$ is the boundary of the resolution. Also define

$K^{i}:P_{p\,q}\to P_{p-i\,q-1+i}\mbox{ by }K^{i}=gdF^{i-1}$

for $i\geq 1$. We also set $K^{0}=\partial:P_{p\,q}\to P_{p-1\,q}$.

######  Lemma 14.

Let $\partial$ denote the boundaries $\partial:P_{p\,q}\to P_{p\,q-1}$ and
$\partial:E_{p}\otimes C_{q}\to E_{p}\otimes C_{q-1}$. Then $\partial
F^{0}=F^{0}\partial$ and for $m\geq 1$ we have

$\displaystyle\partial F^{m}+(-1)^{m+1}F^{m}\partial$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{m-1}(-1)^{k}F^{k}K^{m-k}-dF^{m-1}.$

###### Proof.

The proof by induction is straightforward. ∎

######  Lemma 15.

For $m\geq 0$ we have

$\displaystyle\sum_{s=0}^{m}(-1)^{s}K^{m-s}K^{s}$ $\displaystyle=$
$\displaystyle 0.$

###### Proof.

For $m=0$ this means $\partial\partial=0$, so assume the statement holds for
$m\geq 0$. Then

$\displaystyle\sum_{s=0}^{m+1}(-1)^{s}K^{m-s}K^{s}$ $\displaystyle=$
$\displaystyle gdF^{m}\partial+(-1)^{m+1}\partial
gdF^{m}+\sum_{s=1}^{m}(-1)^{s}K^{m-s}K^{s}$ $\displaystyle=$
$\displaystyle(-1)^{m+1}\left(gd\sum_{k=0}^{m-1}(-1)^{k}F^{k}K^{m-k}-gddF^{m-1}\right)+$
$\displaystyle+\sum_{s=1}^{m}(-1)^{s}gdF^{m-s-1}K^{s}$ $\displaystyle=$
$\displaystyle 0$

by Lemma 14 and since $dd=0$. ∎

Now define a chain complex $TP$ by $TP_{k}=\bigoplus\limits_{p+q=k}P_{p\,q}$
and $\delta:TP_{k}\to TP_{k-1}$ by

$\displaystyle\delta$ $\displaystyle=$
$\displaystyle\sum_{s=0}^{\infty}(-1)^{p}K^{s}$

where $(-1)^{p}$ refers to $P_{p\,k-p}$. By Lemma 15 we get that
$\delta\delta=0$. Also define $TE_{k}=\bigoplus\limits_{p+q=k}E_{p}\otimes
C_{q}$ with $\delta=(-1)^{p}(\partial+d)$. We get a chain map $F:TP\to TE$ by
setting

$\displaystyle F$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}(-1)^{k}F^{k}$

That $F$ is indeed a chain map follows from Lemmata 14 and 15.

Using the filtrations $(TP^{(m)})_{k}=\bigoplus\limits_{p+q=k,p\leq
m}P_{p\,q}$ and
$(TE^{(m)})_{k}=\bigoplus\limits_{p+q=k,p\leq m}E_{p}\otimes C_{q}$ we see
that $F$ induces a chain homotopy equivalence $\bar{F}:TP^{(m)}/TP^{(m-1)}\to
TE^{(m)}/TE^{(m-1)}$ and by a spectral sequence argument $F$ is a chain
homotopy equivalence. Another spectral sequence argument gives a homology
isomorphism from $TE$ to $C$. As $TP_{k}$ is finitely generated free over
${\mathbb{Z}}G_{\xi}$, we now get that $C$ is of finite $n$-type over
${\mathbb{Z}}G_{\xi}$.

Now assume that $S(G,N)\subset\Sigma^{n}(C)$.

Let $X_{i}$ be a ${\mathbb{Z}}G$-basis of $C_{i}$ for $i\leq n$, a finite set
by assumption. Given $c\in C_{i}$, we can therefore write

$\displaystyle c$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{m_{i}}n^{i}_{j}x^{i}_{j}$

with $n^{i}_{j}\in{\mathbb{Z}}G$ and

$\displaystyle\partial c$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{m_{i-1}}n^{i-1}_{j}x_{j}^{i-1}$

here the $x^{i}_{j}$ denote the elements of $X_{i}$. For $y\in{\mathbb{Z}}G$
we denote ${\rm supp}\,y$ as the elements of $G$ with nonzero coefficient and
for $c\in C_{i}$ as above we let

$\displaystyle{\rm supp}\,c$ $\displaystyle=$
$\displaystyle\bigcup_{j=1}^{m_{i}}{\rm
supp}\,n_{j}^{i}\cup\bigcup_{j=1}^{m_{i-1}}{\rm supp}\,n_{j}^{i-1}$

In particular, we get ${\rm supp}\,\partial c\subset{\rm supp}\,c$. The
support depends on the chosen basis, but we fix the basis once and for all.

We also denote $\pi:G\to G/N=Q$. By choosing an inner product on
$Q_{\mathbb{R}}=\mathbb{R}\otimes Q$ we get a norm $\|\cdot\|$ on
$Q_{\mathbb{R}}$ and we can think of $S(G,N)\subset Q_{\mathbb{R}}$ as the
unit sphere in this normed vector space. We extend the norm to $C$ by setting

$\displaystyle\|c\|$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cl}\max\\{\|\pi(g)\|\,|\,g\in{\rm
supp}\,c\\}&c\not=0\\\ 0&c=0\end{array}\right.$

Notice that we set $\|0\|=0$ despite the fact that $\|0\|=-\infty$ in [7]. For
$\xi\in S(G,N)\subset Q_{\mathbb{R}}$ we also obtain a valuation $v_{\xi}$ by
setting

$\displaystyle v_{\xi}(c)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cl}\min\\{\langle\pi(g),\xi\rangle\,|\,g\in{\rm
supp}\,c\\}&c\not=0\\\ \infty&c=0\end{array}\right.$

Let us also set for $a,b\in C$

$\displaystyle{\rm diam}(a,b)$ $\displaystyle=$
$\displaystyle\max\\{\|\pi(g)-\pi(h)\|\,|\,g\in{\rm supp}\,a,h\in{\rm
supp}\,b\\}$

with the convention that ${\rm diam}(a,b)=0$ if $a$ or $b$ is zero. Finally,
for $r>0$ and $c\in C$ we let

$\displaystyle B_{r}(c)$ $\displaystyle=$ $\displaystyle\\{d\in C\,|\,{\rm
diam}(c,d)\leq r\\}.$

Given $r>0$, as $S(G,N)\subset\Sigma^{n}(C)$ we can find for every $\xi\in
S(G,N)$ a chain map $\varphi^{\xi}:C\to C$ and a chain homotopy
$H^{\xi}:1\simeq\varphi^{\xi}$ such that
$v_{\xi}(\varphi^{\xi}c)-v_{\xi}(c)\geq 2r$ for all $c\in C$. Furthermore,
there is an open neighborhood $U_{\xi}$ of $\xi$ in $S(G,N)$ such that

$\displaystyle v_{\eta}(\varphi^{\xi}c)-v_{\eta}(c)$ $\displaystyle\geq$
$\displaystyle r$

for all $\eta\in U_{\xi}$ and $c\in C_{l}$, $l\leq n$.

As $S(G,N)$ is compact, finitely many of the $U_{\xi}$ suffice to cover
$S(G,N)$, so let $(U_{i},\varphi^{i},H^{i})$ for $i=1,\ldots,k$ be triples
where $U_{i}$ cover $S(G,N)$, $\varphi^{i}:C\to C$ chain map with

$\displaystyle v_{\eta}(\varphi^{i}c)-v_{\eta}(c)$ $\displaystyle\geq$
$\displaystyle r$

for all $\eta\in U_{i}$ and all $c\in C_{l}$, $l\leq n$, and
$H^{i}:1\simeq\varphi^{i}$ a chain homotopy.

As we deal only with finitely many chain homotopies, there is a $M\geq 0$ such
that

$\displaystyle v_{\eta}(c)-v_{\eta}(H^{i}c)$ $\displaystyle\leq$
$\displaystyle M$

for all $\eta\in U_{i}$ and all $c\in C_{l}$, $l\leq n$. Furthermore, by
replacing the chain homotopy $H^{i}$ by
$H^{i}-\varphi^{i}H^{i}:1\simeq(\varphi^{i})^{2}$ we can increase $r>0$
without increasing $M>0$. Therefore we can assume that

(6) $\displaystyle r$ $\displaystyle>$ $\displaystyle 3Mn$

Finally, there exists an $L>0$ with

$\displaystyle{\rm diam}(x,x)$ $\displaystyle\leq$ $\displaystyle L$
$\displaystyle{\rm diam}(x,\varphi^{i}x)$ $\displaystyle\leq$ $\displaystyle
L$ $\displaystyle{\rm diam}(x,H^{i}x)$ $\displaystyle\leq$ $\displaystyle L$

for all $i=1,\ldots,k$ and all $x\in X_{l}$, $l\leq n$, as there are only
finitely many such conditions.

Notice that $C$ is a free ${\mathbb{Z}}N$ chain complex, and a basis is given
by $TX_{i}=\\{tx\,|\,t\in T,x\in X_{i}\\}$, where $T\subset G$ is a subset
such that $\pi|T$ induces a bijection from $T$ to $Q$.

######  Lemma 16.

If $S(G,N)\subset\Sigma^{n}(C)$, there exist constants $r>0$, $M>0$, $A>0$, a
${\mathbb{Z}}N$-chain map $\psi:C\to C$ and a ${\mathbb{Z}}N$-chain homotopy
$K:1\simeq\psi$ such that for $m\leq n$ we have $\psi_{m}(z)=z$ if $\|z\|\leq
A$, and for $tx\in TX_{m}$ with $\|tx\|>A$ we get

$\displaystyle\|\psi_{m}(tx)\|$ $\displaystyle\leq$ $\displaystyle\|tx\|-r.$

Furthermore $\|K_{m}(z)\|\leq\|z\|+M$ for all $z\in C_{m}$.

###### Proof.

In order to define $\psi$ and $K$, we define them on $TX_{m}$. Let $r>0$ and
$M>0$ be as above Lemma 16.

######  Lemma 17.

Let $tx\in TX_{m}$ with
$\|tx\|\geq\max\\{\frac{3}{4}r+\frac{L^{2}}{r},\frac{L^{2}}{M}\\}$. Let $i$ be
such that $\xi_{t}/\|\xi_{t}\|\in U_{i}$, where
$\xi_{t}(g)=\langle\pi(g),\pi(t)\rangle$ for $g\in G$. Then

$\displaystyle\|\varphi^{i}(tx)\|$ $\displaystyle\leq$
$\displaystyle\|tx\|-\frac{1}{2}r$ $\displaystyle\|H^{i}(tx)\|$
$\displaystyle\leq$ $\displaystyle\|tx\|+\frac{3}{2}M.$

###### Proof.

Let $g\in{\rm supp}\,tx$ satisfy $\|\pi(g)\|=\|tx\|$. As ${\rm
diam}(tx,\varphi^{i}tx)\leq L$, we get for $h\in{\rm supp}\,\varphi^{i}(tx)$

$\displaystyle\|\pi(h)\|^{2}$ $\displaystyle\leq$
$\displaystyle(\|\pi(g)\|-r)^{2}+L^{2}$ $\displaystyle=$
$\displaystyle\|\pi(g)\|^{2}-2r\|\pi(g)\|+r^{2}+L^{2}$ $\displaystyle\leq$
$\displaystyle\|\pi(g)\|^{2}-r\|\pi(g)\|-\frac{3}{4}r^{2}-L^{2}+r^{2}+L^{2}$
$\displaystyle=$ $\displaystyle(\|\pi(g)\|-\frac{r}{2})^{2}$

So $\|\varphi^{i}(tx)\|\leq\|tx\|-\frac{r}{2}$. Similarly, using
$\|\pi(g)\|\geq\frac{L^{2}}{M}$, we get $H^{i}(tx)\|\leq\|tx\|+\frac{3}{2}M$.
∎

If $\|tx\|>L$, then $\pi({\rm supp}\,tx)\subset Q_{\mathbb{R}}-\\{0\\}$.
Denote $p:Q_{\mathbb{R}}-\\{0\\}\to S(G,N)$ the standard retraction. Using the
Lebesgue number of the covering $U_{1}\cup\ldots\cup U_{k}$, we can find a
constant $A^{\prime}>0$ with $\|tx\|>A^{\prime}$ implying

(7) $\displaystyle p(\pi({\rm supp}\,B_{L^{\prime}}(tx)))$
$\displaystyle\subset$ $\displaystyle U_{i}$

for some $i$, where
$L^{\prime}=\max\\{\frac{3}{4}r+\frac{L^{2}}{r},\frac{L^{2}}{M}\\}+(2n+1)L$.
Let $A=A^{\prime}+L^{\prime}$.

Define $K_{0}:C_{0}\to C_{1}$ by $K_{0}(tx)=0$ if $\|tx\|\leq A$, and
$K_{0}(tx)=H^{i}_{0}(tx)$ if $\|tx\|>A$, where $i$ is the smallest number with
(7) satisfied. Also define $\psi_{0}:C_{0}\to C_{0}$ by

$\displaystyle\psi_{0}(tx)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ccc}tx&\mbox{if}&\|tx\|\leq A\\\
\varphi^{i}(tx)&\mbox{if}&\|tx\|>A\end{array}\right.$

where $i$ is again the smallest number such that (7) holds. It follows that
$\partial K_{0}=1-\psi_{0}$. By Lemma 17 we get

$\displaystyle\|\psi_{0}(tx)\|$ $\displaystyle\leq$
$\displaystyle\|tx\|-\frac{r}{2}$ $\displaystyle\|K_{0}(tx)\|$
$\displaystyle\leq$ $\displaystyle\|tx\|+\frac{3}{2}M$

for $\|tx\|>A$. Also

$\displaystyle{\rm diam}(\psi_{0}(tx),tx)$ $\displaystyle\leq$ $\displaystyle
L$ $\displaystyle{\rm diam}(K_{0}(tx),tx)$ $\displaystyle\leq$ $\displaystyle
L$

for all basis elements $tx$.

Assume now by induction that we have $K_{j}:C_{j}\to C_{j+1}$,
$\psi_{j}:C_{j}\to C_{j}$ with

$\displaystyle\partial K_{j}+K_{j-1}\partial=1-\psi_{j}$

and

(9) $\displaystyle\|\psi_{j}(tx)\|$ $\displaystyle\leq$
$\displaystyle\|tx\|-\frac{r}{2}+j\frac{3}{2}M$ (10)
$\displaystyle\|K_{j}(tx)\|$ $\displaystyle\leq$
$\displaystyle\|tx\|+\frac{3}{2}M(j+1)$

for $\|tx\|>A$ and

(11) $\displaystyle{\rm diam}(\psi_{j}(tx),tx)$ $\displaystyle\leq$
$\displaystyle(2j+1)L$ (12) $\displaystyle{\rm diam}(K_{j}(tx),tx)$
$\displaystyle\leq$ $\displaystyle(2j+1)L$

for all basis elements $tx$, for all $j\leq m-1$ with $m\leq n$.

Then define $K_{m}:C_{m}\to C_{m+1}$ by $K_{m}(tx)=0$ if $\|tx\|\leq A$ and

$\displaystyle K_{m}(tx)$ $\displaystyle=$ $\displaystyle
H^{i}_{m}(tx)-H^{i}_{m}K_{m-1}\partial(tx)$

for $\|tx\|>A$, where $i$ satisfies (7). It is straightforward to check that

$\displaystyle(\partial K_{m}+K_{m-1}\partial)(tx)$ $\displaystyle=$
$\displaystyle tx-\varphi^{i}_{m}(tx)+\varphi^{i}_{m}(K_{m-1}\partial
tx)-H^{i}_{m-1}\psi_{m-1}\partial tx$

for $\|tx\|>A$. So define $\psi_{m}:C_{m}\to C_{m}$ by $\psi_{m}(tx)=tx$ for
$\|tx\|\leq A$ and

$\displaystyle\psi_{m}(tx)$ $\displaystyle=$
$\displaystyle\varphi^{i}_{m}(tx)-\varphi_{m}^{i}(K_{m-1}\partial
tx)+H^{i}_{m-1}\psi_{m-1}\partial tx$

if $\|tx\|>A$. It follows that $\partial K_{m}+K_{m-1}\partial=1-\psi_{m}$. As

$\|\varphi^{i}_{m}(tx)-\varphi_{m}^{i}(K_{m-1}\partial
tx)+H^{i}_{m-1}\psi_{m-1}\partial tx\|\,\,\,\leq\hskip 56.9055pt$
$\max\\{\|\varphi^{i}_{m}(tx)\|,\|\varphi_{m}^{i}(K_{m-1}\partial
tx)\|,\|H^{i}_{m-1}\psi_{m-1}\partial tx\|\\}$

it is easy to see that (9) also holds for $j=m$. Here we use that
${\rm diam}(K_{m-1}\partial tx,tx)\leq(2m-1)L+L$, so that Lemma 17 still
applies to all basis elements occuring in $K_{m-1}\partial tx$ by the choice
of $A$.

Similarly, (10), (11) and (12) hold for $j=m$. For $m>n$ we set $K_{m}=0$ and
define $\psi_{m}$ such that

$\displaystyle\partial K_{m}+K_{m-1}\partial=1-\psi_{m}.$

As the identity is a chain map, we get that $\psi$ is a chain map. Replacing
$r$ by $\frac{r}{2}-\frac{3}{2}nM$ and $M$ by $\frac{3}{2}M(n+1)$ we get Lemma
16. Note that the new $r>0$ by (6). ∎

######  Lemma 18.

If $S(G,N)\subset\Sigma^{n}(C)$, then there exists a free ${\mathbb{Z}}N$
chain complex $D$ with $D_{i}$ finitely generated for $i\leq n$, and
${\mathbb{Z}}N$-chain maps $a:D\to C$, $b:C\to D$ with $ab$ chain homotopic to
the identity on $C$. Also, $a:D_{i}\to C_{i}$ can be assumed to be inclusion
for $i\leq n$, and $D_{i}=C_{i}$ for $i>n$.

###### Proof.

Let $r,M,A>0$, $\psi$ and $K$ as provided by Lemma 16. If $l$ is a positive
integer, $B\geq 0$ and $z\in C_{m}$ with $m\leq n$, we have

(13) $\|\psi^{l}(z)\|\leq A+B\hskip 22.76228pt\mbox{if}\hskip
22.76228pt\|z\|\leq A+l\cdot r+B$

We define a ${\mathbb{Z}}N$-chain homotopy $\Phi:C\to C_{+1}$ as follows. For
$s\leq n$, set

$\displaystyle\Phi_{s}(tx)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{l}K_{s}\psi^{j}tx-K_{s}\psi^{j}\Phi_{s-1}\partial
tx$

where $l$ is an integer such that $\|tx\|\in(A+l\cdot r,A+(l+1)r]$, and for
$s>n$ we simply set $\Phi_{s}=0$. We get a ${\mathbb{Z}}N$-chain map
$\zeta:C\to C$ by setting $\zeta=1-\partial\Phi-\Phi\partial$, in particular,
$\zeta$ is chain homotopy equivalent to the identity.

Using induction, we see that

$\displaystyle\|\Phi_{s}(tx)\|$ $\displaystyle\leq$
$\displaystyle\|tx\|+(s+1)M$

for $s\leq n-1$. It follows that

$\displaystyle\|\Phi_{s}(z)\|$ $\displaystyle\leq$ $\displaystyle\|z\|+(s+1)M$

for all $z\in C_{s}$.

We claim that for $s\leq n$ we get

(14) $\displaystyle\|\zeta(z)\|$ $\displaystyle\leq$ $\displaystyle A+s\cdot
M$

for all $z\in C_{s}$.

This holds for $s=0$ by (13), as $\zeta(tx)=\psi^{l+1}(tx)$ if
$\|tx\|\in(A+l\cdot r,A+(l+1)r]$.

Now notice that

$\displaystyle(\partial\Phi_{s}+\Phi_{s-1}\partial)(tx)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{l}(K_{s}\psi^{j}tx-K_{s}\psi^{j}\Phi_{s-1}\partial
tx)+\Phi_{s-1}\partial(tx)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{l}((\partial
K_{s}\psi^{j}+K_{s-1}\partial\psi^{j})(tx)$
$\displaystyle-\sum_{j=0}^{l}(\partial
K_{s}\psi^{j}\Phi_{s-1}+K_{s-1}\partial\psi^{j}\Phi_{s-1})(\partial tx)$
$\displaystyle+\sum_{j=0}^{l}(K_{s-1}\psi^{j}\partial\Phi_{s-1}\partial tx-
K_{s-1}\partial\psi^{j}tx)+\Phi_{s-1}\partial tx$ $\displaystyle=$
$\displaystyle tx-\psi^{l+1}tx-\Phi_{s-1}\partial
tx+\psi^{l+1}\Phi_{s-1}\partial tx$
$\displaystyle-\sum_{j=0}^{l}K_{s-1}\partial\psi^{j}tx+\Phi_{s-1}\partial tx$
$\displaystyle+\sum_{j=0}^{l}(K_{s-1}\psi^{j}\partial tx-
K_{s-1}\psi^{j}\zeta\partial tx)$ $\displaystyle=$ $\displaystyle
tx-\psi^{l+1}tx+\psi^{l+1}\Phi_{s-1}\partial
tx-\sum_{j=0}^{l}K_{s-1}\psi^{j}\zeta\partial tx$

It follows that

$\displaystyle\zeta_{s}(tx)$ $\displaystyle=$
$\displaystyle\psi^{l+1}tx-\psi^{l+1}\Phi_{s-1}\partial
tx+\sum_{j=0}^{l}K_{s-1}\psi^{j}\zeta_{s-1}\partial tx$

Using induction, we see again by (13), that (14) holds.

Define a chain complex $D$ by

$\displaystyle D_{s}$ $\displaystyle=$ $\displaystyle\\{z\in
C_{s}\,|\,\|z\|\leq A+nM\\}$

for $s\leq n$. For $s>n$ we let $D_{s}=C_{s}$. The boundary map
$\partial_{D}:D_{n+1}\to D_{n}$ is given by
$\partial_{D}=\zeta\circ\partial_{C}$. We can define a chain map $b:C\to D$ by
using $\zeta$ for $s\leq n$ and the identity for $s>n$ and a chain map $a:D\to
C$ by using inclusion for $s\leq n$ and $\zeta$ for $s>n$. Then $ab=\zeta:C\to
C$ is chain homotopic to the identity. As only finitely many $q\in Q$ satisfy
$\|q\|\leq A+nM$, we get that $D_{s}$ is finitely generated free over
${\mathbb{Z}}N$, compare [7, 5.4]. This finishes the proof of Lemma 18. ∎

To finish the proof of Theorem 8, note that by Lemma 18 the chain complex $C$
is dominated over ${\mathbb{Z}}N$ by a chain complex $D$ with $D_{i}$ finitely
generated free for $i\leq n$. By a standard construction, compare Ranicki [27,
§3] or Wall [33], there exists a ${\mathbb{Z}}N$ chain complex $E$ chain
homotopy equivalent to $C$ with $E_{i}$ finitely generated free for $i\leq n$.
∎

######  Corollary 19.

Let $C$ be a finitely generated free ${\mathbb{Z}}G$-chain complex with
$C_{i}=0$ for $i>n$. Then $C$ is ${\mathbb{Z}}N$-chain homotopy equivalent to
a finitely generated projective ${\mathbb{Z}}N$-chain complex $D$ with
$D_{i}=0$ for $i>n$ if and only if $S(G,N)\subset\Sigma^{n}(C)$.

###### Proof.

If $C$ is ${\mathbb{Z}}N$-chain homotopy equivalent to a finitely generated
projective ${\mathbb{Z}}N$-complex, it is, by definition, of finite $n$-type
over ${\mathbb{Z}}N$, hence $S(G,N)\subset\Sigma^{n}(C)$ by Theorem 8.

If $S(G,N)\subset\Sigma^{n}(C)$, then by Lemma 18 there is a chain complex
$D^{\prime}$, finitely generated free over ${\mathbb{Z}}N$ with
$D_{i}^{\prime}=0$ for $i>n$ which dominates $C$. By [27, Prop.3.1], the
required chain complex $D$ exists. ∎

##  8\. Movability of homology classes

Recall from Proposition 7 that the vanishing and non-vanishing of Novikov-
Sikorav homology groups in the universal cover case determine
$\Sigma^{k}(X,{\mathbb{Z}})$. We now want to take a closer look at the
homology of other coverings, in particular coverings with abelian covering
transformation group.

In this section, $R$ is a ring, although we have mainly the cases
$R={\mathbb{Z}}$ and $R={\mathbf{k}}$ a field in mind.

Let $p:\overline{X}\to X$ be a regular cover, where $X$ is again a finite
connected CW-complex. Denote $G=\pi_{1}(X)/\pi_{1}(\overline{X})$. Recall that
$S(X,\overline{X})$ consists of those $\xi\in S(X)$ with $p^{\ast}\xi=0$. For
such $\xi$ we can define a height function $h:\overline{X}\to{\mathbb{R}}$ and
neighborhoods $N\subset\overline{X}$ of infinity with respect to $\xi$ as in
the case of the universal covering.

######  Definition 14.

A homology class $z\in H_{q}(\overline{X};R)$ is said to be movable to
infinity in $\overline{X}$ with respect to $\xi\in S(X,\overline{X})$, if $z$
can be realized by a singular cycle in any neighborhood $N$ of infinity with
respect to $\xi$.

In other words, $z\in H_{q}(\overline{X};R)$ is movable to infinity with
respect to $\xi$, if $z$ is an element of

$\bigcap_{N}{\rm Im}(H_{q}(N;R)\to H_{q}(\overline{X};R))$

where the intersection is taken over all neighborhoods of infinity with
respect to $\xi$.

Note that we have an inverse system of $R$-modules given by
$\\{H_{q}(\overline{X},N;R)\leftarrow H_{q}(\overline{X},N^{\prime};R)\\}$
which runs over neighborhoods $N^{\prime}\subset N$ of infinity with respect
to $\xi$. So $z$ is movable to infinity with respect to $\xi$ if and only if

$z\in{\rm
Ker}(H_{q}(\overline{X};R)\to\lim\limits_{\leftarrow}H_{q}(\overline{X},N;R)).$

Just as with integer coefficients, we can define a Novikov-Sikorav ring
$\widehat{RG}_{\xi}$ with coefficients in an arbitrary ring $R$. As this ring
can be expressed as an inverse limit, standard methods give the following
exact sequence, see [15] or [19] for details.

(15)
$0\rightarrow{\lim\limits_{\leftarrow}}^{1}H_{q+1}(\overline{X},N;R)\rightarrow
H_{q}(X;\widehat{RG}_{-\xi})\rightarrow\lim\limits_{\leftarrow}H_{q}(\overline{X},N;R)\rightarrow
0$

Let us give a simple criterion for $z\in H_{i}(\overline{X};R)$ to be movable
to infinity.

######  Definition 15.

An element $\Delta\in RG$ is said to have $\xi$-lowest coefficient 1, if
$\Delta=1-y$, with $y=\sum a_{j}g_{j}\in RG$ and such that each $g_{j}\in G$
satisfies $\xi(g_{j})>0$.

Such a $\Delta$ is invertible over $\widehat{RG}_{-\xi}$. So if $\Delta\cdot
z=0\in H_{q}(\overline{X};R)$, we get that the image of $z$ in
$H_{q}(X;\widehat{RG}_{-\xi})$ is zero, and by (15), $z$ is movable to
infinity.

Under certain conditions, this is in fact necessary for movability to
infinity. The following theorem is taken from [15, 17].

######  Theorem 9.

Let $X$ be a finite connected CW-complex and $p:\overline{X}\to X$ a regular
covering with covering transformation group $G\cong{\mathbb{Z}}^{r}$. Let
$\xi\in S(X,\overline{X})$ induce an injective homomorphism
$G\to{\mathbb{R}}$. Let $R$ be either ${\mathbb{Z}}$ or a field. For $z\in
H_{q}(\overline{X};R)$, the following are equivalent:

  1. (1)

$z$ is movable to infinity with respect to $\xi$.

  2. (2)

$i_{\ast}(z)=0\in H_{q}(X;\widehat{RG}_{-\xi})$, where
$i_{\ast}:H_{q}(\overline{X};R)\cong H_{q}(X;RG)\to
H_{q}(X;\widehat{RG}_{-\xi})$ is change of coefficients.

  3. (3)

There is $\Delta\in RG$ with $\xi$-lowest coefficient 1 such that $\Delta\cdot
z=0$.

In the case that $R$ is a field, condition (3) is equivalent to the existence
of a non-zero $\Delta\in RG$ with $\Delta\cdot z=0\in H_{q}(\overline{X};R)$,
as we can find $r\in R$ and $g\in G$ such that $\Delta\cdot rg$ has
$\xi$-lowest coefficient 1.

Notice that $\Delta\in RG$ having $\xi$-lowest coefficient 1 is an open
condition in $\xi\in S(X,\overline{X})$; in particular, if $z$ is movable to
infinity with respect to $\xi$, it is also movable to infinity with respect to
nearby $\xi^{\prime}$, under the conditions of Theorem 9.

The equivalence of (1) and (2) is obtained by showing that the
${\lim\limits_{\leftarrow}}^{1}$-term in (15) vanishes. This is done in [15,
17] under the conditions of Theorem 9. Using Usher [31, Thm.1.3] one can see
that the ${\lim\limits_{\leftarrow}}^{1}$ term in (15) vanishes in fact for
any abelian covering and any Noetherian ring $R$.

Theorem 9 is an important ingredient in obtaining lower bounds for
${\rm{cat}}(X,\xi)$ and ${\rm{cat}}^{1}(X,\xi)$ via cup-lengths. We refer the
reader to [16, 17] for details.

##  9\. Function spaces of paths to infinity

If $\gamma:[0,\infty)\to X$ is a map, we can lift it to a map
$\tilde{\gamma}:[0,\infty)\to\tilde{X}$ into the universal cover. We want to
look at those maps, which lift to paths to $\infty$ with respect to a given
$\xi$, compare Definition 7. This does not depend on the particular lift of
$\gamma$.

We set

$\mathcal{C}_{\xi}(X)\,=\,\\{\gamma:[0,\infty)\to X\,|\,\tilde{\gamma}\mbox{
is a path to infinity with respect to }\xi\\}.$

It is equipped with the following topology: For $a,b\in[0,\infty)$ and $U$
open in $X$ let

$\displaystyle W(a,b;U)$ $\displaystyle=$
$\displaystyle\\{\gamma\in\mathcal{C}_{\xi}(X)\,|\,\gamma([a,b])\subset U\\}$

and for $a,A\in[0,\infty)$ let

$\displaystyle W(a,A)$ $\displaystyle=$
$\displaystyle\\{\gamma\in\mathcal{C}_{\xi}(X)\,|\,\forall t\geq
a\,\,\,h_{\xi}\tilde{\gamma}(t)-h_{\xi}\tilde{\gamma}(0)>A\\}.$

These sets form a subbasis for the topology of $\mathcal{C}_{\xi}(X)$. Notice
that the sets $W(a,b;U)$ provide the compact-open topology on
$\mathcal{C}_{\xi}(X)$ while the sets $W(a,A)$ give a ”control at infinity”.

The evaluation $e:\mathcal{C}_{\xi}(X)\to X$ given by $e(\gamma)=\gamma(0)$ is
a fibration and for $x_{0}\in X$ we have the fiber

$\displaystyle\mathcal{M}_{\xi}$ $\displaystyle=$
$\displaystyle\\{\gamma\in\mathcal{C}_{\xi}(X)\,|\,\gamma(0)=x_{0}\\},$

compare [21].

######  Remark 4.

If we consider $\mathcal{M}_{\xi}$ with the compact-open topology, we get that
$\mathcal{M}_{\xi}$ is contractible. To see this, choose a
$\gamma_{\infty}\in\mathcal{M}_{\xi}$. For any $\gamma\in\mathcal{M}_{\xi}$
and $t\in[0,\infty)$, let $\gamma_{t}\in\mathcal{M}_{\xi}$ be given by

$\displaystyle\gamma_{t}(s)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\gamma_{\infty}(s)&0\leq s\leq t\\\
\gamma_{\infty}(2t-s)&t\leq s\leq 2t\\\ \gamma(s-2t)&2t\leq
s\end{array}\right.$

It is easy to see that
$H:\mathcal{M}_{\xi}\times[0,\infty]\to\mathcal{M}_{\xi}$ given by
$H(\gamma,t)=\gamma_{t}$ is continuous in the compact-open topology, and hence
defines a contraction. Of course $H$ is no longer continuous with the $W(a,A)$
open in $\mathcal{M}_{\xi}$.

######  Remark 5.

Let us give another interpretation of the topology on $\mathcal{M}_{\xi}$. Let
$\tilde{X}_{\infty}=\tilde{X}\cup\\{\infty\\}$, that is, we add a point
$\infty$ to $\tilde{X}$; the topology on $\tilde{X}_{\infty}$ is generated by
the open sets of $\tilde{X}$ and sets $N\cup\\{\infty\\}$, where $N$ is an
open neighborhood of $\infty$ with respect to $\xi$ (in the sense of Section
2).

Write

$\displaystyle\mathcal{P}(\tilde{X}_{\infty})$ $\displaystyle=$
$\displaystyle\\{\gamma:[0,\infty]\to\tilde{X}_{\infty}\,|\,\gamma(\infty)=\infty\\},$

which we topologize with the compact open topology. This is a usual path space
with $\infty$ as the basepoint. The space $\mathcal{C}_{\xi}$ of paths to
infinity with respect to $\xi$ can be identified with the subspace of
$\mathcal{P}(\tilde{X}_{\infty})$ consisting of those $\gamma$ with
$\gamma([0,\infty))\subset\tilde{X}$. Similarly, $\mathcal{M}_{\xi}$ can be
identified with a subspace of
$\Omega(\tilde{X}_{\infty})=e^{-1}(\\{\tilde{x_{0}}\\})$, where
$\tilde{x_{0}}\in p^{-1}(\\{x_{0}\\})\subset\tilde{X}$ and
$e:\mathcal{P}(\tilde{X}_{\infty})\to\tilde{X}_{\infty}$ the usual fibration.
Note that $\mathcal{C}_{\xi}$ is a covering space of $\mathcal{C}_{\xi}(X)$
with covering group $\pi_{1}(X)$.

Given $\gamma_{0}\in\mathcal{M}_{\xi}$, we want to examine the homotopy groups
$\pi_{k}(\mathcal{M}_{\xi},\gamma_{0})$ for $k\geq 0$. For this let
$g:(S^{k},\ast)\to(\mathcal{M}_{\xi},\gamma_{0})$ be a map. It gives rise to a
map $\phi_{g}:S^{k}\times[0,\infty)\to X$ by $\phi_{g}(x,t)=g(x)(t)$ and since
$\phi_{g}(x,0)=x_{0}$, a map $\Phi_{g}:{\mathbb{R}}^{k+1}\to X$ such that
$\Phi_{g}(x\cdot t)=\phi(x,t)$ for $x\in S^{k}\subset{\mathbb{R}}^{k+1}$. If
we assume that $\ast\in S^{k}$ corresponds to
$(1,0,\ldots,0)\in{\mathbb{R}}^{k+1}$, we have
$\Phi_{g}(t,0,\ldots,0)=\gamma_{0}(t)$. Furthermore, if we lift $\Phi_{g}$ to
a map $\tilde{\Phi}_{g}:{\mathbb{R}}^{k+1}\to\tilde{X}$, we get that
$h_{\xi}\circ\tilde{\Phi}(x)\to\infty$, as $|x|\to\infty$. A homotopy between
two maps $g_{0},g_{1}:(S^{k},\ast)\to(\mathcal{M}_{\xi},\gamma_{0})$ relative
to the basepoint corresponds to a homotopy
$\Phi:{\mathbb{R}}^{k+1}\times[0,1]\to X$ between $\Phi_{g_{0}}$ and
$\Phi_{g_{1}}$ relative to $[0,\infty)\times\\{0\\}$ and such that
$h_{\xi}\circ\tilde{\Phi}(x,s)\to\infty$ as $|x|\to\infty$ uniformly in
$s\in[0,1]$, for a lifting $\tilde{\phi}$.

Now assume we have a sequence $(N_{i})_{i\geq 0}$ of neighborhoods of $\infty$
with respect to $\xi$ such that $N_{i}\subset N_{i-1}$ for all $i$ and
$\bigcap_{i\geq 0}N_{i}=\emptyset$. Let $\tilde{x}_{0}\in\tilde{X}$ be a
lifting of $x_{0}\in X$. Also, let $\tilde{\gamma}_{0}:[0,\infty)\to\tilde{X}$
be the lifting of $\gamma_{0}$ with $\tilde{\gamma}_{0}(0)=\tilde{x}_{0}$.
Pick a sequence $t_{i}>0$ such that $t_{i+1}>t_{i}$ for all $i\geq 0$ such
that $\tilde{\gamma}_{0}(t)\in N_{i}$ for all $t\geq t_{i}$. This sequence
exists by the definition of $\gamma_{0}\in\mathcal{M}_{\xi}$. We let

$\displaystyle y_{i}$ $\displaystyle=$
$\displaystyle\tilde{\gamma}_{0}(t_{i})\in N_{i}$

be the sequence of basepoints of $N_{i}$ for all $i\geq 0$. We get a natural
homomorphism
$\chi_{i}:\pi_{k}(\tilde{X},N_{i+1},y_{i+1})\to\pi_{k}(\tilde{X},N_{i},y_{i})$
induced by inclusion where the change of basepoint is done using the path
$\tilde{\gamma}_{0}|[t_{i},t_{i+1}]$. This gives rise to an inverse system
$(\chi_{i}:\pi_{k}(\tilde{X},N_{i+1},y_{i+1})\to\pi_{k}(\tilde{X},N_{i},y_{i}))$
for every $k\geq 0$. Note that this is an inverse system of pointed sets for
$k\leq 1$.

For every $i\geq 0$ and $k\geq 0$ define
$\varphi_{i}:\pi_{k}(\mathcal{M}_{\xi},\gamma_{0})\to\pi_{k+1}(\tilde{X},N_{i},y_{i})$
in the following way. Given $g:(S^{k},\ast)\to(\mathcal{M}_{\xi},\gamma_{0})$,
we define $\varphi_{g}:(D^{k+1},S^{k},\ast)\to(\tilde{X},N_{i},y_{i})$ by
$\varphi_{g}(x\cdot t)=\tilde{\Phi}_{g}(\lambda(x)\cdot x\cdot t)$ for $x\in
S^{k}$ and $t\in[0,1]$, where $\lambda:S^{k}\to(0,\infty)$ is a map such that
$\lambda(\ast)=t_{i}$, and for every $x\in S^{k}$ we have $g(x)(t)\in N_{i}$
for all $t\geq\lambda(x)$. It is clear that the homotopy class of
$\varphi_{g}$ does not depend on the particular choice of $\lambda$. Basically
we use the map $\tilde{\Phi}_{g}$ and restrict it to a large enough ball in
${\mathbb{R}}^{k+1}$.

This induces the map
$\varphi_{i}:\pi_{k}(\mathcal{M}_{\xi},\gamma_{0})\to\pi_{k+1}(\tilde{X},N_{i},y_{i})$
and we clearly have $\chi_{i}\varphi_{i+1}=\varphi_{i}$. Thus we get an
induced map

$\displaystyle\varphi:\pi_{k}(\mathcal{M}_{\xi},\gamma_{0})$
$\displaystyle\longrightarrow$
$\displaystyle\lim\limits_{\leftarrow}\,\pi_{k+1}(\tilde{X},N_{i},y_{i})$

which is a group homomorphism for $k\geq 1$ and a map of pointed sets for
$k=0$.

Next we define a map $\psi^{\prime}:\prod_{i\geq
0}\pi_{k+2}(\tilde{X},N_{i},y_{i})\to\pi_{k}(\mathcal{M}_{\xi},\gamma_{0})$,
so let $a_{i}\in\pi_{k+2}(\tilde{X},N_{i},y_{i})$. Represent $a_{i}$ by a map
$g_{i}:(D^{k+2},S^{k+1},\ast)\to(\tilde{X},N_{i},y_{i})$. Furthermore, let
$G^{\prime}:{\mathbb{R}}^{k+1}\to\tilde{X}$ be given by $G^{\prime}(x\cdot
t)=\tilde{\gamma}_{0}(t)$ for $x\in S^{k}$ and $t\in[0,\infty)$. Let
$B(t_{i})$ be a small disc with center at $(-t_{i},0)\in{\mathbb{R}}^{k+1}$
such that the $B(t_{i})$ are pairwise disjoint. We use $(-t_{i},0)$ as the
center since we want to change $G^{\prime}$ on the $B(t_{i})$ without changing
it on $[0,\infty)\times\\{0\\}\subset{\mathbb{R}}^{k+1}$. So homotop
$G^{\prime}$ to a map $G$ relative to ${\mathbb{R}}^{k+1}-\bigcup
B^{\prime}(t_{i})$ such that $G$ is constant to $\tilde{\gamma}_{0}(t_{i})$ on
$B(t_{i})$ for all $i\geq 0$. Here $B^{\prime}(t_{i})$ is a slightly bigger
disc such that they are still pairwise disjoint.

If we restrict the map $g_{i}$ to $S^{k+1}$, we can think of this map as a map
$\tilde{g}_{i}:(D^{k+1},S^{k})\to(N_{i},y_{i})$. Now we can replace $G$ by a
map $\tilde{G}:{\mathbb{R}}^{k+1}\to\tilde{X}$ such that
$\tilde{G}|B(t_{i})\equiv\tilde{g}_{i}$ and $\tilde{G}$ agrees with $G$
everywhere else. Clearly $\tilde{G}$ induces a map
$g:(S^{k},\ast)\to(\mathcal{M}_{\xi},\gamma_{0})$ and this defines a map

$\displaystyle\psi^{\prime}:\prod_{i\geq 0}\pi_{k+2}(\tilde{X},N_{i},y_{i})$
$\displaystyle\longrightarrow$
$\displaystyle\pi_{k}(\mathcal{M}_{\xi},\gamma_{0}).$

Note here that if
$g^{\prime}_{i}:(D^{k+2},S^{k+1},\ast)\to(\tilde{X},N_{i},y_{i})$ also
represents $a_{i}$ for every $i\geq 0$, we get that $g_{i}|S^{k+1}$ is
homotopic to $g^{\prime}_{i}|S^{k+1}$ within $N_{i}$, so the resulting maps
$g$ and $g^{\prime}$ represent the same element in
$\pi_{k}(\mathcal{M}_{\xi},\gamma_{0})$. It is worth pointing out that
$\tilde{G}$ is homotopic to $G$ relative to $[0,\infty)\times\\{0\\}$, but the
resulting function $H:S^{k}\times[0,1]\to\mathcal{M}_{\xi}$ need not be
continuous, since the maps $g_{i}$ are only null homotopic in $\tilde{X}$, but
not necessarily with control.

Let us recall the definition of the derived limit for our inverse system. Two
sequences $(a_{i}),(b_{i})\in\prod_{i\geq 0}\pi_{k+2}(\tilde{X},N_{i},y_{i})$
are called equivalent, if there exists a sequence $(c_{i})\in\prod_{i\geq
0}\pi_{k+2}(\tilde{X},N_{i},y_{i})$ such that $b_{i}=c_{i}\cdot
a_{i}\cdot\chi_{i}(c_{i+1})$ for all $i\geq 0$. Then
${\lim\limits_{\leftarrow}}^{1}\,\pi_{k+2}(\tilde{X},N_{i},y_{i})$ is the set
of equivalence classes. For $k\geq 1$ this has the structure of an abelian
group, but for $k=0$ we only get a pointed set.

It is easy to see that $\psi^{\prime}$ induces a map

$\displaystyle\psi:{\lim\limits_{\leftarrow}}^{1}\,\pi_{k+2}(\tilde{X},N_{i},y_{i})$
$\displaystyle\longrightarrow$
$\displaystyle\pi_{k}(\mathcal{M}_{\xi},\gamma_{0})$

which is a homomorphism for $k\geq 1$ and a map of pointed sets for $k=0$.

######  Proposition 20.

With the above notation there is a short exact sequence

$1\to{\lim\limits_{\leftarrow}}^{1}\,\pi_{k+2}(\tilde{X},N_{i},y_{i})\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}\pi_{k}(\mathcal{M}_{\xi},\gamma_{0})\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}\lim\limits_{\leftarrow}\,\pi_{k+1}(\tilde{X},N_{i},y_{i})\to
1$

which is a short exact sequence of groups in the case $k\geq 1$ and of pointed
sets in the case $k=0$. If $k=0$, $\psi$ is also injective.

The proof is standard and will be omitted, see also [19]. Notice the
similarity between this sequence and (15).

We can use this to give another equivalent definition for
$\xi\in\Sigma^{k}(X)$.

######  Proposition 21.

Let $X$ be a finite connected CW-complex, $\xi\in H^{1}(X;{\mathbb{R}})$ be
nonzero and $k\geq 1$. Then $\xi\in\Sigma^{k}(X)$ if and only if
$\mathcal{M}_{\xi}$ is $(k-1)$-connected.

###### Proof.

If $\mathcal{M}_{\xi}$ is $(k-1)$-connected, then by Proposition 20 we get
that the inverse system $\\{\pi_{l}(\tilde{X},N)\\}$ is pro-trivial for $l\leq
k$ which gives $\xi\in\Sigma^{k}(X)$ by Proposition 2.

To get the other direction, we have to worry about
${\lim\limits_{\leftarrow}}^{1}\,\pi_{k+1}(\tilde{X},N_{i})$. But by the next
lemma $\\{\pi_{k+1}(\tilde{X},N)\\}$ is semi-stable, so the
${\lim\limits_{\leftarrow}}^{1}$-term vanishes. It follows from Proposition 20
that $\mathcal{M}_{\xi}$ is $(k-1)$-connected. ∎

######  Lemma 22.

Let $X$ be a finite connected CW-complex, $\xi\in H^{1}(X;{\mathbb{R}})$ be
nonzero and $h_{\xi}:\tilde{X}\to{\mathbb{R}}$ a height function. If
$\xi\in\Sigma^{k}(X)$, then $\\{\pi_{k+1}(\tilde{X},N)\\}$ is semi-stable.

###### Proof.

Use the homotopy from Proposition 2(4) to push any $k$-sphere in $N$
arbitrarily far away, this gives semi-stability as the pushing may be done in
$N^{\prime}$ slightly bigger than $N$ (depends on $H$ only). ∎

##  Appendix A Sigma invariants of chain complexes

In this appendix we show how $\Sigma^{k}(C)$ is related to criteria involving
chain homotopies on $C$.

If the chain complex $C$ consists of flat $R$-modules, we have the following
criterion.

######  Proposition 23.

Let $C$ be a chain complex of flat $R$-modules and $n$ a non-negative integer.
Then the following are equivalent.

  1. (1)

$C$ is of finite $n$-type.

  2. (2)

For every index set $J$, the natural map
$H_{k}(C,\prod_{J}R)\to\prod_{J}H_{k}(C)$ is an isomorphism for $k<n$ and an
epimorphism for $k=n$.

This is basically [8, Thm.2], but we allow $C$ to be flat and not necessarily
projective so we only have a homology criterion. The proof goes through for
flat modules.

By a filtration of $C$ we mean a family
$\\{C^{\alpha}\\}_{\alpha\in\mathcal{A}}$ of sub-chain complexes where
$\mathcal{A}$ is a directed set, $C^{\alpha}\subset C^{\beta}$ for
$\alpha\leq\beta$ and $C=\bigcup C^{\alpha}$.

Given a filtration, we define $D^{\alpha}=C/C^{\alpha}$.

The analogue of [10, Thm.2.2] is

######  Theorem 10.

Let $C$ be an $R$-chain complex with a filtration
$\\{C^{\alpha}\\}_{\alpha\in\mathcal{A}}$ of finite $n$-type complexes
$C^{\alpha}$ of flat $R$-modules. Then $C$ is of finite $n$-type if and only
if the direct system $\\{H_{i}(D^{\alpha})\\}$ is essentially trivial for
$i\leq n$.

###### Proof.

The proof is similar to the proof of [10, Thm.2.2], but one has to use
Proposition 23. We omit the details. ∎

Let $G$ be a finitely generated group and $\xi:G\to{\mathbb{R}}$ a non-zero
homomorphism. Let $C$ be a free ${\mathbb{Z}}G$-chain complex which is
finitely generated in every degree. Given a valuation on $C$ extending $\xi$,
we can define a subcomplex

$\displaystyle C^{\xi}$ $\displaystyle=$ $\displaystyle\\{x\in C\,|\,v(x)\geq
0\\}$

Valuations are determined by their value on basis elements, so it is easy to
see that $C^{\xi}$ is a finitely generated free chain complex over
${\mathbb{Z}}G_{\xi}$, compare [7, Lemma 3.1]. Given $g\in G$, we can also
look at the subcomplex $gC^{\xi}\subset C$ which is isomorphic to $C^{\xi}$.
Denote $D^{g}=C/gC^{\xi}$.

We use Theorem 10 to get

######  Proposition 24.

Let $C$ be a free ${\mathbb{Z}}G$-chain complex which is finitely generated in
every degree, $v$ a valuation extending $\xi$ and $n$ a non-negative integer.
Then the following are equivalent.

  1. (1)

$\xi\in\Sigma^{n}(C)$.

  2. (2)

The direct system $\\{H_{i}(D^{g})\\}$ is essentially trivial for $i\leq n$.

  3. (3)

There exists a chain map $\varphi:C\to C$ chain homotopic to the identity such
that $v(\varphi(x))>v(x)$ for all non-zero $x\in C_{i}$ with $i\leq n$.∎

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