# Learning Bipedal Walking on a Quadruped Robot via Adversarial Motion Priors

Tianhu Peng1, Lingfan Bao1, Joseph Humphreys1, Andromachi Maria Delfaki2,
Dimitrios Kanoulas2 and Chengxu Zhou2* This work was supported by the Royal
Society [grant number RG\R2\232409] and the UKRI Future Leaders Fellowship
[grant number MR/V025333/1]. Please refer to the video for an overview of our
framework and results at https://youtu.be/JYD1RlrQRWM.For the purpose of open
access, the authors have applied a Creative Commons Attribution (CC BY)
licence to any Author Accepted Manuscript version arising from this
submission.1School of Mechanical Engineering, University of Leeds,
UK.2Department of Computer Science, University College London,
UK.*Corresponding author<EMAIL_ADDRESS>
###### Abstract

Previous studies have successfully demonstrated agile and robust locomotion in
challenging terrains for quadrupedal robots. However, the bipedal locomotion
mode for quadruped robots remains unverified. This paper explores the
adaptation of a learning framework originally designed for quadrupedal robots
to operate blind locomotion in biped mode. We leverage a framework that
incorporates Adversarial Motion Priors with a teacher-student policy to enable
imitation of a reference trajectory and navigation on tough terrain. Our work
involves transferring and evaluating a similar learning framework on a
quadruped robot in biped mode, aiming to achieve stable walking on both flat
and complicated terrains. Our simulation results demonstrate that the trained
policy enables the quadruped robot to navigate both flat and challenging
terrains, including stairs and uneven surfaces.

###### Index Terms:

Legged Robots, Bipedal Locomotion, Deep Reinforcement Learning, Adversarial
Motion Priors

##  I Introduction

Legged robots exhibit superior terrain adaptability compared to their wheeled
and tracked counterparts. Although quadrupedal robots are known for their
stability and agility, bipedal robots offer greater flexibility by freeing the
upper body for complex tasks. This flexibility suggests the potential for
quadrupedal robots to walk in a bipedal gait, using the rear legs for walking
and the front legs for manipulation.

The primary challenge in adapting quadruped robots for bipedal locomotion
stems from their mechanical design constraints. First, unlike typical bipedal
robots that have firm, flat feet, quadruped robots often feature soft, point-
contact feet that inherently lack stability. Second, the rear legs of
quadruped robots are not specifically designed for bipedal walking, their
limited range of motion and underactuation contribute to unnatural and
unstable bipedal gaits. This design mismatch explains why quadruped robots
struggle with bipedal walking modes. This leads to high requirements for
locomotion controllers during bipedal modes.

To achieve bipedal walking for quadruped robots, there are primarily two
approaches: the model-based method and the learning-based method [1]. Model-
based methods are based on highly accurate mathematical models, which have
proven to be effective in executing highly dynamic motions in both quadruped
and bipedal robots. However, these methods lack robustness and generalization
in unseen scenarios, largely due to the difficulty of accurately modeling
ground interactions and contact dynamics. In contrast learning-based methods,
reinforcement learning (RL), provides a more adaptable solution by enabling
the exploration of the robots’ full dynamics and interactions with the
environment, thus offering greater flexibility in controlling complex
locomotive behaviors. Early research in RL on legged robots primarily utilized
unrealistic models within physical simulators[2, 3, 4]. In transitioning to a
practical bipedal robot and learning natural and robust gaits, previous
studies have primarily designed a reference-free learning framework by
designing periodic composition reward [5] or mimicking predefined references
[6, 7, 8].

Reference-free methods explore various gait patterns efficiently, while
reference-based methods leverage prior knowledge to accelerate learning,
resulting in efficient policy exploration and robust locomotion skills. These
methods incorporate expert information and predefined reference trajectories
from motion capture data or trajectory optimization (TO). Generative
Adversarial Imitation Learning [2] and Adversarial motion priors (AMP) [8]
predict state transitions and evaluate the similarity between reference and
agent data, promoting stable gait maintenance. AMP was implemented with a
study of human reference behavior in biped robots [9, 10], combining it with
periodic rewards to promote stable gait maintenance.

Figure 1: Overview of the teacher-student learning framework. (a) The teacher
policy, which leverages privileged data $S^{p}_{t}$,terrain information
$o^{e}_{t}$ and prospective data $o^{p}_{t}$ through RL, aims to maximize a
total reward $r_{t}$ comprising command task reward $r_{t}^{g}$, style reward
$r_{t}^{g}$ based on AMP, and regulation reward $r_{t}^{g}$ for ensuring
safety and smooth motion. (b) The student policy, trained via supervised
learning, seeks to imitate the teacher’s actions $a^{teacher}_{t}$ and
reconstruct the teacher’s latent states $l^{teacher}_{t}$ from historical and
prospective observations $H_{t}:[O_{t-N},O_{t-N-1},...,O_{t-1}]$

To generate appropriate predefined references, several methods are utilized.
Motion capture technology is commonly employed to produce reference data for
various types of legged robots [10, 11, 12]. This technology captures
comprehensive kinematic data from real-world scenarios, enabling versatile
data collection that is not confined to a specific robot model. However,
adapting this data to different robotic platforms often requires a re-
targeting process, which increases both complexity and manual labor. On the
other hand, TO in reduced-order [13] and full dynamics models [7], has been
employed. This approach reduces complexity and eliminates the need for further
re-targeting, making it a more efficient solution for generating references.
Additionally, compared to full dynamics models, reduced-order models can
decrease the computational resources required for optimization and offer
greater generalization across all quadruped robots. Regarding predesigned
references from the optimization method, various models are utilized. Besides
HZD-based full dynamics reference [7], reduced-order dynamics such as Single
Rigid Body Dynamics (SRBD) [14] are also utilized in the training procedure.
Based on the reduced-order model, task-space learning focuses on the foot
setpoints and based velocity [15, 16].

Another significant challenge exists in bridging the sim-to-real gap. To
extend the robustness of locomotion and overcome the sim-to-real gap,
frameworks using the privileged learning paradigm have been introduced [17,
18]. By combining the strengths of AMP in reference-based learning and
privileged learning, there is potential to enable quadrupedal robots to adopt
bipedal walking modes. Similar frameworks have been introduced [11, 9], but
none have been validated on bipedal robots or quadruped robots for bipedal
mode.

Our objective is to train a policy that enables a quadrupedal robot to achieve
bipedal locomotion using only its two rear legs, thus freeing its front legs
for more complex tasks. This capability aims to enhance the robot’s
versatility and functionality in various practical applications. This paper
presents a novel framework that enables quadrupedal robots to achieve robust
and agile blind bipedal locomotion on flat terrain. We adopt a teacher-student
policy framework, where privileged information that the robot cannot directly
access is encoded. The student policy uses historical observation information
to infer this privileged information, thereby enhancing robustness.
Additionally, we integrate the AMP training framework to learn and imitate the
style behaviors of reference data generated through TO based on a SRBD model.
Different from previous work [19] with assistant devices, this comprehensive
training framework equips the policy to support agile bipedal motions in
quadrupedal robots.

In summary, the primary contribution of this paper is to develop a novel
framework (shown in Fig. 1) that allows quadrupedal robots to perform robust
and agile bipedal blind locomotion on flat terrain using only their rear legs.
This bipedal mode frees the front legs for more complex tasks, significantly
enhancing the robot’s versatility and functionality. Besides, We evaluate our
model on the A1 quadrupedal robot using a biped gait model in the Isaac Gym
simulation environment, demonstrating agile and robust movements.

##  II Methodology

###  II-A Reinforcement Learning on Legged Robots

The task of learning legged locomotion poses significant challenges due to the
complex environment and limitations in sensor data. To address this, a
partially observable Markov decision process (POMDP) framework was adopted,
denoted as $(s_{t},a_{t},P,r_{t},p_{0},\gamma)$, where $s_{t}$ represents the
state at time step $t$ ,$a_{t}$ is the action taken by the agent,
$P(s_{t+1}|s_{t},a_{t})$ describes the system dynamics, predicting the next
state based on the current state and action, , $r_{t}(s_{t},a_{t},s_{t+1})$ is
the reward function, quantifying the immediate benefit of taking action
$a_{t}$ in state $s_{t}$ leading to $s_{t+1}$, $p_{0}$ denotes the initial
state distribution, and $\gamma^{t}$ is the discount factor, determining the
importance of future rewards in the decision-making process. The objective of
RL in this context is to identify an optimal policy $\pi_{\theta}$
parameterized by $\theta$ , that maximizes the expected discounted return over
future trajectories. This is formalized by the objective function $J(\theta)$:

$\displaystyle
J(\theta)=\mathbb{E}_{\pi_{\theta}}[\sum^{\infty}_{t=0}\gamma^{t}r_{t}]$ (1)

In the state space, the state $s_{t}^{teacher}$ includes proprioceptive
observation $o_{t}^{p}\in\mathbb{R}^{48}$, privileged state
$s_{t}^{p}\in\mathbb{R}^{45}$ and terrain information
$o^{e}_{t}\in\mathbb{R}^{187}$. The proprioceptive observation
$o_{t}^{p}\in\mathbb{R}^{48}$ encompasses critical data such as the
orientation of the gravity vector, base linear and angular velocity in the
robot’s frame, joint positions and velocities, the previous action
$a_{t-1}\in\mathbb{R}^{12}$ executed by the current policy. The privileged
state $s_{t}^{p}$ contains the information include ground friction
coefficients, ground restitution coefficients, contact forces, external forces
within positions on the robot, and collision state. The privileged state
$s_{t}^{p}\in\mathbb{R}^{45}$ includes additional key details that the
physical robot cannot directly access in the real-world environment. This
information encompasses ground friction and restitution coefficients, contact
and external forces at specific robot positions, collision state information.
The terrain information $o^{e}_{t}\in\mathbb{R}^{187}$ contains the 187 height
measurement sampled from grid around robot base to the ground. In contrast,
the student policy state utilizes a sequence history of proprioceptive
observations $H_{t}:[O_{t-N},O_{t-N-1},...,O_{t}]$ to approximate the
privileged information. By learning from this historical data, the student
policy aims to imitate the inaccessible privileged state, enhancing its
decision-making capabilities in the absence of direct access to certain
environmental variables. Regarding the action space, the policy action $a_{t}$
is a 12 dimensional target joint position offset added to the time-invariant
nominal joint position. This specification guides the joint PD controller,
utilizing fixed gains to compute torque commands effectively for motor
position control.

###  II-B Adversarial Motion Priors and Rewards Design

The AMP framework utilizes adversarial learning to train two neural networks—a
generator and a discriminator—in a competitive setup. The generator produces
motion predictions for the robot, while the discriminator evaluates their
quality and realism. Style rewards are used to measure the similarity between
the demonstrator’s behavior and the robot’s, with higher similarity yielding
more rewards.

Using the reference dataset $D$, the AMP-based style reward function
encourages the robot to replicate the same gait style. According to [20], a
neural network-based discriminator $D_{\varphi}$ predicts whether a state
transition $(S_{t},S_{t+1})$ is from the dataset $D$ or generated by the agent
$A$. Each state $S_{t}^{\text{AMP}}\in\mathbb{R}^{31}$ includes joint
positions, point velocities, base linear velocity, base angular velocity, and
base height relative to the terrain. To avoid mode collapse, the dataset $D$
contains only trot gait motion clips.

The discriminator’s training objective includes a gradient penalty to enforce
smoothness and is defined as:

$\displaystyle\arg\min_{\varphi}$
$\displaystyle\mathbb{E}_{(s_{t},s_{t+1})\sim
D}[(D_{\varphi}(s_{t},s_{t+1})-1)^{2}]$ (2)
$\displaystyle+\mathbb{E}_{(s_{t},s_{t+1})\sim
A}[(D_{\varphi}(s_{t},s_{t+1})-1)^{2}]$
$\displaystyle+\frac{\alpha^{qp}}{2}\mathbb{E}_{(s_{t},s_{t+1})\sim
D}[\|\nabla_{\varphi}D_{\varphi}(s_{t},s_{t+1})\|_{2}^{2}],$

where $\alpha^{qp}$ is a manually specified coefficient ($\alpha^{qp}=10$).

The style reward is defined by:

$\displaystyle r_{t}^{s}[(s_{t},s_{t+1})\sim
A]=\mathbb{\max}[0,1-0.25(d^{score}_{t}-1)^{2}],$ (3)

where $d^{score}_{t}=D_{\varphi}(s_{t},s_{t+1})$ and is scaled to the range
$[0,1]$.

The overall reward function is:

$\displaystyle r_{t}=r_{t}^{g}+r_{t}^{s}+r_{t}^{l},$ (4)

where $r_{t}^{g}$ is the task reward, $r_{t}^{s}$ is the style reward, and
$r_{t}^{l}$ is the regularization reward. Task rewards typically include
tracking base linear and angular velocities:

$r_{t}^{g}=\omega^{v}\exp\left(-|\hat{v}_{t}^{xy}-v_{t}^{xy}|\right)+\omega^{\omega}\exp\left(-|\hat{\omega}_{t}^{z}-\omega_{t}^{z}|\right),$
(5)

where ${\omega}^{v}$ and ${\omega}^{\omega}$ are coefficients, and
$\hat{v}_{t}^{xy}$ and $\hat{\omega}_{t}^{z}$ are the velocity commands.

Regularization rewards promote safe, smooth motion and minimize energy costs,
enhancing the adaptability and efficiency of learned behaviors for real-world
applications. This component contributes to the robustness and effectiveness
of the overall motion.

###  II-C Reference Generation

In our research, we refine the imitation and learning of reference data for
precise bipedal locomotion. Using a single TO formulation from previous work
[13], we generate walking and running gaits for the A1 biped robot, focusing
on its back legs’ trajectories.

We streamline this process with TOWR (Trajectory Optimization for Walking
Robots) [21], which eliminates the need for manual tuning. TOWR generates
dynamically feasible, energy-efficient motions by optimizing smooth and stable
trajectories. To improve imitation fidelity, we integrate inverse kinematics
into TOWR, producing joint space data that closely mimics reference
trajectories.

Our generated trajectories encompass various locomotion patterns, including
forward walking and two distinct running gaits, each lasting 2.4 seconds.
Utilizing TO for motion dataset generation offers several advantages. Firstly,
it enables precise matching of the state space between the simulated agent and
demonstrator, leveraging kinematic dynamics models to refine trajectory
suitability. Moreover, this approach circumvents complexities associated with
other motion re-targeting techniques, ensuring a more seamless and accurate
replication of desired motions.

##  III Framework and Training

###  III-A Learning Framework

####  III-A1 Teacher Policy Architecture

The teacher policy $\pi_{\theta}^{\text{teacher}}$ is trained using Proximal
Policy Optimization [22] with the total reward $r_{t}$ as specified in Section
II-B. During training, the teacher performs a rollout in the environment to
generate a state transition $(s_{t}^{\text{AMP}},s_{t+1}^{\text{AMP}})$. This
state transition is then fed into the discriminator described in Section II-B.

The teacher policy consists of three Multilayer Perceptron (MLP) networks. Two
of these MLPs encode low-dimensional latent representations:
$l_{t}^{e}\in\mathbb{R}^{16}$ for terrain data and
$l_{t}^{p}\in\mathbb{R}^{8}$ for privileged data. Using two separate MLPs for
encoding helps mitigate information loss that often occurs during the
compression process, thereby preserving crucial and necessary information.

The preservation of essential data significantly aids the student policy in
reconstructing the latent representations, facilitating a more efficient and
accurate learning process. The third MLP acts as a low-level network,
utilizing the proprioceptive observation state along with the two encoded
latent representations to generate the teacher’s action in the environment.
The learning framework is shown in Fig.1.

####  III-A2 Student Policy Architecture

The student policy is designed to emulate the teacher policy, replicating
actions without relying on privileged state and terrain information.
Throughout the student training process, a supervised approach is employed,
minimizing two key losses: imitation loss and reconstruction loss. The
imitation loss ensures that the student policy closely mimic the action
$a_{t}^{teacher}\in\mathbb{R}^{12}$ dictated by the teacher’s policy.
Simultaneously, the reconstruction loss encourages the memory encoder within
the student’s policy to faithfully reproduce the latent representation
$l_{t}^{teacher}\in\mathbb{R}^{24}$ consists of the terrain latent
$l_{o}^{t}\in\mathbb{R}^{16}$ and privileged latent
$l_{o}^{t}\in\mathbb{R}^{8}$ employed by the teacher.

The overarching architecture comprises a memory encoder and a low-level MLP
[23] that maintains an identical structure to the teacher’s low-level network.
Memory encoders are implemented by stacking a sequence of 45 historical
observations information $H_{t}:[O_{t-N},O_{t-N-1},...,O_{t-1}]$ into the
input of an MLP.

###  III-B Training and Implementation Details

####  III-B1 Termination

The robot’s locomotion training is governed by specific termination conditions
designed to ensure safety and effectiveness. Episodes are terminated upon
detecting collisions involving the trunk, upper limbs, thighs, and calves,
prioritizing the study’s focus on bipedal locomotion with only two-foot ground
contact.

####  III-B2 Domain Randomization

To facilitate the transfer of learned behaviors from simulation to the real
world, domain randomization has been employed. This approach involves
randomizing various parameters crucial for robot locomotion, such as terrain
friction, base mass, joint PD controller gains, ground friction, restitution,
and perturbations to the robot’s base velocity. During training, sampled
velocity vectors are added to the robot’s current base velocity at random
intervals. The specific randomization variables and their corresponding
uniform distribution ranges are detailed in Table I, enabling robust policy
adaptation and testing in diverse real-world environments.

TABLE I: Randomized Simulation Parameters Parameters | Range | Unit
---|---|---
Link Mass | [0.8, 1.2] | kg
Payload Mass | [0, 3] | kg
Payload Range | [-3] | -
Ground Friction | [0.05, 2.75] | -
Ground Restitution | [0.0, 1.0] | -
Joint $K_{p}$ |  [0.8, 1.2] $\times$ 20 | -
Joint $K_{d}$ |  [0.8, 1.2] $\times$ 0.5 | -
Figure 2: Terrains in Isaac Gym Simulations. Figure 3: Base linear velocity in
the x direction and base angular velocity in yaw for the robot on both uniform
and discrete obstacle terrains.

####  III-B3 Simulation Setup

We trained 500 parallel agents on different types of terrains with increasing
difficulties using the Isaac Gym simulator [24] in overall 26000 iterations
and cost 15.88 hours in total. Fig 2 showed the simulation in different
terrain type. Each RL episode lasts for a maximum of 1000 steps, and
terminates early if it reaches the termination criteria. The control frequency
of the policy is 50 Hz in the simulation. All training’s were performed on a
single NVIDIA RTX 4070 GPU. In A1 locomotion scenarios, actions are
represented by $a_{t}\in R^{12}$ a 12-dimensional vector specifying the
desired positional adjustments for each actuated joint as dictated by the
Proportional-Derivative (PD) controller.

##  IV Results

In our experiments within the Isaac Gym simulation environment, we evaluated
the performance of the updated policy for a quadrupedal robot in biped
locomotion. The results, depicted in Table II, show varying success rates and
tracking accuracy across different terrains and speeds.

TABLE II: Comparison of tracking accuracy and success rate on different terrain and speed Terrain Types |  Uniform |  Wave |  Stepping Stones |  Sloped |  Stairs |  Obstacles
---|---|---|---|---|---|---
0.5 m/s | Acc (%) |  82.76 |  80.84 |  80.06 |  81.37 |  77.67 |  80.88
Succ (%) |  92.3 |  91.76 |  91.6 |  84.21 |  90.09 |  89.55
1.0 m/s | Acc (%) |  81.29 |  79.64 |  77.73 |  80.01 |  71.57 |  73.51
Succ (%) |  90 |  91.59 |  90.01 |  82.14 |  86.14 |  86.59
1.5 m/s | Acc (%) |  76.60 |  77.19 |  73.46 |  78.43 |  64.57 |  71.47
Succ (%) |  86.96 |  85.06 |  88.73 |  79.5 |  84.91 |  84.55
2.0 m/s | Acc (%) |  63.73 |  66.28 |  64.69 |  64.68 |  53.38 |  61.52
Succ (%) |  79.55 |  84.22 |  78.55 |  76.23 |  81.53 |  78.1

The robot generally performs well at lower speeds, with higher success rates
on less challenging terrains. However, success rates and tracking accuracy
decrease as speed and terrain difficulty increase, notably on sloped terrains
and obstacles due to mechanical design limitations.

The uniform terrain and discrete obstacle terrain were specifically chosen for
detailed analysis as they are highly representative of uneven terrains. The
uniform terrain has uneven features, while the discrete obstacle terrain
combines elements of both flat and stepped obstacles, akin to stairs,
providing a comprehensive challenge for evaluating the robot’s locomotion
policy.

Figure 4: Robot’s feet contact forces on both uniform and discrete obstacle
terrains. Figure 5: Snapshots illustrating the response of a robot subjected
to a 100N push force (indicated by a red arrow) applied along the x-axis over
a duration of 0.1 seconds. The sequence shows the robot’s movement and
stabilization process.

Fig. 3 presents the base linear velocity in the x direction and the base
angular velocity in yaw for the robot on both uniform and discrete obstacle
terrains. During the push test, noticeable changes in these velocities occur,
reflecting the robot’s dynamic response to the external force. The robot’s
base linear velocity tracking shows acceptable performance, with the measured
velocities closely following the commanded values. The base angular velocity
tracking, while not as accurate as the linear velocity, still demonstrates the
robot’s ability to follow the commanded trajectory to a reasonable extent. The
large spike observed around the 2-second mark is attributed to the external
push force. Despite this disturbance, the robot quickly regains stability,
indicating robust recovery capabilities of the locomotion policy. This ability
to reject disturbances is crucial for maintaining stable locomotion in
unpredictable environments.

The contact force data, shown in Fig. 4, provides additional insights into the
interaction between the robot’s feet and the terrain. The vertical ground
reaction forces $F_{z}$ were measured for both the rear-left (RL) and rear-
right (RR) legs on discrete obstacles and uniform terrains. The data reveals
asymmetry in the contact forces, indicating that the learned policy does not
exhibit a perfectly symmetrical gait.To address this issue and enhance the
robot’s performance, future work could incorporate a symmetrical reward
function or a reward based on the orientation of base tracking into the
learning algorithm.

The push test results and performance analysis provide additional insights
into the robot’s capabilities and resilience. As illustrated in Fig. 5, the
robot’s response to a 100N push applied along the x-axis over a duration of
0.1 seconds is captured, showcasing the policy’s effectiveness in handling
external disturbances. Despite the significant push, the robot manages to
stabilize itself quickly, demonstrating the robustness and resilience of the
developed locomotion policy.

The results indicate that the legged robot’s locomotion policy exhibits
satisfactory performance in both base linear velocity tracking and recovery
from external disturbances. The ability to maintain stability and follow
commanded trajectories on different terrains, as well as the efficient
recovery from disturbances, underscores the robustness of the developed
policy.

Overall, the experimental outcomes validate the effectiveness of our legged
locomotion policy in achieving stable, efficient, and resilient movement
across various terrains, even under external disturbances. These findings
contribute valuable knowledge towards the development of energy-efficient and
robust legged robots capable of operating in diverse and challenging
environments.

##  V Conclusion and Future work

In this paper, we proposed a novel framework for learning robust, agile, and
natural bipedal locomotion skills for quadruped robots in simulation.
Utilizing a teacher-student learning framework with privileged and terrain
information, we enhanced the robustness of the learned policy and helped
bridge the sim-to-real gap. By integrating adversarial motion imitation, the
learned gait mimics the style and behavior of a TO reference gait. Our results
demonstrate high-performance blind locomotion in a quadruped robot in biped
mode.

Overall, our findings highlight the potential of imitation learning and TO in
achieving agile and robust locomotion across diverse robotic platforms. Future
work will focus on developing more robust biped motion capabilities on uneven
terrain, transferring these capabilities to physical robots, and refining the
transition from quadrupedal to biped mode to enhance legged robots’
versatility.

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§ INTRODUCTION

Thurston at his ICM address in 1974 asked “Given a large supply of some sort of fabrics, what kind of manifolds can be made from it, in a way that the patterns match up along the seams?” And he alluded that “For open manifolds, Gromov's theorem gives a good answer for a wide variety of fabrics. The techniques needed to analyze such a question on a closed manifold are usually different, at least to a casual eye”. In this article, we shall focus on one aspect of this difference between closed and open manifolds that is inspired by a property of diffeomorphism groups called fragmentation property.

But let us first define the basic objects that we would like to study. Recall that manifolds are modeled locally on Euclidean spaces $\bR^n$ along with a subgroup of the homeomorphism group $\tH(\bR^n)$ as the permissible transition functions for a given structure on the manifold. Now a  $C^r$-foliation of codimension $k$ on an $n$-dimensional manifold $M$ is determined by a foliated atlas $\{(U_i, \phi_i)\}_i$ where $\{U_i\}_i$ is an open cover of $M$ and $\phi_i\colon U_i\to \bR^n$ are maps such that for overlapping pairs $U_i$ and $U_j$, the transition function $\phi_j\circ \phi_i^{-1}$ is a $C^r$ diffeomorphism from $\phi_i(U_i\cap U_j)$
to $\phi_j(U_i\cap U_j)$ such that the function $\phi_j\circ \phi_i^{-1}$ is of the form

\[
\phi_j\circ \phi_i^{-1}(x,y)=(f(x,y), g(y)),
\]

where $x$ denotes the first $n-k$ coordinates and $y$ denotes the last $k$ coordinates. If $y$ is fixed, then so is $g(y)$, hence the property of having the first $k$ coordinates being fixed is preserved by the change of coordinates. Therefore the sheets that are given by fixing the first $k$ coordinates glue together globally to form leaves of the foliation. In particular, this structure gives a decomposition of $M$ into $(n-k)$-submanifolds (the leaves).

Foliation charts

Let us give two examples that will show up later in this article. Consider the foliation of $\bR^2$ by straight parallel lines. Since such foliations are invariant under translations in the $x$-axis and $y$-axis, they induce smooth codimension-one foliations on the $2$-torus $T^2=\bR^2/\bZ^2$ (see Figure 2) and they are called linear foliations on torus. Note that if the angle of the lines with the $x$-axis is irrational, the induced linear foliation has no compact leaf.

Linear foliations on $2$-torus

Another classical example is the Reeb foliation which is a smooth codimension-one foliation on the solid torus. The boundary torus is the only compact leaf and the other leaves are diffeomorphic to $ 2$-planes as in the figure below.

Skematic of $2$ dimensional and $3$ dimensional Reeb foliations on an annulus and on solid torus

One can glue two of these foliated solid tori along the torus boundary to obtain smooth codimension-one foliation on $S^3$.

To admit a codimension $k$ foliation, the manifold $M$ must have a plane field of dimension $n-k$ (the tangents to the leaves). A plane field is called integrable if it is given by the tangent-plane field to a foliation.   A classical theorem that is usually ascribed to Frobenius but in fact goes back to the earlier work of A. Clebsch and F. Deahna (<cit.>) gives a necessary and sufficient condition for a plane field to be integrable. Sections of a plane field $\xi\subset TM$ are vector fields. The integrability condition for a plane field $\xi$ is that the Lie bracket of $C^1$ sections of $\xi$ should also be section.

We can think of foliations as plane fields satisfying the integrability condition.  To find a plane field of dimension $n-k$ on $M$ is essentially a homotopy theory question. But the integrability condition is of a different type, in local coordinates it can be described as a system of partial differential equations (hence, it is not an open condition). So after finding a plane field, it is more natural to ask whether we can deform this plane field to become integrable in which case we say it is integrable up to homotopy.  Bott found a topological obstruction for plane fields $\xi\subset TM$ to be integrable up to homotopy, which is now known as the Bott vanishing theorem. He showed that  Pontryagin classes of the quotient bundle $TM/\xi$ must vanish in degrees larger than $2\text{dim}(TM/\xi)$. Using this obstruction, he showed that each complex codimension-one holomorphic subbundle of $T\bC P^n$ when $n$ is odd is not integrable up to homotopy.

“Gromov's theorem” in Thurston's quote refers to the powerful theory of h-principle (homotopy principle) that Gromov developed to reduce the problem of the existence of solutions of differential relations (differential equations and inequalities) in differential geometry to essentially homotopy theory questions. Philosophically, h-principle theorems relate the topology of a geometric object which is more rigid to a homotopy theoretical object that is more flexible to work with. The first famous example is Smale's sphere eversion which says intuitively that one can turn the sphere $S^2$ in $\mathbb{R}^3$ inside out smoothly without cutting or tearing it or creating any crease. More precisely, the space of immersions of $S^2$ into $\bR^3$ is a connected space. Smale showed that this space of immersions is homotopy equivalent to the space of bundle monomorphisms between $TS^2$ and $T\bR^3$, which can be seen to be connected using algebraic topology.

Bott's obstruction shows that the existence of foliations cannot be solely reduced to the homotopy theory question of the existence of plane fields.  So we need more homotopical information to reduce the problem of the existence of foliations to a homotopy theory question. Haefliger introduced in [4] a notion that is now called Haefliger structures on manifolds that are more flexible than foliations in  the following sense and sometimes are called “singular” foliations. In order to pull back a foliation $\mathcal{F}$ on $M$ under the map $f\colon N\to M$, the map $f$ has to be transverse to $\mathcal{F}$ (i.e. $df(TN)+T\mathcal{F}= TM$). However, Haefliger structures are more flexible than foliations since they can be pulled back under all continuous maps. Because of this flexibility, one can study them as objects in algebraic topology.

A $C^r$ Haefliger structure  $\mathcal{H}$ of codimension $k$ on $M$ is given by the triple $(\nu \mathcal{H}, U, \mathcal{F})$ where $\nu\mathcal{H}$ is a $k$-dimensional $C^r$ vector bundle over $M$
which is called the normal bundle of the Haefliger structure $\mathcal{H}$, the open subset $U\subset \nu\mathcal{H}$ around the zero section, and a $C^r$ foliation of codimension $k$ in $U$ that is transverse to the fibers of $\nu\mathcal{H}$ but not necessarily to the zero section. So the intersection
of this foliation with the zero section will be a “singular” foliation. If the Haefliger structure is also transverse to the zero section, then it is called a regular Haefliger structure that comes from a genuine foliation. Two Haefliger structures are considered the same if they have the same normal bundle and if they coincide in a neighborhood of the zero section.

Haefliger structure

So one can think of a Haefliger structure as a germ of a foliated bundle near its zero section. We say two codimension $k$ Haefliger structures $\mathcal{H}_1$ and $\mathcal{H}_2$ on $M$ are concordant, if there is a codimension $k$ Haefliger structure $\mathcal{H}$ on $M\times [0,1]$ that restricts to $\mathcal{H}_i$ on $M\times \{i\}$ for $i=0,1$. A fundamental question is when a Haefliger structure is concordant with a regular one.   Using tools of homotopy theory, Haefliger constructed ([5])  a classifying space $\mathrm{B}\Gamma^r_k$ for codimension $k$ $C^r$-Haefliger structures such that the concordance classes of codimension $k$ $C^r$-Haefliger structures on $M$ are in bijection with continuous maps  $M\to\mathrm{B}\Gamma^r_k$ up to homotopy. There is also a classifying space $\overline{\mathrm{B}\Gamma}^r_k$ that classifies Haefliger structures $\mathcal{H}$ with trivialized normal bundle $\nu\mathcal{H}$ (i.e. $\nu\mathcal{H}$ is a product bundle).

Using Gromov's h-principle and a result of A. Phillips, Haefliger showed for an open manifold $M$, the existence of a codimension $k$ $C^r$ foliation on $M$ can be reduced to the homotopy theory question of the existence of a $C^r$ Haefliger structure $\mathcal{H}$ on $M$ such that its normal bundle $\nu\mathcal{H}$ can be embedded into the tangent bundle $TM$.

The existence and classification of foliations on closed manifolds were much harder and there was a string of constructions by Reeb,  Lickorish, Novikov, Wood, Lawson, Durfee, and Tamura in particular cases (see [7] and references therein). Even the case of finding codimension-one foliations on odd-dimensional spheres was stubborn for many years until Durfee and Tamura found constructions for all odd-dimensional spheres. Thurston proved ([19]) dramatic general results for the existence and classification of foliations on closed manifolds. In particular, he showed that the concordance classes of foliations on a closed manifold $M$ are in bijection with homotopy classes of Haefliger structures $\mathcal{H}$ together with concordance classes of bundle monomorphisms $\nu\mathcal{H}\to TM$.

These spectacular h-principle theorems reduced the subtle question of classifying foliations to the homotopy theory of Haefliger classifying spaces. But the homotopy type of the Haefliger spaces remained mysterious. The main theme of this article is the way Thurston studied the homotopy type of Haefliger classifying spaces in ([18]) by generalizing a result of Mather ([9]) to relate the homotopy groups of $\mathrm{B}\Gamma^r_k$   to the group homology of compactly supported $C^r$-diffeomorphism group $\Diff^r_c(\bR^n)$.

Mather-Thurston's theorem is interesting in at least two ways. From a technical point of view, the original proof by Thurston used what was known about the homotopy type of $\mathrm{B}\Gamma^r_k$ at the time and the idea of fragmentation to prove an intrinsically “compactly supported” statement. Unlike other h-principle theorems, the case of the open manifolds for Mather-Thurston's theorem was very nontrivial and it was later proved by Segal and McDuff ([17, 11]) after they developed their group completion theorem and homology fibration techniques. From a philosophical point of view, it was used in the opposite way than h-principle theorems are normally used. Mather and Thurston applied this theorem to study the homotopy type of a more “flexible” object  $\mathrm{B}\Gamma^r_k$ using algebraic information (like the perfectness or the simplicity) of a more “rigid” object of the discrete group of diffeomorphisms.

There are already many excellent surveys on different aspects of foliations and diffeomorphism groups including dynamical systems, differential topology, differential geometry, and non-commutative geometry. In particular, commentary on foliations in Thurston's collected works ([19]) beautifully summarizes his many contributions to the topic. In this article, we are interested in different aspects of the relationship

\[
\left\{\begin{tabular}{c}
\text{Algebraic properties of}\\ \text{diffeomorphism groups}
\end{tabular}  \right\}\leftrightarrow\left\{\begin{tabular}{c}
\text{Homotopical properties of }\\ \text{foliations and their invariants}
\end{tabular}  \right\}
\]

and its new consequences on invariants of flat bundles.

§ HAEFLIGER-THURSTON'S CONJECTURE

Recall that the space $\mathrm{B}\Gamma^r_k$ classifies codimension $k$ $C^r$-Haefliger structures $\mathcal{H}$ and $\overline{\mathrm{B}\Gamma}^r_k$ classifies  codimension $k$ $C^r$-Haefliger structures whose  normal bundles $\nu\mathcal{H}$ are trivial. They sit in a fibration sequence

\[
\overline{\mathrm{B}\Gamma}^r_k\to \mathrm{B}\Gamma^r_k\xrightarrow{\nu} \mathrm{B}\text{GL}_k(\bR),
\]

where $\nu$ is the map that classifies the normal bundle to Haefliger structures.

The space $\overline{\mathrm{B}\Gamma}^r_k$ is $2k$-connected (i.e. $\pi_i(\overline{\mathrm{B}\Gamma}^r_k)=0$ for $i\leq 2k$).

Haefliger used a result of A. Phillips for open manifolds to prove that $\overline{\mathrm{B}\Gamma}^{r}_k$ is $k$-connected for all $r$.  Thurston  ([18])  used his generalization of Mather's homology isomorphism on compact manifolds to show that  $\overline{\mathrm{B}\Gamma}^{\infty}_k$ is $(k+1)$-connected and shortly after, Mather (<cit.>) proved the same statement for  $\overline{\mathrm{B}\Gamma}^{r}_k$ for all regularities $r$ except  $r=k+1$. Up to $k$-connectedness, the proof is using the “flexibility” of $\overline{\mathrm{B}\Gamma}^r_k$ but for higher homotopy groups, as we shall explain, the only tool known is to use Mather-Thurston's theorem and homological properties of diffeomorphism groups as discrete groups.

As a consequence of Mather's acyclicity result ([8]), it is known that $\overline{\mathrm{B}\Gamma}^0_k$ is contractible. A remarkable theorem of Tsuboi ([20]) also shows that $\overline{\mathrm{B}\Gamma}^1_k$ is contractible. However, for $r>1$, Thurston and R. Roussarie used Godbillon-Vey invariants to show that $\overline{\mathrm{B}\Gamma}^r_k$ is not $(2k+1)$-connected.

Given Thurston's results on regularizing Haefliger structures, <Ref> implies that $n-k$ dimensional plane fields on an $n$-dimensional manifold $M$, where $k$ is at least $(n-1)/2$, are up to homotopy integrable to  $C^r$-foliations. The case of $k=1, n=3$ is known for all regularities except $r=2$ and it goes back to Mather's homology isomorphism. Using Thurston's results, it implies that all $2$-plane fields on closed $3$-manifolds are up to homotopy integrable to a $C^r$ foliation for $r\neq 2$.

Let's focus on the smooth case so we drop the regularity $r$ from the superscripts. Haefliger already proved that $\overline{\mathrm{B}\Gamma}_1$ is simply connected. So, to show that it is $2$-connected, by the Hurewicz theorem it is enough to show that $H_2(\overline{\mathrm{B}\Gamma}_1;\bZ)=0$. This was first proved by Mather as a consequence of a more general result for codimension-one foliations. Here we sketch an argument using geometric ideas that go back to Claude Roger ([16]) and Thurston. Later Gael Meigniez generalized this geometric point of view to higher codimensions in [12].

§.§ Finding a normal form for codimension-one Haefliger structures
For a topological group $G$, we let $G^{\delta}$ denote the same group with the discrete topology. Recall that the classifying space $\mathrm{B}G$ classifies the principal $G$-bundles and the classifying space $\mathrm{B}G^{\delta}$ classifies flat principal $G$-bundles. One way to think about flat bundles is that there exists a foliation on the total space that is transverse to the fibers. The space $\mathrm{B}G^{\delta}$ has $G^{\delta}$ as the fundamental group and has vanishing higher homotopy groups. So a continuous map $g\colon M\to \mathrm{B}G^{\delta}$ induces a representation $\rho\colon \pi_1(M)\to G^{\delta}$. Now let $\widetilde{M}$ denote the universal cover of $M$ and consider the horizontal foliation on $\widetilde{M}\times G$ whose leaves are $\widetilde{M}\times \{x\}$ for $x\in G$. Since this foliation is invariant under the diagonal action of $\pi_1(M)$ on  $\widetilde{M}\times G$, it induces a foliation on the quotient $\widetilde{M}\times_{\pi_1(M)} G$. So  we obtain a principal $G$ bundle

\[
G\to \widetilde{M}\times_{\pi_1(M)} G\to M,
\]

with a foliation on the total space which is transverse to the fibers. So in this sense, $\mathrm{B}G^{\delta}$ classifies flat $G$-bundles, and also the group homology of the group $G^{\delta}$ is the same as the homology of the classifying space $\mathrm{B}G^{\delta}$.

To prove that $H_2(\overline{\mathrm{B}\Gamma}_1;\bZ)=0$, first we construct a map

\begin{align}\label{H_2}
H_1(\BcdDiff \bR;\bZ)&\xrightarrow{}H_2(\overline{\mathrm{B}\Gamma}_1;\bZ),
%H_1(\BcdDiff \bR;\bZ)&\xrightarrow{\cong}H_1(\BdDiffo {S^1};\bZ),
\end{align}

which turns out to be an isomorphism.
To define this so-called “suspension” map, first note that the group $\cdDiff \bR$ is isomorphic to its subgroup $\cdDiff {(0,1)}$ and it is easy to see that the inclusion induces a group homology isomorphism. So it is enough to define the map $H_1(\BcdDiff {(0,1)};\bZ)\to H_2(\overline{\mathrm{B}\Gamma}_1;\bZ)$. Let $I$ be the closed unit interval and consider the horizontal foliation on $I\times I$. Given $f\in \cdDiff {(0,1)}$, the horizontal foliation on $I\times I$ induces a foliation $\mathcal{H}_f$ on

\[
S^1\times I=\frac{I\times I}{\{0\}\times x \sim \{1\}\times f(x)}\cdot
\]

The foliation $\mathcal{H}_f$ is horizontal near the two boundary components. So by contracting the two circle boundary components, we obtain a singular foliation (aka Haefliger structure) $\hat{\mathcal{H}}_f$ on $S^2$ (see  Figure 5) with two center singularities.

The Haefliger structure $\hat{\mathcal{H}}_f$ obtained by the suspension map

So the surjection of the suspension map <ref> is equivalent to the following.
Each codimension-one Haefliger structure on $S^2$ is concordant with a Haefliger structure $\hat{\mathcal{H}}_f$ given by the suspension of a diffeomorphism $f\in \Diff_c {((0,1))}$.

Note that an oriented codimension-one Haefliger structure  on $S^2$ is induced by a map $S^2\to \overline{\mathrm{B}\Gamma}_1$ so it is an element  in $\pi_2(\overline{\mathrm{B}\Gamma}_1)=H_2(\overline{\mathrm{B}\Gamma}_1;\bZ)$. This element can be represented by a germ of a foliation $\mathcal{F}$ on $S^2\times \bR$ near the zero section $S^2\times \{0\}$. On each foliation chart $U_i$, we have a submersion $f_i\colon U_i\to \bR$ (i.e. $Df_i(x)\neq 0$ for all $x\in U_i$) such that $\mathcal{F}|_{U_i}$ is given by the pullback of point foliation on $\bR$ via $f_i$. Since $\mathcal{F}$ is not necessarily transverse to the zero section $S^2\times\{0\}$, the restrictions of functions $f_i$ to the zero section have singularities. One can change $\mathcal{F}$ in its concordance class to make all the singularities Morse singularities. So we end up having either center or saddle singularities.

Center and saddle singularities

Let $\mathcal{H}$ be a Haefliger structure on $S^2$ with a trivialized normal bundle that has only Morse singularities. Then there is a standard strategy to remove singularities by pairing each saddle singular point to a center. Claude Roger observed that one can choose a path between a saddle and a center and change the Haefliger structure up to concordance only in a neighborhood of the path to remove both singularities. To prove <Ref>, we continue this process until we are left with two center singularities on $S^2$, which is guaranteed by the Hopf index theorem. Such a Haefliger structure as in Figure 5 is induced by the suspension of a diffeomorphism.

§.§ Bordism of Haefliger structures and the simplicity trick
Given the normal form in <Ref> for codimension-one Haefliger structures on $S^2$, to prove $H_2(\overline{\mathrm{B}\Gamma}_1;\bZ)=0$, it is enough to show that for each $f\in \Diff_c {((0,1))}$ the Haefliger structure  $\hat{\mathcal{H}}_f$ on $S^2$ can be extended to a $3$-manifold bounding $S^2$. Surprisingly the only proof known for this geometric question is to use a deep result of Herman that the group of orientation preserving diffeomorphisms of the circle $\Diff_0{(S^1)}$ is a perfect group meaning that it is equal to its commutator subgroup. This is against the philosophy of h-principle though, since one might expect that it ought to be easier to compute $H_2(\overline{\mathrm{B}\Gamma}_1;\bZ)$ than to understand the more rigid object of diffeomorphism groups!

Given Herman's theorem, if we show that  $\hat{\mathcal{H}}_f$ is cobordant to zero (i.e. $\hat{\mathcal{H}}_f$ on $S^2$ can be extended to a $3$-manifold bounding $S^2$) for all $f\in \Diff_c {((0,1))}$, we conclude that $\overline{\mathrm{B}\Gamma}_1$ is $2$-connected. Roger used the perfectness of $\Diff_0{(S^1)}$ to show that the null-bordism for $\hat{\mathcal{H}}_f$ exists. But let us mention a trick due to Thurston that builds a specific null-bordism. Choosing an “economical” bordism between Haefliger structures will show up in the last section again.

The perfectness of $\Diff_0(S^1)$  implies that it is a simple group by Epstein's theorem ([1]), meaning it has no nontrivial normal subgroup.  Thurston's trick uses the simplicity of $\Diff_0{(S^1)}$ to show that $\hat{\mathcal{H}}_f$ is cobordant to zero. The first step is to show that $\hat{\mathcal{H}}_f$ is cobordant to a foliation on the torus $S^1\times S^1$ that is transverse to the circles $\{x\}\times S^1$ as follows.

Bordism of Haefliger structures

Let $W$ be a $3$-manifold obtained by gluing $D^2\times I$ to $S^2\times I$ as in Figure 7. Thus $\partial W$ is diffeomorphic to $S^2\coprod S^1\times S^1$. A Haefliger structure on $W$ is given by the product Haefliger structure $\hat{\mathcal{H}}_f\times I$ on $S^2\times I$ and the codimension one foliation given by leaves $D^2\times \{x\}$ on $D^2\times I$. Note that the boundary of $W$ is the union of a sphere and a torus. This gives a bordism between $\hat{\mathcal{H}}_f$ on its sphere boundary component and a foliation on the torus boundary component that is transverse to $\{x\}\times S^1$ for all $x\in S^1$. Hence, this foliation $\mathcal{F}_{\phi}$ on the torus is induced by suspending the horizontal foliation on the cylinder $S^1\times I$ by a diffeomorphism $\phi\in \Diff_0{(S^1)}$.

The idea of Thurston is to consider the subset of the circle diffeomorphisms $\phi$ for which the suspension $\mathcal{F}_{\phi}$ can be extended to a foliation on a solid torus. Then it is not hard to see that this subset is in fact a normal subrgroup of $\Diff_0{(S^1)}$. Given that $\Diff_0{(S^1)}$ is a simple group, it is enough to show that this contains at least one non-trivial diffeomorphism. Suppose that $\phi$ is a nontrivial rotation of the circle. Thurston extends $\mathcal{F}_{\phi}$ into a solid torus by standard construction of spinning leaves around an embedded torus inside the solid torus and filling in the torus leaf by a Reeb foliation as in Figure 8. We conclude that $\hat{\mathcal{H}}_f$ is cobordant to a foliation on the torus that is nullcobordant; therefore $\hat{\mathcal{H}}_f$ is indeed nullcobordant.

Part of a solid torus with spinning leaves and a Reeb component

As we shall see in the next section, it is a consequence of a more general theorem of Mather that $\Diff^r_0{(S^1)}$ is simple for $r\neq 2$. So the above construction also for $r\neq 2$ shows that $C^r$ codimension-one Haefliger structures on $S^2$ are cobordant to the trivial Haefliger structure.

Is $\Diff^2_0{(S^1)}$ a simple group?

By Mather's theorem <ref> below, If one could find a geometric construction without using perfectness or simplicity of diffeomorphism groups to show that suspension Haefliger structures $\hat{\mathcal{H}}_f$ for all $f\in \Diff_c^r {((0,1))}$ is null-cobordant, it would give a geometric proof of the simplicity of $\Diff_0^r{(S^1)}$ in all regularities. But so far, no such construction is known without using the algebraic properties of these groups, which is against the h-principle philosophy!

Mather proved a more general theorem for codimension-one foliations that also gives a non-geometric proof that $\overline{\mathrm{B}\Gamma}_1$ is $2$-connected. Consider the classifying space $\BcdDiff \bR$ and the trivial bundle $\BcdDiff \bR\times \bR\to \BcdDiff \bR$. This is a “universal compactly supported foliated $\bR$-bundle”  in the sense that for each foliated $\bR$-bundle $M\times \bR\to M$ over a manifold $M$ where the foliation on $M\times \bR$ is horizontal outside of a compact subset, is a pullback of this universal bundle via a map $M\to \BcdDiff \bR$. Hence, there is classifying map $\BcdDiff \bR\times \bR\to \overline{\mathrm{B}\Gamma}_1$. The adjoint of this map [Recall that the adjoint of a continuous map $X\times Y\to Z$ is a continuous map $X\to \text{Map}(Y,Z)$.] is a map  $\BcdDiff \bR\to \Omega \overline{\mathrm{B}\Gamma}_1$ where $\Omega \overline{\mathrm{B}\Gamma}_1$ is the  based loop space of $ \overline{\mathrm{B}\Gamma}_1$.
The adjoint map

\[
\BcdrDiff \bR\to \Omega \overline{\mathrm{B}\Gamma}^r_1
\]

induces a homology isomorphism.

In particular, Mather's theorem implies that the suspension map (<ref>)

\[
H_1(\BcdDiff \bR;\bZ)\to H_1(\Omega \overline{\mathrm{B}\Gamma}_1;\bZ)\cong H_2(\overline{\mathrm{B}\Gamma}_1;\bZ),
\]

is in fact an isomorphism. Mather's theorem also has a version for $\drDiffo {S^1}$ which implies that there is an isomorphism

\[
H_1(\BcdrDiff \bR;\bZ)\xrightarrow{\cong}H_1(\BdrDiffo {S^1} ;\bZ).
\]

Since $H_1(\mathrm{B}G^{\delta};\bZ)$ is isomorphic to the abelianization $G/[G,G]$, Herman's perfectness theorem implies that $H_1(\BdDiffo {S^1};\bZ)=0$. Therefore, Mather's homology isomorphism implies that $H_2(\overline{\mathrm{B}\Gamma}_1;\bZ)=0$.

§ MATHER-THURSTON'S THEOREM
Mather's original proof of <Ref> is quite involved. He showed that the delooping of $\BcdDiff \bR$ using  Quillen's delooping techniques in Algebraic K-theory is homotopy equivalent to $\overline{\mathrm{B}\Gamma}_1$. This step was not easy to generalize to higher dimensions. But Thurston found a generalization using a method of fragmentation for compactly supported diffeomorphisms.

To state his theorem, let $M$ be an $n$-dimensional smooth manifold and let $\overline{\BDiff_c^r(M)}$ be the homotopy fiber of the map

\[
\eta\colon \BcdrDiff M\to \BDiff^r_c(M),
\]

which classifies trivialized $M$-bundles equipped with foliations on the total space transverse to the fibers such that the foliation is compactly supported i.e. it is horizontal outside of some compact subset of the total space. Let $\Diff_{c,0}^r(M)$ be the identity component of $\Diff_{c}^r(M)$ and let $\widetilde{\Diff}_{c,0}^r(M)$ be its universal cover. It turns out that the fundamental group of $\overline{\BDiff_c^r(M)}$ is isomorphic to $\widetilde{\Diff}_{c,0}^r(M)^{\delta}$. Now the natural map $\widetilde{\Diff}_{c,0}^r(M)^{\delta}\to \Diff_{c,0}^r(M)$ gives a representation of $\pi_1(\overline{\BDiff_c^r(M)})$ into $\Diff_{c,0}^r(M)$ that induces a trivialized $M$-bundle $\overline{\BDiff_c^r(M)}\times M\to \overline{\BDiff_c^r(M)}$; we call it the universal foliated trivialized $M$-bundle. This universal foliation on the total space induces a diagram commutative
up to homotopy

\[
\begin{tikzpicture}[node distance=1.8cm, auto]
\node (A) {$\overline{\BDiff_c^r(M)}\times M$};
\node (B) [right of=A, node distance=3cm] {$\mathrm{B}\text{GL}_n(\bR),$};
\node (C) [above of= B ] {$\mathrm{B}\Gamma_n^r$};
\draw [->] (A) to node {$\theta$}(B);
\draw [->] (C) to node {$\nu^r$}(B);
\draw [->] (A) to node {$\alpha$}(C);
\end{tikzpicture}
\]

where $\alpha$ classifies the codimension $n$ foliation on the total space. The map $\theta$ classifies the normal bundle to the foliation and since the foliation is transverse to the fibers, this map is induced by the map $\tau_M\colon M\to  \mathrm{B}\text{GL}_n(\bR)$ which classifies the tangent bundle of $M$.

The adjoint of the map $\alpha$ induces a map from $\overline{\BDiff_c^r(M)}\times M$ to the space of lifts of $\tau_M$ to $\mathrm{B}\Gamma_n^r$. One can think about this space of lifts as follows. Let $\tau_M^*(\nu^r)$ be the bundle induced by pulling back the fibration $\nu^r$ via $\tau_M$. Fix a section $s_0$ of this bundle. For any other section $s$, we can define the support of $s$ to be the closure of the set of points $x\in M$ where $s(x)\neq s_0(x)$. Let $\text{Sect}_c(\tau_M^*(\nu^r))$ be the subspace of compactly supported sections of the bundle $\tau_M^*(\nu^r)\to M$.

Therefore, the adjoint of $\alpha$ gives a map

\[
f\colon\overline{\BDiff_c^r(M)}\to \text{Sect}_c(\tau_M^*(\nu^r)).
\]

The map $f$ is an acyclic map; in particular, it induces a homology isomorphism with integer coefficients.

A non-compactly supported version of this theorem also holds. But surprisingly, even the case of an open ball was proved later by McDuff and Segal. As we shall explain, Thurston's fragmentation method only works for the compactly supported version.

Thurston gave a few applications of this theorem. To mention two of them, note that the target of the map $f$ above, which is a section space of a fiber bundle is more amenable to homotopy theory. The fiber of this bundle is $\overline{\mathrm{B}\Gamma}^{r}_n$, which was known by Haefliger to be $n$-connected. Using obstruction theory, one can see that
and in particular it does not depend on $M$. Therefore, $H_1(\overline{\BDiff_c^r(M)};\bZ)$  does not depend on $M$ either. Recall that  $\pi_1(\overline{\BDiff_c^r(M)})$ is isomorphic to $\widetilde{\Diff}_{c,0}^r(M)^{\delta}$ which was known by a deep result of Herman and Mather to be perfect for the case of the torus $M=T^n$ and $r\neq n+1$. Therefore, the universal cover of $\Diff_{c,0}^r(M)$ is perfect for all $M$ and $r\neq n+1$ which implies the same for $\Diff_{c,0}^r(M)$ itself. As a result of Epstein, the perfectness for $\Diff_{c,0}^r(M)$ implies that these groups are simple.

Another application related to the theme of this article is that the perfectness result implies that $\pi_{n+1}(\overline{\mathrm{B}\Gamma}^{r}_n)=0$, so given Haefliger's result that $\overline{\mathrm{B}\Gamma}^{r}_n$ is $n$-connected, the connectivity of $\overline{\mathrm{B}\Gamma}^{r}_n$ is improved by one for $r\neq n+1$.

Using Mather-Thurston's theorem, one can show that Haefliger-Thurston's conjecture is equivalent to the following bundle theoretic statement.
The map

\[
\eta\colon \BcdrDiff M\to \BDiff^r_c(M),
\]

induces an isomorphism on homology in degrees less than $n+1$  and is a surjection on homology in degree $n+1$.

In low regularities when $r=0$ using the acyclicity result of Mather and for $r=1$ using the acyclicity result of Tsuboi, we know that $\eta$  in fact induces a homology isomorphism in all degrees. But for $r>1$, the homological degrees in the conjecture cannot be improved.

Geometrically, this conjecture is equivalent to the following. For every $C^r$ $M$-bundle $M\to E\to B$ where $B$ is a closed manifold and $\text{dim}(B)\leq \text{dim}(M)$, there exists a bordism  $W$ from $B$ to another manifold $B'$ and a $C^r$ $M$-bundle $M\to K\to W$ such that when it is restricted to $B$, it is isomorphic to $E\to B$ and when it is restricted to $B'$, it is a flat $M$-bundle i.e. it is induced by a representation $\pi_1(B')\to \Diff_c^{r}(M)^{\delta}$.

For non-compactly supported diffeomorphisms of open manifolds, the range of isomorphism in the conjecture is expected to be better. For example, for $M=\bR^n$, Segal proved that the natural map

\[
\BdrDiff {\bR^n}\to \mathrm{B}\Gamma^r_n,
\]

induces a homology isomorphism. It is easy to see that Segal's theorem implies that the map

\[
\BdrDiff {\bR^n}\to \BDiff^r(\bR^n),
\]

induces a homology isomorphism already in degrees less than $n+1$. Linearizing diffeomorphisms implies that $\Diff^r(\bR^n)\simeq \text{GL}_n(\bR)$ for $r>0$, so as a consequence, we know the homology of $\BdrDiff {\bR^n}$ up to degree $n$. In particular, Segal's theorem has the following corollary.

The group $\drDiffo {\bR^n}$ is perfect for all $r$.

This proof is in line with the h-principle philosophy and works for all regularities. McDuff and Schweitzer later found an algebraic proof for the perfectness of $\drDiffo M$ using “Ling's factorization” when $M$ is the interior of a compact manifold with a boundary.

McDuff also proved the analog of Mather-Thurston's theorem for volume-preserving diffeomorphisms and Segal's theorem for the volume-preserving case for $n>2$. Curiously, the case and $n=2$ is still open.
Let  $\drDiff {\bR^2, \omega}$ be the group of volume preserving diffeomorphisms of $\bR^2$ with the standard volume form $\omega$. Is $\drDiff {\bR^2, \omega}$ perfect? Is the natural map from $\BdrDiff {\bR^2, \omega}$ to the corresponding Haefliger space $ \mathrm{B}\Gamma^{\omega}_2$ a homology isomorphism?

The case of volume-preserving homeomorphisms is also very interesting.  There has been a recent breakthrough in proving the failure of the perfectness of volume-preserving homeomorphisms of the $2$-disk by Daniel Cristofaro-Gardiner, Vincent Humilière, and Sobhan Seyfaddini, which is beyond the scope of this article.

§.§ The curious case of Piecewise Linear homeomorphisms
The subgroups of piecewise linear (PL for short) homeomorphisms of the line have been a rich source for interesting finitely generated groups with surprising algebraic and dynamical properties. However, not much is known about the algebraic properties of the PL homeomorphisms of higher dimensional PL manifolds. Although they are more combinatorial in nature, the analytical tools for diffeomorphism groups and the Mather infinite repetition trick for homeomorphisms ([8]) are not available for PL homeomorphisms. So the following basic question due to Epstein ([1]) is still open.
(Epstein) Let $M$ be a PL manifold. Is $\text{PL}_0(M)$, the group of PL homeomorphisms of $M$ that are isotopic to the identity, a simple group?

By Epstein's result, perfectness and simplicity are equivalent in this case and he proved that $\text{PL}_c(\bR)$ and $\text{PL}_0(S^1)$ are perfect by observing that in dimension one,  PL homeomorphisms are generated by certain “typical elements” and those typical elements can be easily written as commutators. To generalize his argument to higher dimensions, he suggested the following approach.

Let $B$ be a ball in $\bR^n$. It is PL homeomorphic to $S^{n-2}\star [0,1]$, the join of $S^{n-2}$ with $[0,1]$. Note that for PL manifolds $M$ and $N$, a PL homeomorphism of $N$ extends naturally to a PL homeomorphism of the join $M\star N$.  A glide homeomorphism of the ball $B$ is a PL homeomorphism that is induced by the extension of a compactly supported PL homeomorphism of $(0,1)$ to a PL homeomorphism of $S^{n-2}\star [0,1]$. For a PL $n$-manifold $M$, a  glide homeomorphism $h\colon M\to M$ is the extension by the identity of a glide homeomorphism supported in a PL embedded ball $B\hookrightarrow M$.
However, it is not known if ${\textnormal{\text{PL}}}_{c,0}(M)$ is generated by glide homeomorphisms in all dimensions.

On the other hand, Greenberg started the program to study the Haefliger classifying space for PL foliations and he described them inductively in terms of the codimension of the foliation and observed interesting connections to algebraic $K$-theory of real numbers.  Greenberg's works (see [3] and references therein)  describe an inductive method to build $\overline{\mathrm{B}\Gamma}_n^{\textnormal{\text{PL}}}$ from classifying spaces of matrix groups made discrete. In codimension one, he shows that $\overline{\mathrm{B}\Gamma}_1^{\textnormal{\text{PL}}}$ is homotopy equivalent to $K(\bR_+,1)*K(\bR_+,1)$ the join of two Eilenberg-MacLane spaces where $\bR_+$ is multiplicative group of the positive reals. In higher codimensions, he describes a complicated diagram whose homotopy colimit is homotopy equivalent to $\overline{\mathrm{B}\Gamma}_n^{\textnormal{\text{PL}}}$. Each space in Greenberg's diagram is related to classifying spaces of PL foliations of lower codimensions or classifying spaces of certain matrix groups made discrete. Already in codimension $2$, his model is difficult to do calculations with; in particular one cannot read off from the lower homotopy groups of $\overline{\mathrm{B}\Gamma}_2^{\textnormal{\text{PL}}}$. I used Suslin's work on the algebraic $K$-theory of $\bR$ to prove the analog of Haefliger-Thurston's conjecture for codimension $2$ PL foliations.
(Nariman [14])
The classifying space $\overline{\mathrm{B}\Gamma}_2^{\textnormal{\text{PL}}}$ is  $4$-connected.

Greenberg ([3]) and Gelfand-Fuks independently conjectured that the analog of the Mather-Thurston theorem holds for PL foliations. I used Thurston's fragmentation method in [15] to prove Mather-Thurston's theorem in the PL case.  Using this version of Mather-Thurston for PL foliations of codimension $2$ and the fact that $\pi_3(\overline{\mathrm{B}\Gamma}_2^{\textnormal{\text{PL}}})=0$ solves  Epstein's question about PL homeomorphisms in dimension $2$.

(Nariman [14])
Let $\Sigma$ be an oriented compact surface, possibly with a boundary. The identity component ${\textnormal{\text{PL}}}_0(\Sigma,\text{rel }\partial)$ is a simple group as a discrete group.

This is a homotopy theoretical proof of the simplicity of ${\textnormal{\text{PL}}}_0(\Sigma,\text{rel }\partial)$ which is more in line with the h-principle philosophy and it is unlike the smooth case. Still, it would be interesting to prove the perfectness and simplicity of ${\textnormal{\text{PL}}}_0(\Sigma,\text{rel }\partial)$  group-theoretically.

§ IMMERSION THEORETIC METHOD VS FRAGMENTATION METHOD
Thurston apparently had three different proofs for <Ref>. He gave lectures at Harvard on one of his proofs and Mather worked out the details and wrote it down. He said in an email to the author “The lectures were sketchy and it was really hard to write up the proof. I spent 14 months on it”. McDuff and Segal  used their group completion theorem and homology fibration techniques to give another proof in the spirit of immersion theory. Their approach has the advantage that it also works for open manifolds.

Recently Gael Meigniez ([12]) found a geometric proof similar to the proof of the isomorphism <ref> by C.Roger in which he changes Haefliger structure up to concordance to obtain a “normal” form with specified models for their singularities and he then does surgery on singularities to make the Haefliger structure cobordant to a foliated bundle (see Figure 7).

We shall sketch the main idea of how Thurston proved that the natural map

\begin{equation}\label{T}
\overline{\BDiff_c^r(\bR^n)}\to \Omega^n \overline{\mathrm{B}\Gamma}_n^r,
\end{equation}

where $\Omega^n \overline{\mathrm{B}\Gamma}_n^r$ is an $n$-fold loop space, is a homology isomorphism. This is in fact the main step in proving a compactly supported version of Mather-Thurston's theorem for all manifolds.

Let us first recall the immersion theoretic method that is used in Smale-Hirsch's theorem. Suppose $F_1, F_2\colon (\mathsf{Mfld}^{\partial}_n)^{op}\to \mathsf{S}$ are functors from the category of smooth $n$-manifolds (possibly with nonempty boundary) with smooth embeddings as morphisms to a convenient category of spaces $\mathsf{S}$.  To prove a natural transformation $T\colon F_1\to F_2$  induces a weak homotopy equivalence (i.e. it induces isomorphisms on all homotopy groups) $F_1(N)\simeq F_2(N)$ for all $N\in \mathsf{Mfld}^{\partial}_n$, we fix a handle decomposition of $N$ and induct on the number of handles as follows.

First, we show that $F_1(\bR^n)\to F_2(\bR^n)$ is a weak equivalence. This is normally an easy step by “zooming in” arguments. Now suppose we know that $F_1(M)\to F_2(M)$ is a weak equivalence and  $N$ is a manifold obtained from $M$ by attaching a $k$-handle $D^k\times D^{n-k}$. So $M\cap (D^k\times D^{n-k})$ is $A^k\times D^{n-k}$ where $A^k$ is a neighborhood of $S^{k-1}$ in $D^k$. In the case of Smale-Hirsch's theorem for $i=1, 2$, we have pull-back squares

\begin{equation}\label{d1}
\begin{gathered}
\begin{tikzpicture}[node distance=3cm, auto]
\node (A) {$F_i(N)$};
\node (B) [right of=A] {$F_i(M)$};
\node (C) [below of=A, node distance=1.2cm] {$ F_i(D^k\times D^{n-k})$};
\node (D) [below of=B, node distance=1.2cm] {$F_i(A^k\times D^{n-k}),$};
%  \draw[->, dashed] (C) to node  {$$} (B);
\draw[->] (C) to node {$$} (D);
\draw [->] (A) to node {}(C);
\draw [->] (A) to node {$$} (B);
\draw [->] (B) to node {$$} (D);
\end{tikzpicture}
\end{gathered}
\end{equation}
where the maps are given by restrictions. If the horizontal maps are Serre fibrations then the squares become {\it homotopy} pull-back squares. Therefore, by induction knowing the weak equivalence for $M$, $D^k\times D^{n-k}$ and $A^{k}\times D^{n-k}$, we can compare the two homotopy pull-back squares and conclude that $F_1(N)\to F_2(N)$ is a weak-equivalence.

Normally in h-principle theorems, the functor $F_2$ has nice homotopical properties, for example it is a mapping space or a section space of a bundle over the manifold. In these cases, it is easy to see that the restriction maps similar to the diagram \ref{d1} are Serre fibrations. And the hard step is to prove the same for a more geometric functor $F_1$. In the case of Mather-Thurston, the base case was nontrivial and it was proved by Segal that $F_1(\bR^n)\to F_2(\bR^n)$ is a homology isomorphism. McDuff showed that the restriction maps for $F_1$ in Mather-Thurston's theorem are {\it homology} fibrations.

Thurston proved the compactly supported version without knowing Segal's theorem for the base case. His proof is inspired by the following property of diffeomorphism groups which is called fragmentation.
\begin{prop*} Let $M$ be a compact manifold and  let $\{ U_i\}_{i}$ be a finite open cover of $M$. Then any element $f\in \Diff^r_0(M)$ can be written as a composition of diffeomorphisms $f_j$ such that  $f_j$ is compactly supported in some element of the cover  $\{ U_i\}_{i}$.
\end{prop*}
To illustrate the idea, let us focus on section spaces of fiber bundles and how Thurston fragmented them. Let $\pi\colon E\to M$ be a Serre fibration over an $n$-dimensional Riemannian manifold $M$ with a non-zero injectivity radius. Let  $s_0$ be the fixed section of this bundle as a base section. So with respect to this base section, we define $\text{Sect}_c(\pi)$ to be the space of compactly supported sections of $\pi$ with compact open topology.

We want to give a filtration on $\text{Sect}_c(\pi)$ that has a ``nice'' filtration quotients. Fix a positive $\epsilon$ smaller than the injectivity radius. Let $\text{Sect}_{\epsilon}(\pi)$ denote the subspace of sections $s$ such that the support of $s$ can be covered by $k$ geodesically convex balls of radius $2^{-k}\epsilon$ for some positive integer $k$; we call this subset $\epsilon$-supported sections.

There is a filtration on  $\text{Sect}_{\epsilon}(\pi)$ whose $k$-th level is the subset of sections whose supports can be covered by at most $k$ balls. And the reason for the choice of $2^{-k}\epsilon$ is that the filtration quotients behave nicely. For example, if two balls of radius $\epsilon/4$ intersect, then they can be contained in a ball of radius $\epsilon/2$. So a section in the second filtration quotient has support in two disjoint balls. This is homotopically useful for studying filtration quotients.

Thurston proved the following which can be improved (see [15]) to prove what is now called {\it non-abelian Poincar\' e duality}.
\begin{thm}[Thurston's fragmentation] If the fiber of $\pi$ is at least $n$-connected, the inclusion
\[
\text{\textnormal{Sect}}_{\epsilon}(\pi)\hookrightarrow \text{\textnormal{Sect}}_c(\pi)
\]
is a weak homotopy equivalence.
\end{thm}
Since the fiber of $\pi$ is $n$-connected, the space $\text{\textnormal{Sect}}_c(\pi)$ is connected so each section can be deformed to the base section, which lies in $\text{\textnormal{Sect}}_{\epsilon}(\pi)$. To prove the theorem, one has to deform a family of compactly supported sections to a family of $\epsilon$-supported sections. To see the main idea, let us see how we can obtain such a deformation for a one-parameter family of sections in $\text{\textnormal{Sect}}_c(\pi)$.

Let $f\colon [0,1]\to \text{\textnormal{Sect}}_c(\pi)$ be a one-parameter family. Consider the adjoint of this map $F\colon [0,1]\times M\to E$. We homotope $F$ in two steps so that at each time coordinate we get an $\epsilon$-supported section.

Let $\{\mu_i\}_{i=1}^N$ be a partition of unity with respect to an open cover of $M$. We define a {\it fragmentation homotopy} with respect to this partition of unity. Let $\nu_j$ be the function
\[
\nu_j(x)=\sum_{i=1}^j\mu_i(x).
\]

\[
H_1:[0,1]\times M\to [0,1]\times M,
\]
\[
\]
\[
\]
Since $H_1(x,t)$ preserves the $x$ coordinate, we can define a straight line homotopy $H_t\colon [0,1]\times M\to [0,1]\times M$ from the identity to $H_1$.
\begin{figure}[h]\label{frag}
\[
\begin{tikzpicture}
\def\rectanglepath{-- ++(3cm,0cm)  -- ++(0cm,3cm)  -- ++(-3cm,0cm) -- cycle}
\draw (0,0) \rectanglepath;
% \draw (6.5,0) \rectanglepath;
\draw (4.5,0) \rectanglepath;

\draw [decorate,
decoration = {calligraphic brace, mirror}] (0.1,0.1) --  (0.1,1.9);

\draw [decorate,
decoration = {calligraphic brace, mirror}] (1.1,1.6) --  (1.1,2.2);

\draw [decorate,
decoration = {calligraphic brace, mirror}] (2.1,1.8) --  (2.1,2.9);

\draw (1,0)--(1,3);
\draw (2,0)--(2,3);
\draw (0,2)--(1,2);
\draw (1, 1.5)--(2, 1.5);
\draw (1,2.3)--(2, 2.3);
\draw (2, 1.7)--(3,1.7);
\node (X) at (1.5,-0.5) {$\Delta^1$};
\node (Y) at (6.3,-0.5) {$\Delta^1$};
\node (Z) at (-0.5,1.5) {$M$};
%\node (W) at (4.9,1.5) {$M$};
\node (R) at (0.58,1.) {\scalebox{0.5}{$\text{supp}(\mu_1)$}};
\node (R) at (1.58,1.9) {\scalebox{0.5}{$\text{supp}(\mu_2)$}};
\node (R) at (2.6,2.37) {\scalebox{0.5}{$\text{supp}(\mu_3)$}};
\draw [ fill=gray] (0,2) rectangle (1,3);
\draw [fill=gray] (1,0) rectangle (2,1.5);
\draw [ fill=gray] (1,2.3) rectangle (2,3);
\draw [ fill=gray] (2,0) rectangle (3,1.7);
\draw [ultra thick] (4.5,2) to [out=-20,in=150] (7.5,1.5);
\draw [ultra thick] (4.5,2.3) to [out=-30,in=150] (7.5,1.7);
\draw [ultra thick] (4.5,0) -- (4.5,3);
\draw [ultra thick] (7.5,0) -- (7.5,3);
\draw [->](3.4,1.5) -- (4.,1.5) node[midway,above] {$H_1$};
% \end{scope}
\end{tikzpicture}
\]

\caption{Fragmentation map for $N=3$ and $q=1$. The thick lines are the images of $\{0\}\times M, \{1/3\}\times M, \{2/3\}\times M$ and $ \{1\}\times M$ under the map $H_1$.}
\end{figure}
As in Figure \ref{frag}, the map $H_1$ is defined so that the gray area is mapped to a subcomplex $L$ of  $[0,1]\times M$ (the bold lines in Figure \ref{frag}) in the target which is of dimension $n=\text{dim}(M)$.

Using the fact that $\overline{\mathrm{B}\Gamma}_n^r$  is at least $n$-connected and $L$ is $n$-dimensional subcomplex, it is standard by the obstruction theory, to change $F\colon [0,1]\times M\to E$ up to homotopy to a map $G\colon [0,1]\times M\to E$ such that the restriction of $G$ to the subcomplex $L$ is the same as $\text{id}\times s_0$. Then $F$ is homotopic to $ G\circ H_1$ and one can see that $ G\circ H_1\colon [0,1]\times M\to E$ gives a one-parameter family that at each time is $\epsilon$-supported section. This idea also works for higher dimensional parameter space to prove fragmentation for section spaces. Note that in Mather-Thurston's theorem the fiber of the bundle $\tau_M^*(\nu^r)$ is $\overline{\mathrm{B}\Gamma}_n^r$ which is at least $n$-connected so the above theorem applies and we obtain a filtration on the section space.

On the other hand, recall that the support for foliated trivialized bundle $B\times M\to B$ is the maximal closed subset $K$ of the fiber $M$ such the foliation on $B\times (M\backslash K)\to B$ is horizontal foliation. And one can similarly prove a fragmentation property for the classifying spaces that classify foliated trivialized bundles to obtain a filtration. One can also fragment the Haefliger space $\overline{\mathrm{B}\Gamma}_n^r$ which classifies germs foliations by ``thickening'' its model as follows. The set of $k$-simplices of the semi-simplicial set that realizes to $\overline{\mathrm{B}\Gamma}_n^r$ is the set of germs of foliated bundle $\Delta^k\times \bR^n\to \Delta^k$ near the zero section $\Delta^k\times \{0\}$. We can thicken this model to define the set of $k$-simplices to be the set of germs of foliated bundle $\Delta^k\times \bR^n\to \Delta^k$ near the disk bundle $\Delta^k\times D^n$. This model is also homotopy equivalent to also $\overline{\mathrm{B}\Gamma}_n^r$ and now one considers the support as a subset of the disk $D^n$ and the fragmentation idea applies. Thurston considered the  map
\begin{equation}\label{N}
f\colon\overline{\mathrm{B}\Gamma}_n^r\xrightarrow{\simeq} \text{Map}(D^n, \overline{\mathrm{B}\Gamma}_n^r),
\end{equation}
that is induced by constant maps. So now both $\overline{\mathrm{B}\Gamma}_n^r$ and $\text{Map}(D^n, \overline{\mathrm{B}\Gamma}_n^r)$ satisfy fragmentation property and the map between them is filtration preserving.
\[
\begin{tikzcd}
X_1 \arrow{d}{f_1}\arrow[r,hook] &X_2 \arrow{d}{f_2}\arrow[r,hook] &\cdots \arrow[r,hook] & \overline{\mathrm{B}\Gamma}_n^r \arrow{d}{\simeq}\\
Y_1\arrow[r,hook] &Y_2 \arrow[r,hook] &\cdots \arrow[r,hook] & \text{Map}(D^n, \overline{\mathrm{B}\Gamma}_n^r).
\end{tikzcd}
\]
It turns out that $X_1$ and $Y_1$ are homology isomorphic to the $n$-fold suspensions $\Sigma^n(\overline{\BDiff_c^r(\bR^n)}) $ and $\Sigma^n(\Omega^n \overline{\mathrm{B}\Gamma}_n^r)$ respectively and proving that $f_1$ induces a homology isomorphism implies that
\[
\Sigma^n(\overline{\BDiff_c^r(\bR^n)})\to \Sigma^n(\Omega^n \overline{\mathrm{B}\Gamma}_n^r),
\]
also induces a homology isomorphism which implies the same for the map \ref{T}. Because suspension only shifts the homological degree.

The key is to observe that the map between filtration quotients
\[
\overline{f_k}\colon X_k/X_{k-1}\to Y_k/Y_{k-1},
\]
induces a homology isomorphism in a range that increases linearly in $k$ and the fact that in the limit we have the weak equivalence \ref{N}, a standard argument of comparison of spectral sequences implies that $f_1$ induces a homology isomorphism.

\section{Equivariant Mather-Thurston} Recall that for a topological group $G$, the universal principal $G$-bundle is $\mathrm{E}G\to\mathrm{B}G$, where $\mathrm{E}G$ is a contractible space on which $G$ acts freely. Given a map $f\colon X\to \mathrm{B}G$, one can choose a model for $F$, the homotopy fiber of the map $f$, on which $G$ acts. For example, the pullback $f^*(\mathrm{E}G)$ is a model for the homotopy fiber of $f$ on which $G$ acts naturally. It is standard to see that the homotopy quotient of this action
\[
f^*(\mathrm{E}G)\hcoker G\coloneqq \frac{f^*(\mathrm{E}G)\times \mathrm{E}G}{G},
\]
is homotopy equivalent to $X$. Therefore, we can choose a model for $\overline{\BDiff_c^r(M)}$, which is the homotopy fiber of
\[
\eta\colon \BcdrDiff M\to \BDiff^r_c(M),
\]
on which $\Diff_c^r(M)$ acts and the homotopy quotient $\overline{\BDiff_c^r(M)}\hcoker \Diff_c^r(M)$ is weakly equivalent to $ \BcdrDiff M$.

There is also a natural model for $\text{Sect}_c(\tau_M^*(\nu^r))$ in Mather-Thurston's theorem \ref{MT} on which $\Diff_c^r(M)$ acts as follows. Let $\gamma_n$ be the tautological $n$-dimensional vector bundle over $\mathrm{B}\text{GL}_n(\bR)$. Let $\mathrm{Bun}(\mathrm{T}M, (\nu^r)*\gamma_n)$  be the space of bundle maps $\mathrm{T}M\to (\nu^r)^*\gamma_n$ i.e. continuous maps that are linear isomorphisms on each fiber, equipped with the compact open topology. It is easy to define a map
\[
\text{Sect}_c(\tau_M^*(\nu^r))\to \mathrm{Bun}(\mathrm{T}M, (\nu^r)^*\gamma_n),
\]
which is a weak homotopy equivalence. The group $\Diff_c^r(M)$ acts on $\mathrm{Bun}(\mathrm{T}M, (\nu^r)^*\gamma_n)$ by precomposition with the differential of diffeomorphisms.  In my thesis, I observed that
\begin{thm}
The map in Mather-Thurston's theorem \ref{MT} can be made  $\Diff_c^r(M)$-equivariant.
\end{thm}
Hence, the classifying space $ \BcdrDiff M$ is weakly equivalent to $\mathrm{Bun}(\mathrm{T}M, (\nu^r)^*\gamma_n)\hcoker \Diff_c^r(M)$ which I used  to study the stable homology of surface diffeomorphism groups. Let me mention one consequence of this equivariance about invariants of flat bundles.

Recall that oriented circle bundles are classified by their Euler class. And we have $\BDiff_0^r(S^1)\simeq \mathrm{B}S^1$, whose integral cohomology is a polynomial ring on the Euler class. Hence, we have a map
\[
\bQ[e]\cong H^*(\mathrm{B}S^1; \bQ)\to H^*(\BdrDiffo {S^1}; \bQ).
\]
\begin{thm}[Morita]The powers of the Euler class $e^k$ are nontrivial in $H^*(\BdrDiffo {S^1}; \bQ)$ for all $k$.
\end{thm}
If we restrict the holonomy of the bundle to $\text{PSL}_2(\bR)$, this is no longer true and it turns out that $e^2$ vanishes in $H^4(\mathrm{B}\text{PSL}_2(\bR)^{\delta};\bQ)$. Morita used rational homotopy theory to prove this theorem but one can give a simpler proof using the equivariant Mather-Thurston's theorem. The space $\BdrDiffo {S^1}$ is weakly equivalent to $\mathrm{Bun}(\mathrm{T}S^1, (\nu^r)^*\gamma_1)\hcoker \Diff_0^r(S^1)$. Since $\Diff_0^r(S^1)$ acts nontrivially on the tangent vectors of $S^1$, the action on $\mathrm{Bun}(\mathrm{T}S^1, (\nu^r)^*\gamma_1)$ {\it does not} have a fixed point. But  $\Diff_0^r(S^1)$ is homotopy equivalent to the subgroup of rotations $S^1\subset \Diff_0^r(S^1)$. And it turns out that the action of $S^1$ on $\mathrm{Bun}(\mathrm{T}S^1, (\nu^r)^*\gamma_1)$ has fixed points. This implies that
\[
\mathrm{Bun}(\mathrm{T}S^1, (\nu^r)^*\gamma_1)\hcoker S^1\to \mathrm{B}S^1,
\]
has a section that gives a simpler proof of Morita's theorem.
\begin{problem}Let $\Diff^{\omega}_0(S^1)$ denote the orientation preserving analytic diffeomorphisms of $S^1$. Are the powers of the Euler class $e^k$ nontrivial in $H^*(\BDiff_0^{\omega}(S^1)^{\delta};\bQ)$?
\end{problem}
Thurston claimed that $e^3$ vanishes for analytic flat circle bundles but the proof was not written down and Etienne Ghys called this claim the lost theorem of Thurston!

In the above fixed-point argument, it is important that $S^1$ is a Lie group. For higher dimensional sphere bundles one can use torus actions to prove a generalization of Morita's theorem.
%and for a general Lie group $G$, using the same idea, we obtain the following.
%Any principal $G$-bundle is cobordant via $\Diff(G)$-bundles to a flat $G$-bundle (not necessarily flat principal $G$-bundle).

\begin{thm}[Nariman]
The classes $e^k$ for all $k$  are all nontrivial in $H^*(\BdrDiffo {S^{2n-1}};\bQ)$.
\end{thm}

\section{Controlled Mather-Thurston and Milnor-Wood inequalities} Recall that the Mather-Thurston theorem for $C^0$-foliations and Mather's acyclicity result ([8]) imply that
\[
\BdH_0(M)\to \BH_0(M),
\]
induces a homology isomorphism. Geometrically, this means for each $M$-bundle $M\to E\xrightarrow{\pi} B$ whose group structure is $\tH_0(M)$, there exists a  bordism $V$ from $B$ to $B'$ and an $M$ bundle $W\to V$ that extends the bundle $\pi$ and its restriction to $B'$ is a flat $M$-bundle. Intuitively, one expects that the bordism $V$ from $B$ to $B'$ ought to make the fundamental group of $B$ more and more complicated until $B'$ could accommodate a flat bundle with the same characteristic numbers as the bundle $\pi$.

For example for $M=S^1$, a topological oriented $S^1$-bundle $E\xrightarrow{p} \Sigma_g$ over $\Sigma_g$ a closed oriented surface of genus $g$ is classified by its Euler class $e(p)$. The Milnor-Wood inequality in this case gives the necessary and sufficient condition under which the circle bundle $p$ admits a flat structure, namely
\[
|\langle e(p), [\Sigma_g]\rangle|\leq 2g-2,
\]
where the left-hand side is the evaluation of the class $e(p)$ on the fundamental cycle $[\Sigma_g]$ and it is called the Euler number of the circle bundle $p$. In a more modern language, the Euler class for flat $S^1$-bundles is bounded in the sense of Gromov. So fixing the Euler number of a circle bundle, in order to make the bundle flat up to bordism, we have to increase the genus of the base so that it satisfies the Milnor-Wood inequality.

Inspired by physics, Mike Freedman in [2] took a constructive point of view and demanded two types of control on the bordism $V$ whose existence is guaranteed by the Mather-Thurston theorem. One of them is to formalize the idea that the bordism $V$ is directed from $B$ to $B'$ to increase ``the complexity'' of $\pi_1(B)$ and the other is to control the holonomy of the bundle on $B'$.

\begin{defn}
A  semi-s-cobordism  is a manifold triple $(V, B, B')$ with $\partial V=B\coprod -B'$ where $-B'$ means $B'$ with the reverse orientation, so that the inclusion $\iota\colon B\hookrightarrow V$ is a ``{\it simple}'' homotopy equivalence.
\end{defn}
The directionality of $V$ in the definition is that we do not assume that $\iota'\colon B'\hookrightarrow V$ is a homotopy equivalence. However, the definition implies that there is a (simple) deformation retraction $r\colon V\to B$ and the map $r\circ \iota'\colon B'\to B$ is a degree one map which implies that the map $\pi_1(B')\to \pi_1(B)$ is surjective.

Freedman uses low dimensional techniques to construct certain ``homological'' solid tori and the Bing double construction to do surgeries to construct the bordism $V$ in dimension $4$.
\begin{thm}[Freedman [2]] Suppose $B$ is a closed $3$-manifold and $M\to E\xrightarrow{\pi} B$ is an $M$-bundle. Then there exists a  {\it semi-s-cobordism} $V$ from $B$ to $B'$ and an $M$ bundle $W\to V$ that extends the bundle $p$ and its restriction to $B'$ is a flat $M$-bundle.
\end{thm}
Freedman conjectured that not only this statement holds in all dimensions but also even more strongly when $M$ is a Riemannian manifold, for every positive $\epsilon$, one can choose the semi-s-cobordism $V$ so that the holonomy of the flat $M$ bundle on $B'$ on a generating set of $\pi_1(B')$ is at most $\epsilon$ away from the isometry group of $M$ in the sup norm on the group $\tH_0(M)$.

In fact in higher dimensions, we later learned that the existence of such semi-s-cobordism was known using homological surgery techniques.
\begin{thm}[Hausmann and Vogel [6]] Let $M$ and $B$ be two closed topological manifolds where $\text{dim}(B)\geq 5$ and let $\xi$ be an $M$-bundle over $B$. Then this $\xi$ extends as an $ M$ bundle over $B$ to a semi-s-cobordism such that at the other end, it is a $C^0$-foliated $ M$ bundle.
\end{thm}

Demanding the bordism to be semi-s-cobordism is a qualitative control and the control on the holonomy which is related to the Milnor-Wood inequality is more quantitative. Given the existence of semi-s-cobordism reduction to $C^0$-foliated bundles is known in all dimensions except $4$, one interesting unknown case is the following.

\begin{problem}
Suppose $S^3\to S^7\to S^4$ is a ``generalized'' Hopf fibration. Does there exist a homology $4$-sphere $H$ and a degree one map $H\to S^4$ such that the pull-back of the generalized Hopf fibration is a flat $S^3$-bundle?
\end{problem}
In fact in the 80s, Ghys asked if the Milnor-Wood inequality holds for oriented flat $S^3$-bundles.
\begin{quest}[Ghys]
Let $M^4$ be a compact orientable $4$-manifold and $ \pi_1(M) \xrightarrow{\rho} \Homeo_\circ(\bS^3)$ be a representation. Is it true that the Euler number of the flat $S^3$-bundle associated with $\rho$ over $M$ is bounded by a number depending only on $M$?
\end{quest}
Monod and I answered Ghys' question negatively ([13]) by proving the following result.
\begin{thm}(\cite[Theorem 1.8]{MR4588567})
The fourth bounded cohomology of the group $\tdH_0(S^3)$  vanishes.
\end{thm}
This, in particular, implies that all non-trivial classes in $H^4(\BdH_0(S^3);\bR)$ are unbounded so the Euler class for oriented flat $S^3$ bundle is not bounded. Hence, in Freedman's conjecture, we might have the semi-s-cobordism to flat bundles in higher dimensions, but we may not be able to quantitatively control the topology of $B'$. However, the unboundedness of the Euler class seems to suggest that one might be able to flatten $S^3$-bundles over $4$-manifolds up to bordism without changing the simplicial volume of the base much.

Our proof of the failure of the Milnor-Wood inequality for $S^3$-bundles is not constructive and it leads to both conceptual and computational questions (\cite[Section 7]{MR4588567}) but most importantly we lack geometric intuition to construct examples to exhibit the failure of Milnor-Wood in this case.

\subsection*{Acknowledgments} I am grateful to Laszlo Lempert, Mehdi Yazdi, Fran\c cois Laudenbach, and Mehrdad Shahshahani for their helpful comments on the first draft of this article to improve its readability.  The author was partially supported by  NSF CAREER Grant DMS-2239106 and Simons Foundation Collaboration Grant (855209). I would like to thank Allison May for the opening figure, Patrick Massot for the figure $8$, and my father for helping me with other figures.

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