# Sequential Decision Making on Unmatched Data using Bayesian Kernel
Embeddings

Diego Martinez-Taboada
Department of Statistics & Data Science
Carnegie Mellon University
<EMAIL_ADDRESS>
&Dino Sejdinovic
Department of Statistics
University of Oxford
<EMAIL_ADDRESS>
Work primarily done at the University of Oxford and finished at Carnegie
Mellon University.

###### Abstract

The problem of sequentially maximizing the expectation of a function seeks to
maximize the expected value of a function of interest without having direct
control on its features. Instead, the distribution of such features depends on
a given context and an action taken by an agent. In contrast to Bayesian
optimization, the arguments of the function are not under agent’s control, but
are indirectly determined by the agent’s action based on a given context. If
the information of the features is to be included in the maximization problem,
the full conditional distribution of such features, rather than its
expectation only, needs to be accounted for. Furthermore, the function is
itself unknown, only counting with noisy observations of such function, and
potentially requiring the use of unmatched data sets. We propose a novel
algorithm for the aforementioned problem which takes into consideration the
uncertainty derived from the estimation of both the conditional distribution
of the features and the unknown function, by modeling the former as a Bayesian
conditional mean embedding and the latter as a Gaussian process. Our algorithm
empirically outperforms the current state-of-the-art algorithm in the
experiments conducted.

##  1 INTRODUCTION

Uncertainty quantification is inherent to sequential decision making problems,
where an agent sequentially explores a set of possible actions while seeking
to maximize the associated reward (Alagoz et al. (2010), Sutton and Barto
(2018)). The algorithms designed for such problems are based on an
exploration-exploitation trade-off: the agent tries to exploit its knowledge
on the actions that have yielded high rewards, but it also seeks to explore
actions that count with little information (Macready and Wolpert (1998),
Audibert et al. (2009)).

The multi-armed bandit (MAB) is one of the sequential decision problems that
has received the most attention in the literature (Vermorel and Mohri (2005),
Slivkins et al. (2019)). The classical MAB problem is defined by a tuple
$\langle\mathcal{A},F_{R}\rangle$, where $\mathcal{A}$ is the state of actions
and $F_{R}$ the reward distribution. At time step $t\in\mathbb{N}$, the agent
decides to take an action $a_{t}\in\mathcal{A}$. A reward $r_{a_{t}}$ drawn
from $r_{a_{t}}\sim F_{R}(\cdot|a_{t})$ follows. The MAB aims at maximizing
the cumulative reward

$J_{T}=\mathbb{E}\left[\sum_{t=1}^{T}r_{a_{t}}\right],$ (1)

where $a_{t}$ are the actions sequentially taken. The contextual multi-armed
bandit problem (CMAB) generalizes the MAB by including a state space (or
context space) $\mathcal{S}$ (Lu et al. (2010), Zhou (2015)). Formally, the
CMAB problem is determined by a tuple
$\langle\mathcal{S},\mathcal{A},F_{R}\rangle$, and it aims at maximizing

$J_{T}=\mathbb{E}\left[\sum_{t=1}^{T}r_{s_{t},a_{t}}\right],$ (2)

where $r_{s_{t},a_{t}}\sim F_{R}(\cdot|s_{t},a_{t})$, and $s_{t}$ are
sequentially given.

However, one may be interested in optimizing the expectation of a function
$f(r)$, instead of $r$ itself. In other words, $r_{t}$ may be seen as an
intermediate reward that mediates between the agent’s actions $a_{t}$ and the
ultimate reward $f(r_{t})$. In various fields of application, it is usual to
study a function of the outcome, rather than the outcome itself. For instance,
the conditional value at risk (CVaR), which has seen extensive use in
financial portfolio optimization (Rockafellar et al. (2000), Zhu and Fukushima
(2009)), considers $f(r)=r1_{r\leq r_{\alpha}}$. The median and other
quantiles may be of interest for noisy data (Even-Dar et al. (2002),
Altschuler et al. (2018)).

In more general settings, the function $f$ may be itself unknown. Noisy data
of the form

$(r_{i},f(r_{i})+\xi_{i})_{i=1}^{M}$ (3)

may be the only available information on such dependency. In this case,
uncertainty raises from the lack of information between the traditional CMAB
variables $r,a,s$, but also from the unknown function $f$.

Please note that this scenario opens the door for considering multiple sources
of data. In contrast to the usual matched scenario, where the sequential data
is of the form $(s_{i},a_{i},r_{i},f(r_{i})+\xi_{i})_{i=1}^{t}$, it may be of
interest to consider an unmatched scenario, where a second data set of the
form $(r_{j},f(r_{j})+\xi_{j})_{j=1}^{M}$ is given on top of the sequential
data. Algorithms that handle unmatched data sets have currently been
attracting more and more interest in the so called data fusion problem (Meng
et al. (2020)).

For instance, a new movie recommendation system might obtain a multivariate
$r_{j}$ consisting on: whether the user showed interest in the recommendation,
the number of cliques after the recommendation screen, the time spent reading
the description of the recommendation, etc. The final reward
$f(r_{j})+\xi_{j}$ is given by the number of minutes that the client ended up
watching the content for. A rich data set $(r_{j},f(r_{j})+\xi_{j})_{j=1}^{M}$
might be obtained from other platforms that have been longer in the market.
Not considering this second data set would imply a substantial loss of
information which would to suboptimal approaches.

In this work, we revisit the problem of sequentially maximising the
conditional expectation of a function in the CMAB framework. Our contributions
are two-folded:

  * •

We design a novel algorithm, namely Contextual Bayesian Mean Process Upper
Confidence Bound, which considers two sources of uncertainty derived from the
lack of information on $r,a,s$, as well as the unknown function $f$. It allows
for considering both matched and unmatched data sets.

  * •

We empirically show that the Bayesian Mean Process Upper Confidence Bound
outperforms the state-of-the-art Conditional Mean Embeddings Upper Confidence
Bound (Chowdhury et al. (2020)) in the experimental settings considered.

##  2 RELATED WORK

Multiple algorithms have been proposed for addressing the CMAB problem. The
contextual Gaussian process upper confidence bound (CGP-UCB), which motivates
the algorithm proposed in this work, was first presented in Krause and Ong
(2011). It generalizes the context-free Gaussian process upper confidence
bound (GP-UCB) introduced in Srinivas et al. (2009). The CGP-UCB attains
sublinear contextual regret in many real life applications i.e. it is able to
compete with the optimal mapping from contexts to actions. However, please
note that there exist prominent alternatives to CGP-UCB such as Thompson
sampling (Agrawal and Goyal (2013)). Some algorithms designed for the MAB
problem (Chowdhury and Gopalan (2017)) may also be adapted to the CMAB
framework.

The problem of sequentially maximising the conditional expectation has not
received as much attention in the literature, although a number of closely
related problems have been presented in different forms. Oliveira et al.
(2019) proposed a sequential decision problem where both the reward and
actions are noisy (execution and localisation noise), with applications in
robotics and stochastic simulations. The contextual combinatorial multi-armed
bandit problem (Chen et al. (2013), Qin et al. (2014)) may be understood as
the problem of sequentially maximixing the conditional expectation of a
function, where the chosen super arm may be seen as a multivariate binary
action, and the dependency between the scores of the individual arms and the
reward as the unknown function.

For the problem of sequentially maximizing the expectation of a function,
Chowdhury et al. (2020) proposed the Conditional Mean Embeddings Upper
Confidence Bound (CME-UCB) algorithm. The CME-UCB uses the conditional mean
embedding for modeling the conditional distribution of the features. In
contrast, alternative approaches for such conditional density estimation scale
poorly with the dimension of the underlying space (Grunewalder et al. (2012)).

The conditional mean embedding, key element of the CME-UCB algorithm, extends
the concept of kernel mean embeddings to conditional distributions (Muandet et
al. (2017)). In terms of Bayesian procedures, Flaxman et al. (2016) proposed a
Bayesian approach for learning mean embeddings, and Chau et al. (2021a)
generalized the approach for Bayesian conditional mean embeddings (BayesCME).
Such Bayesian learning of mean embeddings requires the concept of nuclear
dominance (Lukić and Beder (2001)), which allows for defining a GP with
trajectories in a Reproducing Kernel Hilbert Space with probability 1.

Furthermore, conditional mean processes are now well established (Chau et al.
(2021b)), which study the integral of a Gaussian process (GP) with respect to
a conditional distribution. Building on the Bayesian conditional mean
embedding and the conditional mean process, Chau et al. (2021a) proposed the
BayesIMP algorithm for tackling a data fusion problem in a causal inference
setting. Such algorithm considers uncertainty derived from two data samples,
with observations of a mediating variable included in both data samples.

##  3 BACKGROUND

###  3.1 RKHS

The concept of Reproducing Kernel Hilbert Spaces (RKHS) is widely used in
statistics and machine learning (Hofmann et al. (2008), Gretton et al. (2012),
Sejdinovic et al. (2013)).

######  Definition 1 (Reproducing Kernel Hilbert Space)

Let $\mathcal{X}$ be a non-empty set and let $\mathcal{H}$ be a Hilbert space
of functions $f:\mathcal{X}\to\mathbb{R}$ with inner product
$\langle\cdot,\cdot\rangle_{\mathcal{H}}$. A function
$k:\mathcal{X}\times\mathcal{X}\to\mathbb{R}$ is called a reproducing kernel
of $\mathcal{H}$ if it satisfies

  * •

$k(\cdot,x)\in\mathcal{H}\quad\forall x\in\mathcal{X}$,

  * •

(the reproducing property) $\langle
f,k(\cdot,x)\rangle_{\mathcal{H}}=f(x)\quad\forall x\in\mathcal{X},\forall
f\in\mathcal{H}.$

If $\mathcal{H}$ has a reproducing kernel, then it is called reproducing
kernel Hilbert space (RKHS).

Based on the Cauchy-Schwarz inequality, the reproducing property implies that
the map $F_{x}:f\in\mathcal{H}\to f(x)$ is a continuous linear form for all
$x$ in $\mathcal{X}$. Therefore, the reproducing property introduces some
degree of smoothness relative to the Hilbert space inner product of the
functions considered (Gretton (2013)).

Notation: Given a non-empty set $\mathcal{X}$, we refer to the considered
kernel associated to $\mathcal{X}$ as $k_{x}$. Given $x\in\mathcal{X}$, the
element $k(\cdot,x)\in\mathcal{H}$ is also referred as $\phi_{x}(x)$,
motivated by the notation used when understood as a feature representation of
$x$. Given a data vector $\textbf{x}=[x_{1},...,x_{n}]^{T}$, feature matrices
are defined by stacking feature maps along the columns $\Phi_{{\bf
x}}:=[\phi_{x}(x_{1}),...,\phi_{x}(x_{n})]$. The Gram matrix is denoted as
$K_{{\bf x}{\bf x}}=\Phi_{{\bf x}}^{T}\Phi_{{\bf x}}$, and the vector of
evaluations as $k_{x{\bf x}}=[k_{x}(x,x_{1}),...,k_{x}(x,x_{n})]$.
Furthermore, $\Phi_{{\bf x}}(x)=k_{x{\bf x}}^{T}$. The notation is analogously
used for the rest of variables used, such as $k_{y}$, y, $\Phi_{\bf{y}}$ or
$K_{\bf{y}\bf{y}}$ for variable $y\in\mathcal{Y}$.

The Kernel Mean Embedding (KME), which allows for distribution representation
based on RHKS, is the cornerstone of the algorithms that will be discussed in
this work.

######  Definition 2 (Kernel Mean Embedding (KME))

Let $\mathcal{P}$ be the set of all probability measures on a measurable space
$(\mathcal{X},\mathcal{B}_{\mathcal{X}})$, and
$k:\mathcal{X}\times\mathcal{X}$ a reproducing kernel with associated RKHS
$\mathcal{H}$ such that $sup_{x\in\mathcal{X}}k(x,x)<\infty$. The kernel mean
embedding (KME) of $\mathbb{P}\in\mathcal{P}$ with respect to $k$ is defined
as the following Bochner integral

$\mu:\mathcal{P}\to\mathcal{H},\quad\mathbb{P}\to\mu_{\mathbb{P}}:=\int
k(\cdot,x)d\mathbb{P}(x).$

Kernel $k$ is said to be characteristic if $\mu$ in injective. In such case,
$\mu_{\mathbb{P}}$ serves as a representation of $\mathbb{P}$. The frequently
used Gaussian (RBF), Matérn and Laplace kernels are characteristic. The notion
of characteristic kernel is an analogy to the characteristic function
(Fukumizu et al. (2007)). The succeeding lemma immediately follows from the
definition of kernel mean embedding.

###### Lemma 1

The kernel mean embedding of $\mathbb{P}$ maps functions $f\in\mathcal{H}$ to
their mean with respect to $\mathbb{P}$ through the inner product:

$\langle\mu_{\mathbb{P}},f\rangle_{\mathcal{H}}=\mathbb{E}_{X\sim\mathbb{P}}[f(X)].$
(4)

The conditional mean embedding $\mu_{Y|X=x}$ considers the kernel mean
embedding of the conditional distribution $\mathbb{P}_{Y|X=x}$ for every
$x\in\mathcal{X}$.

###  3.2 Bayesian Conditional Mean Embedding

A Bayesian learning framework on conditional embeddings was proposed in Chau
et al. (2021a). In such framework, the conditional mean embedding
$\mu_{Y|X=x}(y)$ is modeled by a Gaussian process $\mu_{gp}(x,y)$. A prior is
defined over $\mu_{gp}\sim GP(0,k_{x}\otimes r_{y})$, and the posterior models
$\mu_{Y|X=x}(y)$.

Please note that the paths $\mu_{gp}(x,\cdot)$ should live in the RKHS
associated to kernel $k$, as they model $\mu_{Y|X=x}(\cdot)$. If
$r_{y}=k_{y}$, then the paths live outside the RKHS with probability 1. In
order to ensure that the paths of such Gaussian process $\mu_{gp}(x,y)$ live
almost surely within the RKHS, the prior covariate kernel may be defined as a
nuclear dominant kernel over $k_{y}$ (Lukić and Beder (2001)). Following a
similar structure to Flaxman et al. (2016) and Chau et al. (2021a), we choose
$r_{y}$ to be the convolution of the original kernel with itself. Therefore,
the GP prior over $\mu_{gp}$ is defined as follows:

$\mu_{gp}\sim GP(0,k_{x}\otimes r_{y}),$ (5) $r_{y}(y_{1},y_{2}):=\int
k_{y}(y_{1},u)k_{y}(u,y_{2})\nu(du),$ (6)

where $\nu$ is a finite measure on $\mathcal{Y}$. The following function-
valued regression is then set up:

$\phi_{y}(y_{i})=\mu_{gp}(x_{i},\cdot)+\lambda^{1/2}\epsilon_{i},$ (7)

where $\epsilon_{i}\stackrel{{\scriptstyle iid}}{{\sim}}GP(0,r)$ are noise
functions. The noise hyperparameter of the GP links this framework to the
spectral kernel mean shrinkage estimator (Muandet et al. (2013)). The
posterior mean and covariance for $\mu_{gp}$, the marginal likelihood, and
specific nuclear dominant kernels may be obtained in closed form and we refer
to Chau et al. (2021a) for the respective derivations.

###  3.3 CGP-UCB

In CGP-UCB, the reward sequence is assumed to be a sample from a known GP
distribution. The exploration-exploitation trade-off translates consequently
to a Bayesian optimization problem. At round $t$, the CGP-UCB picks action
$a_{t}$ such that

$a_{t}=argmax_{a\in
A}\;\mu_{t-1}(s_{t},a)+\beta_{t}^{\frac{1}{2}}\sigma_{t-1}(s_{t},a),$ (8)

where $\beta_{t}$ are appropriate constants, and $\mu_{t-1}(\cdot)$ and
$\sigma_{t-1}(\cdot)$ are the posterior mean and standard deviation of the GP
over the joint set $S\times A$ conditioned on the observations
$(s_{1},a_{1},r_{1}),...,(s_{t-1},a_{t-1},r_{t-1})$. The addition

$\mu_{t-1}(s_{t},a)+\beta_{t}^{\frac{1}{2}}\sigma_{t-1}(s_{t},a)$ (9)

is known as the acquisition function. CGP-UCB attains sublinear contextual
regret in many real life applications. In other words, it is able to compete
with the optimal mapping from contexts to actions.

###  3.4 CME-UCB

The Conditional Mean Embeddings Upper Confidence Bound (CME-UCB) algorithm was
proposed in Chowdhury et al. (2020) for addressing the problem of sequentially
maximising the conditional expectation of a function in an RKHS, using
conditional mean embeddings. Chowdhury et al. (2020) defined the UCB
acquisition function

$\mu_{t-1}(a_{t})+\beta_{t}^{\frac{1}{2}}\sigma_{t-1}(a_{t}),$ (10)

where $\beta_{t}$ are appropiate constants,

$\mu_{t-1}(a)=\Phi(a)^{\top}(K_{{\bf a}{\bf a}}+\lambda I)^{-1}K_{{\bf r}{\bf
r}}(K_{{\bf r}{\bf r}}+\lambda_{f}I)^{-1}y,$ (11)

the standard deviation is described (as an application of the Sherman-Morrison
formula) in terms of the Mahalanobis norm of the control features
$\phi(a_{t})$:

$\sigma_{t-1}(a)=\lambda^{-1/2}\sqrt{k_{a}(a,a)-\Phi_{\bf a}(a)^{\top}(K_{{\bf
a}{\bf a}}+\lambda I)^{-1}\Phi_{\bf a}(a)}.$ (12)

Although this algorithm is not meant to deal with contexts and unmatched data,
it is easily generalized. Given $D_{1}=\\{(s_{i},a_{i},r_{i})\\}_{i=1}^{t-1}$,
$D_{2}=\\{(\tilde{r}_{j},y_{j})\\}_{j=1}^{M}$, and context $s_{t}$, it
suffices to consider the following mean and standard deviation:

$\mu_{t-1}(a)=\Phi(s_{t},a)^{\top}(K+\lambda I)^{-1}K_{{\bf r}\tilde{{\bf
r}}}(K_{\tilde{{\bf r}}\tilde{{\bf r}}}+\lambda_{f}I)^{-1}y,$ (13)
$\displaystyle\sigma_{t-1}(a)=\lambda^{-1/2}\sqrt{G_{s_{t},a}},$ (14)

where
$G_{s_{t},a}=\Phi(s_{t},a)^{\top}\Phi(s_{t},a)-\Phi(s_{t},a)^{\top}(K+\lambda
I)^{-1}\Phi(s_{t},a)$, $K=K_{{\bf s}{\bf s}}\odot K_{{\bf a}{\bf a}}$ and
$\Phi(s,a)=\Phi_{\bf{s}}^{T}k_{s}(s,\cdot)\odot\Phi_{\bf{a}}^{T}k_{a}(a,\cdot)$.
Then, $a_{t}$ is chosen to maximize the acquisition function defined in
Equation (9) with the corresponding $\mu_{t-1}$ and $\sigma_{t-1}$. Please
note that the standard deviation $\sigma_{t}$ only depends on contexts and
actions, without taking into consideration any information about $\\{{\bf
r},\tilde{{\bf r}}\\}$.

##  4 PROBLEM STATEMENT

The problem addressed in this work is formally determined by a tuple
$\langle\mathcal{S},\mathcal{A},\mathcal{R},\mathcal{P}_{R},F_{Y}\rangle$,
where:

  * •

$\mathcal{S}$ is the space of contexts.

  * •

$\mathcal{A}$ is the space of actions.

  * •

$\mathcal{R}$ is the state of intermediate rewards.

  * •

$\mathcal{P}_{R}:\mathcal{A}\times\mathcal{S}\times\mathcal{R}\to[0,1]$ is the
intermediate reward distribution.

  * •

$F_{Y}:\mathcal{R}\times\mathbb{R}\to[0,1]$ is the ultimate reward
distribution.

At time step $t\in\mathbb{R}$, a context $s_{t}$ is given and the agent
decides to take an action $a_{t}\in A$. Two subsequent rewards
$r_{s_{t},a_{t}}$ and $y_{r_{t}}$ drawn from
$\mathcal{P}_{R}(\cdot|s_{t},a_{t})$ and $F_{Y}(\cdot|r_{t})$ follow, which we
name intermediate reward and ultimate reward respectively. Please note that we
have not imposed any restriction to the space of intermediate rewards
$\mathcal{R}$, however the ultimate reward is taken as real-valued. The RCMAB
aims at maximizing the cumulative reward

$J_{T}=\mathbb{E}\left[\sum_{t=1}^{T}y_{r_{t}}\right]=\mathbb{E}\left[\sum_{t=1}^{T}y_{r_{s_{t},a_{t}}}\right],$
(15)

where $(s_{1},...,s_{T})$ is sequentially given and $(a_{1},...,a_{T})$ are
the actions taken.

This problem is referred in the literature as the problem of sequentially
maximizing the conditional expectation of a function. In Chowdhury et al.
(2020), the distribution $F_{Y}$ is expressed as a sum of the conditional mean
and noise such that $y_{t}=f(r_{t})+\zeta_{t}$. In contrast to Chowdhury et
al. (2020), we consider the intermediate reward
$r_{t}|a_{t},s_{t}\sim\mathcal{P}_{R}(a_{t},s_{t})$ to depend on both a
context and action, rather than only an action i.e. we consider the
contextualized version.

We highlight the importance of accounting for the whole distribution of the
intermediate reward $\mathcal{P}_{R}$ in such framework. Even if we assume a
simple deterministic relationship between intermediate and ultimate reward,
the expectation of the ultimate reward cannot be maximized in terms of the
expectation of the intermediate reward.

One may consider the reduced sample $(s_{t},a_{t},y_{t})_{t=1}^{T}$ by
dropping the mediating variables $r_{i}$. However, information is disregarded,
which could translate to a drop in the performance of algorithms (Chowdhury et
al. (2020)). Furthermore, dropping $r_{i}$ does not allow to consider a
second, unmatched data set $(r_{j},y_{j})_{j=1}^{M}$. Hence such approach
limits the power of the sequential decision making algorithms by disregarding
information in case a second, unmatched data set is available.

We call attention to the different scopes of this work and Chau et al.
(2021a). While in Chau et al. (2021a) estimated interventional distributions
are used to warm-start Bayesian optimization, the problem setting is itself
static. By considering a multi-armed bandit framework rather than a causal
inference problem, we make use of the uncertainty quantification proposed in
Chau et al. (2021a) for addressing a sequential exploration-exploitation
trade-off. Furthermore, an extra term to be conditioned on (the context) is
needed in this case, however no intervention needs to be accounted for.

##  5 ALGORITHM DESIGN

We propose a novel algorithm for tackling the problem of sequentially
maximising the expectation of a function. It combines the ideas of the
BayesIMP algorithm proposed in (Chau et al. (2021a)) and the CGP-UBC. The mean
and variance for each action are modeled following the ideas of BayesIMP.
Then, the optimization of the CGP-UCB acquisition function follows, as the
uncertainty estimates considered by the GPs can be used for such exploitation-
exploration trade-off.

Formally, let $D_{1}=(s_{t},a_{t},r_{t})_{t=1}^{T}$ be the sequential sample
obtained from the interaction of the agent with the system, and let
$D_{2}=(r_{j},y_{j})_{j=1}^{T+M}$ be the sample containing observations of the
intermediate and ultimate rewards. As highlighted in the previous section, the
whole distribution $\mathcal{P}_{R}(\cdot|s,a)$ ought to be modeled. As only
the expectation of the final reward is of interest in the framework, there is
no need to model the whole distribution $F_{Y}(\cdot|r_{t})$. It suffices to
estimate its expectation $\mathbb{E}[y_{r_{t}}]$, which we denote
$f(r_{t})=\mathbb{E}[y_{r_{t}}]$.

Based on these considerations and motivated by the BayesIMP algorithm (Chau et
al. (2021a)), we propose the following approach for the problem:

  * •

Contextual Bayesian Mean Process UCB (CBMP-UCB): Expectation $f(r_{t})$ is
trained as a GP, and the conditional mean embedding of
$\mathcal{P}_{R}(\cdot|s,a)$ is taken as a GP by considering the Bayesian
conditional mean embedding $\mu_{R|s,a}$. A nuclear dominant kernel is needed
in both $f$ and $\mu_{R|S=s,A=a}$ so that the inner product $\langle
f,\mu_{R|s,a}\rangle$ can be considered almost surely. Although the process
$\langle f,\mu_{R|s,a}\rangle$ is not a GP, CBMP-UCB picks action $a_{t}$
following Equation (8) by taking $\mu_{t-1}(\cdot)$ and $\sigma_{t-1}(\cdot)$
as the mean and stardard deviation of $\langle f,\mu_{R|s_{t},a}\rangle$ (i.e.
moment matching is used to construct a GP out of $\langle
f,\mu_{R|s,a}\rangle$ for posterior inference).

The means and standard deviations, $\mu_{t-1}(\cdot)$ and
$\sigma_{t-1}(\cdot)$, can be obtained in closed form. They have been
developed in Chau et al. (2021a) for a causal data fusion problem. As such
problem requires adjusting for confounding variables, the closed forms
expressions exhibited in Chau et al. (2021a) are slightly more complex,
although an extra term to be conditioned on (the context) is needed in this
case. The following theorem could be interpreted as a simplified version of
the BayesIMP theorem from Chau et al. (2021a). It is stated here for
completeness with notation adapted for our setting and we refer to Chau et al.
(2021a) for its proof.

###### Proposition 1

Let $D_{1}=\\{(s_{i},a_{i},r_{i})\\}_{i=1}^{t-1}$,
$D_{2}=\\{(\tilde{r}_{j},y_{j})\\}_{j=1}^{M}$ be two unmatched datasets and
$k_{s},k_{a},k_{r}$ the kernels associated to variables $s,a,k$ respectively,
and $r_{r}$ a nuclear dominant kernel of $k_{r}$. We denote $K=K_{{\bf s}{\bf
s}}\odot K_{{\bf a}{\bf a}}$,
$\Phi(s,a)=\Phi_{\bf{s}}^{T}k_{s}(s,\cdot)\odot\Phi_{\bf{a}}^{T}k_{a}(a,\cdot)$,
and $\hat{{\bf r}}=({\bf r},\tilde{{\bf r}})$. If $f$ and $\mu_{R|s_{t},a}$
are modeled as GP, then $g=\langle f,\mu_{R|s_{t},a}\rangle$ has the following
mean $m_{3}$ and variance $\kappa_{3}$:

$\displaystyle m_{3}(a)$ $\displaystyle=E_{a}K_{{\bf
r}\bf\hat{r}}K_{\bf\hat{r}\bf\hat{r}}^{-1}R_{\bf\hat{r}\bf\tilde{r}}(R_{\bf\tilde{r}\bf\tilde{r}}+\lambda_{f}I)^{-1}y,$
(16) $\displaystyle\kappa(a,a^{\prime})$
$\displaystyle=E_{a}\Theta_{1}^{\top}\hat{R}_{\bf\hat{r}\bf\hat{r}}\Theta_{1}E^{\top}_{a^{\prime}}+$
(17)
$\displaystyle+\Theta_{2}^{(a)}F_{aa^{\prime}}-\Theta_{2}^{(b)}G_{aa^{\prime}}+$
(18)
$\displaystyle+\Theta_{3}^{(a)}F_{aa^{\prime}}-\Theta_{3}^{(b)}G_{aa^{\prime}},$
(19)

where $E_{a}=\Phi(s_{t},a)^{\top}(K+\lambda I)^{-1}$,
$F_{aa^{\prime}}=\Phi(s_{t},a)^{\top}\Phi(s_{t},a)$,
$G_{aa^{\prime}}=\Phi(s_{t},a)^{\top}(K+\lambda
I)^{-1}\Phi(s_{t},a^{\prime})^{\top}$,
$\Theta_{1}=K_{\bf\hat{r}\bf\hat{r}}^{-1}R_{\bf\hat{r}{\bf r}}R_{{\bf r}{\bf
r}}^{-1}K_{{\bf r}{\bf r}}$,
$\Theta_{2}^{(a)}=\Theta_{4}^{\top}R_{\bf\hat{r}\bf\hat{r}}\Theta_{4}$,
$\Theta_{2}^{(b)}=\Theta_{4}^{\top}R_{\bf\hat{r}{\bf r}}R_{{\bf r}{\bf
r}}^{-1}R_{{\bf r}\bf\hat{r}}\Theta_{4}$,
$\Theta_{3}^{(a)}=tr(K_{\bf\hat{r}\bf\hat{r}}^{-1}R_{\bf\hat{r}\bf\hat{r}}K_{\bf\hat{r}\bf\hat{r}}^{-1}\bar{R}_{\bf\hat{r}\bf\hat{r}})$,
$\Theta_{3}^{(b)}=tr(R_{\bf\hat{r}{\bf r}}R_{{\bf r}{\bf r}}^{-1}R_{{\bf
r}\bf\hat{r}}K_{\bf\hat{r}\bf\hat{r}}^{-1}\bar{R}_{\bf\hat{r}\bf\hat{r}}K_{\bf\hat{r}\bf\hat{r}}^{-1})$,
$\Theta_{4}=K_{\bf\hat{r}\bf\hat{r}}^{-1}R_{\bf\hat{r}\bf\tilde{r}}(K_{\bf\tilde{r}\bf\tilde{r}}+\lambda_{f})^{-1}y$.
$\bar{R}_{\bf\hat{r}\bf\hat{r}}$ is the posterior covariance of f evaluated at
$\hat{{\bf r}}$.

Interestingly and as explained in Chau et al. (2021a), the covariance of this
closed form can be interpreted as the sum of covariances associated to $D_{1}$
and $D_{2}$, plus an interaction term between the two.

Please note that CMBP-UCB takes into account uncertainties derived from both
$D_{1}$ and $D_{2}$ by considering a GP for modeling both expectation $f$ and
conditional mean embedding $\mu_{R|s,a}$. In contrast, we recall that the CME-
UCB algorithm (Chowdhury et al. (2020)) does not take into consideration any
information of $\\{{\bf r},\hat{{\bf r}}\\}$ in the calculation of the
standard deviation.

CBMP-UCB brings into play several hyperparameters: the lengthscales associated
to the kernels, the noises of the GP, and regularization terms. Hyperparameter
optimization can be conducted as often as possible, taking into consideration
the computational resources available.

##  6 EXPERIMENTS

Figure 1: Mean and 0.05 quantiles of the cumulative rewards over 100 trials
for settings A, B, C, and D with m = 0. Figure 2: Mean and 0.05 quantiles of
the cumulative rewards over 100 trials for settings A, B, C, and D with m =
50.

The main goal of the experiments conducted is to compare the performance of
CBMP-UCB and CME-UCB. For this purpose, four experimental settings (A, B, C,
D) are considered. Settings were selected to explore a range of different
dependencies.

Setting A considers a toy example with a one dimensional intermediate reward.
Settings B and C contemplate a more complex, two-dimensional intermediate
reward. In setting D, the dimension of the intermediate reward is raised to
five. Furthermore, the conditional expectations differ in all four scenarios.

  * i.

Setting A:

    * –

$\tilde{r}_{j}\sim Uniform(-2,2)$.

    * –

$y_{j}|r_{j}\sim 1.5\sin{r_{j}}+1+0.05\mathcal{N}(0,1)$.

    * –

$s_{i}\sim Uniform(-3,3.25)$.

    * –

$r|s,a\sim\sin(\pi s)+\cos(\pi a)+0.25\mathcal{N}(0,1)$.

    * –

$\mathcal{A}=[-3,3.25]$, discretized in 61 points.

    * –

$\mathcal{S}=[-3,3.25]$, discretized in 61 points.

  * ii.

Setting B:

    * –

$\tilde{r}_{j}\sim Uniform(-2,2)^{2}$.

    * –

$y_{j}|r_{j}=(r_{j}^{0},r_{j}^{1})\sim 1.5(\tanh{(5\pi r_{j}^{0})}+\cos{(\pi
r_{j}^{0})})+1+\mathcal{N}(0,1)$.

    * –

$s_{i}\sim Uniform(-3,3.25)$.

    * –

$r|s,a\sim[\sin{(s+a)}+0.25\mathcal{N}(0,1),\sin{(s-a)}+0.25\mathcal{N}(0,1)]$.

    * –

$\mathcal{A}=[-3,3.25]$, discretized in 61 points.

    * –

$\mathcal{S}=[-3,3.25]$, discretized in 61 points.

  * iii.

Setting C:

    * –

$\tilde{r}_{j}\sim Uniform(-2,2)^{2}$.

    * –

$y_{j}|r_{j}=(r_{j}^{0},r_{j}^{1})\sim 1.5(\sin{(\pi r_{j}^{0})}+\sin{(\pi
r_{j}^{1})})+1+\mathcal{N}(0,1)$.

    * –

$s_{i}\sim Uniform(-3,3.25)$.

    * –

$r|s,a\sim[\sin{(s)}+0.25\mathcal{N}(0,1),\cos{(a)}+0.25\mathcal{N}(0,1)]$.

    * –

$\mathcal{A}=[-3,3.25]$, discretized in 61 points.

    * –

$\mathcal{S}=[-3,3.25]$, discretized in 61 points.

  * iv.

Setting D:

    * –

$\tilde{r}_{j}\sim Uniform(-2,2)^{5}$.

    * –

$y_{j}|r_{j}=(r_{j}^{0},r_{j}^{1},r_{j}^{2},r_{j}^{3},r_{j}^{4})\sim
1.5(\sin{(\pi r_{j}^{0})}+\sin{(\pi r_{j}^{1})}+\sin{(\pi
r_{j}^{2})}+\sin{(\pi r_{j}^{3})}+\sin{(\pi r_{j}^{4})})+1+\mathcal{N}(0,1)$.

    * –

$s_{i}\sim Uniform(-3,3.25)$.

    * –

$r|s,a\sim[\sin{(s)},\cos{(a)},\sin{(a)},\cos{(s)},\sin{(s)}+\cos{(s)}]+0.25\mathcal{N}(0,I_{5})$.

    * –

$\mathcal{A}=[-3,3.25]$, discretized in 61 points.

    * –

$\mathcal{S}=[-3,3.25]$, discretized in 61 points.

Kernels $k_{s}$, $k_{a}$ and $k_{r}$ were taken as RBF kernels for the CBMP-
UCB algorithm. For the implementation of the CME-UCB algorithm, the choice of
the kernels considered was motivated by Chowdhury et al. (2020): $k_{s}$ and
$k_{a}$ were taken as Mátern kernels, and $k_{r}$ as an RBF kernel.

For each setting, we considered both a matched and an unmatched scenario,
where $m=0$ and $m=50$ respectively. We run 100 trials with different random
seeds for the CME-UCB and CBMP-UCB algorithms for 100 rounds.

Figure 1 exhibits the results of the matched case, in terms of the means and
the $0.05$ quantiles. We observe that the mean of the cumulative reward of
both CBMP-UCB and CME-UCB is very similar in the one-dimensional intermediate
reward experiment (setting A), although the variance of CME-UCB seems smaller.
In contrast, CBMP-UCB outperforms CME-UCB in the other three settings
considered, where the dimension of the intermediate reward is higher. In
setting B and setting C, both algorithms present a similar variance, although
CBMP-UCB clearly outperforms CME-UCB. The variance of the cumulative reward of
both algorithms goes up in setting D, which is explained by the higher
complexity of the intermediate reward. Although the CBMP-UCB algorithm shows
slightly more variance, it still outperforms CME-UCB.

Figure 2 displays the results of the unmatched case, where $m=50$. The results
look very similar to those in Figure 1. The main difference between the
performance of the algorithms with $m=0$ and $m=50$ is shown in setting D, in
which additional information on the dependence between the five dimensional
intermediate reward and the ultimate reward is crucial, given the higher
complexity of the space of intermediate rewards. CBMP-UCB seems to be
benefiting more from the initial information on such dependence when $m=50$,
given that the difference in performance is greater in this scenario. It is
consistent with the fact that CBMP-UCB uses information of $D_{2}$ for the
variance estimates, in contrast to CME-UCB.

Please note that, in general, CBMP-UCB outperforms CME-UCB in the settings
considered where the intermediate reward is complex. Given the fact that only
CBMP-UCB factors in the uncertainty arising from $D_{2}$, the illustrations of
the two methods back the idea that considering uncertainties stemming from the
two data sets is desirable, at least when the behaviour of the intermediate
reward is complex enough. Considering uncertainty derived from $D_{2}$ does
not prove to be useful in Experiment A, where the potential exploration of the
one dimensional intermediate reward is limited by nature. Experiments B, C and
D show a significant improvement in the performance if taking into account the
uncertainty stemming from $D_{2}$. Such reasoning is consistent with the
theoretical background, as not considering the uncertainty inherited by the
lack of information on $f$ may imply a lack of exploration of regions of the
space of intermediate rewards, which may be especially detrimental when the
distribution of intermediate rewards is complex.

##  7 CONCLUSION

We have designed a novel algorithm CBMP-UCB for tackling the problem of
sequentially maximising the expectation of a function based on the ideas of
BayesIMP (Chau et al. (2021a)) and CGP-UCB (Krause and Ong (2011)), allowing
for contexts and unmatched data sets. In the experiments considered, the CBMP-
UCB has surpassed the overall performance of the baseline CME-UCB algorithm
(Chowdhury et al. (2020)), especially when the dimensionality of the
intermediate reward is raised. Such conclusion is consistent with the
theoretical background, as disregarding the uncertainty stemming from the lack
of information on $f$ may imply an insufficient exploration of the space of
intermediate rewards.

There are several lines of research that could naturally follow this work.
First of all, we have mentioned that hyperparameter optimization may be
conducted as often as desired, given the computational limitations. We
highlight that conducting hyperparameter optimization severely slows the
algorithm, especially when having abundant data. Trying to determine a
strategy on the rate of hyperparameter optimization could be addressed in
future work.

Furthermore, only the expectation of the ultimate reward has been considered
in the proposed framework. However, one may instead be interested in other
properties of the distribution of the ultimate reward, for which modeling the
full distribution via its embedding as a GP could be explored. Considering
several embeddings would potentially enable generalizing the methods so that
they allow for a nested sequence of intermediate rewards, which could be
explored in the future.

Lastly, we highlight that all the algorithms presented require the inversion
of data matrices. Similarly to most kernel methods, the algorithms suffer from
scalability issues inherited by this inversion. Such problem inevitably
manifests when the agent has interacted for many rounds with its environment
in the RCMAB framework. Large-scale approximations to kernel matrices are now
well established (Li et al. (2021)) and can be explore in our context. A deep
learning approach (Collier and Llorens (2018)) is another potential direction
following this work.

#### Acknowledgements

Diego Martinez-Taboada gratefully acknowledges the support provided by the
Barrie Foundation.

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# A Spectral Sequence for locally free isometric Lie group actions

Paweł Raźny  Institute of Mathematics
Faculty of Mathematics and Computer Science
Jagiellonian University in Cracow<EMAIL_ADDRESS>
###### Abstract.

We present a spectral sequence for free isometric Lie algebra actions (and
consequently locally free isometric Lie group actions) which relates the de
Rham cohomology of the manifold with the Lie algebra cohomology and basic
cohomology (intuitively the cohomology of the orbit space). In the process of
developing this sequence we introduce a new description of the de Rham
cohomology of manifolds with such actions which appears to be well suited to
this and similar problems. Finally, we provide some simple applications
generalizing the Wang long exact sequence to Lie algebra actions of low
codimension.

###### Key words and phrases:

Lie group actions; Lie algebra actions; foliations; basic cohomology;
isometries

###### 2010 Mathematics Subject Classification:

53C12

##  1\. Introduction

The main goal of this paper is to provide a spectral sequence for isometric
Lie group actions connecting the de Rham cohomology of the manifolds with the
Lie algebra cohomology (describing the cohomological behaviour in the
direction tangent to the orbits) and the basic cohomology (i.e. the cohomology
of the orbit space) of the foliation induced by the orbits (describing the
cohomological behaviour in the direction transverse to the orbits). This is
achieved by the following Theorem:

######  Theorem 1.1.

Let $(M^{n+s},g)$ be a compact manifold with an isometric locally free action
of an $s$-dimensional connected Lie group $G$. Then, there is a spectral
sequence $E^{p,q}_{r}$ with:

  * •

$E^{p,q}_{2}=H^{p}(M/\mathcal{F},H^{q}(\mathfrak{g}))$, where $\mathfrak{g}$
is the Lie algebra of $G$ and $\mathcal{F}$ is the foliation generated by the
fundamental vector fields of the action $\xi_{1},...,\xi_{s}$.

  * •

$E^{p,q}_{r}$ converges to $H^{\bullet}_{dR}(M)$.

The above theorem can be easily deduced from its version for Lie algebra
actions by simply passing to the Lie algebra action corresponding to the given
Lie group action.

######  Theorem 1.2.

Let $(M^{n+s},g)$ be a compact manifold with a Killing free action of an
$s$-dimensional Lie algebra $\mathfrak{g}$. Then, there is a spectral sequence
$E^{p,q}_{r}$ with:

  * •

$E^{p,q}_{2}=H^{p}(M/\mathcal{F},H^{q}(\mathfrak{g}))$, where $\mathcal{F}$ is
the foliation generated by the fundamental vector fields of the action
$\xi_{1},...,\xi_{s}$.

  * •

$E^{p,q}_{r}$ converges to $H^{\bullet}_{dR}(M)$.

These results can be viewed as a common generalization of the following
results:

  1. (1)

The special case of Leray-Serre spectral sequence for principal $G$-bundles
for a compact connected Lie group $G$. Here the orbit space is a manifold and
hence for a principal $G$-bundle $P$ the second page is simply
$H^{p}_{dR}(P/G,H^{q}(\mathfrak{g}))$ while the sequence converges to
$H^{\bullet}_{dR}(P)$.

  2. (2)

The Lie algebra version of the Lyndon-Hochschild-Serre spectral sequence for
$\mathfrak{h}$ a Lie algebra of a compact connected group $H$ and
$\mathfrak{g}$ an ideal in $\mathfrak{h}$. In this case the second page is
$H^{p}(\mathfrak{h}/\mathfrak{g},H^{q}(\mathfrak{g}))$ while the sequence
converges to $H^{\bullet}(\mathfrak{h})\cong H^{\bullet}_{dR}(H)$. Note that
for this spectral sequence to work the subgroup $G\subset H$ corresponding to
$\mathfrak{g}$ need not be compact and hence it can be applied to some cases
not covered by the previous point.

  3. (3)

The Gysin long exact sequence for flows of nowhere vanishing Killing vector
fields which was thoroughly studied in the case of Sasakian manifolds (cf.
[6]) and recently studied in the general case (cf. [16]) and even generalized
to $\mathbb{S}^{3}$-actions (cf. [15]). These sequences (under the additional
assumption that the action is locally free in the case of [15]) can be derived
from our spectral sequence in a standard way.

  4. (4)

The spectral sequence for almost $\mathcal{K}$-structures used in [14] to
generalize results from $K$-contact and Sasakian manifold to this setting. In
particular, the topological invariance of basic Betti numbers (cf. Theorem
7.4.14 from [6] for the Sasakian case), invariance under deformations of basic
Hodge numbers (cf. [13] for the Sasakian case), and the classification of
harmonic forms in terms of primitive basic harmonic forms (cf. Theorem 7.4.13
from [6] for the Sasakian case).

We prove Theorem 1.2 in several steps. Firstly, in section 3 we choose a new
metric $g^{\prime}$ on $M$ so that the action remains isometric and the metric
admits orthonormal fundamental vector fields $\\{\xi_{1},...,\xi_{s}\\}$.
Then, in section 4 we study the behaviour of the forms
$\\{\eta_{1},...,\eta_{s}\\}$ dual to $\\{\xi_{1},...,\xi_{s}\\}$ under the
exterior derivative (Proposition 4.1). This enables us to establish a new
cochain complex $\Omega_{\mathfrak{g}}^{\bullet}(M)\subset\Omega^{\bullet}(M)$
for such actions which is spanned by forms $\eta_{i_{1}}...\eta_{i_{q}}\alpha$
with $\alpha$ basic. This in turn allows us to easily compute the second page
of the spectral sequence arising from the restriction of the standard
filtration associated to a foliation (cf. [1, 2, 3, 17]) to
$\Omega_{\mathfrak{g}}^{\bullet}(M)$. Sections 5 and 6 finish the proof by
establishing the following theorem, which we believe to be of independent
interest.

######  Theorem 1.3.

Let $(M^{n+s},g)$ be a compact manifold with a Killing free action of an
$s$-dimensional Lie algebra $\mathfrak{g}$. Then, under the above notation,
the inclusion $\Omega_{\mathfrak{g}}^{\bullet}(M)\subset\Omega^{\bullet}(M)$
induces an isomorphism in cohomology.

In the final section of the paper we give some applications of our spectral
sequence to low codimension actions, arriving at long exact sequences
generalizing the Wang long exact sequence.

Acknowledgements The author would like to thank Professor Martin Saralegui-
Aranguren for his insightful remarks and sharing details of his work on
related topics.

##  2\. Preliminaries

###  2.1. Foliations

We provide a quick review of transverse structures on foliations.

######  Definition 2.1.

A codimension $q$ foliation $\mathcal{F}$ on a smooth $n$-manifold $M$ is
given by the following data:

  * •

An open cover $\mathcal{U}:=\\{U_{i}\\}_{i\in I}$ of M.

  * •

A q-dimensional smooth manifold $T_{0}$.

  * •

For each $U_{i}\in\mathcal{U}$ a submersion $f_{i}:U_{i}\rightarrow T_{0}$
with connected fibers (these fibers are called plaques).

  * •

For all intersections $U_{i}\cap U_{j}\neq\emptyset$ a local diffeomorphism
$\gamma_{ij}$ of $T_{0}$ such that $f_{j}=\gamma_{ij}\circ f_{i}$

The last condition ensures that plaques glue nicely to form a partition of $M$
consisting of submanifolds of $M$ of codimension $q$. This partition is called
a foliation $\mathcal{F}$ of $M$ and the elements of this partition are called
leaves of $\mathcal{F}$.

We call $T=\coprod\limits_{U_{i}\in\mathcal{U}}f_{i}(U_{i})$ the transverse
manifold of $\mathcal{F}$. The local diffeomorphisms $\gamma_{ij}$ generate a
pseudogroup $\Gamma$ of transformations on $T$ (called the holonomy
pseudogroup). The space of leaves $M/\mathcal{F}$ of the foliation
$\mathcal{F}$ can be identified with $T/\Gamma$.

######  Definition 2.2.

A smooth form $\omega$ on $M$ is called transverse if for any vector field
$X\in\Gamma(T\mathcal{F})$ (where $T\mathcal{F}$ denotes the bundle tangent to
the leaves of $\mathcal{F}$) it satisfies $\iota_{X}\omega=0$. Moreover, if
additionaly $\iota_{X}d\omega=0$ holds for all $X$ tangent to the leaves of
$\mathcal{F}$ then $\omega$ is said to be basic. Basic $0$-forms will be
called basic functions henceforth.

Basic forms are in one to one correspondence with $\Gamma$-invariant smooth
forms on $T$. It is clear that $d\omega$ is basic for any basic form $\omega$.
Hence, the set of basic forms of $\mathcal{F}$ (denoted
$\Omega^{\bullet}(M/\mathcal{F})$) is a subcomplex of the de Rham complex of
$M$. We define the basic cohomology of $\mathcal{F}$ to be the cohomology of
this subcomplex and denote it by $H^{\bullet}(M/\mathcal{F})$. A transverse
structure to $\mathcal{F}$ is a $\Gamma$-invariant structure on $T$. Among
such structures the following is most relevant to our work:

######  Definition 2.3.

$\mathcal{F}$ is said to be Riemannian if $T$ has a $\Gamma$-invariant
Riemannian metric. This is equivalent to the existence of a Riemannian metric
$g$ (called the transverse Riemannian metric) on
$N\mathcal{F}:=TM/T\mathcal{F}$ with $\mathcal{L}_{X}g=0$ for all vector
fields $X$ tangent to the leaves.

This structure enables the construction of a transverse version of Hodge
theory (see [8]). Firstly, we recall a special class of Riemannian foliations
on which the aforementioned theory is greatly simplified:

######  Definition 2.4.

A codimension $q$ foliation $\mathcal{F}$ on a compact connected manifold $M$
is called homologically orientable if $H^{q}(M/\mathcal{F})=\mathbb{R}$. A
foliation $\mathcal{F}$ on a manifold $M$ is called homologically orientable
if its restriction to each connected component of $M$ is homologically
orientable.

We will see in the final section that there is a large class of foliations
considered in this paper which are homologically orientable and hence we shall
restrict our attention to this case throughout the rest of this subsection.
Let $\mathcal{F}$ be a homologically orientable Riemannian foliation on a
compact manifold $M$. One can use the transverse Riemannian metric to define
the basic Hodge star operator
$*_{b}:\Omega^{k}(M/\mathcal{F})\to\Omega^{q-k}(M/\mathcal{F})$ pointwise.
This in turn allows us to define the basic adjoint operator:

$\delta_{b}=(-1)^{q(r+1)+1}*_{b}d*_{b}.$

######  Remark 2.5.

While we choose this to be the definition of $\delta_{b}$, it is in fact an
adjoint of $d$ with respect to an appropriate inner product on forms induced
by the transverse metric $g$. However, the definition of this inner product is
quite involved and not necessary for our purpose. Although, we shall state
some of the classical results of basic Hodge theory which use this inner
product in their proof. For details see [8].

Using $\delta_{b}$ we can define the basic Laplace operator via:

$\Delta_{b}=d\delta_{b}+\delta_{b}d.$

As it turns out this operator has some nice properties similar to that of the
classical Laplace operator. In particular, it is transversely elliptic in the
following sense:

######  Definition 2.6.

A basic differential operator of order $m$ is a linear map
$D:\Omega^{\bullet}(M/\mathcal{F})\rightarrow\Omega^{\bullet}(M/\mathcal{F})$
such that in local coordinates $(x_{1},...,x_{p},y_{1},...,y_{q})$ (where
$x_{i}$ are leaf-wise coordinates and $y_{j}$ are transverse ones) it has the
form:

$D=\sum\limits_{|s|\leq
m}a_{s}(y)\frac{\partial^{|s|}}{\partial^{s_{1}}y_{1}...\partial^{s_{q}}y_{q}}$

where $a_{s}$ are matrices of appropriate size with basic functions as
coefficients. A basic differential operator is called transversely elliptic if
its principal symbol is an isomorphism at all points of $x\in M$ and all non-
zero, transverse, cotangent vectors at x.

In particular, this implies the following important result from [8]:

######  Theorem 2.7.

Let $\mathcal{F}$ be a Riemannian homologically orientable foliation on a
compact manifold $M$. Then:

  1. (1)

$H^{\bullet}(M/\mathcal{F})$ is isomorphic to the space of basic harmonic
forms $Ker(\Delta_{b})$. In particular, it is finitely dimensional.

  2. (2)

The basic Hodge star induces an isomorphism between $H^{k}(M/\mathcal{F})$ and
$H^{q-k}(M/\mathcal{F})$ given by taking the class of the image through
$*_{b}$ of a harmonic representative.

We finish this section by recalling the spectral sequence of a Riemannian
foliation.

######  Definition 2.8.

We put:

$F^{k}_{\mathcal{F}}\Omega^{r}(M):=\\{\alpha\in\Omega^{r}(M)\text{ }|\text{
}\iota_{X_{r-k+1}}...\iota_{X_{1}}\alpha=0,\text{ for
}X_{1},...,X_{r-k+1}\in\Gamma(T\mathcal{F})\\}.$

An element of $F^{k}_{\mathcal{F}}\Omega^{r}(M)$ is called an $r$-differential
form of filtration $k$.

The definition above in fact gives a filtration of the de Rham complex. Hence,
via known theory from homological algebra we can cosntruct a spectral sequence
as follows:

  1. (1)

The $0$-th page is given by
$E_{0}^{p,q}=F_{\mathcal{F}}^{p}\Omega^{p+q}(M)/F_{\mathcal{F}}^{p+1}\Omega^{p+q}(M)$
and $d^{p,q}_{0}:E_{0}^{p,q}\to E^{p,q+1}_{0}$ is simply the morphism induced
by $d$.

  2. (2)

The $r$-th page is given inductively by:

$E_{r}^{p,q}:=Ker(d^{p,q}_{r-1})/Im(d^{p-r,q+r-1}_{r-1})=\frac{\\{\alpha\in
F^{p}_{\mathcal{F}}\Omega^{p+q}(M)\text{ }|\text{ }d\alpha\in
F_{\mathcal{F}}^{p+r}\Omega^{p+q+1}(M)\\}}{F^{p+1}_{\mathcal{F}}\Omega^{p+q}(M)+d(F_{\mathcal{F}}^{p-r+1}\Omega^{p+q-1}(M))}$

  3. (3)

The $r$-th coboundary operator $d_{r}:E_{r}^{p,q}\to E^{p+r,q-r+1}_{r}$ is
again just the map induced by $d$ (due to the description of the $r$-th page
this has the target specified above and is well defined).

Furthermore, since the filtration is bounded this spectral sequence converges
and its final page is isomorphic to the cohomology of the cochain complex (in
this case the de Rham cohomology of $M$).

######  Remark 2.9.

The above spectral sequence can be thought of as a generalization of the
Leray-Serre spectral sequence in de Rham cohomology to arbitrary Riemannian
foliations (as opposed to fiber bundles).

###  2.2. Lie algebras and Lie groups

In this subsection we give a brief overview of properties of Lie groups, Lie
algebras and their actions which will be useful in the rest of the article.
Throghout this section let $G$ be an $s$-dimensional Lie group and let
$\mathfrak{g}$ be the Lie algebra of left invariant vector fields on $G$ which
can be classically identified with $T_{e}G$ (where $e\in G$ denotes the
identity element). Choose a basis $\\{\tilde{\xi}_{1},...,\tilde{\xi}_{s}\\}$
of $\mathfrak{g}$ and let $\\{Z_{1},...,Z_{s}\\}$ be a set of right invariant
vector fields on $G$ such that $(Z_{i})_{e}=(\tilde{\xi}_{i})_{e}$ then it is
well known that:

  1. (1)

$Z_{i}$ are the fundamental vector fields of the left action of $G$ on itself,

  2. (2)

$[\tilde{\xi}_{i},\tilde{\xi}_{j}]_{e}=-[Z_{i},Z_{j}]_{e}$ for all
$i,j\in\\{1,...,s\\}$,

  3. (3)

$[\tilde{\xi}_{i},Z_{j}]=0$ for all $i,j\in\\{1,...,s\\}$.

In particular, this justifies the following convention:

######  Definition 2.10.

We say that an $\mathbb{R}$-linear map $A:\mathfrak{g}\to\Gamma(TM)$ is an
action of the Lie algebra $\mathfrak{g}$ on a smooth manifold $M$ if
$A([\tilde{\xi},\tilde{\zeta}])=-[A(\tilde{\xi}),A(\tilde{\zeta})]$ for all
$\tilde{\xi},\tilde{\zeta}\in\mathfrak{g}$. Given a basis
$\\{\tilde{\xi}_{1},...,\tilde{\xi}_{s}\\}$ of $\mathfrak{g}$ we say that the
vector fields $\\{A(\tilde{\xi}_{1}),...,A(\tilde{\xi}_{s})\\}$ are the
corresponding fundamental vector fields. We say that a Lie algebra action is
free if $\\{A(\tilde{\xi}_{1}),...,A(\tilde{\xi}_{s})\\}$ are linearly
independent at each point. We say that a Lie algebra action is Killing (or
isometric) if $\\{A(\tilde{\xi}_{1}),...,A(\tilde{\xi}_{s})\\}$ are Killing
vector fields.

######  Remark 2.11.

A Lie group action of $G$ on $M$ induces a Lie algebra action of
$\mathfrak{g}$ on $M$ with the same fundamental vector fields. This action is
free if and only if the corresponding Lie group action is locally free (i.e.
the isotropy groups of the action are discrete) and it is Killing if and only
if $G$ acts by isometries.

Assuming that $G$ is compact any orbit of a smooth action of $G$ on a manifold
$M$ is equivarianlty diffeomorphic to $G/G_{x}$ where $G_{x}$ is the isotropy
group at any point $x$ in the given orbit. Moreover, two orbits $Gx$ and $Gy$
are equivariantly diffeomorphic if and only if $G_{x}$ is conjugate to $G_{y}$
(in this case we say that the two orbits are of the same type; cf. [7]).
Taking $K_{i}$ to be the sum of all orbits of the given type we get a Whitney
stratification of $M$ (in the sense of [12]) with a finite number of strata.
Moreover, we can order the strata so that if $H_{i}\supset H_{j}$ are some
representatives of the conjugacy classes of subgroups of $G$ corresponding to
$K_{i}$ and $K_{j}$ then $i\geq j$. For later use we are going to need the
following property of tubular neighbourhoods of strata:

######  Proposition 2.12.

Given a strata $K_{i}$ of the action of a compact group $G$ on $M$ there
exists an invariant tubular neighbourhood $U$ of $K_{i}$ contained in
$N=\bigcup\limits^{i}_{j=1}K_{j}$ such that $N\backslash\overline{U}$ is
equivariantly diffeomorphic to $N\backslash K_{i}$.

###### Proof.

This is a simple consequence of the existence of invariant tubular
neighbourhoods applied to $K_{i}\subset N$. ∎

We are also going to need the folowing version of the Slice Theorem (cf. [7]):

######  Theorem 2.13.

Let $G$ be a compact Lie group acting on a manifold $M$. For each orbit $Gx$
there exists a tubular neighbourhood $U$ equivariantly diffeomorphic to
$G\times_{G_{x}}N_{x}(Gx)$ such that the image $S$ of
$\\{e\\}\times_{G_{x}}N_{x}(Gx)$ through this diffeomorphism is a
$G_{x}$-invariant manifold passing through $x$ (where we consider the normal
space $N_{x}(Gx)$ of the orbit of $x$ at $x$ with the action of $G_{x}$ given
by derivatives at $x$). In this case we say that $S$ is a slice through $x$
and that $U$ is a neighbourhood admitting a slice.

It is well known (cf. [11]) that the automorphism group $Aut(G)$ (with compact
open topology) of a connected Lie group is a Lie group itself. This group is
relevant to our work due to the following result which summarizes the
properties of dense subgroups in the case of interest to us.

######  Theorem 2.14 (Theorem 2.2.1 [9]).

Let $f:G\to L$ be an injective morphism of Lie groups such, that the image
$f(G)$ is dense. If the inner automorphism group $I(G)$ is a closed subgroup
of $Aut(G)$ then $L=\overline{f(G)}=\overline{f(Z(G))}\cdot f(G)$, where
$Z(G)$ denotes the center of $G$.

######  Remark 2.15.

In this paper we usually treat the group $G$ as a subgroup of its closure and
use the symbol $G$ for both the group with its subgroup and Lie group
topology. Hence, for example in the above Theorem we drop $f$ from the
assertion leaving $\overline{G}=\overline{Z(G)}\cdot G$. This abuse of
notation simplifies the notation somewhat without creating much confusion and
we will leave comments on the topology considered whenever required.

Finally, we wish to recall the construction of Lie algebra cohomology. The
cochain complex,
$(\bigwedge\text{}^{\bullet}\mathfrak{g}^{*},d_{\mathfrak{g}})$ where,

$d_{\mathfrak{g}}\alpha(X_{0},...,X_{n}):=\sum_{1\leq i<j\leq
n}(-1)^{i+j}\alpha([X_{i},X_{j}],X_{0},...,\hat{X}_{i},...,\hat{X}_{j},...,X_{n}),$

is called the Chevalley–Eilenberg complex and its cohomology
$H^{\bullet}(\mathfrak{g})$ the Lie algebra cohomology of $\mathfrak{g}$.
While this cohomology theory has found many applications throughout
mathematics its most important property for our purpose is the following
string of isomorphisms (for a compact connected Lie group $G$):

$H^{\bullet}_{dR}(G)\cong H^{\bullet}_{G}(G)\cong H^{\bullet}(\mathfrak{g}),$

where the middle term is the cohomology of the cochain complex
$\Omega_{G}^{\bullet}(G)$ of left invariant forms on $G$. The first
isomorphism is induced simply by the inclusion of forms and can be proven by
finding a cochain map that induces the inverse mapping in cohomology
(averaging a form over $G$). The second isomorphism follows simply from
comparing $d_{\mathfrak{g}}$ with the exterior derivative $d$ restricted to
left invariant forms.

##  3\. Free Killing Lie algebra actions

Let us assume that $(M^{n+s},g)$ is a compact Riemannian manifold with a free
Killing Lie algebra action of an $s$-dimensional Lie algebra $\mathfrak{g}$
with a choosen basis $\tilde{\xi}_{1},...,\tilde{\xi}_{s}$ and corresponding
fundamental vector fields $\xi_{1},...,\xi_{s}$. In this section we will show
how to construct a new metric $g^{\prime}$ on $M$ so that the action of
$\mathfrak{g}$ admits orthonormal fundamental fields
$\\{\xi^{\prime}_{1},...,\xi^{\prime}_{s}\\}$ and remains a Killing action.
Firstly, let us notice that the setup implies that:

  1. (1)

The splitting $TM:=T\mathcal{F}\oplus T\mathcal{F}^{\perp}$ is preserved by
the corresponding group action which in turn implies that it is preserved by
the operation $[\xi_{i},\bullet]$.

  2. (2)

The above splitting induces a splitting of $g$ into the transverse part
$g_{\perp}$ and the tangent part $g_{T}$.

  3. (3)

The transverse part is well known to induce a transverse Riemannian structure
for the corresponding foliation and hence $\mathcal{L}_{\xi_{i}}g_{\perp}=0$.

  4. (4)

From the previous point we conclude that
$\mathcal{L}_{\xi_{i}}g_{T}=\mathcal{L}_{\xi_{i}}(g_{\perp}+g_{T})=\mathcal{L}_{\xi_{i}}g=0$.

Since the action of $\mathfrak{g}$ is Killing the subgroup $G\subset Diff(M)$
generated by the flows of $\\{\xi_{1},...,\xi_{s}\\}$ is contained in the
group of isometries $Isom(M,g)$ which is well known to be a compact (finitely
dimensional) Lie group and hence, the closure $\overline{G}$ of $G$ in
$Diff(M)$ is compact as well. This in turn implies that there is a bi-
invariant metric $\tilde{g}^{\prime}$ on $G$ given by the restriction of a bi-
invariant metric on $\overline{G}$ (which exists by compactness). Put
$\tilde{g}^{\prime}_{ij}:=\tilde{g}^{\prime}(\tilde{\xi}_{i},\tilde{\xi}_{j})$.
This induces a new metric $g^{\prime}_{T}$ on $T\mathcal{F}$ defined by the
formula $g^{\prime}_{T}(\xi_{i},\xi_{j})=\tilde{g}^{\prime}_{ij}$. Finally, we
put $g^{\prime}:=g_{\perp}+g^{\prime}_{T}$.

######  Lemma 3.1.

The vector fields $\\{\xi_{1},...,\xi_{s}\\}$ are Killing with respect to
$g^{\prime}$.

###### Proof.

We consider the tensor:

$(\mathcal{L}_{\xi_{i}}g^{\prime})(X_{1},X_{2}):=\mathcal{L}_{\xi_{i}}g^{\prime}(X_{1},X_{2})-g^{\prime}([\xi_{i},X_{1}],X_{2})-g^{\prime}(X_{1},[\xi_{i},X_{2}]).$

Firstly, let us assume that $X_{1},X_{2}\in T\mathcal{F}^{\perp}$. In this
case the brackets in the expression above are also elements of
$T\mathcal{F}^{\perp}$ and hence we see that
$(\mathcal{L}_{\xi_{i}}g^{\prime})(X_{1},X_{2})=(\mathcal{L}_{\xi_{i}}g_{\perp})(X_{1},X_{2})=0$.
Secondly, for $X_{1},X_{2}\in T\mathcal{F}$ a similar argument shows that
$(\mathcal{L}_{\xi_{i}}g^{\prime})(X_{1},X_{2})=(\mathcal{L}_{\xi_{i}}g^{\prime}_{T})(X_{1},X_{2})=0$
(where the second equality follows since the given metric restricted to orbits
is induced by the bi-invariant metric on $G$).
Finally, for $X_{1}\in T\mathcal{F}^{\perp}$ and $X_{2}\in T\mathcal{F}$ each
term on the right hand side vanishes. ∎

Finally, we find vector fields $\\{\xi^{\prime}_{1},...,\xi^{\prime}_{s}\\}$
by taking the fundamental fields corresponding to a basis
$\\{\tilde{\xi}^{\prime}_{1},...,\tilde{\xi}^{\prime}_{s}\\}$ of
$\mathfrak{g}$ orthogonal with respect to $\tilde{g}^{\prime}$.

######  Remark 3.2.

Throughout the paper we will assume at various points that the action of
$\mathfrak{g}$ admits an orthonormal frame as above. However, it is important
to note that via applying apriori the above construction our spectral sequence
can be constructed for an arbitrary free Killing Lie algebra action.

##  4\. The spectral sequence of a free Killing Lie algbera action

Throuoghout this section let $(M,g)$ be an $(n+s)$-dimensional compact
Riemannian manifold with a free Killing action of a Lie algebra $\mathfrak{g}$
with orthonormal fundamental vector fields $\xi_{1},...,\xi_{s}$. Let us note
that the decomposition $TM=T\mathcal{F}\oplus T\mathcal{F}^{\perp}$ induces a
decomposition of the cotangent bundle:

$T^{*}M=Ann(T\mathcal{F})\oplus Ann(T\mathcal{F}^{\perp})\cong
T^{*}\mathcal{F}^{\perp}\oplus T^{*}\mathcal{F},$

where $Ann(E_{1})\subset E_{2}^{*}$ denotes the annihilator of the vector
subbundle $E_{1}\subset E_{2}$. This in turn induces a bigrading on
differential forms:

$\Omega^{r}(M)=\bigoplus\limits_{p+q=r}(\bigwedge\text{}^{p}Ann(T\mathcal{F})\otimes\bigwedge\text{}^{q}Ann(T\mathcal{F}^{\perp})).$

Furthermore, note that $Ann(T\mathcal{F}^{\perp})$ is pointwise spanned by
forms $\eta_{i}:=g(\xi_{i},\bullet)$ which allows us to write a form $\alpha$
of bidegree $(p,q)$ as:

$\alpha=\sum_{1\leq i_{1}\leq...\leq i_{q}\leq
s}\eta_{i_{1}}...\eta_{i_{q}}\alpha_{i_{1}...i_{q}},$

for some transverse $p$-forms $\alpha_{i_{1}...i_{q}}$. We now wish to study
how the operator $d$ behaves with respect to this bigradation. To this end we
use the well known formula:

$(d\alpha)(X_{0},...,X_{r})=\sum_{i}(-1)^{i}\mathcal{L}_{X_{i}}\alpha(X_{0},...,\hat{X}_{i},..,X_{r})+\sum_{i<j}(-1)^{i+j}\alpha([X_{i}.X_{j}],X_{0},...,\hat{X}_{i},...,\hat{X}_{j},...,X_{r}).$

Using this formula we see that the operator $d$ splits into $4$ components:

$d=d^{(-1,2)}+d^{(0,1)}+d^{(1,0)}+d^{(2,-1)}.$

Moreover, due to involutivity of $T\mathcal{F}$ it can be easilly seen that
$d^{(-1,2)}=0$. It remains to show how the three remaining components act on
$\eta_{i}$:

######  Proposition 4.1.

Under the above notation:

  1. (1)

The cochain complex
$(\bigwedge^{\bullet}span(\eta_{1},...,\eta_{s}),d^{(0,1)})$ is isomorphic to
the Lie algebra cochain complex of $\mathfrak{g}$ (with $-d_{\mathfrak{g}}$).

  2. (2)

$d^{(1,0)}\eta_{i}=0$ for each $i\in\\{1,...,s\\}$.

  3. (3)

$d^{(2,-1)}\eta_{i}$ is basic for each $i\in\\{1,...,s\\}$.

###### Proof.

For the first point, let $\tilde{\xi}_{i}$ denote the vector in $\mathfrak{g}$
that maps to $\xi_{i}$ through the Lie algebra action. Due to our setup, the
Riemannian metric $g$ induces a scalar product $\tilde{g}$ on $\mathfrak{g}$
for which $\\{\tilde{\xi}_{1},...,\tilde{\xi}_{s}\\}$ is an orthonormal basis.
Put $\tilde{\eta}_{i}:=\tilde{g}(\tilde{\xi}_{i},\bullet)$. Then it is easy to
check that the algebra morphism $A$ defined by $A\eta_{i}=\tilde{\eta}_{i}$
induces the desired isomorphism of chain complexes.
For the second point, we choose a vector field
$X\in\Gamma(T\mathcal{F}^{\perp})$ and compute for any $i\in\\{1,...,s\\}$:

$(d\eta_{j})(X,\xi_{i})=\mathcal{L}_{X}\eta_{j}(\xi_{i})-\mathcal{L}_{\xi_{i}}\eta_{j}(X)-\eta_{j}([X,\xi_{i}])=\eta_{j}(\mathcal{L}_{\xi_{i}}X)=0.$

For the third part, we shall work locally around a point $x_{0}$. We start by
lifting the Lie algebra action to a Lie group action of the group $G$
generated by the flows of $\\{\xi_{1},...,\xi_{s}\\}$. Next we choose a small
open neigbourhood $U_{0}$ of $x_{0}$ and let $T\subset U_{0}$ be a disc
transverse to the orbits and passing through $x_{0}$. We also choose a small
neighbourhood $V$ of $e$ in $G$, such that $V\times T$ is diffeomorphic
through the action to a small neighbourhood $U\subset U_{0}$ of $x_{0}$. We
can now use this diffeomorphism to define vector fields $X_{1},...,X_{s}$ by
pushing forward the left-invariant vector fields restricted to $V$, while in
such coordinates $\xi_{1},...,\xi_{s}$ are pushforwards of right invariant
vector fields, which implies $\mathcal{L}_{X_{i}}\xi_{j}=0$ for all
$i,j\in\\{1,...,s\\}$. On the other hand, we get $\mathcal{L}_{X_{i}}g=0$,
since $\mathcal{L}_{\xi_{i}}g_{\perp}=0$ (for all $i\in\\{1,...,s\\}$) implies
$\mathcal{L}_{X_{i}}g_{\perp}=0$ (for all $i\in\\{1,...,s\\}$) and we have:

$0=\mathcal{L}_{X_{i}}g(\xi_{j},\xi_{k})=g([X_{j},\xi_{j}],\xi_{k})+g(\xi_{j},[X_{i},\xi_{k}])+(\mathcal{L}_{X_{i}}g)(\xi_{j},\xi_{k})=(\mathcal{L}_{X_{i}}g)(\xi_{j},\xi_{k}).$

This means that $\mathcal{L}_{X_{i}}\eta_{j}=0$ for $i,j\in\\{1,...,s\\}$,
since $\mathcal{L}_{X_{i}}g=0$ and $[X_{i},\xi_{j}]=0$. This in turn implies
that:

$\mathcal{L}_{X_{i}}d\eta_{j}=d\mathcal{L}_{X_{i}}\eta_{j}=0.$

However, due to the decomposition of forms and the second point we have:

$d^{(2,-1)}\eta_{j}=d\eta_{j}-d^{(0,1)}\eta_{j}.$

Using the first point and the above consideration we see that:

$\mathcal{L}_{X_{i}}d^{(2,-1)}\eta_{j}=\mathcal{L}_{X_{i}}(d\eta_{j}-d^{(0,1)}\eta_{j})=0.$

This in turn coupled with the transversality of $d^{(2,-1)}\eta_{j}$ proves
that it is in fact basic (when restricted to $U$). Since $x_{0}$ was arbitrary
this ends the proof. ∎

This has a useful corollary which is the first step to defining our spectral
sequence.

######  Corollary 4.2.

The graded vector subspace
$\Omega^{\bullet}_{\mathfrak{g}}(M)\subset\Omega^{\bullet}(M)$ spanned by
differential forms of the form $\eta_{i_{1}}...\eta_{i_{k}}\alpha$ where
$\alpha$ is basic, is in fact a subcomplex of $\Omega^{\bullet}(M)$.

###### Proof.

This follows directly from the previous Proposition since it implies that in
the expression:

$d(\eta_{i_{1}}...\eta_{i_{k}}\alpha)=\sum_{j}(-1)^{j-1}(d^{(0,1)}\eta_{i_{j}}+d^{(2,-1)}\eta_{i_{j}})\eta_{i_{1}}...\hat{\eta}_{i_{j}}...\eta_{i_{k}}\alpha+(-1)^{k}\eta_{i_{1}}...\eta_{i_{k}}d\alpha,$

every summand is an element of $\Omega^{\bullet}_{\mathfrak{g}}(M)$ whenever
$\eta_{i_{1}}...\eta_{i_{k}}\alpha\in\Omega^{\bullet}_{\mathfrak{g}}(M)$ (by
Proposition 4.1). ∎

With this we can now define a filtration on
$\Omega^{\bullet}_{\mathfrak{g}}(M)$ by restricting the filtration from
Definition 2.8 to such forms (in a fashion similar to [14]). In other words we
put:

$F^{k}_{\mathcal{F}}\Omega^{r}_{\mathfrak{g}}(M):=\\{\alpha\in\Omega^{r}_{\mathfrak{g}}(M)\text{
}|\text{ }\iota_{X_{r-k+1}}...\iota_{X_{1}}\alpha=0,\text{ for
}X_{1},...,X_{r-k+1}\in\Gamma(T\mathcal{F})\\}.$

In particular, we can again construct a spectral sequence as follows:

  1. (1)

The $0$-th page is given by
$E_{0}^{p,q}=F_{\mathcal{F}}^{p}\Omega^{p+q}_{\mathfrak{g}}(M)/F_{\mathcal{F}}^{p+1}\Omega^{p+q}_{\mathfrak{g}}(M)$
and $d^{p,q}_{0}:E_{0}^{p,q}\to E^{p,q+1}_{0}$ is simply the morphism induced
by $d$.

  2. (2)

The $r$-th page is given inductively by:

$E_{r}^{p,q}:=Ker(d^{p,q}_{r-1})/Im(d^{p-r,q+r-1}_{r-1})=\frac{\\{\alpha\in
F^{p}_{\mathcal{F}}\Omega^{p+q}_{\mathfrak{g}}(M)\text{ }|\text{ }d\alpha\in
F_{\mathcal{F}}^{p+r}\Omega^{p+q+1}_{\mathfrak{g}}(M)\\}}{F^{p+1}_{\mathcal{F}}\Omega^{p+q}_{\mathfrak{g}}(M)+d(F_{\mathcal{F}}^{p-r+1}\Omega^{p+q-1}_{\mathfrak{g}}(M))}$

  3. (3)

The $r$-th coboundary operator $d_{r}:E_{r}^{p,q}\to E^{p+r,q-r+1}_{r}$ is
again just the map induced by $d$.

Furthermore, since the filtration is bounded this spectral sequence converges
and its final page is isomorphic to the cohomology
$H^{\bullet}_{\mathfrak{g}}(M)$ of the cochain complex
$\Omega^{\bullet}_{\mathfrak{g}}(M)$. We shall show in the subsequent section
that this is in fact isomorphic to the de Rham cohomology of $M$. We end this
section by computing the second page of this spectral sequence.

######  Proposition 4.3.

Under the above notation $E_{2}^{p,q}\cong
H^{p}(M/\mathcal{F},H^{q}(\mathfrak{g}))$.

###### Proof.

By $1)$ of Proposition 4.1 the first page $E_{1}^{p,q}$ is equal to
$\Omega^{p}(M/\mathcal{F})\otimes H^{q}(\mathfrak{g})$. Using $2)$ of
Proposition 4.1 we see that $d_{1}^{p,q}$ is simply the operator $d$ on basic
forms with values in $H^{q}(\mathfrak{g})$. ∎

##  5\. A splitting of de Rham cohomology

Throughout this section let $(M^{n+s},g)$ be a Riemannian manifold on which
there is a free Killing Lie algebra action. Without loss of generality (see
Section 3) we can assume that this action admits orthonormal fundamental
fields $\\{\xi_{1},...,\xi_{s}\\}$. Denote by $G$ the subgroup of $Isom(M,g)$
generated by the flows of $\xi_{1},...,\xi_{s}$. The closure $\overline{G}$ of
$G$ in $Isom(M,g)$ is a finitely dimensional compact Lie group since it is a
closed subgroup of $Isom(M,g)$ which itself is well known to be finitely
dimensional and compact. This allows us to define a left invers
$a:\Omega^{\bullet}(M)\to\Omega_{\mathfrak{g}}^{\bullet}(M)$ to the inclusion
$i:\Omega_{\mathfrak{g}}^{\bullet}(M)\to\Omega^{\bullet}(M)$ by the formula:

$a(\eta_{i_{1}}...\eta_{i_{k}}\alpha):=\eta_{i_{1}}...\eta_{i_{k}}\tilde{a}(\alpha),$

where $\tilde{a}(\alpha)$ denotes the averaging of the form $\alpha$ over the
action of $\overline{G}$. We denote the kernel of $a$ by
$\Omega_{0}^{\bullet}(M)$.

######  Proposition 5.1.

Under the above notation $a$ is a chain map inducing a splitting of the chain
complex:

$\Omega^{\bullet}(M)=\Omega_{\mathfrak{g}}^{\bullet}(M)\oplus\Omega_{0}^{\bullet}(M),$

and consequently in cohomology:

$H_{dR}^{\bullet}(M)=H_{\mathfrak{g}}^{\bullet}(M)\oplus H_{0}^{\bullet}(M).$

###### Proof.

Let us start by noting that this is in fact well defined as the averaging of a
transverse form remains transverse (since the action of $\overline{G}$
preserves $T\mathcal{F}$ due to the fact that the action of $G$ preserves
$T\mathcal{F}$) while being basic follows by invariance of the resulting form
with respect to the action of $\overline{G}$ and hence $G$.
Secondly, let us prove that this is in fact a cochain map using Proposition
4.1:

$\displaystyle da(\eta_{i_{1}}...\eta_{i_{k}}\alpha)$ $\displaystyle=$
$\displaystyle\sum_{l=1}^{k}(-1)^{l-1}\eta_{i_{1}}...\hat{\eta}_{i_{l}}...\eta_{i_{k}}d\eta_{i_{l}}\tilde{a}(\alpha)+(-1)^{k}\eta_{i_{1}}...\eta_{i_{k}}d\tilde{a}(\alpha)$
$\displaystyle=$
$\displaystyle\sum_{l=1}^{k}(-1)^{l-1}\eta_{i_{1}}...\hat{\eta}_{i_{l}}...\eta_{i_{k}}(d^{(0,1)}\eta_{i_{l}}+d^{(2,-1)}\eta_{i_{l}})\tilde{a}(\alpha)+(-1)^{k}\eta_{i_{1}}...\eta_{i_{k}}d\tilde{a}(\alpha)$
$\displaystyle=$
$\displaystyle\sum_{l=1}^{k}(-1)^{l-1}\eta_{i_{1}}...\hat{\eta}_{i_{l}}...\eta_{i_{k}}(d^{(0,1)}\eta_{i_{l}}\tilde{a}(\alpha)+\tilde{a}((d^{(2,-1)}\eta_{i_{l}})\alpha))+(-1)^{k}\eta_{i_{1}}...\eta_{i_{k}}\tilde{a}(d\alpha)$
$\displaystyle=$ $\displaystyle a(d(\eta_{i_{1}}...\eta_{i_{k}}\alpha)).$

The splitting of chain complexes now follows from the fact that $a$ is a left
inverse to $i$. ∎

The previous Proposition implies that the spectral sequence $E^{p,q}_{r}$
converges to $H_{dR}^{\bullet}(M)$ if and only if $H^{\bullet}_{0}(M)=0$.
Moreover, this splitting behaves well on $\overline{G}$-invariant forms:

######  Proposition 5.2.

Under the above notation the restriction of $a$ to $\overline{G}$-invariant
forms $\Omega^{\bullet}_{\overline{G}}(M)$ is a chain map inducing a splitting
of the chain complex:

$\Omega_{\overline{G}}^{\bullet}(M)=\Omega_{\mathfrak{g},\overline{G}}^{\bullet}(M)\oplus\Omega_{0,\overline{G}}^{\bullet}(M),$

and consequently in cohomology

$H_{dR}^{\bullet}(M)\cong
H_{\overline{G}}^{\bullet}(M)=H_{\mathfrak{g},\overline{G}}^{\bullet}(M)\oplus
H_{0,\overline{G}}^{\bullet}(M).$

In particular:

$H_{\mathfrak{g},\overline{G}}^{\bullet}(M)\cong
H_{\mathfrak{g}}^{\bullet}(M),\quad H_{0,\overline{G}}^{\bullet}(M)\cong
H_{0}^{\bullet}(M).$

###### Proof.

We start by noting that for each $i\in\\{1,...,s\\}$ the operator
$\mathcal{L}_{\xi_{i}}$ preserves the decomposition
$\Omega^{\bullet}(M)=\Omega_{\mathfrak{g}}^{\bullet}(M)\oplus\Omega_{0}^{\bullet}(M)$.
For forms in both $\Omega_{\mathfrak{g}}^{\bullet}(M)$ and
$\Omega_{0}^{\bullet}(M)$ this is a simple observation due to the formula
$\mathcal{L}_{\xi_{i}}=d\iota_{\xi_{i}}+\iota_{\xi_{i}}d$ and the fact that
both $d$ and $\iota_{\xi_{i}}$ preserve such forms. This implies that a form
is $\overline{G}$-invariant if and only if both its components with respect to
the splitting
$\Omega_{\mathfrak{g}}^{\bullet}(M)\oplus\Omega_{0}^{\bullet}(M)$ are
$\overline{G}$-invariant.
From the previous paragraph we can infer that $a$ takes
$\overline{G}$-invariant forms to $\overline{G}$-invariant forms since
$a(\alpha_{\mathfrak{g}}+\alpha_{0})=\alpha_{\mathfrak{g}}$ which is invariant
whenever $\alpha_{\mathfrak{g}}+\alpha_{0}$ is invariant. From this and the
fact that $a$ is a left inverse to $i$ (as chain maps) we have the splittings
of $\overline{G}$-invariant forms and cohomology stated in the Proposition.
Finally, the isomorphisms:

$H_{\mathfrak{g},\overline{G}}^{\bullet}(M)\cong
H_{\mathfrak{g}}^{\bullet}(M),\quad H_{0,\overline{G}}^{\bullet}(M)\cong
H_{0}^{\bullet}(M),$

are simply the morphisms induced by inclusions
$\Omega_{\mathfrak{g},\overline{G}}^{\bullet}(M)\subset\Omega_{\mathfrak{g}}^{\bullet}(M)$
and $\Omega_{0,\overline{G}}^{\bullet}(M)\subset\Omega_{0}^{\bullet}(M)$. It
is clear that these are isomorphisms since it is well known that the inclusion
of $\overline{G}$-invariant forms induces an isomorphism in cohomology
$H_{dR}^{\bullet}(M)\cong H_{\overline{G}}^{\bullet}(M)$ for compact connected
Lie groups. ∎

##  6\. Vanishing of $H^{\bullet}_{0}(M)$

We finish the proof of Theorem 1.3 (and consequently Theorem 1.2) by proving
that under its assumptions the group $H^{\bullet}_{0}$ is trivial. In order to
do this we note that $H_{0}^{\bullet}(U)$ is well defined for any saturated
open subset $U$ of $M$. Moreover, the splittings defined in the previous
sections also work for such a set $U$. We will use this in order to compute
$H_{0}^{\bullet}(M)$ applying the following version of Mayer-Vietoris Theorem:

######  Lemma 6.1.

Let $(M,g)$ be a compact Riemannian manifold with a free Killing action of
$\mathfrak{g}$. Moreover, let $U$ and $V$ be open saturated subsets. Then the
following short sequences of chain complexes is exact:

$0\to\Omega_{0}^{\bullet}(U\cup
V)\to\Omega_{0}^{\bullet}(U)\oplus\Omega_{0}^{\bullet}(V)\to\Omega_{0}^{\bullet}(U\cap
V)\to 0.$

###### Proof.

The only non-tirival part is the surjectivity of the last non-trivial chain
map. This is proved via the same argument as in the proof of the Mayer-
Vietoris Theorem in [5] after noting that we can construct a partition of
unity subordinate to $\\{U,V\\}$ consisting of basic functions (take any
partition of unity suboordinate to $\\{U,V\\}$ and average it over
$\overline{G}$). ∎

To apply this lemma effectively we first need to conduct some local
computation. We start with the following lemma:

######  Lemma 6.2.

Under the above notation, $\overline{G}=G\cdot\overline{Z(G)}$ (where the
latter group is the closure of the center of $G$ in $\overline{G}$). Moreover,
(with respect to any bi-invariant metric on $\overline{G}$)
$\mathfrak{g}^{\perp}$ is a Lie subalgebra of $\overline{\mathfrak{g}}$ (Lie
algebra of $\overline{G}$) corresponding to a group $A\subset\overline{Z(G)}$.

###### Proof.

We need to prove that the inner automorphisms $I(G)\subset Aut(G)$ form a
closed set in order to use Theorem 2.14. Let $\mathfrak{g}^{\perp}$ be the
subspace perpendicular to $\mathfrak{g}$ in $\overline{\mathfrak{g}}$ with
respect to a bi-invariant metric on $\overline{G}$. Then a left invariant
basis $\\{\tilde{\xi}_{1},...,\tilde{\xi}_{s}\\}$ of $\mathfrak{g}$ can be
extended to a basis of $\overline{\mathfrak{g}}$ using left invariant vector
fields $\\{\tilde{\mu}_{1},...,\tilde{\mu}_{l}\\}$ in $\mathfrak{g}^{\perp}$.
Moreover, such vector fields commute with each $\overline{\xi}_{i}$, since due
to the fact that the flows of these vector fields preserve the decomposition
$\mathfrak{g}^{\perp}\oplus\mathfrak{g}$, we have
$[\tilde{\mu}_{i},\tilde{\xi_{j}}]\in\mathfrak{g}\cap\mathfrak{g}^{\perp}=\\{0\\}$
and $[\tilde{\mu}_{i},\tilde{\mu_{j}}]\in\mathfrak{g}^{\perp}$. From this, we
can write every element of $\overline{G}$ as $ag$ with $g\in G$ and $a\in A$
such that $\overline{G}=G\cdot A$ and such that all elements in $A$ commute
with elements in $G$. We can conclude from this that $I(\mathfrak{g})\cong
I(G)$ is an embedded subgroup of $I(\overline{\mathfrak{g}})\cong
I(\overline{G})$ (since any element $\xi\in\mathfrak{g}$ acts trivially on
$\mathfrak{g}^{\perp}$ and hence
$e^{ad_{\xi}}|_{\mathfrak{g}^{\perp}}=Id_{\mathfrak{g}^{\perp}}$). Moreover,
the embedding $E:I(G)\to I(\overline{G})$ is in fact bijective (since
conjugation by $ag$ has to agree with conjugation by $g$ on $G$ and
consequently the same is true on $\overline{G}$). Hence, $I(G)$ is isomorphic
as Lie groups with $I(\overline{G})$ which implies that $I(G)$ is compact (in
particular, closed in $Aut(G)$). Hence, by Theorem 2.14 we have
$\overline{G}=G\cdot\overline{Z(G)}$. In particular, $A\subset\overline{Z(G)}$
since it commutes with all the elements of $G$. ∎

This enables us to provide the following result:

######  Proposition 6.3.

Let $(M,g)$ be a compact Riemannian manifold with a free Killing action of
$\mathfrak{g}$ and let $V$ be a saturated open subset admitting a slice
through some orbit contained in $V$. Then $H_{0}(V)=0$.

###### Proof.

It is sufficient to show that each class in $H^{*}_{dR}(V)$ has a
representative in $\Omega_{\mathfrak{g},\overline{G}}(V)$ due to Proposition
5.1. Since $V$ is a neighbourhood admitting a slice through some point $x\in
V$ its cohomology is the same as that of
$\overline{G}x\cong\overline{G}/\overline{G}_{x}$. Moreover, a form in
$\Omega_{\mathfrak{g},\overline{G}}(\overline{G}/\overline{G}_{x})$ can be
extended to a form in $\Omega_{\mathfrak{g},\overline{G}}(V)$ using the Slice
Theorem so that it represents the corresponding cohomology class.
Let $T\supset G$ denote a connected Lie group (with Lie algebra
$\mathfrak{t}$) which forms a transversal for the foliation induced by the
action of $\overline{G}_{x}$ on $\overline{G}$ (with Lie algebras respectively
$\overline{\mathfrak{g}}_{x}$ and $\overline{\mathfrak{g}}$) formed by taking
the Lie group corresponding to $\mathfrak{g}\oplus
span(\tilde{\mu}_{i_{1}},...,\tilde{\mu}_{i_{r}})$ for an appropriate subset
$\\{\tilde{\mu}_{i_{1}},...,\tilde{\mu}_{i_{r}}\\}$ of a basis of
$\mathfrak{g}^{\perp}$. We consider $T$ with its Lie group topology (the
subgroup topology would also work for the argument). We shall show that every
cohomology class in $H^{\bullet}_{dR}(\overline{G}/\overline{G}_{x})$ has a
representative that lifts to a $T$-bi-invariant form on $T$. Firstly, let us
note that $(\overline{G}_{x})_{0}$ is abelian since if there where any
nonvanishing bracket of elements from $\overline{\mathfrak{g}}_{x}$ it would
have to be contained (by Lemma 6.2) in $\mathfrak{g}\subset\mathfrak{t}$ which
gives a contradiction since $\mathfrak{t}$ and $\overline{\mathfrak{g}}_{x}$
meet transversaly (and are of complementary dimensions). We put,
$\Gamma_{T}:=T\cap(\overline{G}_{x})_{0}\cap\overline{Z(G)}\subset T$ and show
that $T/\Gamma_{T}$ is a compact Lie group. To this end we first need to show
that $(\overline{G}_{x})_{0}\cap T/\Gamma_{T}$ is finite. To see this assume
to the contrary this is not the case and consider the compact group
$T/(T\cap\overline{Z(G)})\cong\overline{G}/\overline{Z(G)}$ (the natural
mapping between the two is well known to be a continuous bijection and
computing the differential at identity proves that it is in fact an
isomorphism of Lie groups). Consequently, the space:

$T/(T\cap\overline{Z(G)})\bigg{/}((\overline{G}_{x})_{0}\cap
T)/\Gamma_{T}\cong\overline{G}/\overline{Z(G)}\bigg{/}(\overline{G}_{x})_{0}/((\overline{G}_{x})_{0}\cap\overline{Z(G)}),$

has to be a manifold (due to the right side) despite being covered by a
compact manifold with infinite fiber (left side). Hence, we get our desired
contradiction.
Since, $T$ covers $\overline{G}/(\overline{G}_{x})_{0}$ and the group
$(\overline{G}_{x})_{0}\cap T/\Gamma_{T}$ is finite, we conclude that
$T/\Gamma_{T}$ is a finite cover of:

$\overline{G}/(\overline{G}_{x})_{0}\cong T/((\overline{G_{x}})_{0}\cap
T)\cong(T/\Gamma_{T})/((\overline{G}_{x})_{0}\cap T/\Gamma_{T}),$

where again we can conclude that the natural mappings between the above spaces
are diffeomorphism since they are bijective continuous and the differential at
identity (and hence at any other point by translation by elements of $T$ is an
isomorphism). Hence, due to the compactness of the later space $T/\Gamma_{T}$
is compact itself. Hence:

$T/\Gamma_{T}\to T/((\overline{G_{x}})_{0}\cap T)\to T/(\overline{G_{x}}\cap
T)\cong\overline{G}/\overline{G}_{x},$

forms a tower of finite coverings. Using that finite coverings induce
injections in de Rham cohomology as well as the fact that $T/\Gamma_{T}$ is a
compact connected Lie group with Lie algebra $\mathfrak{t}$ we deduce that to
each cohomology class on $\overline{G}/\overline{G}_{x}$ we can associate a
unique $T$-bi-invariant form on $T$. Finally, let us note that each such form
descends to a form on $\overline{G}/\overline{G}_{x}\cong
T/(\overline{G}_{x}\cap T)$ representing the initial cohomology class.
Using the previous paragraph, let $\alpha$ be a $T$-bi-invariant closed
$k$-form on $T$. Let $\\{X_{1},...,X_{s}\\}$ be a basis of left invariant
vector fields tangent to $G\subset T$ coinciding at identity with the basis
$\\{\overline{\xi}_{1},...,\overline{\xi}_{s}\\}$ of right invariant vector
fields. Put
$\overline{\eta}_{i}:=\pi^{*}\eta_{i}|_{\overline{G}x}=(\pi^{*}g_{T})(\overline{\xi}_{i},\bullet)$.
Note that $\mathcal{L}_{X_{j}}\overline{\eta}_{i}=0$ since $\pi^{*}g_{T}$ is
bi-invariant. This gives:

$0=\mathcal{L}_{X_{j}}\alpha=\mathcal{L}_{X_{j}}\alpha_{0}+\sum_{1\leq
i_{1}<...<i_{l}\leq
s}\overline{\eta}_{i_{1}}....\overline{\eta}_{i_{l}}\mathcal{L}_{X_{j}}\alpha_{i_{1},...,i_{l}}.$

But since $G$ is a dense subgroup (in $\overline{G}\supset T$) the bundle $TG$
tangent to the layers of $G$ is preserved by multiplication on both sides and
consequently $\mathcal{L}_{X_{j}}\alpha_{0}$ and
$\mathcal{L}_{X_{j}}\alpha_{i_{1},...,i_{l}}$ are again normal. This means
that the above equality holds only if $\mathcal{L}_{X_{j}}\alpha_{0}=0$ and
$\mathcal{L}_{X_{j}}\alpha_{i_{1},...,i_{l}}=0$. Since this holds for all
$X_{j}$ the forms $\alpha_{0}$ and $\alpha_{i_{1},...,i_{l}}$ are all basic
(with respect to the orbits of the left $G$-action). Consequently, if $\alpha$
represents a pullback of a form on $\overline{G}/\overline{G}_{x}$ then this
form belongs to $\Omega_{\mathfrak{g},\overline{G}}(\overline{G}x)$. By
extending this form to $V$ the proof is finished. ∎

With this we are able to prove the following Proposition which finishes the
proof of Theorems 1.3 and 1.2.

######  Proposition 6.4.

For any compact manifold $(M^{n+s},g)$ with a Killing free action of an
$s$-dimensional Lie algebra $\mathfrak{g}$, the vector spaces
$H_{0}^{\bullet}(M)$ are all trivial.

###### Proof.

We are going to proceed with the proof by induction on the number of strata.
More precisely, we prove the following statement by induction:

_Statement_. Let $N^{n+s}$ be a saturated open submanifold of some compact
manifold $M^{n+s}$ with a free Killing $\mathfrak{g}$-action. Moreover, let us
assume that $N$ is $\overline{G}$-equivariantly (where the closure is taken in
$Isom(M,g)$) diffeomorphic to an interior of a submanifold with boundary
$\tilde{N}$ of $M$. If, the action of $\overline{G}$ on $N$ has at most $k$
distinct strata then $H^{\bullet}_{0}(N)=0$.

For $k=1$ we can use the Slice Theorem to see that $N/\overline{G}$ is a
manifold. We can take $\mathcal{U}$ to be the cover of $N$ consisting of
preimages of sets from a finite good open cover of $N/\overline{G}$. This
implies that $\mathcal{U}$ is an open saturated cover of $N$ such that any
intersection of sets from that cover admits a slice. We can now use Lemma 6.1
and Proposition 6.3 to conclude that $H^{\bullet}_{0}(N)=0$ (by induction on
the cardinality of the cover).
For the induction step let us assume that the statement is true for all
positive integers up to $k$ and let us prove the statement for $k+1$. Let
$K_{k+1}$ be the final strata of $N$ (in particular the orbits in it are not
quotients of orbits from any other orbit type). Moreover, let us take some
equivariant tubular neighbourhood $U$ of $K_{k+1}$. It can be seen from the
induction hypothesis that $H^{\bullet}_{0}(K_{k+1})=0$ since, one can see
using the slice Theorem that $\overline{K}_{k+1}$ is an immersed (since it
might have intersections) compact submanifold in $M$ and consequently
$K_{k+1}$ satisfies the assumptions of the induction hypothesis (where the
place of $M$ is taken by a compact manifold which is the source of the
immersion with image $\overline{K}_{k+1}$). This implies that,
$H^{\bullet}_{0}(U)=0$ (since otherwise a restriction of the non-trivial class
from $U$ would be a nontrivial class on $K_{k+1}$).
We wish to now prove the same for $N\backslash K_{k+1}$. To this end note that
by induction hypothesis $H^{\bullet}_{0}(N\backslash\overline{U})=0$. On the
other hand, $N\backslash\overline{U}$ is equivariantly diffeomorphic by
Proposition 2.12 to $N\backslash K_{k+1}$ and consequently by applying Lemma
6.1 to $U$ and $N\backslash K_{k+1}$ we get $H^{\bullet}_{0}(N)=0$ (noting
that $H^{\bullet}_{0}(U\cap(N\backslash K_{k+1}))=0$, since $U\cap(N\backslash
K_{k+1})$ satsfies the assumptions of the induction hypothesis). ∎

##  7\. Applications to low codimension actions

In this section we provide some sample applications for our spectral sequence
for low codimension actions. These results for the most part can be thought of
as a generalization of the Wang long exact sequence. We start however with a
simple consequence of the spectral sequence in general (which is a
genralization of a similar result from [14]).

######  Proposition 7.1.

Let $(M^{n+s},g)$ be a compact orientable manifold with a Killing free action
of an $s$-dimensional Lie algebra $\mathfrak{g}$. Then the foliation induced
by the action is homologically orientable.

###### Proof.

It is well known (cf. [8]) that the top basic cohomology of a Riemannian
foliation on a compact manifold is either $0$ or $\mathbb{R}$. In this case it
cannot be $0$ since then we could compute from our spectral sequence that
$H^{n+s}_{dR}(M)\cong E^{n,s}_{2}=0$ which is a contradiction with the
orientability of $M$. Hence, the top basic cohomology is isomorphic to
$\mathbb{R}$ which means that the foliation is homologically orientable. ∎

######  Theorem 7.2.

Let $(M^{1+s},g)$ be a compact connected oriented Riemannian manifold with a
free Killing action of an $s$-dimensional Lie algebra $\mathfrak{g}$. Then:

$H^{\bullet}_{dR}(M)\cong H^{\bullet}(\mathfrak{g})\oplus
H^{\bullet-1}(\mathfrak{g}).$

In particular, no such action exists when $M$ is simply connected (or even
when $H^{1}_{dR}(M)=0$).

###### Proof.

Since $E^{0,q}_{2}$ and $E^{1,q}_{2}$ are the only non-zero columns on the
second page we conclude that the sequence degenerates at this page. Due to
Proposition 7.1 we can use the basic version of Poincaré duality to conclude
from $E_{2}^{p,q}\cong E^{p,q}_{\infty}$ that the cohomology of $M$ is as
prescribed. Moreover, $H^{0}(\mathfrak{g})\neq 0$ together with the above
splitting implies that $H^{1}_{dR}(M)\neq 0$. ∎

######  Theorem 7.3.

Let $(M^{2+s},g)$ be a compact connected oriented Riemannian manifold with a
free Killing action of an $s$-dimensional Lie algebra $\mathfrak{g}$ and
$H^{1}_{dR}(M)=0$. Then we have a long exact sequence:

$...\to H^{k}_{dR}(M)\to H^{k}(\mathfrak{g})\to H^{k-1}(\mathfrak{g})\to
H^{k+1}_{dR}(M)\to...$

.

###### Proof.

We use the well known fact that the morphism $i^{*}:H^{1}(M/\mathcal{F})\to
H^{1}_{dR}(M)$ induced by the inclusion of forms is a monomorphism to conclude
that $H^{1}(M/\mathcal{F})=0$ and hence only the $0$-th and the $2$-nd column
of $E_{2}^{p,q}$ are nonzero. Moreover, due to Proposition 7.1 we can use the
basic version of Poincaré duality to conclude that $E^{0,q}_{2}\cong
H^{q}(\mathfrak{g})$ and $E^{2,q}_{2}\cong H^{q}(\mathfrak{g})$. Hence, we can
use the derivative on the second page to arrive at the above long exact
sequence. ∎

######  Theorem 7.4.

Let $(M^{3+s},g)$ be a compact connected oriented Riemannian manifold with a
free Killing action of an $s$-dimensional Lie algebra $\mathfrak{g}$ and
$H^{1}_{dR}(M)=0$. Then we have a long exact sequence:

$...\to H^{k}_{dR}(M)\to H^{k}(\mathfrak{g})\to H^{k-2}(\mathfrak{g})\to
H^{k+1}_{dR}(M)\to...$

Moreover, $0\cong
H^{1}(\mathfrak{g})\cong\mathfrak{g}/[\mathfrak{g},\mathfrak{g}]$.

###### Proof.

We again use the well known fact that the morphism
$i^{*}:H^{1}(M/\mathcal{F})\to H^{1}_{dR}(M)$ induced by the inclusion of
forms is a monomorphism to conclude that $H^{1}(M/\mathcal{F})=0$. Moreover,
we can use Proposition 7.1 to conclude from the basic version of Poincaré
duality that $H^{2}(M/\mathcal{F})=0$ and hence, that $E^{0,q}_{2}\cong
H^{q}(\mathfrak{g})$ and $E^{3,q}_{2}\cong H^{q}(\mathfrak{g})$ are the only
non-zero columns. Hence, we can use the derivative on the third page to arrive
at the above long exact sequence. The final assertion follows from the fact
that there are no non-zero coboundary operators with source or target on
$E^{0,1}_{r}$ for $r\geq 2$ and consequently $H^{1}(\mathfrak{g})\cong
E^{0,1}_{2}\cong E^{0,1}_{\infty}\cong H^{1}_{dR}(M)=0$. ∎

######  Remark 7.5.

The assumption $0\cong\mathfrak{g}/[\mathfrak{g},\mathfrak{g}]$ is pretty
restrictive and prevents a lot of groups from acting locally freely and
isometricaly with codimension $3$ on compact orientable manifolds with trivial
first cohomology. Among these it is worth mentioning that $\mathbb{R}^{s}$
cannot act in such a way on such manifolds. This can be viewed as a
generalization of the fact that there are no nowhere vanishing vector fields
on simply connected compact orientable $4$-manifolds (which follows
immediately from the Euler characteristic).

It is also worth noting that some similar results can be achieved from the
study of our spectral sequence for actions of higher codimension if we make
additional assumptions on the vanishing of low degree cohomology of $M$ and
Lie algebra cohomology of $\mathfrak{g}$.

Statements and Declarations The authors declare no competing interests.

Availability of data and material Data sharing not applicable to this article
as no datasets were generated or analysed during the current study.

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