<EMAIL_ADDRESS>
# Rapid Object Annotation

Misha Denil

###### Abstract

In this report we consider the problem of rapidly annotating a video with
bounding boxes for a novel object. We describe a UI and associated workflow
designed to make this process fast for an arbitrary novel target.

###### Contents

  1. 1 Introduction
  2. 2 Annotation tool
    1. 2.1 Viewport
    2. 2.2 Timeline
    3. 2.3 Autotrack
    4. 2.4 Sparklines
    5. 2.5 Smartjump
    6. 2.6 Extreme clicking
  3. 3 Experiments
    1. 3.1 Target objects
    2. 3.2 Videos
    3. 3.3 Annotation styles
  4. 4 Results
    1. 4.1 Annotation time
    2. 4.2 Label quality
  5. 5 Methods
    1. 5.1 CenterNet
    2. 5.2 Feature based tracking
    3. 5.3 Feature caching
  6. 6 Discussion
  7. A User interface controls

##  1 Introduction

We envision the following workflow to train a detector for a novel object:

  1. 1.

Record a video of the target object from many angles.

  2. 2.

Annotate the video frames to identify the target object.

  3. 3.

Use the annotations to fine tune a detection model.

In terms of human time and effort the second task is by far the most
expensive. Recording a video is easily done using a cell phone, and fine
tuning a model on a new data set is a standard task in machine learning, and
the setup time can be amortized by standardizing data formats. The remaining
challenge is to make annotation of the collected video easy and fast. We focus
on optimizing the use of human time in this process.

We design an annotation tool that provides assistance to the operator by
taking advantage of a pretrained objectness prior to identify the extent of
objects and to propagate labels between frames. We evaluate the tool by
annotating several videos and measuring the number of high quality bounding
boxes obtained per second of human annotation time.

##  2 Annotation tool

The annotation UI is shown in Figure 1. The UI is composed of three main
components:

  * •

The viewport shows a single video frame, as well as any annotations and
predictions associated with that frame.

  * •

The timeline situates the frame in time. It shows an indicator for the current
frame shown in the viewport, as well as markers to indicate frames for which
there are annotations.

  * •

The sparklines show indicators about the relationship of adjacent frames of
the track to each other. This is useful for identifying when object identity
or position may have been lost.

The rest of this section describes the tool, but the easiest way to get a feel
how it works is to watch a video of it being used.

Figure 1: Example of the annotation UI.

###  2.1 Viewport

The viewport displays a frame, as well as descriptor locations and boxes. Each
frame can have one or zero annotated locations associated with it. Annotated
locations (and their associated boxes) are shown in a different color than
predictions so that they can be easily distinguished.

In Figure 2 we show why it is useful to display more than just the descriptor
location in the viewport. In this case the descriptor location (at the bottom
of the yellow face of the thermometer) appears reasonable for the target
object, but the bounding box produced by the descriptor at this location shows
that the discriptor content is more likely associated with the table than the
target object.

Figure 2: Example of why it is useful to show the bounding boxes. The
descriptor location in this example seems reasonable for the target object,
but the predicted bounding box shows that the descriptor content is not.

###  2.2 Timeline

The timeline allows navigation through the video and shows indicators for
which frames have been annotated. A cursor above the tineline shows the
location of the current frame in the video, and clicking on the timeline moves
the current frame to the location of the click. Indicators under the timeline
show the locations of annotated frames.

###  2.3 Autotrack

Autotracking allows a set of sparsely annotated frames in a video to be
extended to a label for every frame. The mechanism for propagating labels is
described in Section 5.2.

###  2.4 Sparklines

Sparklines appear under the timeline and are aligned with it. They show
information about how the track produced by autotracking changes from frame to
frame. Specifically there are two lines, blue and red, which show the change
in descriptor location and change in box area from frame to frame,
respectively. These lines can be used to quickly identify potential parts of
the video where automatic tracking has failed, allowing the operator to focus
attention on inspecting these areas for further annotation.

###  2.5 Smartjump

Smartjump is a feature that allows automatically jumping to the frame where
the current track shifts the most. A large jump in track location often
indicates loss of tracking, making this a good frame to annotate.

###  2.6 Extreme clicking

Extreme clicking (Papadopoulos et al., 2017) creates axis aligned bounding
boxes from a set of “extreme” points. To create a bounding box the operator
clicks four times, once each at the two horizontal and two verical extremes of
the target object. The resulting annotation is the smallest axis aligned
bounding box that contains the selected points. We use extreme clicking in
this report as the baseline exisiing method for bounding box annotation to
compare against, and also for the creation of ground truth data to evaluate
the lables produced by our tool.

##  3 Experiments

###  3.1 Target objects

We use three semi-arbitrarily chosen objects as the targets for the
experiments. An example of each object is shown in Figure 3.

Figure 3: The three target objects we consider in this report. From left to
right they are an infra-red thermometer, a pair of pliers, and a clock.

The infra-red thermometer was chosen because it is a sort of familiar looking
object that does not appear in the label set of MSCOCO, and is also quite
unlikely to have appeared unlabeled in MSCOCO incidentally. The pliers were
chosen because they are articulated and also highly non-convex, which
potentially makes labeling harder. The alarm clock was chosen because I happen
to have one in my apartment.

###  3.2 Videos

For each target object we collect two videos. The videos are trimmed to 30
seconds and resampled at 10 fps so that each one provides exactly 300 frames.

For each object the first video is taken with the camera held in hand and the
object sitting on a desk. The camera is moved around by the operator to
collect different views of the object, and occasionally the operator reaches
into the frame to reorient the object (for example by flipping it over).

The second video of each object is captured by a stationary mounted camera.
The operator stands in the camera frame and holds the object in view, moving
it around so that it is captured from multiple viewpoints.

###  3.3 Annotation styles

We use the tool to annotate several videos with different sets of features
enabled. In the following we distinguish between a _label_ and an
_annotation_. A label is a point and a bounding box (in some configurations we
record only a point) associated with a frame. An annotation is a label
produced by human input. We collect labels for all frames in all videos.
Depending on the annotation style this may be a result of much fewer
annotations.

#### XClick

Frames are presented in a random order, and bounding boxes and center points
are annotated in each frame using extreme clicking (Papadopoulos et al.,
2017). Data produced by extremely clicking is considered “ground truth” when
such a thing is appropriate. In this configuration the video timeline and
sparklines are hidden.

#### Click

Frames are presented in a random order and the operator is asked to click on
the center of the target object in each frame. The selected location is shown
in the viewport, but no additional feedback is given. The operator can change
their selected point by clicking again. In this configuration the video
timeline and sparklines are hidden.

#### Boxes

Frames are presented in a random order and the operator is asked to click on
the center of the target object in each frame. The selected location in the UI
along with a predicted bounding box for the selected point. The operator can
change their selected point by clicking again. In this configuration the video
timeline and sparklines are hidden.

#### Autotrack

Frames are presented using the video timeline. The operator can navigate
through frames using the standard timeline controls. The operator can click a
point in the image to annotate a point in the current frame, and can also
press r to request the system fill in a track for unannotated frames.
Predicted boxes are shown for the track but not for the annotated points. In
this configuration the sparklines are hidden.

#### Autotrack-boxes

The same configuration as Autotrack but predicted boxes are also shown for the
annotated points.

#### Autotrack-boxes-sparklines

The same configuration as Autotrack-boxes but the sparklines are no longer
hidden.

#### Autotrack-boxes-sparklines-smartjump

The same configuration as Autotrack-boxes-sparklines but jumping nativgates to
the frame with the largest jump in track position, instead of to a random
unannotated frame.

##  4 Results

###  4.1 Annotation time

Figure 4: Annotation times under different tools settings, aggregated across
videos and objects. The red vertical lines indicate “real time.” The black
vertical lines show the time per box reported elsewhere for xclick.

The distributions of annotation times for each configuration are shown in
Figure 4. We show both the time per annotation and time per label, which are
in general not the same. For xclick, click and boxes every frame is annotated
so the time per annotation is equal to the time per label. For the other
settings annotation proceeds until the operator is satisfied that every frame
has a label, which leads to different numbers of annotations for each video.
The fraction of annotated frames for each run is shown in Figure 5.

We can compare ourselves to numbers reported elsewhere on xclick for bounding
box annotation. Our average xclick times are slightly faster than they report,
likely because our tasks are slightly simpler (with exactly one object per
frame and no need to indicate object class).

Setting | Annotation Time
---|---
xclick | 3:30:45
autotrack | 0:50:44
boxes | *0:42:03
click | 0:40:31
autotrack-boxes | 0:37:02
autotrack-boxes-spark | 0:34:33
autotrack-boxes-spark-jmp | 0:32:27
Total | 7:28:05

Figure 5: Left: Fraction of annotated frames for each method where annotating
less than 100% of frames is possible. Right: Total time spent annotating in
the different styles. Each annotation style counts time to annotate 6 videos,
except for “boxes” counts time to annotate 2 videos. If we had annotated all 6
videos using boxes we would expect it to have taken 2:32:12.

All annotation methods were performed on all videos, with the exception of
boxes, which was only done for the two thermometer videos. The decision to not
include boxes was made because it is quite time consuming to annotate every
frame and it was clear early on that this approach fairs poorly in the time
per label metric. The following table shows time spent annotating videos in
each configuration.

###  4.2 Label quality

In addition to timing statistics we also examine the quality of the labels
produced by our tool. In order to do this we treat the bounding boxes from
xclick as ground truth bounding boxes (and the bounding box centers as ground
truth points). XClick bounding boxes are most likely to be reliable, given
that every frame is annotated by an operator (i.e. all labels are annotations)
and the annotation method has been shown already to be highly reliable.

In Figure 6 (left) we show the discrepancy between ground truth point track
and point tracks produced by our tool. Discrepancy is measured in normalized
pixel distance, that is a metric where the image is covered by the coordinates
$[0,1]\times[0,1]$, making the units insensitive to the image size.

In Figure 6 (right) we show the agreement between the ground truth box track
and the box tracks produced by our tool. Note that “click” does not appear in
the box track agreement figure because that annotation style does not produce
a box track. Agreement is measured as the intersection over union of the label
and ground truth box, which gives unitless values independent of the image
size. We also mark the value $0.88$ on the figure, which was shown by
Papadopoulos et al. (2017) to be the approximate limit of average inter-
annotator agreement on Pascal VOC data. We should not expect to achieve
average IoU values higher than this threshold.

Figure 6: Agreement between xclick annotations and different annotation
configurations using our tool. Left: Discrepancy in the point tracks (center
points of the labeled bounding boxes) measured in normalized pixel coordinates
(lower is better; 0 is perfect match). Right: Discrepancy in the box tracks,
measured in units of intersection over union (higher is better; 1 is perfect
match).

The standard metrics for evaluating detectors involve computing a precision-
recall curve by varying a detection threshold and then computing the area
under that curve (with some embellishments, so it’s not precisely the
AUC111https://medium.com/@jonathan_hui/map-mean-average-precision-for-object-
detection-45c121a31173). Since we are evaluating annotations rather than
detections there is no detection threshold upon which to base an AUC curve.
Instead we use a threshold on IoU to determine if each bounding box is
correct, and compute the accuracy of the box tracks produced by each
annotation method. Following Papadopoulos et al. (2017) we evaluate using IoU
thresholds of 0.7 and 0.5, and show the resulting accuracies in Figure 7.

Finally we combine the time and accuracy metrics to measure the proportion of
high quality bounding boxes per unit time. We arbitrarily pick an IoU
threshold of 0.7 to be a “high quality” bounding box and in Figure 8 we show
the number of such bounding boxes produced per second using the different
annotation styles.

It seems reasonable to expect about 0.75 high quality bounding boxes per
second using our annotation method, which should be contrast with 0.14
bounding boxes per second using XClick. This corresponds to a 5.3x increase in
annotation speed beyond XClick, and is likely to be a slight over estimate
because it assumes that every XClick box is high quality.

Figure 7: Accuracy of the different annotation methods when evaluated as a
detector with xclick boxes as ground truth. We do not compute average
precision scores since there is always exactly one “detection” for each ground
truth instance. Instead we compute accuracy of the labels by thresholding IoU
with the xclick boxes to determine if a label is correct. Red lines shown the
value of the corresponding metric for extreme clicking as evaluated on Pascal
VOC (see Papadopoulos et al. (2017), Table 1; we show the stronger value from
Pascal 2007 and Pascal 2012 in both cases). Note that the x-axis has a
different scale between the two plots.

##  5 Methods

###  5.1 CenterNet

CenterNet (Zhou et al., 2019) is a model for “Object detection, 3D detection
and pose estimation.” It was later extended by the same authors to a tracking
model (Zhou et al., 2020). The tracking model is not discussed in this report,
we do something much more naive for tracking here. Incorporating ideas from
the centernet tracking paper is probably a good idea for future work.

The key idea of centernet is to represent an object as a single keypoint. The
keypoint is located at the object center, and has an associated feature vector
(which I will henceforth call a descriptor) that is trained to predict
properties of the object. The network produces a dense descriptor map for
every location in an image, and at training time only the descriptors that
correspond to ground truth detections are supervised. This situation is
diagrammed in Figure 9.

Figure 8: Number of high quality bounding box labels per second of annotation
time. The x-axis is in units of high quality boxes per second.

In addition to bounding boxes, the descriptors are also used to produce class
specific heatmaps for the different categories of objects that the detector
recognizes. At test time the model searches for peaks in the class specific
heatmaps to choose detection centers, and returns bounding boxes regressed
from the descriptors at the corresponding locations. The heatmaps are class
specific, and are therefore tied to the original label set used to train the
model. However, the model uses a single set of weights for bounding box
prediction across all classes, making the bounding box predictions class
agnostic.

###  5.2 Feature based tracking

We work in continuous normalized image coordinates. Normalized means that the
domain $[0,1]\times[0,1]$ covers all valid locations in the image. Continuous
means that we treat pixel data as regularly sampled grid of measurements from
an underlying continuous signal where the values are sampled at the center of
the pixel. The ostensible continuous signal is recovered from pixel values
using bilinear interpolation for interior values, and constant extrapolation
at the boundaries. We apply this logic both to pixel values and to the dense
descriptor maps produced by centernet.

Figure 9: CenterNet supervision. The backbone network analyzes an image and
produces a dense descriptor map for each location in the image. Locations
corresponding to ground truth object centers are used to predict bounding
boxes and the resulting boxes are supervised using a standard bounding box
regression loss.

This is a convenient coordinate system to work in, if we have an image $I$
with a dense descriptor map $\phi(I)$, then the the image at location $p$ has
value $I[p]$ and the corresponding descriptor is $\phi(I)[p]$, regardless of
the relative sizes of the image and descriptor maps.

Suppose we have two images $I_{t}$ and $I_{t^{\prime}}$ with $t^{\prime}-t>1$
and corresponding point annotations $p_{t}$ and $p_{t^{\prime}}$. From this we
obtain two annotated descriptors $d_{t}=\phi(I_{t})[p_{t}]$ and
$d_{t^{\prime}}=\phi(I_{t^{\prime}})[p_{t^{\prime}}]$ by sampling the dense
descriptor maps at each annotated point. Using the annotated descriptrs we can
predict a descriptor track for intermediate frames $t<\tau<t^{\prime}$ as

$\displaystyle\hat{d}_{\tau}=(1-\frac{\tau-t}{t^{\prime}-t})d_{t}+\frac{\tau-t}{t^{\prime}-t}d_{t^{\prime}}$

which is just linear interpolation in descriptor space.

The interpolated descriptors are turned back into locations by computing

$\displaystyle\hat{p}_{\tau}=\arg\min_{p}||\phi(I_{\tau})[p]=\hat{d}_{\tau}||$

which corresponds to nearest neighbour matching of the interpolated descriptor
to the corresponding image frame descriptor map. The predicted locations can
further be turned into boxes by sampling the box predictions in a similar way.

###  5.3 Feature caching

The biggest obstacle to making effective use of predictions from the centernet
model during annotation is speed. A latency of a quarter second to obtain a
bounding box from a point is extremely noticeable and frustrating when you are
trying to refine a selected location to get the best bounding box possible. To
make the UI work at interactive speeds we pre-process every frame in each
video using the centernet backbone and cache the descriptor maps as well as
the bounding box predictions for each point in each frame. Caching features
allows us to avoid re-runing the network to get a new box prediction, and
makes the experience substantially smoother for the operator.

##  6 Discussion

Using the tool described in this report we can reduce the time to annotate a
300 frame video of a single target object by at least a factor of 5.3 compared
to annotating each frame using extreme clicking. This turns a 30-40 minute
annotation task into one that can be completed in less than 10 minutes.
Although the labels obtained in this way appear to be lower quality they do
not appear to be substantially so. Annotating every frame with assistance from
the objectness prior of centernet increases the annotation quality over tracks
produced by the autotrack feature at the cost of dramatically increased
annotation time (although still faster than extreme clicking). It remains to
be seen if the quality is sufficient to train detectors to re-identifiy the
target object in new videos.

UI latency was the biggest challenge in making the centernet predictions
useful during annotation. Refining a predicted bounding box by refining the
selected descriptor is extremely annoying when there is noticeable latency
between clicking and seeing the result. Latency also reduces the number of
refinements that can be tried before it would have been faster to draw the box
directly.

In the current implementation the autotracking feature still takes 0.5-1s to
generate a new track for each frame, and this latency limits the usefulness of
the sparklines and the smartjump feature (since smartjump relies on the track
to function).

The centernet model we used is trained on MSCOCO which has people and tables
as categories. I noticed anecdotally that the model sometimes gets distracted
by these objects e.g. when the target novel object is near the center of a
table the model will prefer to identify the table. I observed similar behavior
with a person in frame manipulating the target object as well. This could
likely remedied by removing people and tables from the original dataset and
retraining the centernet (or perhaps adding a separate head that is
specialized to these classes).

The extreme clicking UI would beenfit from a large crosshair cursor to help
distinguish near ties for extreme points. The authors of the extreme clicking
paper mention that this is not useful, but I think it would have been helpful
for annotating the pliers especially, where it was often necessairy to make
judgements about the relative positions of highly separated points. This
feature is easy to implement so I see no reason not to include it in the
future.

It is harder than you might expect to flim an an object from many angles with
a handheld camera while keeping it consistently in frame. The tool could
benefit from the ability to exclude frames from the video so that frames can
be excluded from annotation (and later from the training set for the
downstream detector). I actually implemented a basic exclusion feature before
running the experiments in this report, but I found it too awkward to use, so
further thought is required on how to present it.

A variant I would like to try in the future is using extreme clicking to
annotate boxes and selecting descriptors to match the operator provided boxes
instead of having the operator pick descriptor locations directly. This would
slow down selecting easy cases a bit but would likely speed up tricky
annotation cases, and would allow the tool to identify when there is no good
descriptor match for the desired annotation.

## References

  * Papadopoulos et al. (2017) Dim P Papadopoulos, Jasper RR Uijlings, Frank Keller, and Vittorio Ferrari.  Extreme clicking for efficient object annotation.  In _Proceedings of the IEEE international conference on computer vision_ , pages 4930–4939, 2017.
  * Zhou et al. (2019) Xingyi Zhou, Dequan Wang, and Philipp Krähenbühl.  Objects as points.  _arXiv preprint arXiv:1904.07850_ , 2019.
  * Zhou et al. (2020) Xingyi Zhou, Vladlen Koltun, and Philipp Krähenbühl.  Tracking objects as points.  _arXiv preprint arXiv:2004.01177_ , 2020.

##  Appendix A User interface controls

The application controls are largely the same across different annotation
styles, although some controls are only available when the corresponding
feature is enabled (e.g. timeline navigation requires the timeline be
enabled).

Regardless of the annotation style left clicking in the viewport always
selects the corresponding point, and middle clicking anywhere in the viewport
clears any annotation (or partial annotation) in the current frame.

When the timeline is active frames can be selected with the mouse by clicking
on the timeline. The current frame cursor above the timeline indicates the
position of the current frame in the video, and interacting with the timline
(clicking, dragging, etc) behaves as you would expect a video scrubber to
beahve.

There are several keyboard actions to support navigation and annotation, which
are shown in the following table:

T | A | Key | Behavior
---|---|---|---
✓ |  | left | Shift backwards by 1 frame
✓ |  | right | Shift by 1 frame
✓ |  | shift+left | Shift backwards by 10 frames
✓ |  | shift+right | Shift forwards by 10 frames
✓ |  | ctrl+left | Shift backwards to the nearest annotated frame
✓ |  | ctrl+right | Shift forwards to the nearest annotated frame
|  | f | Jump to an unannotated frame
✓ | ✓ | r | Refresh the track

The T and A columns indicate if the timeline or autotrack must be enabled for
the key to be active. Requirements are inclusive, so refreshing the track is
only possible when both the timeline and autotrack to be enabled. The behavior
of the jump key changes from random jumping to smart jumping when smartjump is
enabled.


# On the “Einstein-Gauss-Bonnet Gravity in Four Dimension”

Li-Ming Cao${}^{a\,,b}$111e-mail address<EMAIL_ADDRESS>Liang-Bi
Wub222e-mail address<EMAIL_ADDRESS>aPeng Huanwu Center for
Fundamental Theory, Hefei, Anhui 230026, China  b Interdisciplinary Center for
Theoretical Study and Department of Modern Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China

###### Abstract

To ensure the existence of a well defined linearized gravitational wave
equation, we show that the spacetimes in the so-called “Einstein-Gauss-Bonnet
gravity in four dimension” have to be locally conformally flat.

††preprint:  ICTS-USTC/PCFT-21-11

##  I introduction

Recently, a novel Einstein-Gauss-Bonnet gravity in four dimension has been
proposed Glavan:2019inb . The action of the theory is based on

$S=\int
d^{D}x\sqrt{-g}\left(R-2\Lambda+\frac{\hat{\alpha}}{D-4}L_{\text{GB}}\right)\,,$
(1)

where $R$ is the $D$-dimensional Ricci scalar, and $\Lambda$ is the
cosmological constant. The Gauss-Bonnet term is given by

$\displaystyle
L_{\text{GB}}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\,.$
(2)

Here, the usual Gauss-Bonnet coupling constant $\alpha$ has been replaced by
$\hat{\alpha}/(D-4)$. Then the so-called four dimensional Einstein-Gauss-
Bonnet theory is defined by considering the limit $D\to 4$. In
Ref.Glavan:2019inb , it is also claimed that under this limit the Gauss-Bonnet
invariant gives rise to non-trivial contributions to gravitational dynamics.
This is quite different from the content of the well-known Lovelock theorem
Lovelock:1972vz . In view of the importance, it is necessary to investigate
the theory from various perspectives. For example, the well-posedness of the
initial value problem of the theory.

To investigate the local well-posedness of the initial value problem of a
gravitational theory, one should pay attention to the principle symbol of the
linearized perturbation equation Reall:2021voz ; Reall:2014pwa ;
Papallo:2017qvl ; Reall:2014sla . Consider the equations of motion linearized
around a background solution, we get

$\displaystyle
P^{\mu\nu\rho\sigma\alpha\beta}\partial_{\alpha}\partial_{\beta}\delta
g_{\rho\sigma}+\cdots=0\,,$ (3)

where the ellipses denotes the terms with lower than 2-derivatives acting on
$\delta g_{\mu\nu}$. Let $\xi_{\mu}$ be an arbitrary covector, then
$P^{\mu\nu\rho\sigma\alpha\beta}\xi_{\alpha}\xi_{\beta}$ is the so-called the
principle symbol. Hyperbolicity and causality of a gravitational theory is
determined by the principal symbol, so it is natural to require that the
principle symbol should have a well defined behavior in this novel Einstein-
Gauss-Bonnet gravity. Otherwise, we have no well defined linearized
gravitational wave equations.

##  II Principle symbol

Suppose we have a $D$-dimensional spacetime. Based on the above ideas, the
principal symbol should be well defined under the limit $D\rightarrow 4$. For
a diffeomorphism invariant theory, based on the discussion of symmetry of the
tensor $P^{\mu\nu\rho\sigma\alpha\beta}$, Reall has proved that the principle
symbol has a form Reall:2021voz ,

$\displaystyle
P^{\mu\nu\rho\sigma}(\xi)=C^{\mu(\rho|\alpha\nu|\sigma)\beta}\xi_{\alpha}\xi_{\beta}\,,$
(4)

where for Einstein-Gauss-Bonnet gravity in $D>4$ dimension we have

$\displaystyle
C_{\mu\rho\alpha}{}^{\nu\sigma\beta}=\frac{1}{2}\delta_{\mu\rho\alpha}^{\nu\sigma\beta}+\hat{\alpha}\Big{[}\frac{D+1}{(D-1)(D-2)}\delta_{\mu\rho\alpha}^{\nu\sigma\beta}R-\frac{2}{D-2}A_{\mu\rho\alpha}{}^{\nu\sigma\beta}+\frac{1}{2(D-4)}W_{\mu\rho\alpha}{}^{\nu\sigma\beta}\Big{]}\,.$
(5)

In the above equation, $\delta_{\mu\rho\alpha}^{\nu\sigma\beta}$ is a
generalized Kronecker-delta tensor, while
$A_{\mu\rho\alpha}{}^{\nu\sigma\beta}$ and
$W_{\mu\rho\alpha}{}^{\nu\sigma\beta}$ are defined as follows

$A_{\mu\rho\alpha}{}^{\nu\sigma\beta}=\left|\begin{array}[]{ccc}R_{\mu}^{\nu}&R_{\mu}^{\sigma}&R_{\mu}^{\beta}\\\
\delta_{\rho}^{\mu}&\delta_{\rho}^{\sigma}&\delta_{\rho}^{\beta}\\\
\delta_{\alpha}^{\nu}&\delta_{\alpha}^{\sigma}&\delta_{\alpha}^{\beta}\end{array}\right|+\left|\begin{array}[]{ccc}\delta_{\mu}^{\nu}&\delta_{\mu}^{\sigma}&\delta_{\mu}^{\beta}\\\
R_{\rho}^{\mu}&R_{\rho}^{\sigma}&R_{\rho}^{\beta}\\\
\delta_{\alpha}^{\nu}&\delta_{\alpha}^{\sigma}&\delta_{\alpha}^{\beta}\end{array}\right|+\left|\begin{array}[]{ccc}\delta_{\mu}^{\nu}&\delta_{\mu}^{\sigma}&\delta_{\mu}^{\beta}\\\
\delta_{\rho}^{\mu}&\delta_{\rho}^{\sigma}&\delta_{\rho}^{\beta}\\\
R_{\alpha}^{\nu}&R_{\alpha}^{\sigma}&R_{\alpha}^{\beta}\end{array}\right|\,,$
(6)

and

$\displaystyle
W_{\mu\rho\alpha}{}^{\nu\sigma\beta}=36\delta_{[\mu}{}^{[\nu}C_{\rho\alpha]}{}^{\sigma\beta]}\,,$
(7)

where $R_{\mu\nu}$ is the Ricci tensor of the spacetime, and
$C_{\rho\alpha\sigma\beta}$ is the Weyl tensor. It is not hard to find that
$W_{\mu(\rho|\alpha\nu|\sigma)\beta}$ is vanished when $D=4$. So the last term
in the righthand of Eq.(5) is $0/0$ type when $D=4$. To ensure the existence
of the limit $D\rightarrow 4$, we have to impose a condtion

$\displaystyle W_{\mu(\rho|\alpha\nu|\sigma)\beta}=0\,,$ (8)

for all $D>4$. However, in the case with $D>4$, by contracting the two indices
$\rho$ and $\sigma$ in Eq.(8), we obtain

$\displaystyle C_{\mu\alpha\nu\beta}=0\,.$ (9)

This means that the spacetimes of the theory have to be locally conformally
flat. Recently, based on the equations of motion, people have realized that
the theory is well defined only when

$\displaystyle\mathcal{L}_{\mu\nu}=C_{\mu\alpha\beta\gamma}C_{\nu}{}^{\alpha\beta\gamma}-\frac{1}{4}g_{\mu\nu}C_{\alpha\beta\gamma\delta}C^{\alpha\beta\gamma\delta}$
(10)

is vanished for $D>4$, and a well defined four dimensional Einstein-Gauss-
Bonnet theory can be generated as $D\to 4$ Gurses:2020rxb ; Gurses:2020ofy ;
Arrechea:2020gjw ; Arrechea:2020evj . Here, with a similar logic, we obtained
a more restrictive condition, i.e., Eq.(9), on the metrics of the theory.

##  III Examples

To make the problem more transparent, let us consider a spacetime manifold
$\mathcal{M}^{D}\cong M^{m}\times N^{n}$ with a metric

$g_{\mu\nu}dx^{\mu}dx^{\nu}=g_{ab}(y)dy^{a}dy^{b}+r^{2}(y)\gamma_{ij}(z)dz^{i}dz^{j}\,,$
(11)

where $D=m+n$, and the coordinates $\\{x^{\mu}\\}$ is given by
$\\{y^{1}\,,\cdots y^{m}\,;z^{1}\,,\cdots\,,z^{n}\\}$. The tuple
$(M^{m}\,,g_{ab})$ forms a $m$-dimensional Lorentzian manifold if $m>1$, and
$(N^{n},\gamma_{ij})$ is an $n-$dimensional Riemann manifold. This Riemann
manifold $(N^{n},\gamma_{ij})$ is assumed to be an Einstein manifold, i.e.,

$\displaystyle\hat{R}_{ij}=(n-1)K\gamma_{ij}\,,$ (12)

where $\hat{R}_{ij}$ is the Ricci tensor of $(N^{n},\gamma_{ij})$, and $K$ is
the sectional curvature of the space. The metric compatible covariant
derivatives associated with $g_{\mu\nu}$, $g_{ab}$, and $\gamma_{ij}$ are
denoted by $\nabla_{\mu}$, $D_{a}$, and $\hat{D}_{i}$, respectively. For the
metric (11), the tensor perturbation can be put into a form Cao:2021sty

$\displaystyle\Big{(}P^{ab}{}_{ij}{}^{kl}D_{a}D_{b}+P^{mn}{}_{ij}{}^{kl}\hat{D}_{m}\hat{D}_{n}+P^{a}{}_{ij}{}^{kl}D_{a}+V_{ij}{}^{kl}\Big{)}\Big{(}\frac{h_{kl}}{r^{2}}\Big{)}=-\frac{2\kappa_{D}^{2}}{r^{2}}\delta
T_{ij}\,,$ (13)

where

$\displaystyle
P^{ab}{}_{ij}{}^{kl}=P^{ab}\delta_{i}{}^{k}\delta_{j}{}^{l}-\frac{4\alpha}{r^{2}}g^{ab}\hat{C}_{i}{}^{k}{}_{j}{}^{l}\,,$
(14) $\displaystyle
P^{a}{}_{ij}{}^{kl}=P^{a}\delta_{i}{}^{k}\delta_{j}{}^{l}-4\alpha(n-2)\frac{D^{a}r}{r}\frac{\hat{C}_{i}{}^{k}{}_{j}{}^{l}}{r^{2}}\,,$
(15) $\displaystyle
P^{mn}{}_{ij}{}^{kl}=P^{mn}\delta_{i}{}^{k}\delta_{j}{}^{l}+\frac{4\alpha}{r^{2}}\big{(}\hat{C}_{j}{}^{knl}\delta_{i}{}^{m}+\hat{C}_{i}{}^{knl}\delta_{j}{}^{m}+\hat{C}_{j}{}^{mln}\delta_{i}{}^{k}+\hat{C}_{i}{}^{mln}\delta_{j}{}^{k}-\hat{C}^{mknl}\gamma_{ij}-\hat{C}_{i}{}^{k}{}_{j}{}^{l}\gamma^{mn}\big{)}\,,$
(16)

and

$\displaystyle V_{ij}{}^{kl}$ $\displaystyle=$ $\displaystyle
V\delta_{i}{}^{k}\delta_{j}{}^{l}+\frac{2\hat{C}_{i}{}^{k}{}_{j}{}^{l}}{r^{2}}+\alpha\Bigg{\\{}4\Bigg{[}{}^{m}\\!{R}-2(n-3)\frac{{}^{m}\\!\Box
r}{r}+(n^{2}-7n+16)\frac{K}{r^{2}}-(n-3)(n-4)\frac{(Dr)^{2}}{r^{2}}\Bigg{]}\frac{\hat{C}_{i}{}^{k}{}_{j}{}^{l}}{r^{2}}$
(17)
$\displaystyle-\frac{8}{r^{4}}\hat{C}_{imjn}\hat{C}^{mknl}+\frac{4}{r^{4}}\hat{C}_{mnj}{}^{k}{}\hat{C}^{mn}{}_{i}{}^{l}-\frac{2}{r^{4}}\hat{C}^{mnpl}\hat{C}_{mnp}{}^{k}\gamma_{ij}+\frac{\hat{C}^{mnpq}\hat{C}_{mnpq}}{r^{4}}\delta_{i}{}^{k}\delta_{j}{}^{l}\Bigg{\\}}\,.$

In the above equations, $\alpha=\hat{\alpha}/(D-4)$, and we have defined

$\displaystyle
P^{ab}=g^{ab}+2(n-2)\alpha\left\\{2\frac{D^{a}D^{b}r}{r}+\left[(n-3)\frac{K-(Dr)^{2}}{r^{2}}-2\frac{\prescript{m}{}{\Box}r}{r}\right]g^{ab}\right\\}-4\alpha\cdot{}^{m}\\!G^{ab}\,,$
(18) $\displaystyle
P^{mn}=\Bigg{\\{}1+2\alpha\left[{}^{m}\\!{R}-\frac{2(n-3){}^{m}\\!{\Box}r}{r}+(n-3)(n-4)\frac{K-(Dr)^{2}}{r^{2}}\right]\Bigg{\\}}\frac{\gamma^{mn}}{r^{2}}\,,$
(19) $\displaystyle P^{a}$ $\displaystyle=$ $\displaystyle
n\frac{D^{a}r}{r}+2(n-2)\alpha\Bigg{\\{}4\frac{D^{a}D^{b}r}{r}+\Big{[}{}^{m}\\!{R}-2(n-1)\frac{{}^{m}\\!{\Box}r}{r}$
(20)
$\displaystyle+(n-2)(n-3)\frac{K-(Dr)^{2}}{r^{2}}\Big{]}g^{ab}\Bigg{\\}}\frac{D_{b}r}{r}-8\alpha\cdot{}^{m}\\!G^{ab}\frac{D_{b}r}{r}\,,$

and

$\displaystyle V$ $\displaystyle=$
$\displaystyle{}^{m}\\!{R}-2(n-1)\frac{{}^{m}\\!{\Box}r}{r}+\frac{n(n-3)K}{r^{2}}-\frac{(n-1)(n-2)(Dr)^{2}}{r^{2}}-2\Lambda$
(21) $\displaystyle+$
$\displaystyle\alpha\Bigg{\\{}{}^{m}\\!L_{GB}+8(n-1)\cdot{}^{m}\\!G^{ab}\frac{D_{a}D_{b}r}{r}-4(n-1)(n-2)\frac{(D^{a}D^{b}r)(D_{a}D_{b}r)}{r^{2}}$
$\displaystyle+$ $\displaystyle
4(n-1)(n-2)\left(\frac{{}^{m}\\!{\Box}r}{r}\right)^{2}+2n(n-3)\frac{K\cdot{}^{m}\\!R}{r^{2}}-2(n-1)(n-2)\frac{(Dr)^{2}\cdot{}^{m}\\!{R}}{r^{2}}$
$\displaystyle-$ $\displaystyle
4n(n-3)^{2}\frac{K\cdot{}^{m}\\!{\Box}r}{r^{3}}+4(n-1)(n-2)(n-3)\frac{(Dr)^{2}\cdot{}^{m}\\!{\Box}r}{r^{3}}$
$\displaystyle-$ $\displaystyle
2n(n-3)^{2}(n-4)\frac{K\cdot(Dr)^{2}}{r^{4}}+(n-3)(n-4)(n^{2}-3n-2)\frac{K^{2}}{r^{4}}$
$\displaystyle+$
$\displaystyle(n-1)(n-2)(n-3)(n-4)\left[\frac{(Dr)^{2}}{r^{2}}\right]^{2}\Bigg{\\}}\,.$

In the above equations, ${}^{m}\\!{\Box}=g^{ab}D_{a}D_{b}$ is the Laplace-
Beltrami operator on $(M^{m}\,,g_{ab})$, ${}^{m}\\!{R}$ and ${}^{m}\\!G^{ab}$
are the Ricci scalar and Einstein tensor of $(M^{m}\,,g_{ab})$ respectively,
and ${}^{m}\\!L_{GB}$ is the Gauss-Bonnet term which is nontrivial only when
$m>3$. The symbol $\hat{C}_{ijkl}$ is the Weyl tensor of the Riemann manifold
$(N^{n},\gamma_{ij})$.

First, let us consider the case $m=1$, i.e., $D=1+n$, and the metric (11)
becomes

$g_{\mu\nu}dx^{\mu}dx^{\nu}=-dt^{2}+r^{2}(t)\gamma_{ij}dz^{i}dz^{j}\,.$

In this case, the non-trivial components of the Weyl tensor of this spacetime
can be written as

$\displaystyle C_{ijkl}=r^{2}\hat{C}_{ijkl}\,,$ (22)

Obviously, from Eqs.(14), (15), (16), and (17), the limit $D\rightarrow 4$ or
$n\rightarrow 3$, can be done only in the case $\hat{C}_{ijkl}=0$. This means
the Weyl tensor the spacetime is also vanished according to the relation (22).
Under this limit, from Eqs.(18), (19), (20), and (21), $P^{tt}$, $P^{mn}$,
$P^{t}$, $V$ have following forms

$\displaystyle
P^{tt}=-\Big{[}1+2\hat{\alpha}\Big{(}\frac{K}{r^{2}}+H^{2}\Big{)}\Big{]}\,,$
(23) $\displaystyle
P^{mn}=\Big{[}1+2\hat{\alpha}\Big{(}2\dot{H}+H^{2}-\frac{K}{r^{2}}\Big{)}\Big{]}\frac{\gamma^{mn}}{r^{2}}\,,$
(24) $\displaystyle
P^{t}=-3H+2\hat{\alpha}\Big{[}-2\dot{H}-3H^{2}-\frac{K}{r^{2}}\Big{]}H\,,$
(25)

and

$\displaystyle
V=4\dot{H}+6H^{2}-2\Lambda+2\hat{\alpha}\Big{[}4H^{2}\dot{H}+3H^{4}+\frac{K^{2}}{r^{4}}\Big{]}\,.$
(26)

where “$\cdot$” denotes the derivative with respect to the coordinate $t$, and
$H=\dot{r}/r$ is Hubble parameter. In the case $K=0$, with the above results,
Eq.(13) exactly reduces to the relevant part in Glavan:2019inb . When matter
field is absent, Eq.(13) gives the gravitational wave equations on vacumm. For
example, the gravitational wave equations on Minkowski spacetime, de Sitter
spacetime, and anti de Sitter spacetime.

Second, we consider the case of $m=2$. The components of the Weyl tensor of
the spacetime can be written as Cai:2013cja

$\displaystyle C_{abcd}$ $\displaystyle=$ $\displaystyle
2c_{1}wg_{a[c}g_{d]b}\,,$ $\displaystyle C_{iajb}$ $\displaystyle=$
$\displaystyle-c_{2}wr^{2}g_{ab}\gamma_{ij}\,,$ $\displaystyle C_{ijmn}$
$\displaystyle=$ $\displaystyle
2c_{3}wr^{4}\gamma_{i[m}\gamma_{n]j}+r^{2}\hat{C}_{ijmn}\,,$ (27)

where

$w={}^{2}\\!{R}+2\frac{\prescript{2}{}{\Box}r}{r}+2\frac{K-(Dr)^{2}}{r^{2}}\,,$
(28)

and

$c_{1}=\frac{n-1}{2(n+1)}\,,\quad c_{2}=\frac{n-1}{2n(n+1)}\,,\quad
c_{3}=\frac{1}{n(n+1)}\,.$ (29)

For the metric (11) with $\hat{C}_{ijkl}=0$ (this is also one necessary
condition of the existence of the limit $D\rightarrow 4$), the tensor
perturbation can be put into a form

$\displaystyle\Big{(}P^{ab}D_{a}D_{b}+P^{mn}\hat{D}_{m}\hat{D}_{n}+P^{a}D_{a}+V\Big{)}\Big{(}\frac{h_{ij}}{r^{2}}\Big{)}=-\frac{2\kappa_{D}^{2}}{r^{2}}\delta
T_{ij}\,.$ (30)

Here, $P^{ab}$, $P^{mn}$, $P^{a}$, $V$ have the same forms as in Eqs.(18),
(19), (20), and (21) except that ${}^{m}\\!G_{ab}=0$ and ${}^{m}\\!L_{GB}=0$
when $m=2$. It is easy to find that $P^{ab}$ and $P^{a}$ have well defined
limits under $D\rightarrow 4$ or $n\rightarrow 2$. However, generally,
$P^{mn}$ is degenerate under this limit. Actually, we have

$\displaystyle
P^{mn}=\Bigg{\\{}1+2\alpha\left[w-\frac{2(n-2)\prescript{2}{}{\Box}r}{r}+(n-2)(n-5)\frac{K-(Dr)^{2}}{r^{2}}\right]\Bigg{\\}}\frac{\gamma^{mn}}{r^{2}}\,,$
(31)

To ensure the regularity of “effective metric” $P^{\mu\nu}=(P^{ab}\,,P^{mn})$
of the tensor perturbation equation, we have to impose a condition $w=0$. From
Eqs.(III), this implies that the spacetime has to be (locally) conformally
flat.

Furthermore, the function $V$ (one part of the effective potential of the
theory) can be rewritten as

$\displaystyle V$ $\displaystyle=$
$\displaystyle{}^{2}\\!{R}-2(n-1)\frac{{}^{2}\\!{\Box}r}{r}+\frac{n(n-3)K}{r^{2}}-\frac{(n-1)(n-2)(Dr)^{2}}{r^{2}}-2\Lambda$
(32) $\displaystyle+$
$\displaystyle\alpha(n-2)\Bigg{\\{}-4(n-1)\frac{(D^{a}D^{b}r)(D_{a}D_{b}r)}{r^{2}}+4(n-1)\left(\frac{{}^{2}\\!{\Box}r}{r}\right)^{2}$
$\displaystyle+$ $\displaystyle
2(n-1)\frac{K\cdot{}^{2}\\!R}{r^{2}}-2(n-1)\frac{(Dr)^{2}\cdot{}^{2}\\!{R}}{r^{2}}-4(n^{2}-4n+1)\frac{K\cdot{}^{2}\\!{\Box}r}{r^{3}}$
$\displaystyle+$ $\displaystyle
4(n-1)(n-3)\frac{(Dr)^{2}\cdot{}^{2}\\!{\Box}r}{r^{3}}-2(n^{3}-8n^{2}+17n-2)\frac{K\cdot(Dr)^{2}}{r^{4}}$
$\displaystyle+$
$\displaystyle(n^{3}-8n^{2}+15n+8)\frac{K^{2}}{r^{4}}+(n-1)(n-3)(n-4)\left[\frac{(Dr)^{2}}{r^{2}}\right]^{2}\Bigg{\\}}$
$\displaystyle-$ $\displaystyle\alpha\cdot w\cdot\frac{4K}{r^{2}}\,.$

Obviously, under the limit $D\rightarrow 4$ or $n\rightarrow 2$, the potential
$V$ is regular only in the case $w=0$. This also requires the vanished Weyl
tensor of the spacetime.

However, in this case, due to the Birkhoff type theorems in four dimension,
$h_{ij}$ is trivial [this tensor perturbation is based on the tensor
decomposition on a two dimensional space $(N^{2}\,,\gamma_{ij})$. This is very
different from the gravitational perturbation based the tensor decomposition
on a three dimensional space $(N^{3}\,,\gamma_{ij})$, i.e., the case with
$m=1$.]. This means the solution of the tensor perturbation equation (30) is
trivial. Nevertheless, this can not hinder our discussion on the principle
symbol of the tensor perturbation equations. Of course, to get nontrivial
gravitational radiation in four dimension, one has to consider the background
spacetime without the maximal symmetry of $(N^{2},\gamma_{ij})$.

##  IV Conclusion and discussion

In this paper, to get a well defined principle symbol, we have shown the
spacetimes in the so-called “Einstein-Gauss-Bonnet Gravity in four dimension”
have to be (locally) conformally flat. So, locally, the metric always has a
form $g_{\mu\nu}=\Omega^{2}\eta_{\mu\nu}$. Although the theory is
diffeomorphism invariant when $D>4$, the final four dimensional theory has
preferred spacetimes, and it can not be diffeomorphism invariant. This point
is similar to the conclusions in Gurses:2020rxb ; Gurses:2020ofy ;
Arrechea:2020gjw ; Arrechea:2020evj .

## Acknowledgement

This work was supported in part by the National Natural Science Foundation of
China with grants No.11622543, No.12075232, No.11947301, and No.12047502. This
work is also supported by the Fundamental Research Funds for the Central
Universities under Grant No: WK2030000036.

## References

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# From near-symplectic constructions to trisections of $4$–manifolds

David T Gay

##  1 Chatty Introduction (which you can skip if you want to jump to the math)

Rob Kirby was my PhD supervisor at UC Berkeley; I finished my PhD there in
1999. During my first postdoc position, at the University of Arizona, my
postdoc advisor Doug Pickrell suggested that I invite Rob to give a colloquium
talk, I think that was in the fall of 2001. I remember Rob gave a survey of
the beginning of Heegaard Floer theory and in particular I clearly remember
him saying something about organizing the various spinC structures on a
$4$–manifold into “buckets”. However, the key event of that visit for me was a
full moon night hike that Doug organized at Sabino Canyon. (Part of my point
here in the story is to make sure to give Doug due credit for his role in all
of this. Another point is to emphasize the importance of hiking in Rob’s
mathematical life.) During the hike, we ended up talking about my thesis work
on handle-by-handle symplectic constructions and I remember Rob commenting,
“Oh, so that’s what your thesis was about.” The point is not that he didn’t
read my thesis but that the essential idea of my thesis was somehow obscured
in the whole mechanics of writing, reviewing, graduating, and so forth; I do
know that Rob and I both talked about most of the details of my thesis but
somehow we had never stepped back and thought about what it was really about.
And once Rob understood what it was really about, the ideas started flowing.

In this article I want to tell the (mostly mathematical) story of my
collaboration with Rob Kirby that started during that night hike and ended up
with our “discovery” of trisections of $4$–manifolds [9]. I use “discovery”
cautiously because much of what we discovered was already known and in the end
a lot of our work can be thought of as just finding the right way to organize
the ideas and the questions. Our work started hovering around various
topological constructions related to the vision of extending $4$-dimensional
symplectic tools to a larger class of $4$–manifolds using near-symplectic
structures (inspired by Cliff Taubes). We then followed what seemed like the
natural mathematical trail to the study of broken Lefschetz fibrations
(inspired by Auroux, Donaldson and Katzarkov, Usher and Perutz), then to
thinking more generally about generic maps from $4$–manifolds to surfaces
(inspired by Lekili), which we ended up calling “Morse $2$–functions”. (A lot
of this involved rediscovering for ourselves in our own way of understanding
lots of things already known to singularity theorists and others.) From there
we more or less stumbled on trisections of $4$–manifolds as a particularly
nice outcome of the theory of Morse $2$–functions. At the end I will give a
brief survey of some of the developments surrounding trisections that have
happened since our first paper, emphasizing my work with Rob and Aaron Abrams
[1] on group trisections.

I hope to emphasize throughout the powerful influence of Rob’s unique way of
seeing the mathematical world on my own mathematical development and the
naturalness of the mathematical flow of our collaboration. When I was a
graduate student and asked Rob for some advice about what to think about
mathematically, I don’t remember the exact mathematical context, but I
remember clearly that his advice was simultaneously vague, unhelpful and
deeply meaningful: “Follow your nose.”

##  2 Just the math about what trisections are

For those who have not thought about trisections before, the idea is almost
stupidly simple. A trisection of a smooth, closed, oriented, connected
$4$–manifold $X$ is a decomposition of $X$ into three topologically simple
pieces $X=X_{1}\cup X_{2}\cup X_{3}$ which fit together like three “slices of
a pie” as in Figure 1: Each piece $X_{i}$ is diffeomorphic (after some mild
smoothing of corners) to a regular neighborhood of a bouquet of (zero or more)
circles in $\mathbb{R}^{4}$. (A bouquet of zero circles is a point, one circle
is a circle, two circles is a figure eight, etc.) Each pairwise intersection
$X_{i}\cap X_{j}$ is diffeomorphic to a regular neighborhood of a bouquet of
(zero or more) circles in $\mathbb{R}^{3}$ (a.k.a. a solid handlebody). The
triple intersection $X_{1}\cap X_{2}\cap X_{3}$ is diffeomorphic to a closed,
orientable, connected surface.

Figure 1: Three slices of a $4$–dimensional pie (pink, yellow and turqoise),
their pairwise intersections (red, blue and green), and their triple
intersection (black).

The surprising facts which we discovered between 2011 and 2013 are that every
smooth, closed, oriented, connected $4$–manifold has a trisection (existence),
and in fact it has lots of them; that there is a simple to describe
stabilization operation that increases the complexity of a trisection; and
that any two trisections of a given $4$–manifold have a common stabilization
(uniqueness). Furthermore, this means that every such $4$–manifold can be
described up to diffeomorphism by a trisection diagram which is a diagram
drawn on a closed, oriented surface of genus $g$ involving $g$ red simple
closed curves, $g$ blue simple closed curves, and $g$ green simple closed
curves; see Figure 2 for an interesting example. The surface is the triple
intersection and the curves of each color are curves that bound disks in the
pairwise intersections.

Figure 2: An interesting trisection diagram.

One formally satisfying feature of trisections is that they turn out to be the
right generalization of the notion of a Heegaard splitting of a $3$–manifold
to the world of $4$–manifolds, with the surprise being that in order to
generalize a Heegaard splitting (a decomposition of a $3$–manifold into two
simple pieces) one needs to consider decompositions into three pieces. In fact
Heegaard splittings of a $3$–manifold $M$ are best seen as coming from
particularly nice smooth maps $f:M\to[-1,1]$ (Morse functions with well
organized critical points) and then decomposing $M$ into $f^{-1}([-1,0])$ and
$f^{-1}([0,1])$. Similarly, trisections of a $4$–manifold $X$ are best seen as
coming from particularly nice smooth maps $f:X\to D^{2}$ to the disk, called
Morse $2$–functions, with particularly well organized singularities, and
pulling back the decomposition of the disk into three (fat) pie slices. Also,
Heegaard splittings allow $3$–manifolds to be described by Heegaard diagrams,
which are just like trisection diagrams but only involve curves of two colors.
Finally, note that a trisection diagram gives three Heegaard diagrams, by
considering each pair of two colors.

In principle, a trisection diagram is not much different from a framed link
diagram, but it is better organized, and there are algorithms to turn either
one into the other. In particular, trisection diagrams have an interesting
formal property summarized in the slogan “pairwise boring, triply
interesting”. There is a standard move on a collection of disjoint simple
closed curves on a surface, called a “handle slide”, which involves one curve
“sliding over” and picking up a copy of another curve, as shown in Figure 3.
Handle slides do not change either the $4$–manifold associated to a trisection
diagram or the $3$–manifold associated to a Heegaard diagram. A trisection
diagram $(\Sigma,\alpha.\beta,\gamma)$ has the property that each associated
Heegaard diagram of two colors $(\Sigma,\alpha,\beta)$,
$(\Sigma,\beta,\gamma)$ and $(\Sigma,\gamma,\alpha)$ is equivalent, after some
handle slides and a diffeomorphism, to the boring Heegaard diagram in Figure
4, which is a Heegaard diagram for the connected sum of some number of
$S^{1}\times S^{2}$’s (a boring $3$–manifold). However, all $4$–manifolds (and
thus a very large class of very interesting objects) can be described by these
triples of collections of curves, despite the fact that in pairs they are
boring. The interested reader might enjoy verifying this pairwise standardness
for the diagram in Figure 2.

Figure 3: A handle slide on a surface. Figure 4: The standard, “boring”,
Heegaard diagram, with some number of parallel red-blue pairs on the left and
some number of dual red-blue pairs on the right.

A classical example of the “pairwise boring, triply interesing” slogan, by the
way, is the three component link known as the Borromean rings, shown in Figure
5. This link of three circles has the property than any pair of the three are
unlinked but all together they are nontrivially linked. Here is a meta-
question for the mathematical community at large: Where else in mathematics
does this slogan apply?

Figure 5: The Borromean rings; each pair of rings is boringly unlinked but the
triple is interesting.

##  3 The Quixotic quest to make gauge theory mean something topologically

My entire topological career has been dominated by the incredible twin results
of Freedman [8] that two closed simply connected smooth $4$–manifolds are
homeomorphic if and only if they are homotopy equivalent and of Donaldson [4]
that there exist invariants that can distinguish pairs (known as “exotic
pairs”) of closed, simply connected smooth $4$–manifolds that are homotopy
equivalent but not diffeomorphic. Rob was closely connected with Freedman’s
work, but Donaldson’s work used gauge theory (PDE’s coming from physics), and
Rob was never a gauge theorist. Ever since I have known Rob an important part
of his mathematical quest has been to understand in some “genuinely
topological” how to see that exotic pairs are not diffeomorphic. The Donaldson
invariants were later replaced to some extent with the simpler Seiberg-Witten
invariants [5] and eventually many gauge theoretic results could be proved
using Ozsváth and Szabó’s Heegaard Floer invariants [23], but, still,
underlying it all is the problem of counting solutions to PDE’s. Rob always
wanted a way to see what we were counting explicitly and combinatorially, and
to see from a geometric topologist’s perspective, in a direct way, why these
counts were invariants of smooth structures.

We would like to be able to stare at some diagrammatic, combinatorial
description of a $4$–manifold, “count” something in the diagram, perhaps count
smooth isotopy classes of some objects in the diagram, and then understand
that this is an invariant because one understands the moves needed to relate
any two such diagrammatic descriptions of the same $4$–manifold. I would argue
that we are still far from such a perspective, but that for formal reasons
such a perspective should be out there. The closest we have come that I am
aware of are the handful of $4$–dimensional results coming out of Khovanov
homology, in particular Rasmussen’s computations of slice genus [25] and
subsequently Lambert-Cole’s trisections and Khovanov homology based proof of
the Thom conjecture [19]. In what follows, one thread of the story of my
collaboration with Rob is that we were constantly trying to catch up with
gauge and Floer theory and reinterpret it in ways we non-analytically minded
“mere geometric topologists” could understand and really see.

Generally, however, almost all invariants proceed by first imposing some
additional structure, then counting something, and then showing that the
answer does not depend on the choice of that additional structure. In the
gauge theoretic context this additional structure is a collection of
differential geometric data such as Riemannian metrics, connections on various
bundles, and so forth. In the Floer theoretic setting the additional structure
is generally an almost complex structure, usually with a symplectic structure
lurking in the background. In our hypothetically more “combinatorial”, more
“topologically understandable” visions, there would still be extra structure,
but this would come in the form of a diagram describing a construction of the
manifold or simply a decomposition of the manifold into elementary pieces. For
us, inspired heavily by Morse theory, an intermediate structure has typically
been some kind of smooth map from the manifold in question to another simpler
manifold, e.g. $\mathbb{R}$ or $\mathbb{R}^{2}$ or $S^{2}$. Symplectic
topology has always lurked in the background as “borderline understandable” to
simple minded topologists, and thus you will find the following storyline
progresses through symplectic topology to understanding various classes of
maps from dimension $4$ to dimension $2$ and finally arriving at
decompositions of manifolds and associated diagrams, with an addendum in which
the “diagrams” are eventually replaced with group theory.

There is a great deal also to be said about the extent to which complex
algebraic geometry has inspired smooth $4$–dimensional topology, with smooth
algebraic surfaces being in some sense the prototypical $4$–manifolds. This is
too much for me to competently discuss here but suffice it to say that this
thread very much lurks in the background, and I believe that much of Rob’s
insight into the problems he and I have thought about can be traced back to
his early work with Harer and Kas on the smooth topology of complex surfaces
[17]. (To a complex algebraic geometer, a quick introduction to trisections is
to say that they are the natural generalization of the decomposition of the
complex projective plane into three coordinate charts.)

Perhaps a better reason for this quest than just the fact that we couldn’t
understand gauge theory is that all gauge theoretic methods have failed to say
anything about homotopy $4$–spheres, and thus shed no light on the holy grail,
the smooth $4$–dimensional Poincaré conjecture (or its sister, the smooth
$4$–dimensional Schoenflies problem). Thus, like any good $4$–dimensional
topologist should, we have always held out hope that some day we might stumble
upon some invariant, almost certainly not defined using anything like gauge
theory, which could distinguish smooth homotopy $4$–spheres and hence disprove
the Poincaré conjecture. Owning up to such a dream in public is perhaps not
good form, but in our collaboration we have had moments of (probably
misguided) optimism and enthusiasm in this direction. The truth is that we
have been frequently quite naive about the potential to make hard things easy
to understand, but that our naiveté seems to have served us pretty well.

In what follows I will try to give a fairly thorough account of the path Rob
and I took that brought us to thinking about trisections of $4$–manifolds. Of
course we never knew where we were going in the long run, but we looked at a
sequence of related problems that each raised new questions that, in
hindsight, seemed to lead inexorably towards trisections. All of the questions
we thought about along the way are, in my opinion, still very important and
serve to set trisections in the right context. At each step we were working on
$4$–manifolds equipped with certain auxiliary structures, and I will describe
those auxiliary structures as they arise.

##  4 Near symplectic constructions

The relevant auxiliary structures to discuss here are symplectic and almost
complex structures on $4$–manifolds, building on the foundational work of
Gromov [16]. Symplectic structures are closed, nondegenerate $2$–forms, which
means one can always find local coordinates in which they have (in dimension
$4$) the form $dx_{1}\wedge dy_{1}+dx_{2}\wedge dy_{2}$. (A $2$–form at a
point is an antisymmetric bilinear form on the tangent space at that point
that should be thought of measuring the “signed area” of the parallelogram
spanned by two vectors, and in this case the area of the $1\times 1$ squares
in the $x_{1},y_{1}$ and $x_{2},y_{2}$ planes are $1$ while the $1\times 1$
squares in any of the $x_{1},x_{2}$, $x_{1},y_{2}$, $y_{1},x_{2}$ and
$y_{1},y_{2}$ planes have “area” $0$.) An almost complex structure on a
$4$–manifold is a way to “multiply by $i$” in the tangent spaces, in other
words a smoothly varying linear automorphism $J$ of each tangent space such
that $J^{2}=-\mathop{\rm id}\nolimits$. The structure of a $2$–dimensional
complex manifold on a $4$–manifold induces an almost complex structure on the
tangent spaces, but not every almost complex structure comes from a complex
manifold structure.

A classical way to probe the structure of a complex algebraic variety is to
study enumerative problems for curves in that variety. Remembering that
“curves” over $\mathbb{C}$ are actually real $2$–dimensional surfaces, one can
push these enumerative methods to the softer setting of almost complex
manifolds and study “pseudoholomorphic curves”, which are surfaces in the
$4$–manifold whose tangent spaces are fixed by the almost complex structure,
in other words, their real $2$–dimensional tangent spaces are actually complex
lines with respect to $J$. Gromov’s key contribution was to show that, when an
almost complex structure is dominated by a symplectic structure (meaning that
the $2$–form assigns positive area to a parallelogram spanned by $V$ and $JV$
for any nonzero tangent vector $V$), then one can control families of
pseudoholomorphic curves and get compactness results for their moduli spaces.
This then allows, in special cases, for one to have well-defined counts of
pseudoholomorphic curves and prove that these counts are invariant under
various choices. Note that the pseudoholomorphic condition on an embedding of
a surface is basically a partial differential equation, so looking for
pseudoholomorphic curves is basically counting solutions to a PDE, but somehow
it feels a little more concrete than the PDE’s involved in gauge theory.

Returning now to the project of finding a more “concrete” or “geometric
topologist friendly” understanding of gauge theoretic invariants of smooth
$4$–manifolds, Taubes [27] showed that, when a $4$–manifold supports a
symplectic structure, its Seiberg-Witten (SW) invariants can be calculated by
counting pseudo-holomorphic curves with respect to a generic almost complex
structure dominated by that symplectic structure, in other words, a Gromov
invariant (Gr). This result is generally summarized as SW=Gr. This has a range
of spectacular implications, but when handed a random $4$–manifold, it might
not be so clear whether it has a symplectic structure or not. On the other
hand, Taubes, Honda [18] and others had observed that every $4$–manifold with
$b_{2}^{+}>0$, which means that it contains surfaces which have positive
signed intersection number with any wiggled versions of themselves, supports a
“near symplectic” structure. This is a closed $2$–form which is symplectic
away from a $1$–dimensional locus where it is zero, and along this locus it
vanishes transversely in the appropriate sense, so that its zero locus is a
$1$–manifold in a neighborhood of which the $2$–form has a standard model.
Taubes floated the idea that perhaps suitably defined Gromov invariants in
this setting would recover Seiberg-Witten invariants for arbitrary
$4$–manifolds with $b_{2}^{+}$ positive, and initiated a study [28] of the
behavior of pseudoholomorphic curves in the standard local model near the
vanishing locus. This line of reasoning seems to have finally reached fruition
with Chris Gerig’s work [13], which is also a better resource than this
paragraph for a proper account of the history and the motivation for the idea.

Still, the fact that every $4$–manifold with positive $b_{2}^{+}$ supports a
near symplectic structure was not constructive, however, so if the SW=Gr plan
panned out in the near symplectic setting it was still not clear how useful
that would be if one did not know how to explicitly construct a near
symplectic form on a particular $4$–manifold. After our Sabino Canyon hike,
Rob explained these ideas to me and then Rob and I set out to solve this
problem [10], where “explicitly” meant to us “starting from a framed link
diagram describing a handle decomposition of the $4$–manifold” and then
proceeding in something that might loosely be called an algorithm.

It is important to point out here that in the end it is a bit disingenuous to
describe our construction as explicit because of one key point; at a critical
stage in our construction we had two near-symplectic symplectic structures,
one on one half of the $4$–manifold and one on the other, and we needed to
glue them together along the separating $3$–manifold. There is a standard way
to do this, using a contact structure on the $3$–manifold as the appropriate
gluing boundary data. (If you don’t know what a contact structure is, just
know that it is the correct boundary data for symplectic structures.) In order
to glue, we needed to know that the contact structures coming from the
constructions on the two halves were equal (or could be deformed to be equal).
We knew this thanks to Eliashberg’s result [7] that a certain class of contact
structures, known as overtwisted contact structures, could be classified using
algebraic topological invariants, so we adjusted our construction
appropriately to make sure that the invariants coming from the two sides
matched. However it is not at all clear that Eliashberg’s machinery to go from
knowing that the invariants match to deforming (isotoping) the contact
structures to be equal is “explicit”.

##  5 Broken Lefschetz fibrations

The auxiliary structures of concern here are Lefschetz fibrations; for a good
overview see Gompf’s exposition in the Notices of the AMS [15]. In short, a
Lefschetz fibration on an oriented $4$–manifold $X$ is a smooth map from $X$
to the $2$–sphere $S^{2}$ which has finitely many isolated singularities, each
of which is locally modelled (respecting orientations) on the simplest
singularity that can arise in a holomorphic map from $\mathbb{C}^{2}$ to
$\mathbb{C}$: $(z_{1},z_{2})\mapsto z_{1}^{2}+z_{2}^{2}$. Away from these
singularities, the map thus looks like a surface bundle over a surface, so
locally like $\Sigma\times B^{2}\to B^{2}$ for some surface $\Sigma$. To get a
more topological understanding of what it means to support a sympectic
structure, Donaldson [6] showed that, after blowing up enough times, all
symplectic $4$–manifolds support Lefschetz fibrations. (“Blowing up”, despite
its name, is a mild operation that changes a manifold in a very controlled way
so as to allow two surfaces that intersect to become disjoint, at the expense
of introducing a new surface, called the “exceptional divisor”, that they both
intersect; as long as one keeps track of the exceptional divisor one still has
access to the original manifold and all of its topology.) With this in mind,
Usher [29] showed that one could recover the full Gromov pseudoholomorphic
curve count by a certain count of pseudoholomorphic “multi-sections” of
symplectic Lefschetz fibrations, arguably making Taubes’s SW=Gr result
slightly more meaningful to the average geometric topologist, and giving us a
slightly more topological way to think of what the Seiberg-Witten invariants
are counting.

Inspired by Taubes’s near-symplectic vision discussed in the preceding
section, Auroux, Donaldson and Katzarkov [2] generalized Donaldson’s Lefschetz
fibration result to show that every near-symplectic $4$–manifold, after
blowing up, has the structure of a “broken Lefschetz fibration” (BLF), which
is like a Lefschetz fibration over $S^{2}$ but also allows for a
$1$–dimensional locus of singularities of a particular model, called
indefinite folds. This $1$–dimensional singular set is exactly the same zero
locus where the near symplectic form vanishes. Following our noses, we
naturally wondered next how to construct these BLF’s “explicitly” from, for
example, a handle decomposition of a $4$–manifold. In the end [11] we
discovered that if we allowed both orientation preserving and orientation
reversing local models for the Lefschetz type singularities (we called the
resulting fibrations “broken achiral Lefschetz fibrations”, or BALFs), then we
could construct BALFs on all closed oriented $4$–manifolds.

The construction was similar in spirit to our construction of near-symplectic
forms: First we found a good way to split the given $4$–manifold into two
pieces on each of which we could construct partial fibrations and then we
figured out the appropriate boundary conditions to govern these fibrations on
the separating $3$–manifold. (Experienced low-dimensional topologists will not
be surprised that this boundary condition is the structure of an “open book
decomposition”.) Then we massaged our constructions carefully so as to arrange
that this boundary data agreed and we coud glue the fibrations together. As
before, it would be disengenuous to describe our construction as explicit
because of this last step, which passed from open book decompositions to
contact structures by way of the Giroux correspondence [14] and then appealed
again to Eliasherg’s classification [7] of overtwisted contact structures.

Aside from this caveat about explicitness or the lack thereof, another thread
the reader may be picking up is the increasing mixing of categories here: The
study of almost complex structures, symplectic structures and Lefschetz
fibrations already lies (a little bit uncomfortably) somewhere between complex
algebraic geometry and differential topology. Throwing in circles along which
new types of degenerations arise, and in the fibration case having both
isolated complex type singularities as well as $1$–dimensional decidedly
nonholomorphic singularities, and then allowing the isolated singularities to
flip orientations, really starts to seem like a bit of a stretch and an
awkward mix of perspectives. The next phase in our collaboration started to
clean this up.

##  6 Morse $2$–functions

The auxiliary structures of relevance here are, in a broad sense, stable
smooth maps from manifolds of some dimension to other manifolds of lower
dimension. A map is stable if small perturbations do not change its essential
qualitative features; more precisely, the original map and the perturbed are
equal after pre- and post-composing with isotopies of the domain and range.
Thus the function $y=x^{2}$ is stable in the world of smooth maps from
$\mathbb{R}$ to $\mathbb{R}$ while the function $y=x^{3}$ is not. Stable maps
from smooth manifolds to $\mathbb{R}$ (and sometimes to other $1$–manifolds
such as the circle or a closed interval) are called Morse functions and have
been tremendously useful tools for probing the topology of smooth manifolds in
general, especially when used in conjunction with gradient-like vector fields
to give handle decompositions. Stable maps to dimension $2$ are also fairly
well understood but have not been used as extensively as Morse functions as a
tool to probe the topology of manifolds.

Perutz [24] studied the problem of defining, in the broken Lefschetz fibration
setting, something analogous to the counts of pseudoholomorphic multisections
for Lefschetz fibrations studied by Usher, motivated by the possibility that
these would then again recover Seiberg-Witten invariants, or at least be
invariants even if one did not know that they agreed with pre-existing
invariants. To show directly that these counts, called Lagrangian matching
invariants, are in fact $4$–manifold invariants, and did not depend upon the
choice of BLF, one would need to understand how to move from one BLF on a
given $4$–manifold to another. In other words, one needs not only existence
results for B(A)LFs, but also uniqueness results.

While we were thinking about this (working together at the African Institute
of Mathematical Sciences in Muizenberg, South Africa, probably in 2007) we
found out about Lekili’s insightful observation [22] that if one wanted to
connect two BLF’s by a $1$–parameter family of smooth functions, one should
first observe that Lefschetz singularities are actually not stable in the
world of smooth maps. A small perturbation in the world of smooth maps will
make a Lefschetz singularity into a much more complicated circle of
singularities. Thus a generic $1$–parameter family of smooth functions
connecting two Lefschetz fibrations or BALFs should be expected to pass
through functions that are much more general and do not have any Lefschetz
singularities at all. The local behavior of stable smooth maps from
$4$–manifolds to $2$–manifolds has in fact been understood for a long time and
Lekili’s paper [22] has an excellent appendix that runs through the general
machinery for understanding stable maps between various dimensions [30]
applied in the case of dimensions $4$ and $2$. At least this is where I
learned this material, Rob probably basically had already absorbed most of
this by osmosis over the years and just needed a little reminder from Lekili’s
article.

Lekili essentially answered the question of how to think about local moves
connecting one BALF to another, but the key problem was that in the
intermediate stages one might wander quite far away from the world of BALFs
and move through maps that are nothing like small perturbations of BALFs. In
particular, one might run into what are called definite folds, discussed in
more detail below. Whether one could avoid definite folds is analogous to the
question of whether, in a $1$–parameter family of functions connecting two
given Morse functions with the same number of minima and maxima, one can avoid
introducing extra minima or maxima along the way. In general this can be done,
modulo some obvious counterexamples and low-dimensional exceptions. Because of
the analogy with Morse theory, Rob and I began calling stable maps to
dimension $2$ “Morse $2$–functions”, thinking of them as vaguely like a
$2$–category version of Morse functions, whatever that might mean. (Here we
were very much inspired by conversations with Peter Teichner at MSRI in the
spring of 2010.)

The key idea of a Morse $2$–function is that locally a Morse $2$–function $F$
on an $n$–manifold looks like a generic $1$–parameter family $f_{t}$ of Morse
functions on an $(n-1)$–manifold, so that there are local coordinates so that
$F(t,p)=(t,f_{t}(p))$ where $t$ is a single time parameter, $p$ is an
$(n-1)$–dimensional “spatial” coordinate, and $f_{t}$ is a generic path of
functions connecting two Morse functions. However, globally there is no well-
defined time direction, either in domain or range. Generic $1$–parameter
families of Morse functions will be honest Morse functions for all but
finitely many times, and at finitely many times will experience births or
deaths of pairs of critical points, or the coincidence of two critical points
having the same critical value at an instant. The tracks of the critical
points for the honest Morse functions are called “folds”, and are
$1$–dimensional in domain and range, while the birth/death events are called
“cusps”; the coincidences of having two critical points with the same critical
value are folds whose images in the range cross. The generic behavior of
homotopies between Morse $2$–functions is exactly locally modelled on the
generic behavior of homotopies between homotopies between Morse functions, in
other words the moves of “Cerf theory”.

If the preceding paragraph did not mean much to the reader, the basic idea is
well illustrated with a picture of a Morse $2$–function on a surface. Figure 6
shows a Morse $2$–function on a torus. In fact, any time one draws a picture
of a surface on a piece of paper, one is of course presenting a map from that
surface to $\mathbb{R}^{2}$ and, assuming genericity, this will be a Morse
$2$–function. The folds are literally the places where the surface folds over,
and the cusps are where folds “switch directions” in some sense; these are all
clearly visible in this illustration. The preimage of a nonsingular point is
an even number of points in the torus, and the number of these points jumps by
$2$ when crossing a fold. The only difference in higher dimensions in that
different dimensions and codimensions can fold in opposite directions along a
fold, and the preimages of nonsingular points are higher dimensionsal
submanifolds, not collections of points. In dimension $4$, the preimages of
points are surfaces, and the topology of these surfaces changes as one crosses
a fold.

Figure 6: A Morse $2$–function on a torus.

Note that the index of a fold ranges from $0$ to $n-1$, but is only well-
defined up to switching $k$ with $n-1-k$. A fold of index $0$ (equivalently
$n-1$) is called a “definite fold” and all other folds are called indefinite.
Thus on $4$–manifolds there are only really two types of folds: definite index
$0$ (equivalently $3$) folds and indefinite index $1$ (equivalently $2$)
folds. (On a $2$–manifold, all folds are definite.) One reason definite folds
in dimension $4$ might be undesirable is that they in general lead to
disconnected fibers, since crossing a definite fold in the index $0$ direction
creates a new $S^{2}$–component of the fiber. This (and $S^{2}$–fibers in
general perhaps) is bad from a symplectic geometry and invariant constructing
perspective, as explained to us by Katrin Wehrheim, another major motivator
for us in our early work in this subject.

The upshot is that we showed [12] how to construct indefinite, fiber-connected
Morse $2$–functions on a given $n$–manifold in a given homotopy class of maps,
when $n>3$ and a natural $\pi_{1}$ condition is satisfied, and we also showed
that any two such Morse $2$–functions can be connected by a generic homotopy
maintaining the indefinite and fiber-connected properties, again when $n>3$.
Of course, all along we were primarily interested in the case $n=4$ and,
having settled this question, we now started to think seriously about what to
do with Morse $2$–functions in general on smooth $4$–manifolds, how to put
them into particularly nice forms, and how to use them to appropriately probe
the topology of the manifolds involved.

##  7 Trisections

I started my position at the University of Georgia (UGA) in August 2011. One
thing Rob was always fantastic about was travelling to visit me wherever I
ended up, understanding that my family life with three kids made it harder for
me to up and travel at the drop of a hat. And Rob had a long time connection
to UGA going back to the early days of the Georgia Topology Conference, so he
started visiting me at UGA pretty regularly (and making sure to combine each
such visit with some social time with old retired friends Jim Cantrell and
John Hollingsworth). Over the course of a few visits in late 2011 and early
2012 we were busy drawing pictures of $\mathbb{R}^{2}$–valued Morse
$2$–functions on homotopy $4$–spheres arising from balanced presentations of
the trivial group not known to be Andrews-Curtis trivializable. Our grand hope
of course was to find an invariant that could show that these homotopy
$4$–spheres were not diffeomorphic to $S^{4}$, and we had an idea that would
involve counting certain surfaces with boundary on the folds. (We called these
surfaces “flow charts” because they would be swept out by flow lines of
gradient-like vector fields in local charts; this is a loose end idea that
never panned out for us, just in case the story was seeming too tidy with
suspiciously few false starts and dead ends.)

In the process we developed a very systematic and standardized way to organize
the image of a $\mathbb{R}^{2}$–valued Morse $2$–function and, suddenly, one
day trisections jumped out at us! In hindsight so much of it is obvious but at
the time it seemed somehow unusual and dramatic. The key point is that we were
busy building Morse $2$–functions “from left to right” starting from handle
decompositions of $4$–manifolds, with the $0$– and the $1$–handles at the left
contributing a very standard part of the Morse $2$–function, the $3$– and
$4$–handles at the right contributing a mirrored standard piece, and the
$2$–handles in the middle doing something interesting. Then suddenly
trisections jumped out at us when we realized one day in my office at UGA that
we could split things a little differently by separating the cusps into three
groups.

The basic idea is well illustrated by a half-dimensional cartoon. (I believe
that Rob taught all of his students and collaborators the value of the half-
dimensional analogy when working in dimension $4$.) In Figure 7, we show a
sequence of Morse $2$–functions on the torus, each obtained by performing some
procedure to the preceding one, and ending with one which might seem
unnecessarily complicated. However, if one then trisects the torus using the
last Morse $2$–function, as illustrated in Figure 8 and Figure 9, we see that
in fact the torus is decomposed into three disks (careful inspection shows
that these disks are hexagons), and this is the basic “trisection” of $T^{2}$.
We essentially mimic this procedure with more care in dimension $4$ to produce
trisections of arbitrary $4$–manifolds.

Figure 7: A sequence of Morse $2$–functions on the torus. Figure 8: The last
Morse $2$–function from Figure 7 with three cut lines indicated in black.
Figure 9: A trisection of the torus coming from cutting open along the
preimages of the three black lines in Figure 8.

We released two versions of our paper on the arXiv with quite a gap between
them and a significant difference. One of Rob’s life lessons for me was that
if you prove an existence theorem then you should prove a uniqueness theorem.
Our first version (May 2012) only had the existence result and then it took us
more than a year to nail down the uniqueness result (September 2013).
Somewhere in between those two dates was a hardworking session at the Max
Planck Institute for Mathematics in Bonn, in which Andras Stipsicz had several
helpful comments about how to proceed with uniqueness. Once we had the
uniqueness result it was stupidly obvious, and the (for us) really startling
fact was just how closely it mirrored the uniqueness result for Heegaard
splittings of $3$–manifolds.

Having “discovered” trisections, it did also become clear to us that in many
ways we were obviously rediscovering things others already knew quite well. Of
particular note is the fact that Birman and Craggs [3] essentially already
thought about trisections of $4$–manifolds from a mapping class groups of
surfaces perspective, using the term “triadic $4$–manifolds”. (At least, this
term appears in a section heading in that paper, although it never appears in
the actual textual body of the paper. Perhaps they had an intuition that there
was something more going on with these structures on $4$–manifolds or even, as
is so often the case, knew much more than they wrote in their paper.) More
recently, Oszváth and Szabó were very obviously working with these structures
in the guise of Heegaard triples [23]. Our main existence result is more or
less implicit in Oszváth and Szabó’s work, and one could interpret the thrust
of our work as being to understand the raw topological implications of the
fact that every $4$–manifold can be described by a Heegaard triple, rather
than simply viewing Heegaard triples as a tool to understanding various maps
in Heegaard-Floer theory. However the uniqueness result, and the study of
trisections as actual decompositions of $4$–manifolds, unifies the subject and
sets the stage for many interesting directions of study.

##  8 Group trisections

Many other mathematicians have made striking contributions to the theory of
trisections since our first paper. Each of this contributions is worth a
lengthy discussion, but since we are focusing on Rob Kirby’s work here, I’ll
briefly describe the most important extension to the theory that Rob and I
were involved in, namely the development of “group trisections” [1] in
collaboration with Aaron Abrams.

One of the first things we thought about with trisections was understanding
how much information about a smooth $4$–manifold one retains when applying
algebraic topological functors to a trisection, (to all the pieces and their
intersections and the various inclusion maps). Since Freedman taught us that
in some special cases the algebraic topology of $4$–manifolds seems to
basically record homeomorphism type, we expected somehow that by passing from
smooth trisections to the algebraic topological shadows of trisections we
would pass from smooth information to topological information and then be able
to understand something about how trisections of homeomorphic but not
diffeomorphic manifolds are related. In fact it turned out, after Aaron Abrams
reminded Rob and me of some basic facts from $3$–manifold topology, that the
most basic algebraic topology functor of all, $\pi_{1}$, loses no information
at all! The inclusion maps between the triple intersections, double
intersections, single pieces and total manifold making up a trisection fit
into a cube of spaces and, when the $\pi_{1}$ functor is applied, one gets a
cube of groups involving one surface group, six free groups, and the
fundamental group of the total $4$–manifold and from this cube of groups one
can completely recover the smooth $4$–manifold and its trisection.

We thus defined a “group trisection” of a given group $G$ as a cube of groups
and homomorphisms between them such that, at one vertex we have the
fundamental group of a closed, oriented surface, at the opposite vertex we
have the group $G$, and at the 6 intermediate vertices we have free groups.
The maps all flow from the surface group in the direction of $G$, all the maps
are surjective, and all the six faces of the cube are pushouts. The main
result is that a trisected group gives a trisected $4$–manifold and vice
versa. This is striking; for example, a trisected group carries with it an
integral quadratic form, namely the intersection form of the associated
$4$–manifold.

The crucial piece of $3$–manifold topology that Aaron reminded of us is the
following “standard fact” (for a proof see [21]): Any surjective group
homomorphism from the fundamental group of a genus $g$ surface to a free group
of rank $g$ arises as the map induced by the inclusion of the surface as the
boundary of a genus $g$ handlebody, and the handlebody filling of that surface
is uniquely determined by the group surjection. This, together with Laudenbach
and Poénaru’s result [20] that every diffeomorphism of a connected sum of
$S^{1}\times S^{2}$’s extends across a boundary connected sum $S^{1}\times
B^{3}$’s is really all you need to prove that a trisected $4$–manifold can be
completely recovered from the associated group trisection.

In fact, everything we did is foreshadowed in, and inspired by, John
Stallings’s famous paper [26] “How not to prove the Poincaré conjecture”,
which begins with the memorable line “I have committed the sin of falsely
proving Poincaré’s Conjecture. But that was in another country and until now,
no one has known about it.” One consequence of the dictionary between
trisections of groups and trisections of $4$–manifolds is that the smooth
$4$–dimensional Poincaré conjecture can be reformulated entirely group
theoretically, just as Stallings reformulated the $3$–dimensional Poincaré
conjecture purely group theoretically. I argued unsuccessfully with my
coauthors in favor of titling our group trisections paper “How not to prove
the smooth $4$–dimensional Poincaré conjecture”. I suspect that Rob, in
particular, rejected my proposal because, deep down, he is a fundamentally
optimistic mathematician and, whether the conjecture is true or false, he’s
always willing to consider the possibility that some new idea really might be
the right way forward to settling the question once and for all.

## About the author

David Gay was born in 1968 in Suacoco County in upcountry Liberia, grew up in
Liberia, England, Lesotho and Massachusetts, ended up with a PhD in
Mathematics from UC Berkeley in 1999, and has been a professor at the
University of Georgia since 2011. He has thought about topology and taught
mathematics at many levels around the world, with postdoctoral positions at
the University of Arizona, the Nankai Institute of Mathematics, and the
University of Quebec, a Senior Lectureship at the University of Cape Town,
and, most recently, a visiting position as Hirzebruch Research Chair at the
Max Planck Institute of Mathematics in Bonn in 2019-20.

## References

  * [1] Aaron Abrams, David T. Gay, and Robion Kirby.  Group trisections and smooth 4-manifolds.  Geom. Topol., 22(3):1537–1545, 2018.
  * [2] Denis Auroux, Simon K. Donaldson, and Ludmil Katzarkov.  Singular Lefschetz pencils.  Geom. Topol., 9:1043–1114, 2005.
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