# Perfect fluid warp drive solutions
with the cosmological constant

Osvaldo L. Santos-Pereira<EMAIL_ADDRESS>Physics Institute, Universidade
Federal do Rio de Janeiro, Rio de Janeiro, Brazil  Everton M. C. Abreu
<EMAIL_ADDRESS>Physics Department, Universidade Federal Rural do Rio de
Janeiro, Seropédica, Brazil  Physics Department, Universidade Federal de Juiz
de Fora, Juiz de Fora, Brazil  Multidisciplinary Graduate Program in Applied
Physics, Physics Institute, Universidade Federal do Rio de Janeiro, Rio de
Janeiro, Brazil  Marcelo B. Ribeiro<EMAIL_ADDRESS>Physics Institute,
Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil
Multidisciplinary Graduate Program in Applied Physics, Physics Institute,
Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

(August 27, 2024)

###### Abstract

The Alcubierre metric describes a spacetime geometry that allows a massive
particle inside a spacetime distortion, called warp bubble, to travel with
superluminal global velocities. In this work we advance solutions of the
Einstein equations with the cosmological constant for the Alcubierre warp
drive metric having the perfect fluid as source. We also consider the
particular dust case with the cosmological constant, which generalizes our
previous dust solution (Santos-Pereira et al. 2020) and led to vacuum
solutions connecting the warp drive with shock waves via the Burgers equation,
as well as our perfect fluid solution without the cosmological constant
(Santos-Pereira et al. 2021). All energy conditions are also analyzed. The
results show that the shift vector in the direction of the warp bubble motion
creates a coupling in the Einstein equations that requires off-diagonal terms
in the energy-momentum source. Therefore, it seems that to achieve
superluminal speeds by means of the Alcubierre warp drive spacetime geometry
one may require a complex configuration and distribution of energy, matter and
momentum as source in order to produce a warp drive bubble. In addition, warp
speeds seem to require more complex forms of matter than dust for stable
solutions and that negative matter may not be a strict requirement to achieve
global superluminal speeds.

warp drive, cosmic fluid, Burgers equation, shock waves, Alcubierre geometry

###### pacs:

04.20.Gz; 04.90.+e; 47.40.-x

##  1 Introduction

The warp drive is a mechanism based on General Relativity which in theory
allows for massive particles to be propelled throughout the spacetime with
global superluminal speeds Alcubierre1994 ; Alcubierre2017 . The theory
describes this possibility by means of a localized spacetime distortion,
called warp bubble, that would contain a local lightcone where the particle
would follow special relativity, that is, move locally with speeds smaller
than light. However, the metric is such that the warp bubble moves along a
geodesic that creates an expansion of spacetime behind it and a contraction in
front of it in such a way that an observer outside the warp bubble sees it
moving with superluminal speed.

In the original paper by M. Alcubierre the warp drive metric was established
without solving the Einstein equations Alcubierre1994 . The Einstein tensor
components were calculated and it was noticed that negative energy density
would be required to create the warp bubble, thus violating the weak and
dominant energy conditions.

Ford and Roman FordRoman1996  calculated via quantum inequalities the amount
of negative energy required for the warp drive to be possible, concluding that
it would be a massive amount, impossible to achieve. Pfenning and Ford
Pfenning1997  also concluded that it would be necessary an enormous amount of
energy for the warp drive to be possible. They obtained a quantity ten orders
of magnitude greater than the mass-energy of the entire visible Universe, also
with negative density.

Krasnikov Krasnikov1998  discussed the possibility of a massive particle
moving in space faster than a photon, arguing that this is not possible due to
limitations on globally hyperbolic spacetimes properties with feasible
physical assumptions. He is the creator of a specific spacetime topology with
devices that would allow massive particles to travel between two points in
space with superluminal velocities without the need for tachyons. Everet and
Roman EveretRoman1997  coined the name for this spacetime topology as the
Krasnikov tube. They generalized the metric designed by Krasnikov by proposing
a tube in the direction of the particle’s path, connecting the start and end
point. Inside this tube, the spacetime is flat and the lightcones are opened
to allow the one direction superluminal travel. The Krasnikov tube also
requires huge amounts of negative energy density. Since the tube is designed
to not possess closed timelike curves, it would be theoretically possible to
construct a two way non-overlapping system that could work as a time machine.
The energy-momentum tensor (EMT) for the Krasnikov metric is positive in some
regions. Both the metric and the obtained EMT were thoroughly analyzed in
Refs. Lobo2002 ; Lobo2003 .

Van de Broeck Broeck1999  made a relevant contribution to warp drive theory by
demonstrating that a small modification of the original Alcubierre geometry
would reduce, to a few solar masses, the total negative energy necessary for
the creation of the warp bubble distortion of spacetime. This result have led
van de Broeck to suppose that other geometrical modifications of the
Alcubierre’s geometry for the warp drive could also reduce the amount of
energy necessary to create a warp drive bubble in the same way.

Natario Natario2002  stated that the spacetime contraction and expansion of
the warp bubble is a peculiar consequence of the warp drive metric. Hence, he
designed a spacetime where no contraction or expansion occurs for the warp
drive bubble. Lobo and Visser LoboVisser2004b ; LoboVisser2004  discussed that
the center of the warp bubble proposed by Alcubierre needs to be massless (see
also Refs. White2003, ; White2011, ). They proposed a linearized model for
both Alcubierre and Natario proposals and demonstrated that for small speeds,
the energy stored in the warp fields must be a significant fraction of the
mass of the spaceship inside the warp bubble. Quarra quarra  discussed null
geodesics moving faster-than-light according to far away observers when inside
a region-delimited gravitational wave field. Lee and Cleaver cleaver1 ;
cleaver2  analyzed how external radiation might affect the Alcubierre warp
bubble to turn it unsustainable. They also claimed that a warp field
interferometer could not detect spacetime distortions. Mattingly et al.
cleaver3 ; cleaver4  studied the curvature invariants characteristic of
Natario and Alcubierre warp drives, whereas Mattingly matt  discussed further
curvature invariants for warped spacetimes.

Bobrick and Martire bobrick  claimed that any warp drive spacetime consists of
a shell of regular or exotic material moving inertially with a certain speed,
also reaching at a class of subluminal spherically symmetric warp drives.
Lentz lentz  and Fell and Heisenberg fell  advanced superluminal capable
soliton solutions with positive energy warp drives. Santiago et al. santiago ;
santiago2  argued that only comoving timelike Eulerian observers satisfy the
weak energy condition, whereas this not the case for all timelike observers.
Furthermore, they claimed that all physically reasonable warp drives will
violate the null and weak energy conditions, therefore disputing the claim
advanced by Refs. bobrick ; lentz ; fell  that it would be theoretically
possible to set up positive energy warp drives that satisfy the weak energy
condition.

Motivated by the fact that neither the original paper by Alcubierre, nor the
subsequent ones cited above, did actually solve the Einstein equations using
the warp drive metric, we proceed to investigate possible solutions for a dust
particle energy momentum tensor nos . Our results showed that solutions of the
Einstein equations for the Alcubierre warp drive metric having dust as source
connect them in a particular case the warp drive geometry to the well-known
Burgers equation, which describes the dynamics of the waves moving through an
inviscid fluid. Hence, shock waves appear to be vacuum solutions of the
Einstein equations endowing the warp drive metric nos .

In our second paper nos2  we investigated solutions for the warp drive metric
having the perfect fluid and a special case of anisotropic fluid with heat
flux, but both with zero cosmological constant in the Einstein equations. The
resulting solutions indicate that positive matter densities are possibly
capable of generating superluminal speeds. In our third paper nos3  a charged
dust was used as source EMT for the Alcubierre metric and the Einstein
equations which included the cosmological constant. We obtained solutions
connecting the electric energy density with the cosmological constant and,
again, some solutions were found having positive matter density and satisfying
the energy conditions.

Motivated by the results we obtained in Ref. nos3  we have pondered that even
though the Alcubierre warp drive metric is a vacuum geometry, the warp bubble
would be created by geometry alone, or if a vacuum energy would make it
possible through other material sources of energy and momentum. Hence, in this
paper we went back to the perfect fluid source but included the cosmological
constant in the Einstein equations as an additional flexibility and
geometrical properties for the solutions.

We calculated the Einstein equations and analyzed the null divergence of the
energy momentum tensor together with the validity requirements for the energy
conditions inequalities to be satisfied. We found that the perfect fluid with
the cosmological constant as source for the Alcubierre warp drive results in
four sets of differential equations, two of them are very similar and raise
the possibility for the shift vector to be a complex function in one case,
depending on the $(t,y)$ coordinates, and in another case depending on the
$(t,z)$ coordinates. The other two sets of solutions are identical to each
other and similar to the solution we found in Ref. nos , except that now there
is a cosmological constant coupled with the Burgers equation and, again, the
warp drive is connected to shock waves solutions. Considering that the zero
pressure reduces the perfect fluid to the dust EMT, the solution for this case
is identical to the one we found in Ref. nos , namely, the vacuum solution of
the Einstein equations connecting the warp drive to shock waves.

The plan of the paper is as follows. Section 2 presents a brief review of the
basic equations and concepts of the warp drive theory, and in Section 3 we
solve the Einstein equations and calculated the covariant divergence for the
EMT. Section 4 discusses the energy conditions inequalities and their validity
for the warp drive with $\Lambda\not=0$ and the perfect fluid as a source. In
section 5 we analyze incoherent matter as a source assuming that this is as a
special case considering the perfect fluid with null pressure. In section 6 we
depict our conclusions and final remarks.

##  2 Warp drive basic concepts

###  2.1 Warp drive metric

The warp drive metric is a generic metric in a foliated spacetime given by the
following expression,

${ds}^{2}=-\left(\alpha^{2}-\beta_{i}\beta^{i}\right)\,dt^{2}+2\beta_{i}\,dx^{i}\,dt+\gamma_{ij}\,dx^{i}\,dx^{j}\,\,,$
(2.1)

where $d\tau$ is the proper time lapse, $\alpha$ is the lapse function that
controls the amount of time elapsed between two hypersurfaces of constant time
coordinate, $\beta^{i}$ is the spacelike shift vector and $\gamma_{ij}$ is the
spatial metric for the hypersurfaces. The lapse function $\alpha$ and the
shift vector $\beta_{i}$ are functions of the spacetime coordinates to be
determined, $\gamma_{ij}$ is a positive-definite metric on each one of the
spacelike hypersurfaces and these features make this spacetime globally
hyperbolic. Throughout this paper Greek indices will range from 0 to 3,
whereas the Latin ones indicate the spacelike hypersurfaces and will range
from 1 to 3.

We have the following choices for Eq. (2.1) Alcubierre1994 ,

$\displaystyle\alpha$ $\displaystyle=1,$ (2.2) $\displaystyle\beta_{1}$
$\displaystyle=-v_{s}(t)f\big{[}r_{s}(t)\big{]},$ (2.3)
$\displaystyle\beta_{2}$ $\displaystyle=\beta_{3}=0,$ (2.4)
$\displaystyle\gamma_{ij}$ $\displaystyle=\delta_{ij}.$ (2.5)

Hence, the warp drive metric is given by,

$ds^{2}=-\left[1-v_{s}(t)^{2}f(r_{s})^{2}\right]dt^{2}-v_{s}(t)f(r_{s})\,dx\,dt+dx^{2}+dy^{2}+dz^{2}\,\,,$
(2.6)

where $v_{s}(t)$ is the velocity of the center of the bubble moving along the
curve $x_{s}(t)$, given by,

$v_{s}(t)=\frac{dx_{s}(t)}{dt}\,\,.$ (2.7)

The function $f(r_{s})$ is the warp drive regulating form function. It
describes the shape of the warp bubble, which is given by the expression,

$f(r_{s})=\frac{\tanh\left[\sigma(r_{s}+R)\right]-\tanh\left[\sigma(r_{s}-R)\right]}{2\tanh(\sigma
R)}\,\,,$ (2.8)

where $\sigma$ and $R$ are constants to be determined. The function $r_{s}(t)$
defines the distance from the center of the bubble $[x_{s}(t),0,0]$ to a
generic point $(x,y,z)$ on the surface of the bubble, given by the following
equation,

$r_{s}(t)=\sqrt{\left[x-x_{s}(t)\right]^{2}+y^{2}+z^{2}}\,\,.$ (2.9)

From Eq. (2.9) one can see that the warp bubble is perturbed in a one-
dimensional manner because of the term $x-x_{s}(t)$.

###  2.2 Einstein tensor components

The components of the Einstein tensor with a cosmological constant for the
warp drive metric in Eq. (2.1) are given by the expressions below,

$G_{00}=\Lambda(1-\beta^{2})-\frac{1}{4}(1+3\beta^{2})\left[\left(\frac{\partial\beta}{\partial
y}\right)^{2}+\left(\frac{\partial\beta}{\partial
z}\right)^{2}\right]-\beta\left(\frac{\partial^{2}\beta}{\partial
y^{2}}+\frac{\partial^{2}\beta}{\partial z^{2}}\right),$ (2.10)

$G_{01}=\Lambda\beta+\frac{3}{4}\beta\left[\left(\frac{\partial\beta}{\partial
y}\right)^{2}+\left(\frac{\partial\beta}{\partial
z}\right)^{2}\right]+\frac{1}{2}\left(\frac{\partial^{2}\beta}{\partial
y^{2}}+\frac{\partial^{2}\beta}{\partial z^{2}}\right),$ (2.11)

$G_{02}=-\frac{1}{2}\frac{\partial^{2}\beta}{\partial x\partial
y}-\frac{\beta}{2}\left(2\frac{\partial\beta}{\partial
y}\,\frac{\partial\beta}{\partial x}+\beta\frac{\partial^{2}\beta}{\partial
x\partial y}+\frac{\partial^{2}\beta}{\partial t\partial y}\right),$ (2.12)

$G_{03}=-\frac{1}{2}\frac{\partial^{2}\beta}{\partial x\partial
z}-\frac{\beta}{2}\left(2\frac{\partial\beta}{\partial
z}\,\frac{\partial\beta}{\partial x}+\beta\frac{\partial^{2}\beta}{\partial
x\partial z}+\frac{\partial^{2}\beta}{\partial t\partial z}\right),$ (2.13)

$G_{11}=\Lambda-\frac{3}{4}\left[\left(\frac{\partial\beta}{\partial
y}\right)^{2}+\left(\frac{\partial\beta}{\partial z}\right)^{2}\right],$
(2.14)

$G_{12}=\frac{1}{2}\left(2\frac{\partial\beta}{\partial
y}\,\frac{\partial\beta}{\partial x}+\beta\frac{\partial^{2}\beta}{\partial
x\partial y}+\frac{\partial^{2}\beta}{\partial t\partial y}\right),$ (2.15)

$G_{13}=\frac{1}{2}\left(2\frac{\partial\beta}{\partial
z}\,\frac{\partial\beta}{\partial x}+\beta\frac{\partial^{2}\beta}{\partial
x\partial z}+\frac{\partial^{2}\beta}{\partial t\partial z}\right),$ (2.16)

$G_{23}=\frac{1}{2}\frac{\partial\beta}{\partial
z}\,\frac{\partial\beta}{\partial y},$ (2.17)

$G_{22}=-\Lambda-\frac{1}{4}\left[\frac{\partial^{2}\beta}{\partial t\partial
x}+\beta\frac{\partial^{2}\beta}{\partial
x^{2}}+\left(\frac{\partial\beta}{\partial
x}\right)^{2}\right]-\frac{1}{4}\left[\left(\frac{\partial\beta}{\partial
y}\right)^{2}-\left(\frac{\partial\beta}{\partial z}\right)^{2}\right],$
(2.18)

$G_{33}=-\Lambda-\frac{1}{4}\left[\frac{\partial^{2}\beta}{\partial t\partial
x}+\beta\frac{\partial^{2}\beta}{\partial
x^{2}}+\left(\frac{\partial\beta}{\partial
x}\right)^{2}\right]+\frac{1}{4}\left[\left(\frac{\partial\beta}{\partial
y}\right)^{2}-\left(\frac{\partial\beta}{\partial z}\right)^{2}\right]\,\,,$
(2.19)

where $\beta=-\beta_{1}=v_{s}(t)f(r_{s})$, as in Eq. (2.3). Also noticed that
we incorporated the cosmological constant into the Einstein tensor

$G_{\mu\nu}\to G_{\mu\nu}-\Lambda g_{\mu\nu}$ (2.20)

###  2.3 Energy conditions revisited

The components for the Eulerian (normal) observers’ 4-velocities are given by,

$u^{\alpha}=\left(1,-\beta,0,0\right),\ \ u_{\alpha}=(-1,0,0,0)\,\,.$ (2.21)

Using these results into the Einstein equations,

$T_{\alpha\beta}u^{\alpha}u^{\beta}=\frac{1}{8\pi}G_{\alpha\beta}u^{\alpha}u^{\beta}\,\,,$
(2.22)

results in an expression concerning the energy conditions. From Eqs. (2.21)
and considering that the only non-zero terms of Eq. (2.22) are $G_{00}$,
$G_{01}$ and $G_{11}$, we obtain the following expression,

$T_{\alpha\beta}\,u^{\alpha}u^{\beta}=\frac{1}{8\pi}\left(G_{00}-2\beta
G_{01}+\beta^{2}G_{11}\right).$ (2.23)

Substituting Eqs. (2.10), (2.11) and (2.14) into Eq. (2.23) the result may be
written as,

$T_{\alpha\beta}\,u^{\alpha}u^{\beta}=\Lambda-\frac{v_{s}^{2}}{32\pi}\left[\left(\frac{\partial
f}{\partial y}\right)^{2}+\left(\frac{\partial f}{\partial
z}\right)^{2}\right].$ (2.24)

The bubble radius is given by using Eq. (2.9). So, Eq. (2.24) is given by,

$T_{\alpha\beta}u^{\alpha}u^{\beta}=\Lambda-\frac{v_{s}^{2}}{16\pi}\frac{y^{2}+z^{2}}{r_{s}^{2}}\left(\frac{\partial
f}{\partial r_{s}}\right)^{2}.$ (2.25)

This result is similar to the one found by Alcubierre Alcubierre1994 , with
the difference that Ref. Alcubierre1994  did not consider the cosmological
constant. Considering the results in Ref. Alcubierre1994  we realized that
both the weak and dominant energy conditions would be violated nos  if the
bubble was formed. However, these same energy conditions would be satisfied in
the case of a vacuum solution, which discloses the new result that the warp
drive metric is a vacuum solution for the Einstein equations. Besides, the
Burgers equation is connected to this geometry where shock waves are partial
solutions. Here, with the inclusion of the cosmological constant it may be
possible that the weak and dominant energy conditions could be satisfied if
$\Lambda$ is positive and large enough in Eq. (2.25).

##  3 Matter content energy-momentum tensors

###  3.1 Perfect fluid energy momentum tensor

For Eulerian observers 4-velocity $u^{\alpha}=\left(1,-\beta,0,0\right)$ and
$u_{\alpha}=(-1,0,0,0)$ the perfect fluid EMT for those observers is given by
the expression below,

$T_{\alpha\beta}=(\mu+p)\,u_{\alpha}u_{\beta}+pg_{\mu\nu}\,\,,$ (3.1)

where $\mu$ is a scalar function that represents the matter density, $p$ is
the fluid pressure, and $g_{\mu\nu}$ is the metric tensor. It must be noted
that the dust EMT is a particular case for the perfect fluid with null
pressure.

From the Einstein tensor components, Eqs. (2.10) to (2.19), and the perfect
fluid EMT it is possible to write all the components of Einstein equations.
After some algebraic work we found the following set of equations,

$\frac{4}{3}\Lambda=8\pi\left[T_{00}+2\beta
T_{01}+\left(\beta^{2}-\frac{1}{3}\right)T_{11}\right]=8\pi\left(\mu-\frac{1}{3}p\right)\,,$
(3.2)

$\left(\frac{\partial\beta}{\partial
y}\right)^{2}+\left(\frac{\partial\beta}{\partial
z}\right)^{2}-4\Lambda=-32\pi\left(T_{00}+2\beta
T_{01}+\beta^{2}T_{11}\right)=-32\pi\mu\,\,,$ (3.3)

$\frac{\partial^{2}\beta}{\partial y^{2}}+\frac{\partial^{2}\beta}{\partial
z^{2}}=16\pi(T_{01}+\beta T_{11})=0\,,$ (3.4)

$\left(\frac{\partial\beta}{\partial
y}\right)^{2}+\left(\frac{\partial\beta}{\partial
z}\right)^{2}-\frac{4}{3}\Lambda=-\frac{32}{3}\pi T_{11}=-\frac{32}{3}\pi
p\,,$ (3.5)

$-\frac{\partial}{\partial x}\left(\frac{\partial\beta}{\partial
t}+\frac{1}{2}\frac{\partial}{\partial
x}(\beta^{2})\right)-2\Lambda=8\pi(T_{33}+T_{22})=16\pi p\,,$ (3.6)

$\frac{\partial^{2}\beta}{\partial x\partial y}=-16\pi(T_{02}+\beta
T_{12})=0\,,$ (3.7)

$\frac{\partial^{2}\beta}{\partial x\partial z}=-16\pi(T_{03}+\beta
T_{13})=0\,,$ (3.8)

$\frac{\partial\beta}{\partial y}\frac{\partial\beta}{\partial z}=16\pi
T_{23}=0\,\,,$ (3.9)

###  3.2 Solving the Einstein equations with $\Lambda$ for the perfect fluid

From Eq. (3.9) it follows that either $\partial\beta/\partial z=0$, or
$\partial\beta/\partial y=0$, or both vanish. From Eqs. (3.7) and (3.8) it is
easy to see that $\partial\beta/\partial x$ can also be zero. Those cases
reveals four possibilities, which we will discuss in detail as follows.

Case 1: $\bm{\left[\displaystyle\frac{\partial\beta}{\partial z}=0\right]}$

Case 1a: $\bm{\left[\displaystyle\frac{\partial\beta}{\partial
z}=0\,\,\,\,\text{and}\,\,\,\,\frac{\partial\beta}{\partial x}=0\right]}$

For this case Eqs. (3.2) to (3.9) simplify to,

$\Lambda=6\pi\left(\mu-\frac{p}{3}\right)\,,$ (3.10)

$\left(\frac{\partial\beta}{\partial y}\right)^{2}=4(\Lambda-8\pi\mu)\,\,,$
(3.11)

$\left(\frac{\partial\beta}{\partial y}\right)^{2}=\frac{4}{3}\pi(\Lambda-8\pi
p)\,\,.$ (3.12)

The set of these last equations implies that the shift vector $\beta$ is not
uniquely defined. It is a complex valued function that depends only on $(t,y)$
spacetime coordinates. In the case of the dust EMT as $p=0$, the warp drive
metric is no longer a vacuum solution as it was found in nos , because the
existence of the cosmological constant as another parameter originated a
solution that does not consider shock waves via the Burgers equation.

Case 1b: $\bm{\left[\displaystyle\frac{\partial\beta}{\partial
z}=0\,\,\,\,\text{and}\,\,\,\,\frac{\partial\beta}{\partial y}=0\right]}$

For this case one has to solve the following equations,

$\Lambda=8\pi\mu=8\pi p=0\,,$ (3.13)

$-\frac{\partial}{\partial x}\left(\frac{\partial\beta}{\partial
t}+\frac{1}{2}\frac{\partial}{\partial x}(\beta^{2})\right)=0\,.$ (3.14)

Eq. (3.14) is the Burgers equation that connects the warp drive to shock
waves, as discussed in Ref. nos . The cosmological constant, fluid pressure
and matter density are all equal to zero and the warp drive metric (2.6) is a
vacuum solution for the Einstein equations.

Case 2: $\bm{\left[\displaystyle\frac{\partial\beta}{\partial y}=0\right]}$

Case 2a: $\bm{\displaystyle\left[\frac{\partial\beta}{\partial y}=0\
\text{and}\ \frac{\partial\beta}{\partial x}=0\right]}$

For this configuration, the set of Eqs. (3.2) to (3.9) simplify to

$\Lambda=6\pi\left(\mu-\frac{p}{3}\right)\,,$ (3.15)

$\left(\frac{\partial\beta}{\partial z}\right)^{2}=4(\Lambda-8\pi\mu)\,\,,$
(3.16)

$\left(\frac{\partial\beta}{\partial z}\right)^{2}=\frac{4}{3}\ (\Lambda-8\pi
p)\,.$ (3.17)

The above set of equations are very similar to Case 1a, where the shift vector
is a complex valued function and it is not uniquely defined, but in this case
$\beta$ depends on the $(t,z)$ coordinates.

Case 2b: $\bm{\left[\displaystyle\frac{\partial\beta}{\partial
z}=0\,\,\,\,\text{and}\,\,\,\,\frac{\partial\beta}{\partial y}=0\right]}$

For this case, one has to solve the following equations

$\Lambda=8\pi\mu=8\pi p=0\,,$ (3.18)

$-\frac{\partial}{\partial x}\left(\frac{\partial\beta}{\partial
t}+\frac{1}{2}\frac{\partial}{\partial x}(\beta^{2})\right)=0\,\,,$ (3.19)

which is the same case as Case 1b.

###  3.3 Divergence for the perfect fluid EMT

Calculating the divergence for the perfect fluid EMT, one arrives at the
following equations,

${T^{0\nu}}_{;\nu}=-(\mu+p)\frac{\partial\beta}{\partial
x}-\beta\frac{\partial(p+\mu)}{\partial x}-\frac{\partial\mu}{\partial t}\,,$
(3.20)

${T^{1\nu}}_{;\nu}=\frac{\partial p}{\partial x}\,,$ (3.21)

${T^{2\nu}}_{;\nu}=\frac{\partial p}{\partial y}\,,$ (3.22)

${T^{3\nu}}_{;\nu}=\frac{\partial p}{\partial z}\,.$ (3.23)

Besides, imposing the null divergence condition, Eqs. (3.20) to (3.23) implies
that the pressure $p$ does not depend on the spatial coordinates. Considering
cases 1a and 2a there is another partial differential equation to solve,

$\beta\frac{\partial\mu}{\partial x}+\frac{\partial\mu}{\partial t}=0\,,$
(3.24)

and for Cases 1b and 2b Eq. (3.20) is trivially satisfied since $\mu=p=0$.

##  4 Energy conditions

###  4.1 Weak Energy Conditions

For this case the EMT at each point of the spacetime must obey the inequality

$T_{\alpha\sigma}\,u^{\alpha}u^{\sigma}\geq 0\,\,$ (4.1)

for any timelike vector $\textbf{u}\,(u_{\alpha}u^{\alpha}<0)$ and any null
zero vector $\textbf{k}\,(k_{\alpha}k^{\alpha}=0)$. For an observer with unit
tangent vector v at a certain point of the spacetime, the local energy density
measured by any observer is non-negative HawkingEllis1973 . For the perfect
fluid EMT the expression $T_{\alpha\sigma}\,u^{\alpha}u^{\sigma}$ is

$T_{\alpha\sigma}\,u^{\alpha}u^{\sigma}=\mu\,\,,$ (4.2)

and the weak energy condition from Eq. (4.1) is satisfied if the matter
density $\mu$ is positive. This is also the case for the dust EMT.

###  4.2 Dominant Energy Conditions

For every timelike vector $u_{\alpha}$, the following inequality must be
satisfied,

$T^{\alpha\beta}\,u_{\alpha}u_{\beta}\geq 0,\quad\text{and}\quad
F^{\alpha}F_{\alpha}\leq 0\,\,,$ (4.3)

where $F^{\alpha}=T^{\alpha\beta}u_{\beta}$ is a non-spacelike vector, and the
following condition must also be satisfied

$T^{00}\geq|T^{\alpha\beta}|,\ \text{for each}\ \alpha,\beta\,\,.$ (4.4)

Evaluating the first condition for the perfect fluid EMT we have that,

$T^{\alpha\beta}\,u_{\alpha}u_{\beta}=\mu.$ (4.5)

The other condition $F^{\alpha}F_{\alpha}$ is given by the result

$F^{\alpha}F_{\alpha}=-\mu^{2}\leq 0\,\,.$ (4.6)

Hence, the dominant energy condition is satisfied for $\mu>0$, as can be seen
in Eq. (4.5). Besides, Eq. (4.6) is always satisfied no matter the sign of the
matter density. This condition also holds true if one considers the dust EMT
as a particular case for the perfect fluid with null pressure.

###  4.3 Strong Energy Conditions

For the strong energy condition the expression

$\Bigg{(}T_{\alpha\beta}-\frac{1}{2}T\,g_{\alpha\beta}\Bigg{)}u^{\alpha}u^{\beta}\geq
0$ (4.7)

is true for any timelike vector $u$. Computing the strong energy condition in
Eq. (4.7) yields,

$\Bigg{(}T_{\alpha\beta}-\frac{1}{2}T\,g_{\alpha\beta}\Bigg{)}u^{\alpha}u^{\beta}=\frac{1}{2}(3p+\mu)\,\,,$
(4.8)

and the strong energy condition stated in Eq. (4.7) is satisfied if
$3p+\mu\geq 0$. The same is valid for the dust EMT, considering $p=0$ for the
perfect fluid, if $\mu\geq 0$.

###  4.4 Null Energy Conditions

The null energy conditions are satisfied in the limit of null observers. For
the null vector k the following conditions must be satisfied,

$T_{\alpha\sigma}\,k^{\alpha}k^{\sigma}\geq 0,\qquad\text{for any null
vector}\ k^{\alpha}\,\,.$ (4.9)

Assuming that the following null vector $k^{\alpha}$ is given by,

$k^{\alpha}=(a,b,0,0)\,\,,$ (4.10)

we have that the relation between the components $a$ and $b$ are obtained by
solving $k_{\alpha}k^{\alpha}=0$. The two solutions given by,

$a=\frac{b}{\beta+1}\qquad\mbox{and}\qquad a=\frac{b}{\beta-1}\,\,.$ (4.11)

Then, the null energy condition reads,

$T_{\alpha\sigma}\,k^{\alpha}k^{\sigma}=\left(\frac{b}{\beta\pm
1}\right)^{2}\left(\mu+p\right)\,,$ (4.12)

and the null energy condition may be satisfied if the following conditions are
written as

$\mu+p\geq 0\,\,.$ (4.13)

Eq. (4.13) is also true for the dust EMT if one considers it as a particular
case for the perfect fluid with zero pressure, then the null energy condition
is satisfied for the dust if the matter density is positive.

##  5 Dust as a particular case from the perfect fluid

Table 1 summarizes the results found for the energy conditions for the perfect
fluid with the cosmological constant that are widely known HawkingEllis1973 .
Considering the dust EMT as a particular case for the perfect fluid by
imposing the pressure $p$ to be zero, the energy conditions for the warp drive
metric and the dust EMT would be trivially satisfied, since for this case, the
solution of the Einstein equations is a vacuum solution nos .

Table 1: Summary results for the perfect fluid energy conditions Energy condition | Results
---|---
Weak | $\mu\geq 0$
Strong | $\mu\geq 0$
Dominant | $\mu+3p\geq 0$
Null | $\mu+p\geq 0$

Table 2 summarizes the solutions of the Einstein equations for the perfect
fluid EMT with the cosmological constant and the warp drive metric. As can be
seen there are two types of solutions and each is divided in two sub cases.
Solutions 1b and 2b are identical and require that $\Lambda=p=\mu=0$, where
the two solutions are the ones already found in Ref. nos  for the dust of non
interacting particles EMT. This led to a vacuum solution of the Einstein
equations and the connection between shock waves and the warp drive via the
Burgers equation.

Solutions 1a and 2a in table 2 have structures very similar to the ones of the
same type of equations, but for the solution 1a the shift vector is a function
of both the time and the $y$-spatial coordinates, i.e., $\beta=\beta(y,t)$.
For solution 2a it is a function of both the time and the $z$-spatial
coordinate, i.e., $\beta=\beta(z,t)$.

If we consider the dust solution as a particular case of perfect fluid with
the imposition that the pressure is zero, we have that the four sets of
partial differential equations in Table 2 become a solution for the warp drive
metric and the dust EMT with the cosmological constant. In the case of dust
EMT there is no longer a set of equations 1a and 2a to be solved, only 1b and
2b, that are identical to the ones appearing in Ref. nos . Even with a
cosmological constant the dust EMT seems to be not a stable source of matter,
energy and momentum for the warp drive.

Case |  Condition |  Results
---|---|---
$1)\ \displaystyle{\frac{\partial\beta}{\partial z}=0}$ |  $1a)\ \displaystyle{\frac{\partial\beta}{\partial x}=0}$ |  $\begin{array}[]{ll}\Lambda=6\pi\left(\mu-\frac{p}{3}\right)\\\\[6.0pt] \beta=\beta(y,t)\\\\[6.0pt] \displaystyle{\frac{\partial\beta}{\partial y}=\pm\sqrt{4(\Lambda-8\pi\mu)}}\\\\[8.0pt] \displaystyle{\frac{\partial\beta}{\partial y}=\pm\sqrt{\frac{4}{3}(\Lambda-8\pi p)}}\\\\[8.0pt] \displaystyle{\beta\frac{\partial\mu}{\partial x}+\frac{\partial\mu}{\partial t}=0\ \ \text{(null divergence)}}\\\\[8.0pt] \end{array}$
$1b)\ \displaystyle{\frac{\partial\beta}{\partial y}=0}$ |  $\begin{array}[]{ll}\Lambda=8\pi\mu=8\pi p=0\\\\[6.0pt] \beta=\beta(x,t)\\\\[6.0pt] \displaystyle{\frac{\partial\beta}{\partial t}+\frac{1}{2}\frac{\partial}{\partial x}(\beta^{2})=h(t)}\\\\[6.0pt] \text{Null divergence is trivially satisfied}\\\\[2.0pt] \text{This is the solution found in Ref.\ \cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{nos}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\\[8.0pt] \end{array}$
$2)\ \displaystyle{\frac{\partial\beta}{\partial y}=0}$ |  $2a)\ \displaystyle{\frac{\partial\beta}{\partial x}=0}$ |  $\begin{array}[]{ll}\Lambda=6\pi\left(\mu-\frac{p}{3}\right)\\\\[6.0pt] \beta=\beta(y,t)\\\\[6.0pt] \displaystyle{\frac{\partial\beta}{\partial z}=\pm\sqrt{4(\Lambda-8\pi\mu)}}\\\\[8.0pt] \displaystyle{\frac{\partial\beta}{\partial z}=\pm\sqrt{\frac{4}{3}(\Lambda-8\pi p)}}\\\\[8.0pt] \displaystyle{\beta\frac{\partial\mu}{\partial x}+\frac{\partial\mu}{\partial t}=0\ \ \text{(null divergence)}}\\\\[8.0pt] \end{array}$
$2b)\ \displaystyle{\frac{\partial\beta}{\partial z}=0}$ |  $\begin{array}[]{ll}\Lambda=8\pi\mu=8\pi p=0\\\\[6.0pt] \beta=\beta(x,t)\\\\[6.0pt] \displaystyle{\frac{\partial\beta}{\partial t}+\frac{1}{2}\frac{\partial}{\partial x}(\beta^{2})=h(t)}\\\\[6.0pt] \text{Null divergence is trivially satisfied}\\\\[2.0pt] \text{This is the solution found in Ref.\ \cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{nos}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\\[8.0pt] \end{array}$
Table 2: Summary of all solutions of the Einstein equation with the
cosmological constant and the Alcubierre warp drive metric having the perfect
fluid EMT as mass-energy source. This table is also valid for the dust
particle if considered as a particular case for the perfect fluid with null
pressure.

##  6 Conclusions and final remarks

In this work we investigated how the presence of a cosmological constant would
affect the solutions of the Einstein equations endowed with the Alcubierre
warp drive metric and the perfect fluid EMT as the source. Firstly, we solved
the Einstein equations and obtained two solutions, Cases 1b and 2b, that are
similar to the solutions we found for the dust particle without the
cosmological constant nos , and two other solutions, Case 1a with
$\beta=\beta(t,y)$ and Case 2a with $\beta=\beta(t,z)$, having the following
equation of state relating the cosmological constant $\Lambda$, the matter
density $\mu$ and the fluid pressure $p$: $\Lambda=2\pi(3\mu-p).$

The presence of the cosmological constant allows the shift vector to be a real
valued function as can be seen from Eqs. (3.11) and (3.12) for Case 1a in
Table 2, and Eqs. (3.16) and (3.17) for Case 2a, namely, $\Lambda-8\pi\mu\geq
0$ and $\Lambda-8\pi p\geq 0.$

If we do not consider the cosmological constant, then the shift vector would
become a complex valued function for Cases 1a and 2a in Table 2. The energy
conditions are all satisfied for the perfect fluid if the conditions in Table
1 are satisfied. Solutions 1b and 2b shown in Table 2 connect the Burgers
equation to both the warp drive and the perfect fluid solution. In Ref. nos
we found this intrinsic relationship between the warp drive and shock waves by
solving Einstein equations for the warp drive metric and the dust particle
EMT, but we concluded that there is an impossibility of coupling the dust as a
source in this case. So, the presence of shock waves would imply that the
Alcubierre metric shown in Eq. (2.6) is a vacuum solution for the warp drive.
In Ref. Alcubierre1994  the Einstein equations were not solved, since the
metric was merely guessed with a form function (see Eq. 2.8) that rules the
warp bubble shape.

The results found here led us to a kind of prescription where the warp drive
requires more complex forms of matter than dust in order to obtain stable
solutions. In addition, considering this work and the previous ones of this
series of papers nos ; nos2 ; nos3  it becomes increasingly clear and that
negative matter density may not be a strict requirement to obtain warp speeds.
The shift vector in the direction of the warp bubble movement creates a
coupling in the Einstein equations that requires off-diagonal terms in the EMT
source. In the light of these results we may conjecture that the key for
engineering a superluminal propelling system for interstellar travel could be
understood as a complex distribution of energy, matter and momentum sources
that could stabilize the warp drive geometry, allowing then superluminal
travel.

## Acknowledgments

E.M.C.A. thanks CNPq (Conselho Nacional de Desenvolvimento Científico e
Tecnológico), Brazil’s federal scientific supporting agency, for partial
financial support: grant number 406894/2018-3.

## References

  * (1) M. Alcubierre, The warp drive: hyper-fast travel within general relativity, Class. Quant. Grav. 11 (1994) L73, arXiv:gr-qc/0009013
  * (2) M. Alcubierre and F.S.N. Lobo, Warp drive basics. In: Wormholes, Warp Drives and Energy Conditions, F. Lobo (ed). Fundamental Theories of Physics, vol 189a (2017) 257-279. Springer, Cham., arXiv:2103.05610
  * (3) L.H. Ford and T.A. Roman, Quantum field theory constrains traversable wormhole geometries, Phys. Rev. D 53 (1996) 5496, arXiv:gr-qc/9510071
  * (4) M.J. Pfenning and L.H. Ford The unphysical nature of warp drive, Class. Quant. Grav. 14 (1997) 1743, arXiv:gr-qc/9702026
  * (5) S.V. Krasnikov Hyperfast interstellar travel in General Relativity, Phys. Rev. D 57 (1998) 4760, arXiv:gr-qc/9511068
  * (6) A.E. Everett and T.A. Roman, A superluminal subway: The Krasnikov tube, Phys. Rev. D 56 (1997) 2100, arXiv:gr-qc/9702049
  * (7) F.S.N. Lobo and P. Crawford, Weak energy condition violation and superluminal travel, in Fernández-Jambrina L., González-Romero L.M. (eds) Current Trends in Relativistic Astrophysics. Lecture Notes in Physics, vol 617, (2003). Springer, Berlin, Heidelberg, arXiv:gr-qc/0204038
  * (8) F.S.N. Lobo and P. Crawford, Weak energy condition violation and superluminal travel, Lect. Notes Phys. 617 (2003) 277, arXiv:gr-qc/0204038
  * (9) C. Van Den Broeck, A warp drive with more reasonable total energy, Class. Quant. Grav. 16 (1999) 3973, arXiv:gr-qc/9905084
  * (10) J. Natario, Warp drive with zero expansion, Class. Quant. Grav. 19 (2002) 1157, arXiv: gr-qc/0110086
  * (11) F.S.N. Lobo and M. Visser, Fundamental limitations on ‘warp drive’ spacetimes, Class. Quantum Grav. 21 (2004) 5871, arXiv:gr-qc/0406083
  * (12) F.S.N. Lobo and M. Visser, Linearized warp drive and the energy conditions, Preprint (2004), arXiv:gr-qc/0412065v1
  * (13) H.G. White, A discussion of space-time metric engineering, Gen. Rel. Grav. 35 (2003) 2025
  * (14) H.G. White, Warp field mechanics 101, J. Brit. Interplanetary Society 66 (2011) 242
  * (15) C.J. Quarra, Creating spacetime shortcuts with gravitational waveforms, Preprint (2016), arXiv:1602.01439v1
  * (16) J. Lee and G. Cleaver, Effects of external radiation on an Alcubierre warp bubble, Physics Essays 29 (2016) 201
  * (17) J. Lee and G. Cleaver, The inability of the White-Juday warp field interferometer to spectrally resolve spacetime distortions, Int. J. Modern Phys.: Advances in Theory and Application 2 (2017) 35; arXiv:1407.7772
  * (18) B. Mattingly, A. Kar, M. Gorban, W. Julius, C.K. Watson, M.D. Ali, A. Baas, C. Elmore, J.S. Lee, B. Shakerin, E.W. Davis, and G.B. Cleaver, Curvature invariants for the accelerating Natario warp drive, Particles 3 (2020) 642-659, arXiv:2008.03366
  * (19) B. Mattingly, A. Kar, M. Gorban, W. Julius, C.K. Watson, M. Ali, A. Baas, C. Elmore, J.S. Lee, B. Shakerin, E.W. Davis and G.B. Cleaver, Curvature invariants for the Alcubierre and Natário warp drives, Universe 7 (2021) 21, arXiv:2010.13693
  * (20) B. Mattingly, Curvature invariants for wormholes and warped spacetimes, Preprint (2021), arXiv:2103.14725v1
  * (21) A. Bobrick and G. Martire, Introducing physical warp drives, Class. Quant. Grav. 38 (2021) 105009, arXiv:2102.06824
  * (22) E.W. Lentz, Breaking the warp barrier: Hyper-fast solitons in Einstein-Maxwell-plasma Theory, Class. Quant. Grav. 38 (2021) 075015, arXiv:2006.07125
  * (23) S.D.B. Fell and L. Heisenberg, Positive energy warp drive from hidden geometric structures, Class. Quant. Grav. 38 (2021) 155020, arXiv:2104.06488
  * (24) J. Santiago, S. Schuster and M. Visser, Generic warp drives violate the null energy condition, Preprint (2021), arXiv:2105.03079v1
  * (25) J. Santiago, S. Schuster and M. Visser, Tractor beams, pressor beams and stressor beams in General Relativity, Universe 7 (2021) 271, arXiv:2106.05002
  * (26) O.L. Santos-Pereira, E.M.C. Abreu and M.B. Ribeiro, Dust content content solutions for the Alcubierre warp drive spacetime. Eur. Phys. J. C 80 (2020) 786, arXiv:2008.06560
  * (27) O.L. Santos-Pereira, E.M.C. Abreu and M.B. Ribeiro, Fluid dynamics in the warp drive spacetime geometry. Eur. Phys. J. C 81 (2021) 133, arXiv:2101.11467
  * (28) O.L. Santos-Pereira, E.M.C. Abreu and M.B. Ribeiro, Charged dust solutions for the warp drive spacetime. Gen. Rel. Grav. 53 (2021) 23, arXiv:2102.05119
  * (29) S.W. Hawking and G.F.R. Ellis,The large-scale structure of spacetime, Cambridge University Press, 1973


# MultiLegalPile: A 689GB Multilingual Legal Corpus

###### Abstract

Large, high-quality datasets are crucial for training Large Language Models.
However, so far, there are few datasets available for specialized critical
domains such as law and the available ones are often only for the English
language. We curate and release MultiLegalPile, a 689GB corpus in 24 languages
from 17 jurisdictions. The MultiLegalPile corpus, which includes diverse legal
data sources with varying licenses, allows for pretraining NLP models under
fair use, with more permissive licenses for the Eurlex Resources and Legal mC4
subsets. We pretrain two RoBERTa models and one Longformer multilingually, and
24 monolingual models on each of the language-specific subsets and evaluate
them on LEXTREME. Additionally, we evaluate the English and multilingual
models on LexGLUE. Our multilingual models set a new SotA on LEXTREME and our
English models on LexGLUE. We release the dataset, the trained models, and all
of the code under the most open possible licenses.

Keywords: Machine Learning, ICCL, DMLR, Large, Legal, Dataset, NLP,
Pretraining

FSCS

    Federal Supreme Court of Switzerland
SCI

    Supreme Court of India
ECHR

    European Convention of Human Rights
ECtHR

    European Court of Human Rights
SCOTUS

    Supreme Court of the United States
SPC

    Supreme People’s Court of China
SJP

    Swiss-Judgment-Prediction
ASO

    Almost Stochastic Order
ILDC

    Indian Legal Documents Corpus
US

    United States
EU

    European Union
NLP

    Natural Language Processing
ML

    Machine Learning
LJP

    Legal Judgment Prediction
SJP

    Swiss-Judgment-Prediction
PJP

    Plea Judgment Prediction
BERT

    Bidirectional Encoder Representations from Transformers
LSTM

    Long Short-Term Memory
GRU

    Gated Recurrent Unit
BiLSTM

    Bidirectional Long Short-Term Memory
CNN

    Convolutional Neural Networks
PLM

    pretrained Language Model
LLM

    Large Language Model
CLT

    Cross-Lingual Transfer
HRL

    high resource language
LRL

    low resource language
POS

    Part-of-Speech
SLTC

    Single Label Text Classification
MLTC

    Multi Label Text Classification
NER

    Named Entity Recognition
NLU

    Natural Language Understanding
GNB

    Gaussian Naive Bayes
DT

    Decision Tree
SVM

    Support-vector Machine
RF

     Random Forest
XGBoost

    eXtreme Gradient Boosting
MLIR

    Multilingual Information Retrieval
BGE

    Leading Decision
FSCD

    Federal Supreme Court Decisons
FSCS

    Swiss Federal Supreme Court

MultiLegalPile: A 689GB Multilingual Legal Corpus

Anonymous submission
---

Abstract content

## 1\. Bibliographical References


# Stable diffeomorphism classification of some unorientable 4-manifolds

Arun Debray  Department of Mathematics, University of Texas, Austin, Texas
78712<EMAIL_ADDRESS>
###### Abstract.

Kreck’s modified surgery theory reduces the classification of closed,
connected 4-manifolds, up to connect sum with some number of copies of
$S^{2}\times S^{2}$, to a series of bordism questions. We implement this in
the case of unorientable 4-manifolds $M$ and show that for some choices of
fundamental groups, the computations simplify considerably. We use this to
solve some cases in which $\pi_{1}(M)$ is finite of order 2 mod 4: under an
assumption on cohomology, there are nine stable diffeomorphism classes for
which $M$ is pin+, one stable diffeomorphism class for which $M$ is pin-, and
four stable diffeomorphism classes for which $M$ is neither. We also determine
the corresponding stable homeomorphism classes.

##  0\. Introduction

The classification of closed $4$-manifolds up to diffeomorphism is impossible
in general: a solution would also solve the word problem for groups. Even if
one fixes the fundamental group to avoid this problem, the classification is
still currently intractable. For this reason, topologists study weaker
classifications of $4$-manifolds which are coarse enough to be calculable yet
fine enough to be useful.

Stable diffeomorphism is an example of such an invariant. Two closed
$4$-manifolds $M$ and $N$ are _stably diffeomorphic_ if there are $m,n\geq 0$
such that $M\mathop{\\#}m(S^{2}\times S^{2})$ is diffeomorphic to
$N\mathop{\\#}n(S^{2}\times S^{2})$. This notion of equivalence has
applications to quantum topology: for example, Reutter [Reu20, Theorem A]
shows that the partition functions of 4d semisimple oriented TFTs are
insensitive to stable diffeomorphism along the way to showing that such TFTs
cannot distinguish homotopy-equivalent closed, oriented $4$-manifolds. And
stable diffeomorphism classes are computable: once the fundamental group $G$
is fixed, Kreck [Kre99] shows how to reduce the classification of
$4$-manifolds up to stable diffeomorphism to a collection of bordism
computations, and for many choices of $G$, the classification of closed,
connected, oriented $4$-manifolds with $\pi_{1}(M)\cong G$ up to stable
diffeomorphism has been completely worked out, thanks to work of Wall [Wal64],
Teichner [Tei92], Spaggiari [Spa03], Crowley-Sixt [CS11], Politarczyk [Pol13],
Kasprowski-Land-Powell-Teichner [KLPT17], Pedrotti [Ped17], Hambleton-Hildum
[HH19], and Kasprowski-Powell-Teichner [KPT20].

Researchers interested in topological manifolds also study _stable
homeomorphism_ of topological manifolds, i.e. homeomorphism after connect-
summing with some number of copies of $S^{2}\times S^{2}$. Kreck’s theorem
applies to this case too, reframing the question in terms of bordism of
topological manifolds. Stable homeomorphism classifications are studied by
Teichner [Tei92, §5], Wang [Wan95], Hambleton-Kreck-Teichner [HKT09],
Kasprowski-Land-Powell-Teichner [KLPT17, §§4–5], Hambleton-Hildum [HH19], and
Kasprowski-Powell-Teichner [KPT20, §2.3],

Much less work has been done on unorientable $4$-manifolds, even though the
theory still works and is simpler in some cases, as we explain below. There is
some work in the literature, such as that of Kreck [Kre84], Wang [Wan95],
Kurazono [Kur01], Davis [Dav05], and Friedl-Nagel-Orson-Powell [FNOP19, §12].

The goal of this paper is to compute sets of stable diffeomorphism and stable
homeomorphism classes for a class of unorientable $4$-manifolds, as well as
determining the corresponding complete stable diffeomorphism and homeomorphism
invariants. As a consequence of our Theorem 2.1, for many finite groups $G$,
the classification of stable diffeomorphism or homeomorphism classes of
unorientable $4$-manifolds with $\pi_{1}(M)\cong G$ reduces to the stable
classifications for a smaller $2$-group. For example, we show that the stable
diffeomorphism, resp. homeomorphism classification when
$\pi_{1}(M)\cong\mathbb{Z}/2$ determines the stable diffeomorphism, resp.
homeomorphism classification for some groups $G$ of order $2\bmod 4$. We then
compute these classifications using Kreck’s techniques.

Suppose $G$ is the fundamental group of an unorientable manifold. Then there
is an extension

(0.1)
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}/2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1,}$

where $G\twoheadrightarrow\mathbb{Z}/2$ is defined by classifying loops as
orientation-preserving or orientation-reversing. Therefore $\mathbb{Z}/2$ acts
on $K$.

######  Theorem (Main theorem).

Let $G$ be a finite group of order $2\bmod 4$, and suppose that in (0.1),
$\mathbb{Z}/2$ acts trivially on $H^{*}(BK)$.

  1. (1)

There are fourteen equivalence classes of closed, connected, unorientable
$4$-manifolds $M$ up to stable diffeomorphism: nine for which $M$ is pin$+$,
one for which $M$ is pin$-$, and four for which $M$ is neither.

  2. (2)

There are twenty equivalence classes of closed, connected, unorientable
topological $4$-manifolds $M$ up to stable homeomorphism: ten for which $M$ is
pin$+$, two for which $M$ is pin$-$, and eight for which $M$ is neither.

This is a combination of Theorems 3.1, 3.5, 4.2, and 4.5. In those theorems we
also determine complete stable diffeomorphism/homeomorphism invariants for
these manifolds. The classification for $M$ neither pin$+$ or pin$-$ can be
extracted from work of Davis [Dav05, Theorem 2.3], but the other parts are
new.

We prove these theorems by establishing isomorphisms of bordism groups.
Specifically, Kreck’s modified surgery theory associates to $G$ a set of
symmetry types $\xi\colon B\to B\mathrm{O}$ and expresses the set of stable
diffeomorphism classes in terms of the bordism groups $\Omega_{4}^{\xi}$; we
show that when $\lvert G\rvert\equiv 2\bmod 4$ and the assumption about
$H^{*}(K)$ holds, the Thom spectra of these symmetry types are homotopy
equivalent to the Thom spectra for unoriented, pin$+$, and pin$-$ bordism. In
the smooth case, the bordism groups $\Omega_{4}^{\mathrm{O}}$,
$\Omega_{4}^{\mathrm{Pin}^{+}}$, and $\Omega_{4}^{\mathrm{Pin}^{-}}$ are well-
known. The topological versions of these bordism groups are less well-known,
but Kirby-Taylor [KT90b, §9] compute $\Omega_{4}^{\mathrm{TopPin}^{\pm}}$ and
provide enough information for us to compute $\Omega_{4}^{\mathrm{Top}}$,
which we do in Proposition 4.7.

The argument we use to establish the isomorphism from $\xi$-bordism to a
simpler kind of bordism applies to more general choices of $\pi_{1}(M)$.

######  Theorem 2.1.

Suppose $G$ is a finite group fitting into an extension

(0.2)
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1,}$

where $\lvert K\rvert$ is odd and $P$ is a $2$-group, and suppose $P$ acts
trivially on $H^{*}(BK)$. For any unorientable virtual vector bundle $V\to
BP$, $\varphi$ induces an equivalence of Thom spectra
$(BG)^{\varphi^{*}V}\overset{\simeq}{\to}(BP)^{V}$.

The Pontrjagin-Thom construction turns this equivalence into isomorphisms of
bordism groups from the unorientable symmetry types Kreck associates to $G$ to
the unorientable symmetry types for $P$, which we can use to compute stable
diffeomorphism classes. The proof strongly requires the assumption that $V$ is
unorientable; nothing like this is true in the oriented case.

Our main theorem above covers the case $\lvert G\rvert\equiv 2\bmod 4$. The
next step would be to consider $P\cong\mathbb{Z}/2\times\mathbb{Z}/2$ or
$\mathbb{Z}/4$, which would suffice for many groups $G$ of order $4\bmod 8$.
For these choices of $P$, many of the needed bordism groups have already been
computed in the literature for other applications. For $P\cong\mathbb{Z}/4$,
see Botvinnik-Gilkey [BG97, §5]; for $P\cong\mathbb{Z}/2\times\mathbb{Z}/2$,
see work of Guo-Ohmori-Putrov-Wan-Wang [GOP+20, §7], the author in [KPMT20,
Appendix F] and [Deb21, §4.4], and Wan-Wang-Zheng [WWZ20, Appendix A].

We begin in §1 with a quick review of Kreck’s theorem [Kre99] on stable
diffeomorphism classes of $4$-manifolds within a given $1$-type. In §2, we
study the Thom spectra of unorientable vector bundles over $BG$, where $G$ is
a finite group, proving Theorem 2.1. In §3, we specialize to the case where
$\lvert G\rvert\equiv 2\bmod 4$, determining the three possible normal
$1$-types and computing the sets of stable diffeomorphism classes for them. We
prove Theorems 3.1 and 3.5, which together form the smooth part of the main
theorem above. In Example 3.4, we discuss an example: $\mathbb{RP}^{4}$ is
homeomorphic but not stably diffeomorphic to Cappell-Shaneson’s fake
$\mathbb{RP}^{4}$. This fact was known to Cappell-Shaneson [CS71, CS76] and
the proof using Kreck’s surgery theory is due to Stolz [Sto88]. In §4, we
consider stable homeomorphism classes of topological manifolds with
$\lvert\pi_{1}(M)\rvert\equiv 2\bmod 4$, and prove Theorems 4.2 and 4.5, which
form the topological part of the main theorem above.

### Acknowledgments

I thank my advisor, Dan Freed, for his constant help and guidance. I also
would like to thank Matthias Kreck, Riccardo Pedrotti, and Oscar Randal-
Williams for some helpful conversations related to this paper.

A portion of this work was supported by the National Science Foundation under
Grant No. 1440140 while the author was in residence at the Mathematical
Sciences Research Institute in Berkeley, California, during January–March
2020.

##  1\. Review: normal $1$-types, normal $1$-smoothings, and stable
diffeomorphism classes

We review some standard definitions in this area. We will always assume our
manifolds are closed and connected. Except in §4, we also assume they are
smooth.

######  Definition 1.1.

A _normal $1$-type_ of a manifold $M$ is a fibration $\xi\colon B\to
B\mathrm{O}$ such that there is a lift of the map $\nu\colon M\to BO$
classifying the stable normal bundle of $M$ to a map $\widetilde{\nu}\colon
M\to B$ such that $\xi\circ\widetilde{\nu}=\nu$, $\widetilde{\nu}$ is
$2$-connected, and $\xi$ is $2$-coconnected.

A choice of such a lift is called a _normal $1$-smoothing_ of $M$.

Any two normal $1$-types of a given manifold are homotopy equivalent as spaces
over $B\mathrm{O}$, so we will abuse notation and say “the” normal $1$-type.

The map $\xi\colon B\to B\mathrm{O}$ determines a bordism theory of manifolds
with a lift of the stable normal bundle across $\xi$, which we denote
$\Omega_{*}^{\xi}$; a normal $1$-smoothing of $M$ determines a class in this
bordism group. Different normal smoothings of the same manifold do not always
define the same class in $\Omega_{*}^{\xi}$.

Let $V_{\mathrm{SO}}\to B\mathrm{SO}$, $V_{\mathrm{Spin}}\to B\mathrm{Spin}$,
etc., denote the tautological stable vector bundles over their respective
spaces. We use the convention that maps to $B\mathrm{O}$ are represented by
rank-zero virtual vector bundles, which is why we write $E-\dim E$ in (1.3),
for example.

######  Example 1.2 (Kreck [Kre99, §2, Proposition 2]).

When $M$ is unorientable, Kreck classifies the possible normal $1$-types of
$M$ into two families. Let $M^{\prime}\to M$ be the universal cover of $M$,
which is classified by a map $\theta\colon M\to B\pi_{1}(M)$.

Almost spin:

If $M^{\prime}$ admits a spin structure, $M$ is called _almost spin_. In this
case, $w_{1}(M)=\theta^{*}x_{1}$ and $w_{2}(M)=\theta^{*}x_{2}$ for some
$x_{1},x_{2}\in H^{*}(BG;\mathbb{Z}/2)$. Assume there is a vector bundle $E\to
BG$ such that $w_{i}(E)=x_{i}$ for $i=1,2$.111This will be true for all cases
we consider in this paper, but is not true in general. Then, the normal
$1$-type of $M$ is

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Totally non-spin:

If $M^{\prime}$ does not admit a spin structure, $M$ is called _totally non-
spin_. In this case, $w_{1}(M)=\theta^{*}x$ for some $x\in
H^{1}(BG;\mathbb{Z}/2)$. Let $E\to BG$ be a line bundle with $w_{1}(E)=x$.
Then the normal $1$-type of $M$ is

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Because $S^{2}\times S^{2}$ has trivial stable normal bundle, taking connect
sum with $S^{2}\times S^{2}$ does not change the normal $1$-type of a
$4$-manifold; thus the classification of $4$-manifolds up to stable
diffeomorphism can proceed one normal $1$-type at a time. Moreover, because
$S^{2}\times S^{2}$ is null-bordant, one might conclude that stably
diffeomorphic $4$-manifolds $M$ and $N$ are bordant — or, more precisely, that
$M$ and $N$ admit normal $1$-smoothings which are bordant in
$\Omega_{4}^{\xi}$. So a plausible lower bound for the set of stable
diffeomorphism classes with normal $1$-type $\xi$ would be $\Omega_{4}^{\xi}$
modulo some identifications arising from inequivalent normal $1$-smoothings of
the same underlying manifold. Remarkably, this turns out to be a complete
classification!

######  Theorem 1.5 (Kreck [Kre99, Theorem C; §3, Proposition 4]).

  1. (1)

If $M$ and $N$ are $4$-manifolds of the same normal $1$-type $\xi$ admitting
normal $1$-smoothings which are bordant in $\Omega_{4}^{\xi}$, then $M$ is
stably diffeomorphic to $N$.

  2. (2)

If $\pi_{1}(\xi)$ is finite, every class in $\Omega_{4}^{\xi}$ can be realized
as the normal $1$-smoothing of a $4$-manifold with normal $1$-type $\xi$.

The upshot is that if $\mathrm{Aut}(\xi)$ denotes the group of fiber homotopy
equivalences of $\xi\to B\mathrm{O}$, the set of stable diffeomorphism classes
of $4$-manifolds with normal $1$-type $\xi$ is
$\Omega_{4}^{\xi}/\mathrm{Aut}(\xi)$.

The set of bordism classes of normal $1$-smoothings of a given $4$-manifold is
contained within an $\mathrm{Aut}(\xi)$-orbit of $\Omega_{4}^{\xi}$, so one
effect of the quotient is to identify these as all coming from the same
manifold.

This illustrates the standard way to calculate stable diffeomorphism classes:
determine $\Omega_{4}^{\xi}$, then determine the $\mathrm{Aut}(\xi)$-action.
These bordism groups are the homotopy groups of the Thom spectrum $M\xi$ of
$\xi$, so in the next section we begin the calculation of stable
diffeomorphism classes by simplifying $M\xi$.

##  2\. Simplifying Thom spectra

Theorem 1.5 tells us to investigate the Thom spectra of the normal $1$-types
in Example 1.2. In both cases, the vector bundle is an exterior direct sum, so
the Thom spectra split, as $\mathit{MSpin}\wedge(B\pi_{1}(M))^{V}$ in the
almost spin case and $\mathit{MSO}\wedge(BG)^{V}$ in the totally non-spin
case, where $V$ is a rank-zero unoriented virtual vector bundle. We attack the
problem by simplifying $(B\pi_{1}(M))^{V}$ for some choices of $\pi_{1}(M)$.

######  Theorem 2.1.

Suppose $G$ is a finite group fitting into an extension

(2.2)
$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1,}$

where $\lvert K\rvert$ is odd and $P$ is a $2$-group, and suppose $P$ acts
trivially on $H^{*}(BK)$. For any unorientable virtual vector bundle $V\to
BP$, $\varphi$ induces an equivalence of Thom spectra
$(BG)^{\varphi^{*}V}\overset{\simeq}{\to}(BP)^{V}$.

We’ll prove this in a series of lemmas.

######  Definition 2.3.

Let $H$ be a group, $A$ be an abelian group, and $\alpha\in
H^{1}(BH;\mathbb{Z}/2)$. Using the identification
$H^{1}(BH;\mathbb{Z}/2)\cong\operatorname{Hom}(H,\mathbb{Z}/2)$, let
$A_{\alpha}$ be the $\mathbb{Z}[H]$-module which is the abelian group
$\mathbb{Z}$ with the $H$-action in which $g\in H$ acts by $(-1)^{\alpha(g)}$.

######  Lemma 2.4.

In the situation of Theorem 2.1, both $\widetilde{H}^{*}((BG)^{\varphi^{*}V})$
and $\widetilde{H}^{*}((BP)^{V})$ are $2$-torsion.

###### Proof.

Using Definition 2.3, we define the $\mathbb{Z}[P]$-module $A_{w_{1}(V)}$ and
the $\mathbb{Z}[G]$-module $A_{w_{1}(\varphi^{*}P)}$, which is isomorphic to
the pullback of $A_{w_{1}(V)}$ by $\varphi$. The Thom isomorphism provides
isomorphisms of graded abelian groups

(2.5a) $\displaystyle H^{*}(BP;\mathbb{Z}_{w_{1}(V)})$
$\displaystyle\overset{\cong}{\longrightarrow}\widetilde{H}^{*}((BP)^{V})$
(2.5b) $\displaystyle H^{*}(BG;\mathbb{Z}_{w_{1}(\varphi^{*}V)})$
$\displaystyle\overset{\cong}{\longrightarrow}\widetilde{H}^{*}((BG)^{\varphi^{*}V}),$

so we will prove the lemma using group cohomology – specifically, using the
Lyndon-Hochschild-Serre spectral sequence

(2.6) $E_{2}^{p,q}=H^{p}(BP;(H^{q}(BK;\mathbb{Z}))_{w_{1}(V)})\Longrightarrow
H^{p+q}(BG;\mathbb{Z}_{w_{1}(\varphi^{*}V)}).$

Here it is crucial that $P$ acts trivially on $H^{*}(BK)$; otherwise we would
have a different local coefficient system than
$H^{q}(BK;\mathbb{Z})_{w_{1}(V)}$ in (2.6).

Since $E_{2}^{p,q}\cong H^{p}(BP;M_{q})$ for some $\mathbb{Z}[P]$-module
$M_{q}$, $E_{2}^{p,q}$ is $2$-torsion for $p>1$ by Maschke’s theorem.222We use
Maschke’s theorem as follows: if $G$ is a finite group and $k$ is a field of
characteristic $0$ or characteristic $\ell\nmid\\#G$, the category of
$k[G]$-modules is semisimple. Therefore all positive-degree Ext groups vanish,
in particular $H^{m}(BG;M)\cong\operatorname{Ext}_{k[G]}^{m}(\mathbb{Z},M)$
for any $k[G]$-module $M$ and $m>1$. Combined with the universal coefficient
theorem, this implies that for any $\mathbb{Z}[G]$-module $M$ and $m>1$,
$H^{m}(G;M)$ is torsion ($k=\mathbb{Q}$), and lacks $\ell$-torsion if
$\ell\nmid\\#G$. When $p=0$,

(2.7) $E_{2}^{0,q}\cong
H^{0}(BP;H^{q}(BK)_{w_{1}(V)})\cong(H^{q}(BK)_{w_{1}(V)})^{P}.$

We will show this vanishes. First, $H^{q}(BK)$ is $\mathbb{Z}$ for $q=0$ and
is odd-primary torsion for $q>0$ (by Maschke’s theorem, because $2\nmid\\#K$).
Therefore if $a\in H^{q}(BK)$ and $-a=a$, $a=0$. Since $w_{1}(V)\neq 0$, there
is some $g\in P$ which acts on $\mathbb{Z}_{w_{1}(V)}$ as $-1$, hence also
acts on $H^{q}(BK)_{w_{1}(V)}$ as $-1$, so the subgroup of invariants of
$H^{q}(BK)_{w_{1}(V)}$ is $\\{0\\}$.

Considering the line $q=0$ proves $H^{*}(BP;\mathbb{Z}_{w_{1}(V)})$ is
$2$-torsion. For $H^{*}(BG;\mathbb{Z}_{w_{1}(\varphi^{*}V)})$, we have shown
the $E_{2}$-page is $2$-torsion, so the graded abelian group the spectral
sequence converges to is also $2$-torsion. ∎

######  Lemma 2.8.

With $G$ and $P$ as in Theorem 2.1, $\varphi^{*}\colon
H^{*}(BP;\mathbb{Z}/2)\to H^{*}(BG;\mathbb{Z}/2)$ is an isomorphism of graded
rings.

###### Proof.

Since $K$ has odd order, its mod $2$ cohomology is $\mathbb{Z}/2$ in degree
$0$ and vanishes elsewhere, so the result follows from the Leray-Hirsch
theorem applied to the fibration $BK\to BG\to BP$ induced by (2.2). ∎

###### Proof of Theorem 2.1.

Use the homology Whitehead theorem: if $f\colon X\to Y$ is a map of bounded-
below spectra which induces an isomorphism on rational cohomology and on mod
$p$ cohomology for every prime $p$, then $f$ is a homotopy equivalence. Lemma
2.4 and the universal coefficient theorem imply that if $k=\mathbb{Q}$ or
$k=\mathbb{Z}/p$ for an odd prime $p$,
$\widetilde{H}^{*}((BG)^{\varphi^{*}V};k)$ and $\widetilde{H}^{*}((BP)^{V};k)$
both vanish, so the map between them is vacuously an isomorphism. The sole
remaining case is $p=2$. Since $1\equiv-1\bmod 2$, $(\mathbb{Z}/2)_{w_{1}(V)}$
carries the trivial $P$-action; thus, the Thom isomorphism has the form

(2.9a)
$H^{*}(BP;\mathbb{Z}/2)\overset{\cong}{\longrightarrow}\widetilde{H}^{*}((BP)^{V};\mathbb{Z}/2).$
Analogously, there is a Thom isomorphism (2.9b)
$H^{*}(BG;\mathbb{Z}/2))\overset{\cong}{\longrightarrow}\widetilde{H}^{*}((BG)^{\varphi^{*}V};\mathbb{Z}/2).$

As the Thom isomorphism is functorial with respect to pullbacks of vector
bundles, Lemma 2.8 lifts to imply that

(2.10)
$\varphi^{*}\colon\widetilde{H}^{*}((BP)^{V};\mathbb{Z}/2)\longrightarrow\widetilde{H}^{*}((BG)^{\varphi^{*}V};\mathbb{Z}/2)$

is an isomorphism. ∎

##  3\. The case $\lvert\pi_{1}(X)\rvert\equiv 2\bmod 4$

If $M$ is an unorientable manifold, the description of loops as orientation-
preserving or orientation-reversing defines a surjection
$p\colon\pi_{1}(M)\to\mathbb{Z}/2$, so $\pi_{1}(M)$ cannot have odd order.
Thus the simplest case occurs when $\lvert\pi_{1}(M)\rvert\equiv 2\bmod 4$, so
that $\lvert\ker(p)\rvert$ is odd. For the rest of this section, fix such a
group $G$, and assume that $\mathbb{Z}/2$ acts trivially on $H^{*}(B\ker(p))$.

In this case, Theorem 2.1 applies to show that if $\mathbb{Z}/2$ acts
trivially on $H^{*}(B\ker(p))$ and $V\to B\mathbb{Z}/2$ is any unorientable
virtual vector bundle, the map
$(B\pi_{1}(M))^{p^{*}V}\overset{\simeq}{\to}(B\mathbb{Z}/2)^{V}$ is an
equivalence.

Let $\sigma\to B\mathbb{Z}/2$ denote the tautological line bundle and
$x\coloneqq w_{1}(\sigma)\in H^{1}(B\mathbb{Z}/2;\mathbb{Z}/2)$, so
$H^{*}(B\mathbb{Z}/2;\mathbb{Z}/2)\cong\mathbb{Z}/2[x]$. Because $\ker(p)$ has
odd order, the Leray-Hirsch theorem implies $p^{*}\colon
H^{*}(B\mathbb{Z}/2;\mathbb{Z}/2)\to H^{*}(B\pi_{1}(M);\mathbb{Z}/2)$ is an
isomorphism.

###  3.1. The almost spin case

Example 1.2 shows there are two unorientable normal $1$-types in this case:
$w_{1}(\nu)\neq 0$, so it must be the pullback of $p^{*}x\in
H^{1}(B\pi_{1}(M);\mathbb{Z}/2)$, and for $w_{2}$, we have two choices:
$w_{2}=0$ (the normal bundle is pin$+$) and $w_{2}=p^{*}x^{2}$ (the normal
bundle is pin$-$).

Recall that for a manifold $M$, $M$ is pin$\pm$ (i.e. the tangent bundle is
pin$\pm$) iff the normal bundle is pin$\mp$. A (tangential) pin$+$
$4$-manifold $M$ has a $\mathbb{Z}/16$-valued invariant given by the
$\eta$-invariant of a twisted Dirac operator [Sto88, §4]; let $\eta^{\prime}$
be the invariant assigning to a pin$+$ $4$-manifold $M$ the image of this
$\eta$-invariant in the nine-element set $(\mathbb{Z}/16)/(x\sim-x)$. We will
see in the proof of Theorem 3.1 that all pin$+$ structures on $M$ give the
same value of $\eta^{\prime}$, so we may define it as an invariant of
manifolds which admit a pin$+$ structure, without choosing such a structure.

######  Theorem 3.1.

There are nine stable diffeomorphism classes of unorientable $4$-manifolds
with $\pi_{1}(M)\cong G$ that admit a (tangential) pin$+$ structure, and there
is a single stable diffeomorphism class of manifolds with $\pi_{1}(M)\cong G$
that admit a (tangential) pin$-$ structure. In the pin$+$ case,
$\eta^{\prime}$ is a complete stable diffeomorphism invariant.

###### Proof.

Both choices of $(w_{1},w_{2})$ arise from vector bundles: $(p^{*}x,0)$ from
$p^{*}\sigma$, and $(p^{*}x,p^{*}x^{2})$ from $p^{*}(3\sigma)$. Thus the
normal $1$-types are

(3.2a) $\displaystyle V_{\mathrm{Spin}}\oplus(p^{*}\sigma-1)\colon
B\mathrm{Spin}\times B\pi_{1}(M)$ $\displaystyle\longrightarrow B\mathrm{O}$
(3.2b) $\displaystyle V_{\mathrm{Spin}}\oplus(p^{*}(3\sigma)-3)\colon
B\mathrm{Spin}\times B\pi_{1}(M)$ $\displaystyle\longrightarrow B\mathrm{O},$

and their Thom spectra are
$\mathit{MSpin}\wedge(B\pi_{1}(M))^{p^{*}\sigma-1}$, resp.
$\mathit{MSpin}\wedge(B\pi_{1}(M))^{p^{*}(3\sigma)-3}$. By Theorem 2.1, these
are equivalent to $\mathit{MSpin}\wedge(B\mathbb{Z}/2)^{\sigma-1}$, resp.
$\mathit{MSpin}\wedge(B\mathbb{Z}/2)^{3\sigma-3}$.

######  Theorem 3.3 (Peterson [Pet68, §7], Kirby-Taylor [KT90a, Lemma 6]).

There are equivalences
$\mathit{MSpin}\wedge(B\mathbb{Z}/2)^{\sigma-1}\simeq\mathit{MTPin}^{-}$ and
$\mathit{MSpin}\wedge(B\mathbb{Z}/2)^{3\sigma-3}\simeq\mathit{MTPin}^{+}$.333There
is an important subtlety in the names of these spectra in the literature:
$\mathit{MPin}^{\pm}$ denotes the Thom spectra classifying pin$\pm$ structures
on the stable normal bundle, and $\mathit{MTPin}^{\pm}$ denotes the Thom
spectra classifying pin$\pm$ structures on the stable tangent bundle. There
are equivalences $\mathit{MPin}^{\pm}\simeq\mathit{MTPin}^{\mp}$. Information
on pin$\pm$ bordism is usually stated in terms of $\mathit{MTPin}^{\pm}$.

These bordism groups are known.

  * •

In the case $w_{2}(\nu)=0$,
$\Omega_{4}^{\xi}\cong\Omega_{4}^{\mathrm{Pin}^{-}}\cong 0$ [ABP69, KT90b] —
all $4$-manifolds with this normal $1$-type are stably diffeomorphic.

  * •

When $w_{2}(\nu)=p^{*}x^{2}$,
$\Omega_{4}^{\xi}\cong\Omega_{4}^{\mathrm{Pin}^{+}}\cong\mathbb{Z}/16$ [Gia73,
KT90a, KT90b].

In the latter case we have to determine the $\mathrm{Aut}(\xi)$-action.
$\mathbb{RP}^{4}$ admits two pin$+$ structures, and Kirby-Taylor [KT90b,
Theorem 5.2] choose an isomorphism
$\Omega_{4}^{\mathrm{Pin}^{+}}\overset{\cong}{\to}\mathbb{Z}/16$ sending these
two pin$+$ structures to $\pm 1$. Therefore for any equivalence class
$x\in\Omega_{4}^{\mathrm{Pin}^{+}}$ and any
$g\in\mathrm{Aut}(\mathrm{Pin}^{+})$, $g\cdot x=\pm x$, because we can
represent $x$ as a disjoint union of copies of $\mathbb{RP}^{4}$ with some
pin$+$ structure, and the $\mathrm{Aut}(\mathrm{Pin}^{+})$-orbit of the
$\mathbb{RP}^{4}$s is $\\{\pm 1\\}$. The isomorphism from $\xi$-bordism to
pin$+$ bordism allows us to also deduce that the $\mathrm{Aut}(\xi)$-orbit of
a class $[M]$ in $\Omega_{4}^{\xi}$ is $\\{\pm[M]\\}$. We obtain nine
equivalence classes: $0,\pm 1,\pm 2,\dotsc,\pm 7,8$, detected by the image of
the $\eta$-invariant in $(\mathbb{Z}/16)/(x\sim-x)$. ∎

As a consequence of Kreck’s classification in Example 1.2, we have seen that
all unorientable, almost spin $4$-manifolds $M$ with $\pi_{1}(M)\cong G$ are
either pin$+$ or pin$-$, and that this determines their normal $1$-type. This
is not true for more general $G$.

######  Example 3.4.

Cappell-Shaneson [CS71, CS76] construct a closed, smooth $4$-manifold $Q$ that
is homeomorphic but not diffeomorphic to $\mathbb{RP}^{4}$, and show that $Q$
and $\mathbb{RP}^{4}$ are not stably diffeomorphic. Stolz [Sto88] gives
another proof of this fact by computing the classes of $\mathbb{RP}^{4}$ and
$Q$ in $\Omega_{4}^{\xi}/\mathrm{Aut}(\xi)$. We briefly summarize Stolz’
proof.

Since $\pi_{1}(\mathbb{RP}^{4})\cong\mathbb{Z}/2$ and
$w_{2}(\mathbb{RP}^{4})=0$, the proof of Theorem 3.1 shows
$M\xi\simeq\mathit{MTPin}^{+}$, $\Omega_{4}^{\xi}\cong\mathbb{Z}/16$, and the
set of stable diffeomorphism classes is
$\Omega_{4}^{\xi}/\mathrm{Aut}(\xi)\cong(\mathbb{Z}/16)/(x\sim-x)$. Stolz
[Sto88] chooses an isomorphism
$\Omega_{4}^{\xi}\overset{\cong}{\to}\mathbb{Z}/16$ and shows that it sends
the two pin$+$ structures on $\mathbb{RP}^{4}$ to $\pm 1$ and the two pin$+$
structures on $Q$ to $\pm 9$; therefore $\mathbb{RP}^{4}$ and $Q$ are not
stably diffeomorphic.

###  3.2. The totally non-spin case

######  Theorem 3.5.

There are four stable diffeomorphism classes of unorientable, totally non-spin
$4$-manifolds with $\pi_{1}(M)\cong G$. The Stiefel-Whitney numbers $w_{4}$
and $w_{2}^{2}$ detect these classes.

This theorem can also be extracted from work of Davis [Dav05, Theorem 2.3],
who computes a different set of invariants.

###### Proof.

Example 1.2 shows there is only one unorientable normal $1$-type in this case:
$w_{1}(\nu)\neq 0$, so it must be pulled back from $p^{*}x\in
H^{1}(B\pi_{1}(M);\mathbb{Z}/2)$. Since $p^{*}x=w_{1}(p^{*}\sigma)$, the
normal $1$-type is

(3.6) $V_{\mathrm{SO}}\oplus(p^{*}\sigma-1)\colon B\mathrm{SO}\times
B\pi_{1}(M)\longrightarrow B\mathrm{O}$

and its Thom spectrum is $\mathit{MSO}\wedge(B\pi_{1}(M))^{p^{*}\sigma-1}$,
which by Theorem 2.1 is equivalent to
$\mathit{MSO}\wedge(B\mathbb{Z}/2)^{\sigma-1}$.

######  Lemma 3.7 (Gray [Gra80, §2]).

There is an equivalence
$\mathit{MSO}\wedge(B\mathbb{Z}/2)^{\sigma-1}\simeq\mathit{MO}$.

So $\Omega_{4}^{\xi}\cong\Omega_{4}^{\mathrm{O}}$, and
$\Omega_{4}^{\mathrm{O}}\cong\mathbb{Z}/2\oplus\mathbb{Z}/2$ [Tho54,
Corollaire following Théorème IV.12]. The $\mathrm{Aut}(\xi)$-action is
trivial. To see this, first observe that $\mathrm{Aut}(\mathrm{id}\colon
B\mathrm{O}\to B\mathrm{O})$ is trivial, hence acts trivially on
$\Omega_{4}^{\mathrm{O}}$. Thus the $\mathrm{Aut}(\xi)$-orbit of a class in
$\Omega_{4}^{\xi}$ maps to a single class in $\Omega_{4}^{\mathrm{O}}$, so
$\mathrm{Aut}(\xi)$-orbits are singletons. Therefore any complete bordism
invariant for $\Omega_{4}^{\mathrm{O}}$ also is a complete stable
diffeomorphism invariant for the normal $1$-type $\xi$, such as
$(w_{2}^{2},w_{4})$. ∎

######  Remark 3.8.

If $M$ is pin$+$ or pin$-$, then its double cover is spin, and hence $M$ is
almost spin. So totally non-spin manifolds are neither pin$+$ nor pin$-$.
Therefore the three normal $1$-types that occur when $\pi_{1}(M)\cong G$ and
$M$ is unorientable are the cases pin$+$, pin$-$, and neither pin$+$ nor
pin$-$.

##  4\. Stable homeomorphism classes

In order to classify stable homeomorphism classes of topological
$4$-manifolds, we run the same story, replacing $B\mathrm{O}$ with
$B\mathrm{Top}$, where $\mathrm{Top}_{n}$ is the topological group of
homeomorphisms $\mathbb{R}^{n}\to\mathbb{R}^{n}$ that fix the origin and
$\mathrm{Top}\coloneqq\varinjlim_{n}\mathrm{Top}_{n}$. As in the previous
section, fix a group $G$ finite of order $2\bmod 4$ with a surjective map
$p\colon G\to\mathbb{Z}/2$, and assume that $\mathbb{Z}/2$ acts trivially on
$H^{*}(B\ker(p))$.

Given a topological manifold $M$, there is a map $\nu\colon M\to
B\mathrm{Top}$ called the _stable topological normal bundle_ , so we can
define normal $1$-types, and Kreck’s classification argument still applies in
the topological setting, this time determining stable homeomorphism classes.

######  Lemma 4.1.

Let $M$ be a closed, unorientable $4$-manifold. The possible normal $1$-types
of $M$ are the same as in Example 1.2, except replacing $B\mathrm{O}$ with
$B\mathrm{Top}$, $B\mathrm{SO}$ with $B\mathrm{STop}$, and $B\mathrm{Spin}$
with $B\mathrm{TopSpin}$.

###### Proof.

The proof is very similar to Kasprowski-Land-Powell-Teichner’s determination
of the possible normal $1$-types of topological $4$-manifolds in the
orientable case [KLPT17, Proposition 4.1]. Since the Stiefel-Whitney classes
of a manifold are homotopy invariants, notions of almost spin and totally non-
spin make sense for topological manifolds. In the almost-spin case, we have to
check that a lift $M\to B\mathrm{TopSpin}\times B\pi_{1}(M)$ is $2$-connected:
the proof is the same as in the smooth case, because
$\pi_{2}(B\mathrm{TopSpin})=0$. For the totally non-spin case,
$\pi_{1}(B\mathrm{STop})\cong\mathbb{Z}/2$, detected by $w_{2}$, and since $M$
is totally non-spin, $w_{2}(M)\neq 0$, so the lift is surjective on $\pi_{2}$
just as in the smooth case. ∎

Our arguments below make use of the fact that bordism groups of topological
manifolds are homotopy groups of Thom spectra, which requires a transversality
argument. In dimension $4$, Scharlemann [Sch76] proves the topological
transversality theorem that we need. See Teichner [Tei93, §IV] for more
information.

Let $E_{8}$ denote Freedman’s $E_{8}$ manifold [Fre82]. The obstruction to
admitting a triangulation defines a bordism invariant
$\Omega_{4}^{\mathrm{Top}}\to\mathbb{Z}/2$ [KT90b, §9] which is nonzero on
$E_{8}$.

###  4.1. The almost spin case

There are topological versions of spin and pin$\pm$ structures; see Kirby-
Taylor [KT90b, §9] for details. Kirby-Taylor also produce a homomorphism
$S\colon\Omega_{4}^{\mathrm{TopPin}^{+}}\to\Omega_{2}^{\mathrm{TopPin}^{-}}\cong\Omega_{2}^{\mathrm{Pin}^{-}}\cong\mathbb{Z}/8$
sending a pin$+$ topological $4$-manifold $M$ to the pin$-$ bordism class of a
continuously embedded representative of the Poincaré dual of $w_{1}(M)^{2}$,
which has an induced pin$-$ structure and a unique smooth structure. Let
$S^{\prime}$ be the invariant sending a topological pin$+$ $4$-manifold $M$ to
the image of $S(M)$ in the set $(\mathbb{Z}/8)/(x\sim-x)$.

######  Theorem 4.2.

  1. (1)

There are ten stable homeomorphism classes of unorientable pin$+$ topological
$4$-manifolds with $\pi_{1}(M)\cong G$. These classes are detected by the
invariant $S^{\prime}$ constructed above and the triangulation obstruction.

  2. (2)

There are two stable homeomorphism classes of unorientable pin$-$ topological
$4$-manifolds with $\pi_{1}(M)\cong G$. These classes are detected by the
triangulation obstruction.

###### Proof.

Following the same line of argument as in the proof of Theorem 3.1, the two
normal $1$-types’ Thom spectra are
$\mathit{MTopSpin}\wedge(B\pi_{1}(M))^{p^{*}\sigma-1}$ and
$\mathit{MTopSpin}\wedge(B\pi_{1}(M))^{p^{*}(3\sigma)-3}$, and Theorem 2.1
simplifies these to $\mathit{MTopSpin}\wedge(B\mathbb{Z}/2)^{\sigma-1}$ and
$\mathit{MTopSpin}\wedge(B\mathbb{Z}/2)^{3\sigma-3}$, respectively.

######  Lemma 4.3.

There are equivalences
$\mathit{MTopSpin}\wedge(B\mathbb{Z}/2)^{\sigma-1}\simeq\mathit{MTTopPin}^{-}$
and
$\mathit{MTopSpin}\wedge(B\mathbb{Z}/2)^{3\sigma-3}\simeq\mathit{MTTopPin}^{+}$.

###### Proof.

There are surjective maps $d_{n}\colon\mathrm{Top}_{n}\twoheadrightarrow\\{\pm
1\\}$ given by assigning to a homeomorphism the automorphism it defines on
$H_{n}(\mathbb{R}_{n},\mathbb{R}_{n}\setminus 0)\cong\mathbb{Z}$. These
commute with the inclusions
$\mathrm{Top}_{n}\hookrightarrow\mathrm{Top}_{n+1}$, and passing to the
colimit defines a map $d\colon\mathrm{Top}\twoheadrightarrow\\{\pm 1\\}$. This
is a topological version of assigning an orthogonal matrix its determinant,
classifying whether it preserves or reverses orientation. Given a principal
$\mathrm{Top}$-bundle $P\to M$, let $\operatorname{Det}(P)\to M$ be the line
bundle $P\times_{\mathrm{Top}}\mathbb{R}\to M$, where $\mathrm{Top}$ acts on
$\mathbb{R}$ through $d$. The maps
$\mathrm{Top}_{n}\times\mathrm{O}_{1}\to\mathrm{Top}_{n}\times\mathrm{Top}_{1}\to\mathrm{Top}_{n+1}$
allow us to make sense of “$P\oplus n\operatorname{Det}(P)$” as a principal
$\mathrm{Top}$-bundle.

We abuse notation for a moment to say that a $G$-structure on a principal
$\mathrm{Top}$-bundle $P\to M$ is a reduction of structure group of $P$ from
$\mathrm{Top}$ to $G$. Then, just as in the smooth case, there is a natural
equivalence between the set of $\mathrm{TopPin}^{-}$-structures on $P$ and the
set of $\mathrm{TopSpin}$ structures on $P\oplus\operatorname{Det}(P)$, and
similarly between the set of $\mathrm{TopPin}^{+}$-structures on $P$ and the
set of $\mathrm{TopSpin}$ structures on $P\oplus 3\operatorname{Det}(P)$. The
proof is the same as in the smooth case. These equivalences are the only facts
we need to know about $\mathrm{Pin}^{\pm}$ in order to prove Theorem 3.3 in
the smooth setting, so the argument in the topological setting can proceed in
the same way. ∎

Therefore by Theorem 2.1, our two normal $1$-types are equivalent to
$\mathit{MTTopPin}^{\pm}$. The caveat about switching between pin$+$ and
pin$-$ when one passes between the tangent and normal bundles still applies
here.

######  Theorem 4.4 (Kirby-Taylor [KT90b, Theorem 9.2]).

  1. (1)

$\Omega_{4}^{\mathrm{TopPin}^{-}}\cong\mathbb{Z}/2$, generated by $E_{8}$.

  2. (2)

$\Omega_{4}^{\mathrm{TopPin}^{+}}\cong\mathbb{Z}/8\oplus\mathbb{Z}/2$, with
$\mathbb{RP}^{4}$ generating the $\mathbb{Z}/8$ summand and $E_{8}$ generating
the $\mathbb{Z}/2$ summand.

  3. (3)

The map $\Omega_{4}^{\mathrm{Pin}^{+}}\to\Omega_{4}^{\mathrm{TopPin}^{+}}$ is
identified with a map $\mathbb{Z}/16\to\mathbb{Z}/8\oplus\mathbb{Z}/2$ which
surjects onto the first factor and does not hit $E_{8}$.

  4. (4)

The homomorphism
$S\colon\Omega_{4}^{\mathrm{TopPin}^{+}}\to\Omega_{2}^{\mathrm{TopPin}^{-}}\cong\Omega_{2}^{\mathrm{Pin}-}\cong\mathbb{Z}/8$
sends $\mathbb{RP}^{4}$ to a generator.

Since $\mathbb{Z}/2$ is rigid, we conclude there are two stable homeomorphism
classes in the pin$-$ case, detected by the triangulation obstruction. For the
pin$+$ case, the same line of reasoning in the proof of Theorem 3.1 allows us
to reduce to the case when $\xi$ is a topological pin$+$ structure, so we can
compute the action of $\mathrm{Aut}(\xi)$ on the generators. Since $E_{8}$ is
simply connected, it admits a unique topological pin$+$ structure, so is fixed
by $\mathrm{Aut}(\xi)$. For $\mathbb{RP}^{4}$, every topological pin$+$
structure on arises from a smooth pin$+$ structure, so we can reuse our work
from Theorem 3.1 to conclude the $\mathrm{Aut}(\xi)$-orbit of
$\mathbb{RP}^{4}$ is again $\pm[\mathbb{RP}^{4}]$. Therefore the set of stable
diffeomorphism classes is $((\mathbb{Z}/8)/(x\sim-x))\times\mathbb{Z}/2$,
which has ten elements, and the triangulation obstruction and $S^{\prime}$ are
together a complete invariant. ∎

###  4.2. The totally non-spin case

By Lemma 4.1, there is only one normal $1$-type to worry about.

######  Theorem 4.5.

There are eight stable homeomorphism classes of unorientable, totally non-spin
topological $4$-manifolds with $\pi_{1}(M)\cong G$. The triangulation
obstruction and the Stiefel-Whitney numbers $w_{4}$ and $w_{2}^{2}$ are
together a complete stable homeomorphism invariant.

Again, this can be extracted from a theorem of Davis [Dav05, Theorem 2.3], who
uses a different but equivalent set of invariants.

###### Proof.

Following the same line of reasoning as in Theorem 3.5, Lemma 4.1 tells us we
only have one normal $1$-type, and its Thom spectrum is
$\mathit{MSTop}\wedge(B\mathbb{Z}/2)^{\sigma-1}$.

######  Lemma 4.6.

There is an equivalence
$\mathit{MTop}\simeq\mathit{MSTop}\wedge(B\mathbb{Z}/2)^{\sigma-1}$.

###### Proof.

The proof goes through as in the smooth case, since we have a determinant map
and the fact that for any $\mathrm{Top}$-bundle $P\to M$,
$P\oplus\operatorname{Det}(P)$ is canonically oriented, analogously to the
smooth case. ∎

So we need to calculate $\Omega_{4}^{\mathrm{Top}}$.

######  Proposition 4.7.

$\Omega_{4}^{\mathrm{Top}}\cong(\mathbb{Z}/2)^{\oplus 3}$, with a basis given
by the classes of $\mathbb{RP}^{4}$, $\mathbb{RP}^{2}\times\mathbb{RP}^{2}$,
and $E_{8}$. The Stiefel-Whitney numbers $w_{4}$ and $w_{2}^{2}$ and the
triangulation obstruction are linearly independent on this bordism group.

###### Proof.

Draw the Atiyah-Hirzebruch spectral sequence computing
$\Omega_{4}^{\mathrm{Top}}$ as
$\Omega_{4}^{\mathrm{STop}}((B\mathbb{Z}/2)^{\sigma-1})$. It collapses for
degree reasons in total degree $4$ and below, and the $4$-line of the
$E_{\infty}$-page has order $8$. Therefore it suffices to find three linearly
independent nonzero elements of $\Omega_{4}^{\mathrm{Top}}$, which can be done
by computing $w_{4}$, $w_{2}^{2}$, and the triangulation obstruction on
$\mathbb{RP}^{4}$, $\mathbb{RP}^{2}\times\mathbb{RP}^{2}$, and $E_{8}$. ∎

Just as in the smooth case, $\mathrm{Aut}(\xi)$ acts trivially. ∎

Remark 3.8 also applies in the topological case: the three normal $1$-types
for unorientable topological manifolds with $\pi_{1}(M)\cong G$ are precisely
the cases where $M$ has a topological pin$+$ structure, $M$ has a topological
pin$-$ structure, and $M$ has neither.

## References

  * [ABP69] D. W. Anderson, E. H. Brown, Jr., and F. P. Peterson.  Pin cobordism and related topics.  Comment. Math. Helv., 44:462–468, 1969.
  * [BG97] Boris Botvinnik and Peter Gilkey.  The Gromov-Lawson-Rosenberg conjecture: the twisted case.  Houston J. Math., 23(1):143–160, 1997.
  * [CS71] Sylvain E. Cappell and Julius L. Shaneson.  On four dimensional surgery and applications.  Comment. Math. Helv., 46:500–528, 1971.
  * [CS76] Sylvain E. Cappell and Julius L. Shaneson.  Some new four-manifolds.  Ann. of Math. (2), 104(1):61–72, 1976.
  * [CS11] Diarmuid Crowley and Jörg Sixt.  Stably diffeomorphic manifolds and $l_{2q+1}(\mathbb{Z}[\pi])$.  Forum Math., 23(3):483–538, 2011.  https://arxiv.org/abs/0808.2008.
  * [Dav05] James F. Davis.  The Borel/Novikov conjectures and stable diffeomorphisms of 4-manifolds.  In Geometry and topology of manifolds, volume 47 of Fields Inst. Commun., pages 63–76. Amer. Math. Soc., Providence, RI, 2005.
  * [Deb21] Arun Debray.  Invertible phases for mixed spatial symmetries and the fermionic crystalline equivalence principle.  2021\.  https://arxiv.org/abs/2102.02941.
  * [FNOP19] Stefan Friedl, Matthias Nagel, Patrick Orson, and Mark Powell.  A survey of the foundations of four-manifold theory in the topological category.  2019\.  https://arxiv.org/abs/1910.07372.
  * [Fre82] Michael Hartley Freedman.  The topology of four-dimensional manifolds.  J. Differential Geometry, 17(3):357–453, 1982.
  * [Gia73] V. Giambalvo.  Pin and $\mathrm{Pin}^{\prime}$ cobordism.  Proc. Amer. Math. Soc., 39:395–401, 1973.
  * [GOP+20] Meng Guo, Kantaro Ohmori, Pavel Putrov, Zheyan Wan, and Juven Wang.  Fermionic finite-group gauge theories and interacting symmetric/crystalline orders via cobordisms.  Comm. Math. Phys., 376(2):1073–1154, 2020.  https://arxiv.org/abs/1812.11959.
  * [Gra80] Brayton Gray.  Products in the Atiyah-Hirzebruch spectral sequence and the calculation of $M{\rm SO}_{\ast}$.  Trans. Amer. Math. Soc., 260(2):475–483, 1980.
  * [HH19] Ian Hambleton and Alyson Hildum.  Topological 4-manifolds with right-angled Artin fundamental groups.  J. Topol. Anal., 11(4):777–821, 2019.  https://arxiv.org/abs/1411.5662.
  * [HKT09] Ian Hambleton, Matthias Kreck, and Peter Teichner.  Topological 4-manifolds with geometrically two-dimensional fundamental groups.  J. Topol. Anal., 1(2):123–151, 2009.  https://arxiv.org/abs/0802.0995.
  * [KLPT17] Daniel Kasprowski, Markus Land, Mark Powell, and Peter Teichner.  Stable classification of 4-manifolds with 3-manifold fundamental groups.  J. Topol., 10(3):827–881, 2017.  https://arxiv.org/abs/1511.01172.
  * [KPMT20] Justin Kaidi, Julio Parra-Martinez, and Yuji Tachikawa.  Topological superconductors on superstring worldsheets.  SciPost Phys., 9:10, 2020.  With a mathematical appendix by Arun Debray. https://www.scipost.org/SciPostPhys.9.1.010.
  * [KPT20] Daniel Kasprowski, Mark Powell, and Peter Teichner.  Algebraic criteria for stable diffeomorphism of spin 4-manifolds.  2020\.  https://arxiv.org/abs/2006.06127.
  * [Kre84] M. Kreck.  Some closed $4$-manifolds with exotic differentiable structure.  In Algebraic topology, Aarhus 1982 (Aarhus, 1982), volume 1051 of Lecture Notes in Math., pages 246–262. Springer, Berlin, 1984.
  * [Kre99] Matthias Kreck.  Surgery and duality.  Ann. of Math. (2), 149(3):707–754, 1999.  https://arxiv.org/abs/math/9905211.
  * [KT90a] R. C. Kirby and L. R. Taylor.  A calculation of $\mathrm{Pin}^{+}$ bordism groups.  Commentarii Mathematici Helvetici, 65(1):434–447, Dec 1990.
  * [KT90b] R. C. Kirby and L. R. Taylor.  $\mathit{Pin}$ structures on low-dimensional manifolds, volume 151 of London Math. Soc. Lecture Note Ser., pages 177–242.  Cambridge Univ. Press, 1990.
  * [Kur01] Ichiji Kurazono.  Cobordism group with local coefficients and its application to 4-manifolds.  Hiroshima Math. J., 31(2):263–289, 2001.
  * [Ped17] Riccardo Pedrotti.  Stable classification of certain families of four-manifolds.  Master’s thesis, Max Planck Institute for Mathematics, 2017.
  * [Pet68] F. P. Peterson.  Lectures on Cobordism Theory.  Lectures in Mathematics. Kinokuniya Book Store Co., Ltd., 1968.
  * [Pol13] Wojciech Politarczyk.  4-manifolds, surgery on loops and geometric realization of Tietze transformations.  2013\.  https://arxiv.org/abs/1303.6502.
  * [Reu20] David Reutter.  Semisimple 4-dimensional topological field theories cannot detect exotic smooth structure.  2020\.  https://arxiv.org/abs/2001.02288.
  * [Sch76] Martin G. Scharlemann.  Transversality theories at dimension four.  Invent. Math., 33(1):1–14, 1976.
  * [Spa03] Fulvia Spaggiari.  On the stable classification of Spin four-manifolds.  Osaka J. Math., 40(4):835–843, 2003.
  * [Sto88] Stephan Stolz.  Exotic structures on $4$-manifolds detected by spectral invariants.  Invent. Math., 94(1):147–162, 1988.
  * [Tei92] Peter Teichner.  Topological 4-manifolds with finite fundamental group.  PhD thesis, University of Mainz, 1992.  https://math.berkeley.edu/~teichner/Papers/phd.pdf.
  * [Tei93] Peter Teichner.  On the signature of four-manifolds with universal covering spin.  Math. Ann., 295(4):745–759, 1993.
  * [Tho54] René Thom.  Quelques propriétés globales des variétés differentiables.  PhD thesis, University of Paris, 1954.
  * [Wal64] C. T. C. Wall.  On simply-connected $4$-manifolds.  J. London Math. Soc., 39:141–149, 1964.
  * [Wan95] Zhenghan Wang.  Classification of closed nonorientable $4$-manifolds with infinite cyclic fundamental group.  Math. Res. Lett., 2(3):339–344, 1995.
  * [WWZ20] Zheyan Wan, Juven Wang, and Yunqin Zheng.  Higher anomalies, higher symmetries, and cobordisms ii: Lorentz symmetry extension and enriched bosonic/fermionic quantum gauge theory.  Ann. Math. Sci. Appl., 5(2):171–257, 2020.  https://arxiv.org/abs/1912.13504.