# Oriented and unitary equivariant bordism of surfaces

Andrés Angel  Departmento de Matematicas
Universidad de los Andes
Carrera 1 N. 18A -12, Bogotá, Colombia<EMAIL_ADDRESS>,  Eric
Samperton  Department of Mathematics
University of Illinois at Urbana-Champaign
1409 W. Green Street (MC-382), Urbana, IL 61801, USA<EMAIL_ADDRESS>https://smprtn.pages.math.illinois.edu ,  Carlos Segovia  Instituto de
Matemáticas
UNAM Unidad Oaxaca
Antonio de León #2, altos, Col. Centro
Oaxaca de Juárez, CP. 68000
México<EMAIL_ADDRESS>and  Bernardo Uribe  Departamento de
Matemáticas y Estadística
Universidad del Norte
Km. 5 via Puerto Colombia, Barranquilla, Colombia  Max Planck Institut für
Mathematik
Vivatsgasse 7
Bonn 53115, Germany<EMAIL_ADDRESS><EMAIL_ADDRESS>https://sites.google.com/site/bernardouribejongbloed/

###### Abstract.

Fix a finite group $G$. We study $\Omega^{SO,G}_{2}$ and $\Omega^{U,G}_{2}$,
the unitary and oriented bordism groups of smooth $G$-equivariant compact
surfaces, respectively, and we calculate them explicitly. Their ranks are
determined by the possible representations around fixed points, while their
torsion subgroups are isomorphic to the direct sum of the Bogomolov
multipliers of the Weyl groups of representatives of conjugacy classes of all
subgroups of $G$. We present an alternative proof of the fact that surfaces
with free actions which induce non-trivial elements in the Bogomolov
multiplier of the group cannot equivariantly bound. This result permits us to
show that the 2-dimensional SK-groups (Schneiden und Kleben, or “cut and
paste") of the classifying spaces of a finite group can be understood in terms
of the bordism group of free equivariant surfaces modulo the ones that bound
arbitrary actions.

###### Key words and phrases:

Equivariant bordism, equivariant vector bundle, surface

###### 2020 Mathematics Subject Classification:

57R85, 55N22, 57R75, 57R77

The second author is supported by NSF grant DMS #2038020. The third author is
supported by cátedras CONACYT, Convocatoria PAEP-2018 and Proyecto CONACYT
ciencias básicas 2016, No. 284621. The fourth author acknowledges and thanks
the continuous support of the Alexander Von Humboldt Foundation and of CONACYT
through project CB-2017-2018-A1-S-30345-F-3125. We are indebted to Prof. Peter
Landweber for reading earlier versions of this work and for suggesting changes
which have improved the paper. Thank you Prof. Landweber.

##  1\. Introduction

Equivariant bordism groups have been a subject of ongoing research since the
1960s. Conner, Floyd, Landweber, Stong, Smith and tom Dieck, among others,
laid the foundations for the extraordinary homology and cohomology theories
obtained from equivariant bordism, and found many interesting properties of
these groups. Given a finite group $G$, a particularly important problem is
the explicit calculation of the oriented and complex $G$-equivariant bordism
groups of a point, since they provide the coefficients for the theories. This
turns out to be a complicated task.

Explicit calculations of the equivariant bordism groups for finite abelian
groups [Sto70, Lan72, Oss72] led some to expect that, at least in the unitary
case, equivariant bordism groups are always a free module over the unitary
bordism ring for any finite group $G$ (see [Row80, Page 1], [May96,
§XXVIII.5], [GM97, Conj. 1.2]). This belief was confirmed for general abelian
groups [Löf74] [May96, §XXVIII, Thm. 5.1] and for metacyclic groups [Row80],
and therefore it was conjectured that for any finite group this was the case.
This conjecture remained dormant for some years and it was recalled by the
third author in his 2018 ICM Lecture [Uri18] where he coined it "The evenness
conjecture in equivariant unitary bordism".

When the evenness conjecture holds true for a group $G$, it implies that the
$G$-equivariant unitary bordism ring is torsion-free. In particular, any
unitary manifold with a free action of a finite group that generates a torsion
class in the unitary bordism group of free actions would bound equivariantly.
This has always been the first step for proving the evenness conjecture,
namely, to construct explicit equivariant manifolds whose boundaries are the
desired generators of the equivariant unitary bordism groups of free actions.

In the case of surfaces, the evenness conjecture would imply that all oriented
surfaces with orientation preserving free actions bound equivariantly (note
that if an oriented surface with orientation preserving free action does not
bound equivariantly, then the class of the difference of this surface with
$G$-times the quotient surface induces a non-trivial torsion class in the
reduced $G$-equivariant unitary bordism group). The third author in joint work
with Dominguez [SD20] showed that indeed this was the case for abelian,
dihedral, symmetric and alternating groups. Nevertheless, it fails to be true
in general. It has been recently shown by the second author that there is an
obstruction class for an oriented surface with an orientation preserving free
action to bound equivariantly [Sam20, Sam21], and this obstruction class lies
in the Bogomolov multiplier of the group [Bog87, Kun10] (the Bogomolov
multiplier of a finite group consists of the classes of the Schur multiplier
$H^{2}(G,\mathbb{C}^{*})$ that vanish once restricted to any abelian subgroup;
the homological version of the Bogomolov multiplier is the quotient of the
second integral homology of the group by the classes generated by two
dimensional tori [Mor12]). This result implies that indeed there are torsion
classes in the equivariant unitary bordism groups and therefore that the
evenness conjecture in equivariant unitary bordism is false in general. The
evenness conjecture might then be restated instead as a classification
question, namely: Which finite groups satisfy the evenness conjecture in
equivariant unitary bordism?

In this work we focus on the calculation of the oriented and the unitary
$G$-equivariant bordism groups for compact surfaces. We use the fixed point
construction methods developed in [Row78] to determine the rank of the
equivariant bordism groups, and then, we use the explicit generators of the
equivariant bordism groups for adjacent families in dimension 3 in order to
determine which equivariant surfaces bound. In Theorem 4.17 we present a
generalization to all finite groups of the result shown by the second author
in [Sam20], which states that the obstruction class for equivariantly bounding
an oriented surface with free action is the element in the Bogomolov
multiplier of the group that the surface defines. The Conner-Floyd spectral
sequence will then allow us to determine the torsion group in the equivariant
bordism group of surfaces. Our main result is:

######  Theorem.

4.41. Let $G$ be a finite group and
$\operatorname{Tor}_{\mathbb{Z}}(\Omega^{G}_{2})$ the torsion subgroup of the
unitary or oriented $G$-equivariant bordism of surfaces $\Omega^{G}_{2}$. Then
there is a canonical isomorphism

$\bigoplus_{(K)}\tilde{B}_{0}(W_{K})\cong\operatorname{Tor}_{\mathbb{Z}}(\Omega^{G}_{2})$

where $(K)$ runs over all conjugacy classes of subgroups of $G$,
$W_{K}=N_{G}K/K$ and $\tilde{B}_{0}(W_{K})$ is the homology version of the
Bogomolov multiplier of the group $W_{K}$.

With the torsion group in hand, we describe explicitly in Theorem 4.48 the
$G$-equivariant bordism groups of surfaces, unitary and oriented.

Since there are infinitely many groups with non-trivial Bogomolov multipliers,
we conclude that there are infinitely many groups which do not satisfy the
evenness conjecture in equivariant unitary bordism. On the other hand, there
are also infinitely many groups $G$ whose $G$-equivariant unitary bordism
group of surfaces is a free abelian group, thus implying that these groups may
still satisfy the evenness conjecture for equivariant unitary bordism.

We use our previous calculations to interpret which equivariant surfaces bound
in terms of the $SK$ relation (cutting and pasting from the German Schneiden
und Kleben). The study of invariants under cutting and pasting started with
the characterization by Jänich [Jän69, Jän68] of invariants with the additive
properties of the Euler characteristic and the signature, and it was further
developed with the introduction of the $SK$-groups of a space in the book
[KKNO73]. The $SK$-groups of a space can be understood as the groups of
equivalence classes of manifolds with continuous maps to the space subject to
the equivalence relation given by cutting and pasting. The 2-dimensional
$SK$-groups of $BG$ can be understood in terms of cutting and pasting surfaces
with free $G$-actions. The $SK$-groups of $BG$ were studied in [KKNO73] and
were identified in [Neu75, Thm. 2] with the second integral homology group of
$BG$ modulo the toral classes (as far as we know this is the first reference
where the homological Bogomolov multiplier appears).

We finish this work with the study of two explicit groups of order 64 and 243
respectively whose Bogomolov multipliers are non-trivial. We sketch why both
groups possess non-trivial Bogomolov multipliers and we give explicit
homomorphisms from the fundamental group of a genus 2 surface to both groups
that define the desired surfaces with free actions that do not bound
equivariantly. These constructions allow us to give explicit generators for
the torsion subgroup of the equivariant unitary bordism groups for both
groups.

##  2\. Preliminaries

###  2.1. Equivariant bordism

Let $G$ be a finite group and consider compact manifolds endowed with smooth
actions of the group $G$ preserving either the orientation or the unitary
(tangentially stable complex) structure. Following the notation of Stong
[Sto70], denote by $\Omega^{G}_{*}$ either the bordism ring
$\Omega^{SO,G}_{*}$ of $G$-equivariant oriented manifolds or the bordism ring
$\Omega^{U,G}_{*}$ of $G$-equivariant unitary (tangentially stable complex)
manifolds. Whenever the upper script $SO$ or $U$ is not specified, it means
that the construction and results apply to both homology theories.

###  2.2. Equivariant bordism for families

The study of the equivariant bordism groups led Conner and Floyd to restrict
their attention to manifolds with prescribed isotropy groups [CF64, CF66]. The
allowed isotropy groups are therefore organized in families of subgroups of
$G$ which are closed under conjugation and under taking subgroups. For any
such family of subgroups $\mathcal{F}$ there is a classifying $G$-space
$E\mathcal{F}$ for actions whose isotropy groups lie on $\mathcal{F}$. This
$G$-space is characterized by its properties on fixed points sets, namely, the
fixed point set $E\mathcal{F}^{H}$ is contractible whenever $H\in\mathcal{F}$
and empty otherwise. The construction of $E\mathcal{F}$ can be carried out in
such a way that an inclusion of families
$\mathcal{F}^{\prime}\subset\mathcal{F}$ induces a $G$-cofibration
$E\mathcal{F}^{\prime}\to E\mathcal{F}$ [tD87, §1.6].

The equivariant bordism groups
$\Omega_{*}^{G}\\{\mathcal{F},\mathcal{F}^{\prime}\\}$ for a pair of families
$\mathcal{F}^{\prime}\subset\mathcal{F}$ are the bordism groups of
$G$-equivariant compact manifolds with boundary $(M,\partial M)$ such that the
isotropy groups of $M$ lie in $\mathcal{F}$ and the isotropy groups of its
boundary $\partial M$ lie in $\mathcal{F}^{\prime}$. Following [tD72, p.310]
one may define the bordism of groups for a pair of $G$-spaces $(X,A)$ and a
pair of families as follows

(2.1)
$\Omega^{G}_{*}\\{\mathcal{F},\mathcal{F}^{\prime}\\}(X,A):=\Omega^{G}_{*}(X\times
E\mathcal{F},X\times E\mathcal{F}^{\prime}\cup A\times E\mathcal{F});$

or equivalently, using a more geometrical description [Sto70].

###  2.3. Long exact sequence for families

Whenever three families are related by the inclusions
$\mathcal{F}^{\prime\prime}\subset\mathcal{F}^{\prime}\subset\mathcal{F}$
there is induced a long exact sequence in bordism [CF66, Thm. 5.1]

(2.2)
$\cdots\to\Omega_{*}^{G}\\{\mathcal{F}^{\prime},\mathcal{F}^{\prime\prime}\\}\to\Omega_{*}^{G}\\{\mathcal{F},\mathcal{F}^{\prime\prime}\\}\to\Omega_{*}^{G}\\{\mathcal{F},\mathcal{F}^{\prime}\\}\stackrel{{\scriptstyle\partial}}{{\to}}\Omega_{*-1}^{G}\\{\mathcal{F}^{\prime},\mathcal{F}^{\prime\prime}\\}\to\cdots.$

###  2.4. Conner-Floyd spectral sequence

More generally, associated to the families
$\mathcal{F}_{0}\subset\mathcal{F}_{1}\subset\cdots\subset\mathcal{F}_{k}=\mathcal{F}$
there is a spectral sequence converging to $\Omega^{G}_{n}\\{\mathcal{F}\\}$,
whose filtration is

(2.3)
$F_{p}\Omega^{G}_{n}\\{\mathcal{F}\\}:=\operatorname{Im}(\Omega^{G}_{n}\\{\mathcal{F}_{p}\\}\to\Omega^{G}_{n}\\{\mathcal{F}\\}).$

This spectral sequence is usually called the Conner-Floyd spectral sequence,
its first page is given by

(2.4)
$E^{1}_{p,q}\cong\Omega^{G}_{p+q}\\{\mathcal{F}_{p},\mathcal{F}_{p-1}\\},$

and the differentials are induced by the boundary maps. The first page of this
spectral sequence might be difficult to calculate, but whenever the pair of
families $\mathcal{F}_{p-1}\subset\mathcal{F}_{p}$ are adjacent (see below for
the definition), fixed point methods together with the classification of the
normal bundles can make them computable in terms of non-equivariant bordism
groups.

###  2.5. Equivariant bordism for adjacent families

A pair of families $\mathcal{F}^{\prime}\subset\mathcal{F}$ are called
adjacent whenever they differ by the conjugacy class $(K)$ of a subgroup $K$,
in other words $\mathcal{F}-\mathcal{F}^{\prime}=(K)$. A manifold $(M,\partial
M)$ in $\Omega^{G}_{n}\\{\mathcal{F},\mathcal{F}^{\prime}\\}$ is cobordant to
the $G$-equivariant tubular neighborhood of the fixed point set $M^{K}$. The
space $M^{K}$ becomes a free $W_{K}:=N_{G}K/K$ space and the tubular
neighborhood can be reconstructed from a specific $W_{K}$-equivariant twisted
bundle over $M^{K}$. Hence if $M^{K}$ is of dimension $n-k$ and $M^{K}/W_{K}$
is connected, its tubular neighborhood can be recovered from a map $M^{K}\to
C_{N_{G}K,K}(k)$ where $C_{N_{G}K,K}(k)$ is a $W_{K}$-space which classifies
the $G$-equivariant tubular neighborhoods of rank $k$ around $K$-fixed points.
In the unitary case there is a decomposition in terms of non-equivariant
unitary bordism groups [Uri18, Thm. 2.8]

(2.5)
$\Omega^{U,G}_{n}\\{\mathcal{F},\mathcal{F}^{\prime}\\}:=\bigoplus_{2k\leq
n}\Omega^{U}_{n-2k}\left(C_{N_{G}K,K}(k)\times_{W_{K}}EW_{K}\right),$

and a similar one in the case of oriented bordisms [Á09, Thm. 2.11] This
localization theorem will become very useful in what follows once we apply it
for the study of the equivariant bordism groups of surfaces.

###  2.6. $G$ fixed points

For every subgroup $K$ of $G$ denote by $\mathcal{A}K$ the family of all
subgroups of $K$ and its conjugates in $G$, and denote by $\mathcal{P}K$ the
family $\mathcal{A}K-(K)$ of all _proper_ subgroups of $K$, and its conjugates
in $G$. The localization map at the fixed points of the whole group action

(2.6) $\Omega_{*}^{G}\to\Omega_{*}^{G}\\{\mathcal{A}G,\mathcal{P}G\\}$

together with the decomposition into non-equivariant bordisms groups presented
in (2.5), has been a powerful tool for determining the equivariant bordism
groups for abelian groups (see for instance [tD70, Löf74, Han05]). In this
particular case, the bordism groups
$\Omega_{*}^{G}\\{\mathcal{A}G,\mathcal{P}G\\}$ are isomorphic to the non-
equivariant bordism groups of products of complex Grassmannians in the unitary
case, and of products of real, complex and quaternionic Grassmannians in the
oriented case.

###  2.7. Rowlett spectral sequence

We still need another spectral sequence suited for understanding the
equivariant bordism groups of pairs of families. This spectral sequence was
constructed by Rowlett in [Row78, Prop. 2.1] and therefore its name. Consider
a pair of families $\mathcal{F}^{\prime}\subset\mathcal{F}$ that are also
families of subgroups of the normal subgroup $K$ of $G$. Then it is easy to
see that the classifying map $M/K\to EW_{K}$ of the free $W_{K}=G/K$ action of
the quotient induces an isomorphism of bordism groups
$\Omega_{*}^{G}\\{\mathcal{F},\mathcal{F}^{\prime}\\}\stackrel{{\scriptstyle\cong}}{{\to}}\Omega_{*}^{G}\\{\mathcal{F},\mathcal{F}^{\prime}\\}(EW_{K})$.
The space $EW_{K}$ can be constructed as a CW-complex whose $n$-skeleton
$(EW_{K})^{n}$ is constructed from $(EW_{K})^{n-1}$ by attaching a finite
number of copies of $W_{K}\times B^{n}$ with $W_{K}$ acting trivially on the
$n$-dimensional balls. One may filter
$\Omega_{*}^{G}\\{\mathcal{F},\mathcal{F}^{\prime}\\}(EW_{K})$ by the images
under the inclusion of the skeletons
$\Omega_{*}^{G}\\{\mathcal{F},\mathcal{F}^{\prime}\\}((EW_{K})^{n})$ and
therefore one obtains a spectral sequence converging to
$\Omega_{*}^{G}\\{\mathcal{F},\mathcal{F}^{\prime}\\}$ whose first page
becomes:

$\displaystyle E^{1}_{p,q}$
$\displaystyle\cong\Omega_{p+q}^{G}\\{\mathcal{F},\mathcal{F}^{\prime}\\}((EW_{K})^{p},(EW_{K})^{p-1})$
(2.7) $\displaystyle\cong
H_{p}((EW_{K})^{p},(EW_{K})^{p-1})\otimes_{W_{K}}\Omega_{q}^{K}\\{\mathcal{F},\mathcal{F}^{\prime}\\},$

and whose second page is:

(2.8) $E^{2}_{p,q}\cong
H_{p}(W_{K},\Omega_{q}^{K}\\{\mathcal{F},\mathcal{F}^{\prime}\\}),$

where the action of an element of $W_{K}$ on an $K$-manifold $M$ consists of
the same manifold $M$ endowed with the conjugate $K$-action. The zero-th
column consists of the $W_{K}$-coinvariants

(2.9)
$E_{0,q}^{2}\cong\left(\Omega_{q}^{K}\\{\mathcal{F},\mathcal{F}^{\prime}\\}\right)_{W_{K}},$

and the edge homomorphism

(2.10) $\Omega_{q}^{K}\\{\mathcal{F},\mathcal{F}^{\prime}\\}\cong
E_{0,q}^{1}\to E_{0,q}^{2}\to
E_{0,q}^{\infty}\to\Omega_{q}^{G}\\{\mathcal{F},\mathcal{F}^{\prime}\\}$

is simply the extension homomorphism factorizing through the coinvariants

(2.11)
$\Omega_{q}^{K}\\{\mathcal{F},\mathcal{F}^{\prime}\\}\to(\Omega_{q}^{K}\\{\mathcal{F},\mathcal{F}^{\prime}\\})_{W_{K}}\to\Omega_{q}^{G}\\{\mathcal{F},\mathcal{F}^{\prime}\\},\
\ \ M\mapsto M\times_{K}G.$

In characteristic zero the spectral sequence collapses on the zero-th column
of the second page. Since in characteristic zero the invariants and the
coinvariants are isomorphic we conclude that the extension homomorphism
induces an isomorphism

(2.12)
$\Omega_{*}^{K}\\{\mathcal{F},\mathcal{F}^{\prime}\\}^{W_{K}}\otimes\mathbb{Q}\stackrel{{\scriptstyle\cong}}{{\to}}\Omega_{*}^{G}\\{\mathcal{F},\mathcal{F}^{\prime}\\}\otimes\mathbb{Q}.$

In order to find the torsion classes in $\Omega_{*}^{G}$ we will construct the
inverse map of the isomorphism (2.12) for every pair of adjacent families of
groups. This map will be simply given by the localization at fixed points and
will be the subject of the next section.

##  3\. Localization at fixed points

For every subgroup $K$ of $G$ let us define the fixed-point homomorphism

(3.1) $f_{K}\circ
r^{G}_{K}:\Omega_{*}^{G}\to\Omega_{*}^{K}\\{\mathcal{A}K,\mathcal{P}K\\}$

as the composition of the restriction homomorphism
$r^{G}_{K}:\Omega_{*}^{G}\to\Omega_{*}^{K}$ together with the localization at
$K$-fixed points

(3.2) $\displaystyle
f_{K}:\Omega_{*}^{K}\to\Omega_{*}^{K}\\{\mathcal{A}K,\mathcal{P}K\\}.$

The composition $f_{K}\circ r^{G}_{K}$ takes a $G$-manifold and maps it to the
tubular neighborhood $N$ of the $K$-invariant points $M^{K}$. Since on the
complement of $N$ in $M$ there are no points with isotropy $K$, the tubular
neighborhood $N$ and $M$ become cobordant in
$\Omega_{*}^{K}\\{\mathcal{A}K,\mathcal{P}K\\}$ [CF66, Lem. 5.2]. Since
$N_{G}K$ acts on the normal bundle $N$ of $M^{K}$, the localization at
$K$-fixed points lands in the $W_{K}$ fixed submodule. Therefore the fixed-
point homomorphism becomes

(3.3) $f_{K}\circ
r^{G}_{K}:\Omega_{*}^{G}\to\Omega_{*}^{K}\\{\mathcal{A}K,\mathcal{P}K\\}^{W_{K}}.$

Also, for every pair of families of subgroups in $G$, we have the localized
fixed-point homomorphism

(3.4)
$\phi_{*}:\Omega_{*}^{G}\\{\mathcal{F},\mathcal{F}^{\prime}\\}\to\bigoplus_{(K)\subset\mathcal{F}-\mathcal{F}^{\prime}}\Omega_{*}^{K}\\{\mathcal{A}K,\mathcal{P}K\\}^{W_{K}}.$

This homomorphism applied to the pair of adjacent families
$\\{\mathcal{A}K,\mathcal{P}K\\}$, composed with the edge homomorphism of the
Rowlett spectral sequence (2.11), gives us the following maps

(3.5)
$\Omega^{K}_{*}\\{\mathcal{A}K,\mathcal{P}K\\}_{W_{A}}\to\Omega^{G}_{*}\\{\mathcal{A}K,\mathcal{P}K\\}\stackrel{{\scriptstyle\phi}}{{\to}}\Omega^{K}_{*}\\{\mathcal{A}K,\mathcal{P}K\\}^{W_{A}}.$

In characteristic zero this composition is an isomorphism and therefore we
obtain the isomorphism

(3.6)
$\phi_{*}:\Omega^{G}_{*}\\{\mathcal{A}K,\mathcal{P}K\\}\otimes\mathbb{Q}\stackrel{{\scriptstyle\cong}}{{\to}}\Omega^{K}_{*}\\{\mathcal{A}K,\mathcal{P}K\\}^{W_{A}}\otimes\mathbb{Q},$

which becomes the inverse of the map in (2.12) for adjacent families.

Applying the Conner-Floyd spectral sequence we see that the fixed-point
homomorphism of (3.4) in characteristic zero becomes an isomorphism, and
therefore we quote:

######  Theorem 3.7.

[Row78, Thm. 1.1] The fixed-point homomorphism in characteristic zero is an
isomorphism

(3.8)
$\phi_{*}\otimes\mathbb{Q}:\Omega_{*}^{G}\otimes\mathbb{Q}\stackrel{{\scriptstyle\cong}}{{\to}}\bigoplus_{(K)}\Omega_{*}^{K}\\{\mathcal{A}K,\mathcal{P}K\\}^{W_{K}}\otimes\mathbb{Q}.$

###  3.1. Kernel of fixed-point homomorphism

In the unitary case, the equivariant bordism group
$\Omega^{U,K}_{*}\\{\mathcal{A}K,\mathcal{P}K\\}$ is isomorphic to the unitary
bordism group of a disjoint union of products of complex Grassmannians [Uri18,
Thm. 2.8]. Therefore, the group
$\Omega_{*}^{U,K}\\{\mathcal{A}K,\mathcal{P}K\\}$ is a free
$\Omega^{U}_{*}$-module on even dimensional generators. Hence, we obtain the
following result:

######  Lemma 3.9.

The group of torsion elements in $\Omega^{U,G}_{*}$ is isomorphic to the
kernel of the fixed-point homomorphism $\phi$ of (3.4):

(3.10)
$\operatorname{Tor}_{\mathbb{Z}}(\Omega^{U,G}_{*})=\operatorname{Ker}(\phi^{U}_{*}).$

Whenever a group $G$ satisfies the evenness conjecture in equivariant unitary
bordism then the fixed-point homomorphism $\phi_{*}^{U}$ is automatically a
monomorphism. This is the case for abelian [Löf74] and metacyclic [Row80]
groups. In the next section we will show that there are groups $G$ such that
the kernel of the fixed-point homomorphism is not trivial in dimension 2, thus
defining torsion elements in $\Omega^{U,G}_{2}$. This fact refutes the
evenness conjecture in the general case.

In the oriented case there are many torsion classes in the bordism ring
$\Omega^{SO}_{*}$, all of order 2 [Wal60, Sto68]. Therefore we will be mainly
interested in the torsion classes of the equivariant bordism group
$\Omega^{SO,G}_{*}$ which are trivial under the fixed-point homomorphism
$\phi^{SO}_{*}$.

A very interesting and more general question associated to the equivariant
oriented case would be the following:

Are there $G$-equivariant oriented manifolds whose bordism class vanishes
under the fixed-point homomorphism $\phi^{SO}$ which do not bound
equivariantly?

In the next section we answer this question for dimension 2. The 3-dimensional
case (with its interesting application to Chern-Simons theory) remain open for
the interested reader.

Note that the equivariant bordism group
$\Omega^{SO,K}_{*}\\{\mathcal{A}K,\mathcal{P}K\\}$ is in general more
difficult to calculate than the unitary one. On the one hand the fixed point
set $M^{K}$ need not be orientable, and on the other, the normal bundles are
classified by products of real, complex and quaternionic Grassmannians.

Since we are mainly interested in the 2 and the 3-dimensional cases, we know
that all fixed points are of codimension 0, 2 or 3. In the case that the fixed
points are of codimension 2, the normal bundle is of complex dimension 1 in
the unitary case and of real dimension 2 in the real case. Therefore denote by

(3.11) $\displaystyle\operatorname{\textup{Hom}_{\textup{inj}}}(K,U(1))$

the 1-dimensional faithful complex representations of the group $K$,
parametrized by monomorphisms from $K$ to $U(1)$. Let $\tau$ denote the
complex conjugation map on this set of representations and denote by
$\operatorname{\textup{Hom}_{\textup{inj}}}(K,U(1))_{\tau}$ the quotient of
$\operatorname{\textup{Hom}_{\textup{inj}}}(K,U(1))$ by the complex
conjugation map $\tau$.

We have the following result:

######  Proposition 3.12.

If $K$ is a finite group, then

(3.13) $\displaystyle\Omega^{SO,K}_{2}\\{\mathcal{A}K,\mathcal{P}K\\}$
$\displaystyle=\left\\{\begin{matrix}\mathbb{Z}/2&\mbox{if}&K\cong\mathbb{Z}/2\\\
\bigoplus_{\operatorname{\textup{Hom}_{\textup{inj}}}(\mathbb{Z}/k,U(1))_{\tau}}\mathbb{Z}&\mbox{if}&K\cong\mathbb{Z}/k,\
\ k>2\\\ 0&\mbox{else,}&\end{matrix}\right.$ (3.14)
$\displaystyle\Omega^{SO,K}_{3}\\{\mathcal{A}K,\mathcal{P}K\\}$
$\displaystyle=\left\\{\begin{matrix}\bigoplus_{\operatorname{\textup{Hom}_{\textup{inj}}}(\mathbb{Z}/k,U(1))_{\tau}}\mathbb{Z}/2&\mbox{if}&K\cong
D_{2k}\\\ \mathbb{Z}/2&\mbox{if}&K\cong A_{4},\mathfrak{S}_{4},A_{5}\\\
0&\mbox{else,}&\end{matrix}\right.$ (3.15)
$\displaystyle\Omega^{U,K}_{2}\\{\mathcal{A}K,\mathcal{P}K\\}$
$\displaystyle=\left\\{\begin{matrix}\Omega_{2}^{U}\oplus\bigoplus_{\operatorname{\textup{Hom}_{\textup{inj}}}(\mathbb{Z}/k,U(1))}\mathbb{Z}&\mbox{if}&K\cong\mathbb{Z}/k\\\
\Omega_{2}^{U}&\mbox{else,}&\end{matrix}\right.$ (3.16)
$\displaystyle\Omega^{U,K}_{3}\\{\mathcal{A}K,\mathcal{P}K\\}$
$\displaystyle=0.$

###### Proof.

Any manifold $M$ in $\Omega^{K}_{*}\\{\mathcal{A}K,\mathcal{P}K\\}$ is
equivalent to the normal bundle $N$ around the fixed point set $M^{K}$.
Whenever $M$ is connected, of dimension 2, and $M\neq M^{K}$, then this normal
bundle is classified by a map

(3.17) $\displaystyle
M^{K}\to\bigsqcup_{\operatorname{\textup{Hom}_{\textup{inj}}}(K,U(1))}BU(1),\
\ \
M^{K}\to\bigsqcup_{\operatorname{\textup{Hom}_{\textup{inj}}}(K,U(1))_{\tau}}BU(1)$

in the unitary and the oriented case respectively (here we are using that
$SO(2)\cong U(1)$). Since $K$ needs to act freely on the fibers of the normal
bundle, the representations need to be faithful and therefore $K$ must be
cyclic (whenever $K$ is not cyclic $M^{K}=\emptyset$).

In the unitary case the number of points for each faithful representation are
counted. In the oriented case we know that points with complex conjugate
faithful representations (inverse orientation) are inverse to each other in
the bordism group; take the rotation action on a 2-dimensional sphere and note
that the fixed points have complex conjugate representations. The only
faithful representation which is its own inverse is the faithful
representation of $\mathbb{Z}/2$. Counting the number of fixed points with a
fixed faithful representation and subtracting the number of points with the
opposite representation, provides the integer invariant. For the group
$K\cong\mathbb{Z}/2$ simply take the congruence modulo 2 of the number of
fixed points.

Whenever $M$ is connected and $M^{K}=M$, then $\partial M=\emptyset$ and the
equivariant bordism class is mapped to $[M]\in\Omega_{2}^{U}$. In the oriented
case we know that $M$ bounds. This finishes the proof of the 2-dimensional
bordism groups.

Now let us see the 3-dimensional case. Whenever $K$ is cyclic, $M$ connected
and $M\neq M^{K}$, the normal bundle of $M^{K}$ is classified by maps as in
(3.17) for the unitary and the oriented case respectively. Since $M^{K}$ is a
compact 1-dimensional manifold and
$\Omega_{1}^{U}(BU(1))=\Omega_{1}^{SO}(BU(1))=0$ we see that $M$ bounds. The
bordism groups $\Omega^{U,K}_{3}\\{\mathcal{A}K,\mathcal{P}K\\}$ are trivial
for any group $K$ which is not cyclic, because for these groups there are no
faithful 1-dimensional complex representations.

In the oriented case, the only non-cyclic groups $K$ with faithful
3-dimensional representations are the dihedral groups $D_{2k}$ and the
symmetry groups of the platonic solids $A_{4}$, $\mathfrak{S}_{4}$ and
$A_{5}$. If $V$ is a real 3-dimensional faithful representation, then the unit
ball $B(\mathbb{R}\oplus V)$ bounds twice the unit ball $B(V)$ of the
representation $V$ in $\Omega^{SO,K}_{3}\\{\mathcal{A}K,\mathcal{P}K\\}$.
Therefore the group $\Omega^{SO,K}_{3}\\{\mathcal{A}K,\mathcal{P}K\\}$ is
isomorphic to a sum of copies of $\mathbb{Z}/2$, one for each real faithful
3-dimensional irreducible representation of $K$. The groups $A_{4}$,
$\mathfrak{S}_{4}$ and $A_{5}$ have only one such representation, and the
dihedral groups $D_{2k}$ have as many as
$\operatorname{\textup{Hom}_{\textup{inj}}}(\mathbb{Z}/k,U(1))_{\tau}$
representations. The $\mathbb{Z}/2$ invariant for these groups is the parity
of the number of fixed points with isomorphic irreducible faithful
representation.

Finally note that whenever $M^{K}=M$, $M$ bounds since
$\Omega^{U}_{3}=\Omega^{SO}_{3}=0$. ∎

As a consequence of the previous result we see that the 2-dimensional bordism
classes of interest have only 0-codimension fixed points.

######  Corollary 3.18.

The equivariant bordism groups $\operatorname{Ker}(\phi_{2}^{SO})$,
$\Omega_{3}^{U,G}$ and $\Omega_{3}^{SO,G}$ ($G$ of odd order in the oriented
case) are generated by equivariant manifolds with only fixed points of
codimension 0. The group $\operatorname{Ker}(\phi_{2}^{U})$ is generated by
surfaces without 0-dimensional fixed points and whose Euler characteristic is
zero (the quotient surface need not be connected). Therefore we have

(3.19)
$\operatorname{Tor}_{\mathbb{Z}}(\Omega^{U,G}_{2})=\operatorname{Ker}(\phi^{U}_{2}),\
\
\operatorname{Tor}_{\mathbb{Z}}(\Omega^{SO,G}_{2})=\operatorname{Ker}(\phi^{SO}_{2}).$

###### Proof.

The range of the fixed-point homomorphism $\phi_{3}^{U}$ is trivial, and the
same happens for $\phi_{3}^{SO}$ whenever $G$ is of odd order. The group
$\operatorname{Ker}(\phi_{2}^{U})$ is generated by manifolds without
codimension 2 fixed points and whose Euler characteristic is zero. In the
2-dimensional oriented case, any oriented $\mathbb{Z}/2$-action on a closed
surface will have an even number of isolated fixed points. Therefore the image
of the fixed-point homomorphism $\phi_{2}^{SO}(\Omega_{2}^{SO,G})$ is a free
$\mathbb{Z}$-module and its kernel consists of classes of manifolds without
codimension 0 fixed points. Note that pairs of points fixed by a group
isomorphic to $\mathbb{Z}/2$ could be removed via an equivariant bordism
attaching a handle. ∎

The presence of the platonic groups $A_{4}$, $\mathfrak{S}_{4}$, $A_{5}$ or
the dihedral groups $D_{2k}$ as subgroups of a general group $G$ makes the
understanding ot the bordism group $\Omega_{3}^{SO,G}$ more interesting. We
need first a definition.

######  Definition 3.20.

Let $M$ be a $G$-manifold $M$ (oriented or unitary). Define the ramification
locus of the $G$-action as the space

(3.21) $\overline{M}:=\bigcup_{K\subset G\atop K\neq\\{1\\}}M^{K},$

where $M^{K}$ denotes the space of fixed points of the subgroup $K$.

Let us start with the dihedral groups.

######  Proposition 3.22.

The equivariant bordism groups $\Omega_{3}^{SO,D_{2k}}$ are generated by
equivariant manifolds whose fixed-points are all of codimension 0 or empty. In
particular the fixed-point homomorphism $\phi_{3}^{SO}$ is trivial.

###### Proof.

Take $M$ a closed oriented $D_{2k}$-equivariant manifold such that $M/D_{2k}$
is connected. Let us first assume first that no element in $D_{2k}$ besides
the identity acts trivially (we could always take the induced action on $M$ of
the group $D_{2k}/L$ where $L$ is the subgroup that acts trivially and
consider $M$ as a $D_{2k}/L$-equivariant manifold). Hence we have that the
ramification locus $\overline{M}$ is the union of 1-dimensional and
0-dimensional manifolds.

Whenever the fixed-point set $M^{D_{2k}}$ is non-empty, it will consist of a
finite number of isolated points. We will argue that the number of fixed-
points with isomorphic normal representations is even, thus implying that the
image of the localization map of (3.2) at $D_{2k}$-fixed points

(3.23) $\displaystyle
f_{D_{2k}}:\Omega_{3}^{SO,D_{2k}}\to\Omega_{3}^{SO,D_{2k}}\\{\mathcal{A}D_{2k},\mathcal{P}D_{2k}\\}$

is trivial, and moreover, that the fixed-point set $M^{D_{2k}}$ could be
removed with an equivariant cobordism by attaching handles around pairs of
fixed points with isomorphic normal representation.

If $x$ belongs to $M^{D_{2k}}$, we claim that there is another fixed point
$x^{\prime}\in M^{D_{2k}}$, such that both have isomorphic representations of
$D_{2k}$ on their normal neighborhoods. The reason for this is the following.
Consider the class $[x]\in\overline{M}/D_{2k}$ on the quotient of the
ramification locus $\overline{M}$. The connected component of the fixed point
set of the cyclic subgroup $\mathbb{Z}/k$ around $x$ defines a path on the
quotient $\overline{M}/D_{2k}$ with the class $[x]$ at one end. Since
$\overline{M}/D_{2k}$ is compact, the other end of this path ends at the class
of the point $[x^{\prime}]$ where we have chosen $x^{\prime}$ to be on the
same connected component as $x$ on the fixed point set $M^{\mathbb{Z}/k}$. The
$D_{2k}$ representations around $x$ and $x^{\prime}$ are isomorphic because
their restrictions to the group $\mathbb{Z}/k$ give representations with
opposite orientations.

Note that whenever $k>2$ the points $x$ and $x^{\prime}$ are different. When
$k=2$ it could be the case that $x=x^{\prime}$, and if this were the case, we
need to notice that around $[x]$ in $\overline{M}/D_{2k}$ we would have a loop
(the path we defined above from $x$ to $x^{\prime}=x$) and an extra path
leaving from it. Following this third path from $x$, we will reach another
point $x^{\prime\prime}$ which will be different from $x$.

We just have shown that the fixed points in $M^{D_{2k}}$ come in pairs with
isomorphic representations. If the isomorphic representation is $V$ and $B(V)$
denotes the unit ball in $V$, this pair of points could removed by the bordism
that adds the handle $[0,1]\times B(V)$ on the normal neighborhoods of the
pair of points.

The previous construction could be carried out on all the fixed points of the
conjugacy classes of subgroups which are of dihedral type, and therefore we
see that $M$ is equivariantly cobordant to a manifold $M^{\prime}$ whose
fixed-points of its dihedral subgroups are empty. Hence the ramification locus
$\overline{M^{\prime}}$ is a 1-dimensional manifold and therefore
$\phi_{3}^{SO}([M])=\phi_{3}^{SO}([M^{\prime}])=0$.

We could then chose as generators of $\Omega_{3}^{SO,D_{2k}}$ manifolds
without 0-dimension and 1-dimensional fixed points. ∎

Propositions 3.12 and 3.22 imply that the fixed-point homomorphism
$\phi_{3}^{SO}$ is trivial on subgroups isomorphic to cyclic or dihedral
groups. Therefore the fixed-point homomorpshism may only be non-trivial when
evaluated on subgroups isomorphic to the platonic groups $A_{4}$,
$\mathfrak{S}_{4}$ and $A_{5}$. To understand the image of $\phi_{3}^{SO}$ for
the platonic groups we first need to define the blow up of a representation.

######  Definition 3.24.

Let $V$ be a finite dimensional real $G$-representation. The blow up
$\gamma(V)$ of $V$ is the total space of the bundle of real lines
$\mathbb{P}(V)$ of $V$:

(3.25) $\displaystyle\gamma(V):=\\{(v,L)\in V\times\mathbb{P}(V)\colon v\in
L\\},$

endowed with the natural $G$ action: $g\cdot(v,L):=(gv,gL)$. Denote
$B(\gamma(V))$ and $S(\gamma(V))$ the unit ball and sphere bundles of
$\gamma(V)$ respectively.

Note that the sphere bundle of $\gamma(V)$ and the sphere of the
representation $S(V)$ are canonically isomorphic

(3.26)
$\displaystyle\rho:S(V)\stackrel{{\scriptstyle\cong}}{{\to}}S(\gamma(V)),\ \
v\mapsto(v,\langle v\rangle).$

Therefore one may glue $B(\overline{V})$, where $\overline{V}$ is $V$ with the
opposite orientation, to $B(\gamma(V))$ along their boundary

(3.27) $\displaystyle Y(V):=B(\overline{V})\cup_{\rho}B(\gamma(V)),$

thus constructing a closed oriented $G$-manifold.

What is interesting about the blow up is the fact that for faithful
3-dimensional oriented real representations $V$, the blow up $\gamma(V)$ only
contains points with cyclic or dihedral isotropy groups. This is a key fact
that will be used in what follows.

######  Proposition 3.28.

Let $G$ be any finite group. Then the fixed-point homomorphism $\phi_{3}^{SO}$
is only non-trivial on subgroups isomorphic to the platonic groups $A_{4}$,
$\mathfrak{S}_{4}$ and $A_{5}$. Moreover, its restriction

(3.29) $\displaystyle\phi_{3}^{SO}:\Omega_{3}^{SO,G}\to\bigoplus_{(K)\atop K\
platonic}\Omega_{3}^{SO,K}\\{\mathcal{A}K,\mathcal{P}K\\}^{W_{K}}$

is surjective.

###### Proof.

Let $(K)$ be a conjugacy class of subgroups of $G$ with $K$ isomorphic to any
of the platonic groups $A_{4}$, $\mathfrak{S}_{4}$ or $A_{5}$. Denote by
$V_{K}$ the 3-dimensional real representation induced by the symmetries of the
respective platonic solid. Note that $V_{K}$ is isomorphic to the
representation with the reverse orientation $\overline{V_{K}}$, and therefore
the closed oriented $K$-manifold $Y(V_{K})$ defined in (3.27) is diffeomorphic
to $B(V_{K})\cup_{\rho}B(\gamma(V_{K}))$.

The localization map at $K$-fixed points of (3.2)

(3.30) $\displaystyle
f_{K}:\Omega_{3}^{SO,K}\to\Omega_{3}^{K}\\{\mathcal{A}K,\mathcal{P}K\\}\cong\mathbb{Z}/2$

maps $Y(V_{K})$ to the normal bundle of its $K$-fixed points $Y(V_{K})^{K}$.
Since the blow up $\gamma(V_{K})$ has no $K$-fixed points, then
$Y(V_{K})^{K}=B(V_{K})^{K}$ and it consists of only one point. Hence
$f_{K}(Y(V_{K}))=B(V_{K})$ with $[B(V_{K})]$ the generator of the group
$\Omega_{3}^{SO,K}\\{\mathcal{A}K,\mathcal{P}K\\}$.

The commutativity of the diagram

(3.35)

where $i_{H}^{L}:\Omega_{*}^{H}\to\Omega_{*}^{L}$, $[M]\mapsto[L\times_{H}M]$
is the induction map for the inclusion of groups $H\subset L$, implies that
the manifold $f_{K}\circ r^{G}_{K}\left(G\times_{K}Y(V_{K})\right)$ generates
the group $\Omega_{3}^{SO,K}\\{\mathcal{A}K,\mathcal{P}K\\}^{W_{K}}$.

Note that whenever $K\subsetneqq K^{\prime}$, we have
$\left(G\times_{K}Y(V_{K})\right)^{K^{\prime}}=\emptyset$. Therefore we
conclude that the images under $\phi_{3}^{SO}$ of the $G$-manifolds
$G\times_{K}Y(V_{K})$, where $(K)$ runs over the conjugacy classes of platonic
subgroups of $G$, provides the desired surjectivity. ∎

Let us see the previous result in an example. Let $G=A_{5}$ and take the
$A_{5}$-manifolds $Y(V_{A_{5}})$ and $A_{5}\times_{A_{4}}Y(V_{A_{4}})$ in
$\Omega_{3}^{SO,A_{5}}$. The image under $\phi_{3}^{SO}$ of these two
manifolds in

(3.36)
$\displaystyle\Omega_{3}^{SO,A_{5}}\\{\mathcal{A}A_{5},\mathcal{P}A_{5}\\}\oplus\Omega_{3}^{SO,A_{4}}\\{\mathcal{A}A_{4},\mathcal{P}A_{4}\\}\cong\mathbb{Z}/2\oplus\mathbb{Z}/2$

is respectively $(1,1)$ and $(0,1)$. The surjectivity of the map (3.29)
follows.

###  3.2. Surfaces without 0-dimensional fixed points

Let us then denote by $\overline{\Omega}_{2}^{G}$ the subgroup of
${\Omega}_{2}^{G}$ generated by manifolds without isolated fixed points, and
whose underlying Euler characteristic is zero in the unitary case. Since by
Corollary 3.19 we have that

(3.37)
$\overline{\Omega}_{2}^{G}=\operatorname{Ker}(\phi_{2})=\operatorname{Tor}_{\mathbb{Z}}(\Omega^{G}_{2}),$

we may study the properties of $\overline{\Omega}_{2}^{G}$ restricted to
families.

######  Lemma 3.38.

Let $\\{\mathcal{F},\mathcal{F}^{\prime}\\}$ be an adjacent pair of families
differing by the conjugacy class $(K)$ of the subgroup $K\subset G$. Then, the
bordism groups of surfaces fit into a split exact sequence

(3.39)
$\overline{\Omega}_{2}^{G}\\{\mathcal{F}^{\prime}\\}\to\overline{\Omega}_{2}^{G}\\{\mathcal{F}\\}\to\overline{\Omega}_{2}^{G}\\{\mathcal{F},\mathcal{F}^{\prime}\\}\to
0,$

with
$\overline{\Omega}_{2}^{G}\\{\mathcal{F},\mathcal{F}^{\prime}\\}\cong\Omega_{2}^{SO}(BW_{K})$
in the oriented case, and
$\overline{\Omega}_{2}^{U,G}\\{\mathcal{F},\mathcal{F}^{\prime}\\}\cong\widetilde{\Omega}_{2}^{U}(BW_{K})$
in the unitary case with $\widetilde{\Omega}_{2}^{U}$ the reduced unitary
bordism groups.

###### Proof.

The $K$-fixed points $M^{K}$ of any surface $M$ in
$\overline{\Omega}_{2}^{G}\\{\mathcal{F},\mathcal{F}^{\prime}\\}$ has no
boundary, and there is a $G$-equivariant homomorphism
$G\times_{N_{G}K}M^{K}\stackrel{{\scriptstyle\cong}}{{\to}}M$, $[(g,m)]\mapsto
gm$. The closed surface $M^{K}$ is endowed with a free action of the group
$W_{K}$, thus inducing a unique map up to homotopy $M^{K}/W_{K}\to BW_{K}$,
and hence
$\overline{\Omega}_{2}^{SO,G}\\{\mathcal{F},\mathcal{F}^{\prime}\\}\cong\Omega_{2}^{SO}(BW_{K})$.
For the unitary case we have that

(3.40)
$\chi(M)=0\Longleftrightarrow\chi(M^{K})=0\Longleftrightarrow\chi(M^{K}/W_{K})=0$

where $\chi$ denotes the Euler characteristic, hence
$\overline{\Omega}_{2}^{U,G}\\{\mathcal{F},\mathcal{F}^{\prime}\\}\cong\widetilde{\Omega}_{2}^{U}(BW_{K})$.

Here we have used the isomorphism

(3.41)
$\widetilde{\Omega}_{2}^{U}(BW_{K})\cong\operatorname{Ker}\left({\Omega}_{2}^{U}(BW_{K})\to{\Omega}_{2}^{U}\right)$

where the forgetful map ${\Omega}_{2}^{U}(BW_{K})\to{\Omega}_{2}^{U}$ simply
takes a framed bordism $[\Sigma\to BW_{K}]$ and maps it to $[\Sigma]$. The
kernel consists of framed bordisms with $\chi(\Sigma)=0$.

The exact sequence and the splitting follow from the fact that the manifolds
in $\overline{\Omega}_{2}^{G}\\{\mathcal{F},\mathcal{F}^{\prime}\\}$ are
closed. The canonical map
$\widetilde{\Omega}_{2}(BW_{K})\to\overline{\Omega}_{2}^{G}\\{\mathcal{F}\\}$,
$\Sigma/W_{K}\mapsto G\times_{N_{G}K}\Sigma$, provides the desired splitting.
∎

##  4\. Bounding equivariant surfaces

In this section we present the main result of this work, which is the
calculation of the groups $\overline{\Omega}_{2}^{G}$. To address this
question we bring the Conner-Floyd spectral sequence of §2.4 associated to the
families of subgroups

(4.1)
$\\{1\\}=\mathcal{F}_{0}\subset\mathcal{F}_{1}\subset\cdots\subset\mathcal{F}_{l}=\mathcal{A}G$

where all the pairs are adjacent, i.e.
$\mathcal{F}_{j}-\mathcal{F}_{j-1}=(K_{j})$ for some conjugacy class of
subgroups $(K_{j})$, and such that the conjugacy classes $(K_{j})$ span all
conjugacy classes of subgroups of $G$ (hence $l+1$ is the number of conjugacy
classes of subgroups of $G$).

We may filter the group $\overline{\Omega}_{2}^{G}$ by the subgroups

(4.2)
$F_{p}\overline{\Omega}_{2}^{G}:=\operatorname{Im}(\overline{\Omega}_{2}^{G}\\{\mathcal{F}_{p}\\}\to\overline{\Omega}_{2}^{G})$

whose associated graded groups are the quotients

(4.3)
$\operatorname{Gr}_{p}\overline{\Omega}_{2}^{G}=F_{p}\overline{\Omega}_{2}^{G}/F_{p-1}\overline{\Omega}_{2}^{G}.$

The Conner-Floyd spectral sequence then implies that we have the exact
sequence

(4.4)
$\overline{\Omega}_{3}^{G}\\{\mathcal{A}G,\mathcal{F}_{p}\\}\stackrel{{\scriptstyle\partial}}{{\to}}\overline{\Omega}_{2}^{G}\\{\mathcal{F}_{p},\mathcal{F}_{p-1}\\}\to\operatorname{Gr}_{p}\overline{\Omega}_{2}^{G}\to
0$

where the surjectivity of the map
$\overline{\Omega}_{2}^{G}\\{\mathcal{F}_{p},\mathcal{F}_{p-1}\\}\to\operatorname{Gr}_{p}\overline{\Omega}_{2}^{G}$
follows from Lemma 3.38.

We therefore need to understand the image of the boundary map

(4.5)
${\Omega}_{3}^{G}\\{\mathcal{A}G,\mathcal{F}_{p}\\}\stackrel{{\scriptstyle\partial}}{{\to}}\overline{\Omega}_{2}^{G}\\{\mathcal{F}_{p},\mathcal{F}_{p-1}\\}$

in order to determine the groups
$\operatorname{Gr}_{p}\overline{\Omega}_{2}^{G}$.

Note that the image of the boundary map of (4.5) is equivalent to the image of
the boundary map

(4.6)
${\Omega}_{3}^{W_{K}}\\{\mathcal{A}W_{K},\\{1\\}\\}\stackrel{{\scriptstyle\partial}}{{\to}}\widetilde{\Omega}_{2}(BW_{K}).$

This follows from the fact that the manifolds of interest will have trivial
actions of the groups in the conjugacy class $(K_{p})$ and then one follows
the same argument as presented in Lemma 3.38. Therefore we obtain the
following result.

######  Lemma 4.7.

Consider the associated graded groups
$\operatorname{Gr}_{*}\overline{\Omega}_{2}^{G}$ of
$\overline{\Omega}_{2}^{G}$ induced by the families of subgroups presented in
(4.1). Then

(4.8)
$\operatorname{Gr}_{p}\overline{\Omega}_{2}^{G}\cong\operatorname{Coker}\left({\Omega}_{3}^{W_{K_{p}}}\\{\mathcal{A}W_{K_{p}},\\{1\\}\\}\stackrel{{\scriptstyle\partial}}{{\to}}\widetilde{\Omega}_{2}(BW_{K_{p}})\right).$

Hence we need to understand which surfaces with free actions equivariantly
bound.

###  4.1. Bounding free actions on surfaces

It turns out that the only free actions on surfaces that equivariantly bound
are those on which the quotient surface is a torus. This result is originally
due to the first author [Sam20, Sam21] whenever the group $G$ does not contain
any subgroup isomorphic to the platonic groups $A_{4},\mathfrak{S}_{4},A_{5}$
or to the dihedral groups $D_{2k}$, and it was the one who triggered the
investigation presented in this work. Here we will produce an alternative
proof, generalizing it for all finite groups. Let us first recall the
definition of the Bogomolov multiplier of a finite group.

The cohomology group $H^{2}(G,\mathbb{C}^{*})$ determines the isomorphism
classes of central $\mathbb{C}^{*}$ group extensions of $G$, and therefore
complex irreducible projective representations of the group $G$ define
elements in $H^{2}(G,\mathbb{C}^{*})$. Schur [Sch04] extensively studied this
cohomology group and therefore it was called the Schur multiplier of $G$
[Kar87].

Bogomolov [Bog87] defined the subgroup $B_{0}(G)$ of the Schur multiplier
consisting of all elements which vanish when restricted to all its abelian
subgroups

(4.9) $B_{0}(G)=\bigcap_{A\subset
G,\mathrm{abelian}}\operatorname{Ker}\left(\mathrm{res}^{G}_{A}:H^{2}(G,\mathbb{C}^{*})\to
H^{2}(A,\mathbb{C}^{*})\right).$

The interest in this group comes among other things from a result Bogomolov
proved in [Bog87, Thm. 3.1] which states that whenever the field of
$G$-invariants $\mathbb{C}[G]^{G}$ of the rational field $\mathbb{C}[G]$ is
rational over $\mathbb{C}$ then the Bogomolov multiplier of the group $G$
vanishes.

Using the fact that for finite groups $H^{2}(G,\mathbb{C}^{*})\cong
H_{2}(G,\mathbb{Z})$, Moravec [Mor12] showed that the Bogomolov multiplier
group $B_{0}(G)$ is isomorphic to the group

(4.10) $\tilde{B}_{0}(G):=H_{2}(G,\mathbb{Z})/M_{0}(G)$

where $M_{0}(G)$ is the subgroup of $H_{2}(G,\mathbb{Z})$ generated by the
images

(4.11) $\operatorname{Im}\left(H_{2}(\mathbb{Z}\times\mathbb{Z},\mathbb{Z})\to
H_{2}(G,\mathbb{Z})\right)$

of all homomorphisms $\mathbb{Z}\times\mathbb{Z}\to G$. This homology version
of the Bogomolov multiplier was then used to calculate $B_{0}(G)$ for several
types of finite groups [Mor12].

In this homological form, the Bogomolov multiplier appeared much earlier in
[Neu75] in connection with $SK$-groups (cutting and pasting of manifolds) and
in [Oli80] as $SK_{1}$ in algebraic K-theory.

Using now the fact that there are canonical isomorphisms

(4.12)
$\widetilde{\Omega}_{2}^{U}(BG)\stackrel{{\scriptstyle\cong}}{{\to}}{\Omega}_{2}^{SO}(BG)\stackrel{{\scriptstyle\cong}}{{\to}}{H}_{2}(BG,\mathbb{Z}),$

we present a generalization of a result which was established by the first
author in [Sam21]. First we need a lemma.

######  Lemma 4.13.

Let $\Sigma$ be an oriented surface with free $G$-action that bounds
equivariantly. Then $\Sigma$ can be extended to an oriented $G$-manifold whose
ramification locus is a 1-dimensional manifold (all the isotropy groups are
all cyclic).

###### Proof.

Let $M$ be an oriented $G$-manifold whose boundary is the surface with free
$G$-action $\Sigma$. The obstruction for the ramification locus $\overline{M}$
to be a 1-dimensional manifold is the presence of points with isotropy groups
isomorphic to dihedral or platonic groups ($A_{4}$, $\mathfrak{S}_{4}$,
$A_{5}$). First we will perform the blow up construction on the normal
neighborhoods of the points whose isotropies are isomorphic to either $A_{5}$,
$\mathfrak{S}_{4}$ or $A_{4}$; this procedure produces a new manifold
$M^{\prime}$ with the same boundary as $M$ but without points with $A_{5}$,
$\mathfrak{S}_{4}$ or $A_{4}$ isotropy. Now, on $M^{\prime}$ we construct a
cobordism by attaching the appropriate equivariant handles around all pairs of
points with dihedral isotropy and isomorphic normal representation. The new
manifold thus constructed $M^{\prime\prime}$ is cobordant to
$M^{\prime\prime}$, has the same boundary as $M$, and has only fixed points
with cyclic isotropy. Hence the ramification locus of $M^{\prime\prime}$ is
smooth.

Let us show with more detail the first step. Take a point $x\in M$ whose
isotropy $G_{x}$ is imorphic to $A_{5}$ (we will start with the larger
isotropy first). Let $N_{x}$ be a normal $G_{x}$-neighborhood of $x$ such that
$N_{x}\cap g\cdot N_{x}=\emptyset$ for all $g\in G-G_{x}$, and let

(4.14)
$\displaystyle\sigma:B(V_{G_{x}})\stackrel{{\scriptstyle\cong}}{{\to}}N_{x}$

be a $G_{x}$-equivariant diffeomorphism with $V_{G_{x}}$ the faithful
representation of $G_{x}$ around $x$. Take $G\cdot N_{x}$ as a $G$-equivariant
neighborhood around the orbit $G\cdot x$ and note that $\sigma$ induces a
$G$-equivariant diffeomorphism
$G\times_{G_{x}}B(V_{G_{x}})\stackrel{{\scriptstyle\cong}}{{\to}}G\cdot
N_{x}$. Construct the blow up $B(\gamma(V_{G_{x}}))$ presented in Definition
3.24 and note that the sphere bundles are $G_{x}$-diffeomorphic to the
boundary of $N_{x}$:

(4.15) $\displaystyle
S(\gamma(V_{G_{x}}))\cong(V_{G_{x}})\stackrel{{\scriptstyle\cong}}{{\to}}\partial
N_{x}.$

Cut $G\cdot N_{x}$ from $M$ and glue $G\times_{G_{x}}B(\gamma(V_{G_{x}}))$
along the boundary using the diffeomorphism $\sigma$. Define the new
$G$-manifold

(4.16) $\displaystyle M^{\prime}:=\left(M-G\cdot
N_{x}\right)\cup_{\partial(G\cdot N_{x})}G\times_{G_{x}}B(\gamma(V_{G_{x}}))$

and note that $M^{\prime}$ has the same boundary as $M$, but with the property
that inside $\partial N_{x}$ there are no more points with isotropy isomorphic
to $A_{5}$. Cutting and pasting the blow ups for every point with isotropy
isomorphic to $A_{5}$, produces a manifold without points whose isotropy is
isomorphic to $A_{5}$. Then this blow up procedure is carried out for points
with isotropy isomorphic to $\mathfrak{S}_{4}$ and then to points with
isotropy isomorphic to $A_{4}$. The resulting manifold $M^{\prime}$ has the
same boundary as $M$, but it does not contain points with isotropy isomorphic
to either $A_{5}$, $\mathfrak{S}_{4}$ or $A_{4}$. The only isotropies that
appear on $M^{\prime}$ are cyclic or dihedral groups.

Now let us do the second step. Here we follow the same procedure as presented
in Proposition 3.22. For a point $x\in\overline{M}$ with isotropy a dihedral
group $K$, there exists another point $x^{\prime}\in\overline{M}$ with the
same isotropy group and isomorphic normal representation. Simply follow the
path on $\overline{M}$ of the fixed point set of the largest cyclic subgroup
of $K$ (if $K$ is $D_{4}$ any subgroup works) and another point $x^{\prime}$
will be reached with the same istropy as $x$ and isomorphic normal
representation. If $x=x^{\prime}$, as it might be the case whenever $K\cong
D_{4}$, simply follow on $\overline{M}$ the third path that leaves $x$ and a
different point $x^{\prime\prime}$ will be reached (see the proof of
Proposition 3.22). Attach an equivariant handle of the form
$[0,1]\times(G\times_{K}B(V))$, where $V$ is the faithful $K$-representation
which is $K$-equivariantly diffeomorphic to the normal neighborhood $N_{x}$ of
$x$. This handle is attached to the $G$-equivariant normal neighborhood of
$G\cdot x$ and $G\cdot x^{\prime}$. Take the cobordant manifold where the
$G$-equivariant normal neighborhoods of $G\cdot x$ and $G\cdot x^{\prime}$ are
removed, and note that in this new manifold there are less points with
isotropy isomorphic to dihedral groups. Repeat the previous procedure until
the points with dihedral isotropy have been removed.

The resulting manifold $M^{\prime\prime}$ has the same boundary as $M$, is
cobordant to $M^{\prime}$, and its ramification locus
$\overline{M^{\prime\prime}}$ is smooth because all its isotropy groups are
cyclic (or trivial).

Here it is important to highlight that $M^{\prime}$ and $M^{\prime\prime}$ are
cobordant, while $M$ and $M^{\prime}$ might be in different cobordism classes.
The fixed-point homomorphism $\phi_{3}^{SO}$ of Proposition 3.28 will
distinguish the different cobordism classes. ∎

We are now ready to show which surfaces with free $G$-actions bound
equivariantly.

######  Theorem 4.17.

Let $G$ be a finite group. Then the oriented and unitary equivariant bordism
of surfaces with free $G$-actions fits into the exact sequence

(4.18)
${\Omega}_{3}^{G}\\{\mathcal{A}G,\\{1\\}\\}\stackrel{{\scriptstyle\partial}}{{\to}}\widetilde{\Omega}_{2}(BG)\to\tilde{B}_{0}(G)\to
0.$

###### Proof.

Let us first show that the image of the boundary map consists of toral classes
in $H_{2}(BG,\mathbb{Z})$, that is, homology classes coming from the image of
maps of tori $S^{1}\times S^{1}\to BG$.

Let $M$ be a 3-dimensional $G$-manifold (oriented or unitary) whose boundary
$\partial M$ has a free $G$-action; in the oriented case take $M$ as shown in
Lemma 4.13 Note that if the ramification locus $\overline{M}$ is not empty,
then it is a smooth oriented 1-dimensional manifold; in the unitary case this
follows from the fact that fixed points of all non-trivial subgroups can only
have complex codimension 0 or 1.

If $\overline{M}$ is empty then $M$ has a free $G$ action and therefore the
boundary surface $(\partial M)/G$ bounds. If $\overline{M}$ is not empty we
may consider the $G$-equivariant tubular neighborhood $N$ of $\overline{M}$ in
$M$. The manifolds $N$ and $M$ define the same bordism class since on $M-N$
the action of $G$ is free, and therefore $\partial M$ and $\partial N$ are
cobordant. The tubular neighborhood $N$ is homeomorphic to the unit ball
bundle $B\nu$ of the normal bundle $\nu$ of $\overline{M}$ in $M$. The sphere
bundle $S\nu$ defines the $S^{1}$-principal bundle $S^{1}\to
S\nu\to\overline{M}$ and since every circle bundle over the circle is
topologically a torus, then the sphere bundle $S\nu$ is homeomorphic to a
disjoint union of 2-dimensional tori. Hence $\partial N$ is a disjoint union
of 2-dimensional tori, and its quotient $\partial N/G$ is a torus (since $M/G$
is connected and $\chi(\partial N/G)=\chi(\partial N)/|G|$). Hence we have now
proved that the image of the boundary map $\partial$ of (4.18) consists only
of toral classes in $H_{2}(BG,\mathbb{Z})$.

Now let us show the converse, namely, that any toral class in
$H_{2}(BG,\mathbb{Z})$ lies in the image of the boundary map of (4.18). Take
any toral class defined by a homomorphism
$\varphi:\mathbb{Z}\times\mathbb{Z}\to G$ and denote by
$A:=\varphi(\mathbb{Z}\times\\{0\\})$ and
$C:=\varphi(\\{0\\}\times\mathbb{Z})$ the cyclic subgroups of $G$ that define
the toral class. Denote by $a:=\varphi(1,0)$ and $c:=\varphi(0,1)$ the
generators of $A$ and $C$ respectively.

Let $N_{G}A$ be the normalizer of $A$ in $G$ and note that $C$ is a subgroup
of the normalizer. Denote by $\iota$ and $\bar{\iota}$ the homomorphism
$\iota:\mathbb{Z}\to N_{G}A$, $\iota(n):=c^{n}$ and the homomorphism to the
quotient $\bar{\iota}:\mathbb{Z}\to W_{A}$. Consider the irreducible
representation $\rho:A\to U(1)$, $\rho(a):=e^{\frac{2\pi i}{|A|}}$, and define
the $U(1)$ extension of $W_{A}$ by the exact sequence of groups

(4.19) $U(1)\to U(1)\times_{A}N_{G}A\to W_{A}$

where $U(1)\times_{A}N_{G}A$ is defined by the equivalence relation
$(\lambda\rho(\alpha),g)\sim(\lambda,\alpha g)$ for all $\alpha\in A$,
$\lambda\in U(1)$ and $g\in N_{G}A$.

Consider the homomorphism $\tilde{\iota}:\mathbb{Z}\to U(1)\times_{A}N_{G}A$,
$\tilde{\iota}(n):=[(1,\iota(c^{n}))]$, and note that its classifying map

(4.20) $B\tilde{\iota}:S^{1}\to B(U(1)\times_{A}N_{G}A)$

factors through the classifying map $B\bar{\iota}:S^{1}\to BW_{A}$.

Let $E:=(B\tilde{\iota})^{*}E(U(1)\times_{A}N_{G}A)$ be the pullback of the
universal bundle and note two things. First, $E$ is a principal
$U(1)\times_{A}N_{G}A$-bundle over the circle $S^{1}$, and therefore it is a
surface. Second, the canonical homomorphism

(4.21) $N_{G}A\to U(1)\times_{A}N_{G}A,\ \ g\mapsto[(1,g)],$

induces a free action of $N_{G}A$ on $E$. Now it is straightforward to notice
that the homology class of the surface $E/N_{G}A\to BN_{G}A\to BG$ in
$\widetilde{\Omega}_{2}(BG)$ agrees with the homology class defined by
$B\varphi_{*}[S^{1}\times S^{1}]$.

We still need to show that the surface $E$ equivariantly bounds. Take the
quotient $F=E/U(1)$ and note that $F$ is homeomorphic to
$(B\bar{\iota})^{*}EW_{A}$; hence $F$ is the principal $W_{A}$-bundle over the
circle that $\bar{\iota}$ defines (see the following commutative diagram).

(4.36)

We have that $E$ is a principal $U(1)$-bundle over $F$ and therefore we may
take the associated complex vector bundle

(4.37) $\mathbb{C}\to\mathbb{C}\times_{U(1)}E\to F.$

The unit bundle $D(\mathbb{C}\times_{U(1)}E)$ is a unitary manifold endowed
with the action of $N_{G}A$, whose boundary, the sphere bundle
$S(\mathbb{C}\times_{U(1)}E)$, is homeomorphic to $E$:

(4.38)
$\partial\left(D(\mathbb{C}\times_{U(1)}E)\right)=S(\mathbb{C}\times_{U(1)}E)\cong
E.$

Therefore we have just proved that

(4.39)
$[G\times_{N_{G}A}D(\mathbb{C}\times_{U(1)}E)]\stackrel{{\scriptstyle\partial}}{{\to}}[E/N_{G}A]=B\varphi_{*}[S^{1}\times
S^{1}],$

thus showing that any toral class in $\widetilde{\Omega}_{2}(BG)$
equivariantly bounds. ∎

Now we can put all the pieces together to understand the torsion of the
equivariant bordism group of surfaces.

###  4.2. Torsion of the equivariant bordism group of surfaces

By Corollary 3.19 we have

(4.40)
$\overline{\Omega}_{2}^{G}=\operatorname{Ker}(\phi_{2})=\operatorname{Tor}_{\mathbb{Z}}(\Omega^{G}_{2}).$

Let us now determine explicitly these torsion subgroups.

######  Theorem 4.41.

Let $G$ be a finite group. Then there is a canonical isomorphism

(4.42)
$\bigoplus_{(K)}\tilde{B}_{0}(W_{K})\cong\operatorname{Tor}_{\mathbb{Z}}(\Omega^{G}_{2})$

where $(K)$ runs over all conjugacy classes of subgroups of $G$,
$W_{K}=N_{G}K/K$ and $\tilde{B}_{0}(W_{K})$ is the homology version of the
Bogomolov multiplier of the group $W_{K}$.

###### Proof.

Denote by
$\operatorname{Gr}_{*}\operatorname{Tor}_{\mathbb{Z}}(\Omega^{G}_{2})$ the
associated graded groups of the $G$-equivariant, unitary or oriented, bordism
groups of surfaces that are induced by the Conner-Floyd spectral sequence of
the families of subgroups of (4.1). Lemma 4.7 and Theorem 4.17 imply that

(4.43)
$\displaystyle\operatorname{Gr}_{p}\operatorname{Tor}_{\mathbb{Z}}(\Omega^{G}_{2})\cong\tilde{B}_{0}(W_{K_{p}}),$

and since all consecutive pairs of families are adjacent, we obtain the graded
isomorphism

(4.44)
$\displaystyle\operatorname{Gr}_{*}\operatorname{Tor}_{\mathbb{Z}}(\Omega^{G}_{2})\cong\bigoplus_{(K)}\tilde{B}_{0}(W_{K}).$

Now, for a fixed conjugacy class of subgroups $(K)$, the canonical map

(4.45) $\displaystyle\widetilde{\Omega}_{2}(BW_{K})\to\Omega_{2}^{G},\ \ \ \
\Sigma/W_{K}\mapsto G\times_{N_{G}K}\Sigma,$

which sends the quotient space of a surface $\Sigma$ by the free
$W_{K}$-action to the surface with $G$-action whose isotropy groups lie in
$(K)$, factors through $\tilde{B}_{0}(W_{K})$, thus producing a canonical
homomorphism

(4.46) $\displaystyle\tilde{B}_{0}(W_{K})$ $\displaystyle\to\Omega_{2}^{G}.$

Bundling up all these homomorphisms we obtain a canonical map

(4.47)
$\displaystyle\bigoplus_{(K)}\tilde{B}_{0}(W_{K})\to\operatorname{Tor}_{\mathbb{Z}}(\Omega^{G}_{2})$

which becomes an isomorphism since it is compatible with the graded
isomorphism of (4.44). ∎

In particular, if $G$ is a group whose Bogomolov multipliers vanish for all
groups $W_{K}$ with $K$ a non-trivial subgroup, then
$\operatorname{Tor}_{\mathbb{Z}}(\Omega^{G}_{2})\cong\tilde{B}_{0}(G)$. This
is the case whenever $G$ is one of the smallest $p$-groups with non-trivial
Bogomolov multiplier. In the last section we present two $p$-groups of this
kind.

We are now ready to provide an explicit calculation of the unitary and
oriented equivariant bordism group of surfaces. Assembling Theorem 3.8,
Proposition 3.12 and Theorem 4.41 we obtain the following result.

######  Theorem 4.48.

Let $G$ be a finite group. Then the unitary and oriented equivariant bordism
of surfaces canonically decompose as follows:

(4.49) $\displaystyle\Omega_{2}^{U,G}\cong$
$\displaystyle\bigoplus_{(K)}\left(\tilde{B}_{0}(W_{K})\oplus\Omega_{2}^{U}\right)\oplus\bigoplus_{(K)\atop
K\
cyclic}\left(\bigoplus_{\operatorname{\textup{Hom}_{\textup{inj}}}(K,U(1))/W_{K}}\mathbb{Z}\right),$
(4.50) $\displaystyle\Omega_{2}^{SO,G}\cong$
$\displaystyle\bigoplus_{(K)}\left(\tilde{B}_{0}(W_{K})\right)\oplus\bigoplus_{(K)\atop
K\ncong\mathbb{Z}/2\
cyclic}\left(\bigoplus_{\operatorname{\textup{Hom}_{\textup{inj}}}(K,U(1))_{\tau}/W_{K}}\mathbb{Z}\right).$

Here $(K)$ runs over the conjugacy classes of subgroups of $G$, $W_{K}$ is the
Weyl group $N_{G}K/K$, $\operatorname{\textup{Hom}_{\textup{inj}}}(K,U(1))$ is
the set of monomorphisms,
$\operatorname{\textup{Hom}_{\textup{inj}}}(K,U(1))_{\tau}$ is the set of
equivalence classes of monomorphisms modulo complex conjugation, and
$\operatorname{\textup{Hom}_{\textup{inj}}}(K,U(1))/W_{K}$ and
$\operatorname{\textup{Hom}_{\textup{inj}}}(K,U(1))_{\tau}/W_{K}$ are the
quotients defined by the induced action of the Weyl group $W_{K}$ on these
sets of homomorphisms.

##  5\. 2-dimensional SK-groups of classifying spaces

Jänich in [Jän69, Jän68] started the study of the characterization of
invariants with the additivity property of the Euler characteristic and the
signature under cutting and pasting of manifolds.

Karras and Kreck in their Diplom thesis extended the ideas of Jänich to
cutting and pasting in the bundle situation. The book [KKNO73] presented and
simplified these results with the definition of the $SK$-groups of a space
(cutting and pasting groups from the German Schneiden und Kleben). Later
Neumann [Neu75] completely calculated the 2-dimensional $SK$-groups of a space
in terms of what is now known as the Bogomolov multiplier of its fundamental
group. We recall in this section the main results of [KKNO73] and [Neu75] that
allow us to relate the $SK$-relation with the equivariant bordism relation on
surfaces with free actions.

The Schneiden und Kleben groups $SK_{*}(X)$ of a space are defined as the
Grothendieck group of the semigroups obtained by defining the $SK$-equivalence
on the class of continuous maps from oriented $n$-dimensional manifolds to $X$
[KKNO73].

The $SK$-relation is defined as follows: given $(M_{i},f_{i})$ with
$f_{i}:M_{i}\rightarrow X$, we say that $(M_{1},f_{1})$ and $(M_{2},f_{2})$
are related by cutting and pasting along $\partial N$ if
$M_{1}=N\cup_{\phi}-N^{\prime}$,$M_{2}=N\cup_{\psi}-N^{\prime}$ and there are
homotopies $f_{1}\mid_{N}\simeq f_{2}\mid_{N}$, $f_{1}\mid_{N^{\prime}}\simeq
f_{2}\mid_{N^{\prime}}$.

The Schneiden und Kleben bordism groups $\overline{SK}_{n}(X)$ of a space are
defined as the quotient of the oriented bordism groups by the equivalence
relation generated by the $SK$-relation:

(5.1) $\overline{SK}_{*}(X)=\Omega^{SO}_{*}(X)/\sim.$

The group $\overline{SK}_{2}(BG)$ can be interpreted as the bordism group of
surfaces with free $G$-actions modulo the $SK$-relations.

The following results summarize the main properties of the $SK$-relation
[KKNO73, Lem. 1.5 & 1.6].

  1. (1)

Any $f:S^{1}\rightarrow X$ is zero in $SK_{1}(X)$ .

  2. (2)

If $M$ fibers over $S^{n}$ with fiber $F$ then for any $f:M\rightarrow X$, in
$SK_{*}(X)$ we have

(5.2) $[M,f]=[S^{n},*][F,f|_{F}].$

  3. (3)

If $[M_{2},f_{2}]$ is obtained from $[M_{1},f_{1}]$ by surgery of type
$(k+1,n-k)$, then in $SK_{*}(X)$

(5.3) $[M_{1},f_{1}]+[S^{n},*]=[M_{2},f_{2}]+[S^{k}\times S^{n-1},*].$

Now, if $I_{*}$ denotes the subgroup of $SK_{*}(X)$ generated by the spheres,
we have:

######  Theorem 5.4.

[KKNO73, Thm. 1.1] For a connected space $X$, there is the following exact
sequence

(5.5) $0\rightarrow I_{*}\rightarrow
SK_{*}(X)\rightarrow\overline{SK}_{*}(X)\rightarrow 0,$

which is moreover split.

The groups $\overline{SK}_{*}(X)$ fit into short exact sequences whose middle
terms are the oriented bordism groups.

######  Theorem 5.6.

[KKNO73, Thm 1.2] Let $F_{n}(X)$ be the submodule of $\Omega^{SO}_{n}(X)$
generated by all elements which have a representative that fibers over
$S^{1}$. Then $F_{n}(X)$ fits into the short exact sequence

(5.7) $0\rightarrow
F_{*}(X)\rightarrow\Omega^{SO}_{*}(X)\rightarrow\overline{SK}_{*}(X)\rightarrow
0.$

This theorem follows from the observations that any manifold that fibers over
$S^{1}$ gives a class that is zero in $\overline{SK}_{*}(X)$, and that the
kernel of the homomorphism $\Omega^{SO}_{*}(X)\rightarrow\overline{SK}_{*}(X)$
consists of mapping tori. The key lemma for the opposite inclusion asserts
that if $(M_{1},f_{1})$ is obtained from $(M_{2},f_{2})$ by cutting and
pasting along $N$, then in $\Omega^{SO}_{*}(X)$ the class of
$(N\cup_{\phi}-N^{\prime},f_{1})-(N\cup_{\psi}-N^{\prime},f_{2})$ is equal to
the mapping torus of the diffeomorphism of $\partial N$, $\phi^{-1}\circ\psi$.
Any mapping torus fibers over $S^{1}$ and any fibration over $S^{1}$ is a
mapping torus.

The $SK$\- groups of a point are given by [KKNO73]:

$\displaystyle SK_{4n+1}(*)=$ $\displaystyle 0,\ \ \ \ SK_{4n+3}(*)=0,$ (5.8)
$\displaystyle SK_{4n+4}(*)=$ $\displaystyle\mathbb{Z}\oplus\mathbb{Z},\text{
basis }[S^{4n+4}],[\mathbb{C}\mathbb{P}^{2n+2}],$ $\displaystyle
SK_{4n+2}(*)=$ $\displaystyle\mathbb{Z},\text{ basis }[S^{4n+2}].$

The $\overline{SK}$\- groups of a point are given by:

(5.9) $\displaystyle\overline{SK}_{n}(*)\cong\begin{cases}0&\text{ $n$ is odd
or $n\equiv 2\mod 4$,}\\\ \mathbb{Z}&\text{ $n\equiv 0\mod 4$. }\end{cases}$

There are augmentation maps

(5.10) $\varepsilon^{X}:SK_{*}(X)\rightarrow SK_{*}(*),\ \ \
\overline{\varepsilon}^{X}:\overline{SK}_{*}(X)\rightarrow\overline{SK}_{*}(*)$

given by ignoring the maps to $X$. For $X$ a connected space, these
augmentation maps give decompositions

(5.11) $SK_{*}(X)\cong SK_{*}(*)\oplus\operatorname{Ker}\varepsilon^{X},\ \ \
SK_{*}(X)\cong\overline{SK}_{*}(*)\oplus\operatorname{Ker}\overline{\varepsilon}^{X},$

where
$\operatorname{Ker}\varepsilon^{X}\cong\operatorname{Ker}\overline{\varepsilon}^{X}$
and the projection
$\eta:SK_{*}(X)\rightarrow\operatorname{Ker}\varepsilon^{X}$ is given by
$\eta(M,f)=[M,f]-[M,*]$.

In dimension $0$ and $1$ the groups $\overline{SK}_{n}(X)$ are trivial. In
dimension $2$ the oriented manifolds that fiber over the circle are tori.
Therefore by Theorem 5.6 we obtain the following result:

######  Theorem 5.12.

[Neu75, Thm. 2] Let $G$ be a discrete group. Then the 2-dimensional
$\overline{SK}$-group of $BG$ is isomorphic to the Bogomolov multiplier of
$G$, i.e.:

(5.13) $\overline{SK}_{2}(BG)\cong\tilde{B}_{0}(G).$

Therefore we have $SK_{2}(BG)\cong\mathbb{Z}\oplus\tilde{B}_{0}(G)$. By
Theorem 4.17, we can reinterpret when a surface $\Sigma$ with a free action of
$G$ is zero in the group $SK_{2}(BG)$: the Euler characteristic of $\Sigma/G$
should be zero and the $G$-manifold $\Sigma-G\times(\Sigma/G)$ is the boundary
of a three dimensional manifold with a $G$-action.

It would be interesting to explore the relation of this work with the higher
dimensional $SK$-groups of classifying spaces.

##  6\. Small groups with non-trivial Bogomolov multiplier

We conclude this work by presenting some explicit examples of groups with non-
trivial Bogomolov multiplier which induce non-trivial torsion subgroups in the
equivariant bordism groups of surfaces. Some of the calculations were done
with the help of the Homological Algebra Programming package for GAP [GAP21].

###  6.1. 2-group of size 64

The smallest groups with non-trivial Bogomolov multiplier are 2-groups of
order 64. There are 9 of them and all are in the same isoclinism class. By
[Mor14, Thm 1.2] they all have isomorphic Bogomolov multipliers and in this
case it is the group $\mathbb{Z}/2$. Among the 9 isoclinic groups we chose to
study the group

(6.1) $C_{8}\rtimes Q_{8}$

which is the semidirect product of the group of quaternions $Q_{8}$ with the
cyclic group $C_{8}$ of order $8$; this group is denoted

SmallGroup(64,182)

in the GAP small groups library. Consider the presentations of the groups
$Q_{8}=\langle a,b\colon a^{2}=b^{2},aba^{-1}=b^{-1}\rangle$ and
$C_{8}=\langle c\colon c^{8}=1\rangle$ and the action of $Q_{8}$ on $C_{8}$
given by the equations

(6.2) $a\cdot c=c^{3},\ \ \ b\cdot c=c^{5},\ \ \ (ab)\cdot c=c^{7}.$

Since $H^{2}(C_{8},\mathbb{C}^{*})=0=H^{2}(Q_{8},\mathbb{C}^{*})$, we know by
the Lyndon-Hochschild spectral sequence that

(6.3) $H^{2}(C_{8}\rtimes Q_{8},\mathbb{C}^{*})\cong
H^{1}(Q_{8},H^{1}(C_{8},\mathbb{C}^{*})).$

Denote
$\widehat{C}_{8}:=\operatorname{\textup{Hom}}(C_{8},\mathbb{C}^{*})=H^{1}(C_{8},\mathbb{C}^{*})$
and let $\widehat{C}_{8}=\langle\rho\colon\rho^{8}=1\rangle$ with
$\rho(c)=e^{\frac{2\pi i}{8}}$. Take the first two terms of the complex
$C^{*}(Q_{8},\widehat{C}_{8})$

(6.4)
$\widehat{C}_{8}\stackrel{{\scriptstyle\delta}}{{\to}}\operatorname{Map}(Q_{8},\widehat{C}_{8})$

and note that

(6.5) $\displaystyle\delta(\rho^{k})(a^{\pm})=\rho^{-2k},\ $ $\displaystyle\
\delta(\rho^{k})(b^{\pm})=\rho^{4k},$
$\displaystyle\delta(\rho^{k})((ab)^{\pm})=\rho^{2k},\ $ $\displaystyle\
\delta(\rho^{k})(a^{2})=\rho^{0}.$

On the other hand, take the 1-cocycle $F:Q_{8}\to\widehat{C}_{8}$ defined by
the equations

(6.6) $\displaystyle F(a^{\pm})=\rho^{2},\ $ $\displaystyle\
F(b^{\pm})=\rho^{0},$ (6.7) $\displaystyle F((ab)^{\pm})=\rho^{2},\ $
$\displaystyle\ F(a^{2})=\rho^{0},$

and note that $F$ does not bound but $F^{2}=\delta(\rho^{2})$. We have
therefore that

(6.8)
$H^{1}(Q_{8},\widehat{C}_{8})\cong\langle[F]\colon[F^{2}]=0\rangle\cong\mathbb{Z}/2.$

Now, any abelian subgroup of $C_{8}\rtimes Q_{8}$ splits as a semi-direct
product of abelian groups $C\rtimes A$ with $C\subset C_{8}$ and $A\subset
Q_{8}$. Since $A$ can only be $\mathbb{Z}/4$ or $\mathbb{Z}/2$, it is now
straightforward to check that $[F]|_{C\rtimes A}=0$. Hence $[F]$ is the
generator of the Bogomolov multiplier of $C_{8}\rtimes Q_{8}$ and we have that

(6.9) $\overline{\Omega}_{2}^{U,C_{8}\rtimes
Q_{8}}\cong\overline{\Omega}_{2}^{SO,C_{8}\rtimes Q_{8}}\cong\mathbb{Z}/2.$

Finally, with the explicit description of $F$ we can define a surface
$\Sigma_{2}$ of genus $2$ which defines the generator of
$\widetilde{\Omega}_{2}^{U}(B(C_{8}\rtimes Q_{8}))$. Consider the presentation
of the fundamental group of the surface

(6.10) $\pi_{1}(\Sigma_{2})=\langle x,y,z,w\colon[x,y][z,w]=1\rangle$

and define the following assignment

(6.11) $\displaystyle\Phi:\pi_{1}(\Sigma_{2})$ $\displaystyle\to C_{8}\rtimes
Q_{8}$ $\displaystyle x\mapsto a,\ \ y\mapsto c,\ $ $\displaystyle\ z\mapsto
ab,\ \ w\mapsto c,$

which induces a surjective homomorphism since

(6.12)
$\Phi([x,y][z,w])=aca^{-1}c^{-1}(ab)c(ab)^{-1}c^{-1}=c^{3}c^{-1}c^{7}c^{-1}=c^{0}.$

The homomorphism $\Phi$ induces a map $B\Phi:\Sigma_{2}\to B(C_{8}\rtimes
Q_{8})$, and from the construction above of $F$, we deduce that
$B\Phi_{*}[\Sigma_{2}]$ generates the group $H_{2}(B(C_{8}\rtimes
Q_{8}),\mathbb{Z})$.

Hence the surface

(6.13) $\widetilde{\Sigma}:=B\Phi^{*}E(C_{8}\rtimes Q_{8})$

is a unitary surface with a free action of $C_{8}\rtimes Q_{8}$ which does not
equivariantly bound.

By Theorem 4.41 the class of $\widetilde{\Sigma}$ is the generator of the
torsion subgroup of ${\Omega}_{2}^{SO,C_{8}\rtimes Q_{8}}$:

(6.14) $\operatorname{Tor}_{\mathbb{Z}}{\Omega}_{2}^{SO,C_{8}\rtimes
Q_{8}}=\langle[\widetilde{\Sigma}]\rangle\cong\mathbb{Z}/2.$

To make sure that the Euler characteristic vanishes, we take the bordism class

(6.15) $[\widetilde{\Sigma}]-[(C_{8}\rtimes
Q_{8})\times\Sigma_{2}]\in\overline{\Omega}_{2}^{U,C_{8}\rtimes
Q_{8}}\cong\mathbb{Z}/2$

and by Theorem 4.41 we conclude that this class is indeed the generator of the
torsion subgroup of ${\Omega}_{2}^{U,C_{8}\rtimes Q_{8}}$:

(6.16) $\operatorname{Tor}_{\mathbb{Z}}{\Omega}_{2}^{U,C_{8}\rtimes
Q_{8}}=\langle[\widetilde{\Sigma}]-[(C_{8}\rtimes
Q_{8})\times\Sigma_{2}]\rangle\cong\mathbb{Z}/2.$

###  6.2. 3-group of size 243

The smallest 3-groups with non-trivial Bogomolov multiplier are of order 243,
and the three of them are isoclinic with Bogomolov multiplier the group
$\mathbb{Z}/3$. We chose to study the group

(6.17) $G:=(C_{9}\rtimes C_{9})\rtimes C_{3}$

which is defined by the presentation

(6.18) $G=\langle a,b,c\colon
a^{3}=c^{3},a^{9}=b^{9}=1,[a,b]=c^{8}b^{6},[b,c]=a^{3},[a,c]=b^{3}c^{6}\rangle.$

The left $C_{9}$ is generated by $c$, the middle $C_{9}$ by $b$ and the right
$C_{3}$ by $ab$, and their corresponding actions are

(6.19) $bcb^{-1}=c^{4},\ \ (ab)b(ab)^{-1}=c^{8}b^{7},\ \
(ab)c(ab)^{-1}=cb^{3}.$

This group corresponds to the small group

SmallGroup(243,30)

in the small groups library of GAP [GAP21].

The second page of the Lyndon-Hochschild spectral sequence has for terms

(6.20) $H^{2}(C_{9}\rtimes C_{9},\mathbb{C}^{*})^{C_{3}}=0,\ \
H^{1}(C_{3},H^{1}(C_{9}\rtimes C_{9},\mathbb{C}^{*}))=\mathbb{Z}/3,\ \
H^{2}(C_{3},\mathbb{C}^{*})=0$

where the middle term encodes the information of the Bogomolov multiplier.

Consider the surface $\Sigma_{2}$ of genus 2 as in (6.10) and define the
assignment

(6.21) $\displaystyle\Phi:\pi_{1}(\Sigma_{2})$ $\displaystyle\to(C_{9}\rtimes
C_{9})\rtimes C_{3}$ $\displaystyle x\mapsto a,\ $ $\displaystyle\ y\mapsto
b^{6},\ \ z\mapsto c,\ \ w\mapsto b,$

which induces a surjective homomorphism since $[a,b^{6}]=a^{3}$, $[c,b]=a^{6}$
and

(6.22) $\Phi([x,y][z,w])=[a,b^{6}][c,b]=1.$

The map $B\Phi:\Sigma_{2}\to B((C_{9}\rtimes C_{9})\rtimes C_{3})$ generates
the Bogomolov multiplier and therefore the surface
$\widetilde{\Sigma}:=(B\Phi)^{*}E((C_{9}\rtimes C_{9})\rtimes C_{3})$
generates the torsion subgroup of the equivariant oriented bordism group of
surfaces

(6.23) $\operatorname{Tor}_{\mathbb{Z}}{\Omega}_{2}^{SO,(C_{9}\rtimes
C_{9})\rtimes C_{3}}=\langle[\widetilde{\Sigma}]\rangle\cong\mathbb{Z}/3.$

In the unitary case we have

(6.24) $\operatorname{Tor}_{\mathbb{Z}}{\Omega}_{2}^{U,(C_{9}\rtimes
C_{9})\rtimes C_{3}}=\langle[\widetilde{\Sigma}]-[(C_{9}\rtimes C_{9})\rtimes
C_{3}\times\Sigma_{2}]\rangle\cong\mathbb{Z}/3.$

The surface $\widetilde{\Sigma}$ is a surface of genus 486 with a free action
of $(C_{9}\rtimes C_{9})\rtimes C_{3}$ which does not equivariantly bound.

## Conflict of interest

On behalf of all authors, the corresponding author states that there is no
conflict of interest.

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# T violation and the dark sector

R. Vilela Mendes
CMAFcIO, Universidade de Lisboa,
C6 - Campo Grande, 1749-016 Lisboa e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>https://label2.tecnico.ulisboa.pt/vilela/

###### Abstract

It is argued, as a working hypothesis, that ”normal” and dark matter
interactions can only be T and CP violating. One way to implement this idea is
to consider that time reversal in dark matter is implemented, not by an
antiunitary operator, but by an unitary operator. It is shown how this occurs
naturally in the context of complex spacetime with an extended symmetry group.

##  1 The interactions of dark matter

There is now extensive evidence for the existence in the Universe of matter
not seen by our instruments. Strong evidence comes from the spiral galaxy
rotation curves measured by visible and radio wavelengths, from the galaxy
clusters matter content being larger than expected from luminosity, from the
virial theorem, from velocity dispersion and gravitational lensing, as well as
from galaxy clusters dynamics, the large scale structure and the cosmic
microwave background radiation anisotropies, etc. Another current cosmological
mystery is the acceleration of the universe expansion, which may be associated
to an essentially constant density of dark energy. This may or may not be
associated to the dark matter phenomena. Or it may simply mean that Einstein’s
equation with the cosmological term is the right classical gravitational
equation. In any case matter in the dark sector, whatever it is, has so far
manifested itself only by gravitational interactions with ordinary matter.
What is so special about gravitational interactions?

A feature that might distinguish gravity from the other interactions at the
quantum level, is the absence of global symmetries. It is of course possible
to construct quantum field theory Lagrangians for matter and gravitational
fields which, at the perturbative level, preserve all symmetries. However it
has for a long time been conjectured [1] [2] that, in an actual quantum theory
of gravity, no global symmetries will be preserved. This conjecture which, in
the past, was mostly based on singularity and black hole physics was somewhat
strengthened in the AdS/CFT holographic context [3]. Admittedly AdS/CFT is not
the final quantum gravity theory, nevertheless it is, so far, the best
understood consistent framework.

Among the symmetries that would not be preserved in quantum gravity is time-
reversal ([3], Sec.7 and Th. 4.2). This suggests as a possible interpretation
(or a working hypothesis) for the elusive nature of the interactions of dark
with ordinary matter, that they must be $T-$violating interactions. If $CPT$
is conserved, at least locally, this might also imply a $CP-$violating nature
for the interactions of dark and ordinary matter.

That only $T$ and $CP$ violating interactions are associated to dark matter is
a general hypothesis that might be realized in different ways. Here a
particular scenario will be explored, in a complex spacetime context, where
normal and dark matter belong to different superselection sectors, having
different symmetry properties.

##  2 T-violation in a complex spacetime scenario

Several authors have studied extensions of gravitational theory by
complexifying the functions that appear in the classical equations. Already
Einstein in 1945 [4] generalized his theory of gravitation by considering a
metric with complex components and Hermitian symmetry, others [5] have
complexified the tangent bundle over the real four-dimensional spacetime
manifold, constructed solutions of the complex Einstein equation [6], or
provided a complex manifold description of massless particles [7] [8], etc.

A more radical approach consisted in assuming that the spacetime manifold
itself should be parametrized but complex coordinates. Several such
explorations have been carried out, see for example [9] [10] and for a more
complete set of references and perspectives see Ref.[11]. Of course, a
question that is left open is, if the spacetime dimensions are indeed indexed
by complex numbers, why we do not feel it in our everyday experience.

Recently this question was revisited [12] for spacetimes where coordinates
take values in higher normed division algebras, complex, quaternion or
octonion. In this extended spacetime, any four independent directions may be
chosen as a basis for a real vector space, that is, for a real slice of the
complex spacetime. Using as consistency condition that the symmetry groups of
the extended spacetime must reduce to the real Lorentz and Poincaré groups in
each real slice, one obtains, for the homogeneous group of the extended
spacetime, the condition

$\Lambda^{{\dagger}}G\Lambda=G$ (1)

that is, a $U\left(1,3,\mathbb{K}\right)$ group,
$\mathbb{K=C},\mathbb{Q},\mathbb{O}$, with $G$ the metric
$\left(1,-1,-1,-1\right)$. For $\mathbb{K=C}$ this has been called the complex
Lorentz group with real metric [13]. It is a $16-$parameter group different
from the $12-$parameter group

$\Lambda^{T}G\Lambda=G$ (2)

used in the analytic continuations of the S-matrix. Nevertheless it is the
condition (1) that insures an identical group in each one of the real slices.

The nature of the matter states, that is implied by this group structure, was
obtained by studying the representations of the semidirect group

$T_{4}\circledS U\left(1,3,\mathbb{C}\right)$ (3)

$T_{4}$ being the complex spacetime translations. The conclusions were:

\- Half-integer spin states are not elementary states for the full complex
group (3). That is, half-integer elementary states, that are irreducible
representations of the real groups, cannot communicate between different real
slices, in the sense that they cannot be rotated from one real slice to
another.

\- Integer-spin states are elementary states of the full group. However these
states have a particular nature when the full group, including reflections, is
taken into account. In the complex Lorentz group (1) parity and time reversal
are continuously connected to the identity. Therefore, in faithful continuous
norm-preserving representations, both parity and time reversal must be
implemented by unitary operators. On the other hand one knows that, in each
real slice, positivity of energy requires an antiunitary time reversal
operator.

In conclusion: In the complex spacetime scenario there are three types of
states: half-integer spin states that are confined to the real slices,
integer-spin positive energy states that are also only associated to the real
slices and finally integer-spin states, with unitary time reversal operator,
that can communicate between real slices. Because of its association to time
reversal violation, this last type will be denoted here as chronoparticles.

The conclusions are qualitatively similar for the normed division algebras
$\mathbb{Q}$ and $\mathbb{O}$ [12]. Also notice that although the extended
Lorentz group does not contain half-integer elementary states, it might still
operate on these states. However, rather than rotating them between real
slices it generates a multiplicity of states (see [12], Appendix D).

It is worthwhile to recall the energy positivity argument that leads to the
antiunitarity of the time reversal operator. When time-reversal is a conserved
symmetry, invariance of the Schrödinger equation

$i\partial_{t}\psi=H\psi$

implies that either the $T$ operator is antiunitary or it anticommutes with
the Hamiltonian. However, if it is unitary and anticommutes with the
Hamiltonian, then

$\left(T\psi,HT\psi\right)=-\left(\psi,H\psi\right)$

and there are negative energy states.

Chronoparticles, as defined above, because they connect different real slices
may establish interactions between matter in different real slices. Because of
the argument above, they might behave as negative energy particles from the
point of view of physics in the real slices. However, the above reasoning does
not really apply because $T$ is not conserved in their interactions.

Because of the unitarity of the $T$ operator, chronoparticles and ordinary
matter in the real slices are in different superselection sectors and in their
interactions $T$ is not a symmetry. The argument [12] goes as follows:

Let $\psi_{o}\in V_{o}$ be a state in the (real slice) ordinary matter space
and $\psi_{c}\in V_{c}$, a chronoparticle (dark matter?) state. There is a
superselection rule operating between these two types of spaces. Consider a
linear superposition of two of these states

$\Phi=\alpha\psi_{o}+\beta\psi_{c}$

with $\alpha,\beta$ real numbers. Now $\Phi$ and $e^{i\theta}\Phi=\alpha
e^{i\theta}\psi_{o}+\beta e^{i\theta}\psi_{c}$ belong to the same ray and
therefore should represent the same state. Applying the time reversal operator
to both $\Phi$ and $e^{i\theta}\Phi$

$\displaystyle T\Phi$ $\displaystyle=$ $\displaystyle\alpha T\psi_{o}+\beta
T\psi_{c}$ $\displaystyle Te^{i\theta}\Phi$ $\displaystyle=$
$\displaystyle\alpha e^{-i\theta}T\psi_{o}+\beta e^{i\theta}T\psi_{c}$ (4)

$T\Phi$ and $Te^{i\theta}\Phi$ belong to different rays, hence $T$ does not
establish a ray correspondence in $V_{o}\oplus V_{c}$ unless $\alpha=0$ or
$\beta=0$, that is, $V_{o}$ and $V_{c}$ belong to different superselection
sectors.

The fact that the states are in different superselection sectors does not mean
that they cannot interact. Whereas the superselection rule result concerns the
structure of the direct sum $V_{o}\oplus V_{c}$, the nature of the
interactions depends on the way the group transformations operate in the
tensor product $V_{o}\otimes V_{c}$. Consider now the $T$ operation acting on
$V_{o}\otimes V_{c}$ and compute its action on a matrix element

$\left(T\left(\psi_{o}^{(1)}\otimes\psi_{c}^{(1)}\right),T\left(\psi_{o}^{(2)}\otimes\psi_{c}^{(2)}\right)\right)=\left(U_{o}\psi_{o}^{(2)}\otimes
U_{c}\psi_{c}^{(1)},U_{o}\psi_{o}^{(1)}\otimes U_{c}\psi_{c}^{(2)}\right)$ (5)

where $U_{o}$ and $U_{c}$ are unitary operators, acting on the degrees of
freedom of the states. From (5) one concludes that there is no choice of
phases that can make $T$ a global unitary or anti-unitary operator in the
tensor product space. Therefore, by Wigner’s theorem, $T$ cannot be a symmetry
in $V_{o}\otimes V_{c}$.

How a particle with a unitary implementation of the $T$ operation may induce
symmetry violations in its interactions with ordinary matter is simple to
illustrate. Let the Fourier decompositions of the quantum fields $\Phi_{o}$
and $\Phi_{c}$, associated to ordinary matter and to an integer-spin
chronoparticle be

$\displaystyle\Phi_{o}\left(x^{0},\overrightarrow{x}\right)$ $\displaystyle=$
$\displaystyle\sum_{\mu}\int
d^{3}k\frac{\epsilon\left(k,\mu\right)}{2\omega\left(k\right)}\left\\{a_{o}\left(k,\mu\right)e^{-ik.x}+b_{o}^{{\dagger}}\left(k,\mu\right)e^{ik.x}\right\\}$
$\displaystyle\Psi_{c}\left(x^{0},\overrightarrow{x}\right)$ $\displaystyle=$
$\displaystyle\sum_{\mu}\int
d^{3}k\frac{\epsilon\left(k,\mu\right)}{2\omega\left(k\right)}\left\\{a_{c}\left(k,\mu\right)e^{-ik.x}+b_{c}^{{\dagger}}\left(k,\mu\right)e^{ik.x}\right\\}$
(6)

$\mu$ being an helicity label and $\epsilon\left(k,\mu\right)$ the
polarization vectors.

With

$\displaystyle P\Phi\left(x^{0},\overrightarrow{x}\right)P^{-1}$
$\displaystyle=$
$\displaystyle\eta_{P}\Phi\left(x^{0},-\overrightarrow{x}\right)$
$\displaystyle T\Phi\left(x^{0},\overrightarrow{x}\right)T^{-1}$
$\displaystyle=$
$\displaystyle\eta_{T}\Phi\left(-x^{0},\overrightarrow{x}\right)$
$\displaystyle C\Phi\left(x^{0},\overrightarrow{x}\right)C^{-1}$
$\displaystyle=$
$\displaystyle\eta_{C}\Phi^{{\dagger}}\left(x^{0},\overrightarrow{x}\right)$
(7)

for both ordinary matter and chronoparticle fields, one obtains

$\displaystyle Pa_{o}\left(k^{0},\overrightarrow{k}\right)P^{-1}$
$\displaystyle=$
$\displaystyle\eta_{P}a_{o}\left(k^{0},-\overrightarrow{k}\right)$
$\displaystyle Pb_{o}^{{\dagger}}\left(k^{0},\overrightarrow{k}\right)P^{-1}$
$\displaystyle=$
$\displaystyle\eta_{P}b_{o}^{{\dagger}}\left(k^{0},-\overrightarrow{k}\right)$
$\displaystyle Ta_{o}\left(k^{0},\overrightarrow{k}\right)T^{-1}$
$\displaystyle=$
$\displaystyle\eta_{T}a_{o}\left(k^{0},-\overrightarrow{k}\right)$
$\displaystyle Tb_{o}^{{\dagger}}\left(k^{0},\overrightarrow{k}\right)T^{-1}$
$\displaystyle=$
$\displaystyle\eta_{T}b_{o}^{{\dagger}}\left(k^{0},-\overrightarrow{k}\right)$
$\displaystyle Ca_{o}\left(k^{0},\overrightarrow{k}\right)C^{-1}$
$\displaystyle=$
$\displaystyle\eta_{C}b_{o}\left(k^{0},\overrightarrow{k}\right)$
$\displaystyle Cb_{o}^{{\dagger}}\left(k^{0},\overrightarrow{k}\right)T^{-1}$
$\displaystyle=$
$\displaystyle\eta_{C}a_{o}^{{\dagger}}\left(k^{0},\overrightarrow{k}\right)$
(8)

for ordinary matter, with $P$ and $C$ unitary and $T$ antiunitary and

$\displaystyle Pa_{c}\left(k^{0},\overrightarrow{k}\right)P^{-1}$
$\displaystyle=$
$\displaystyle\eta_{P}a_{c}\left(k^{0},-\overrightarrow{k}\right)$
$\displaystyle Pb_{c}^{{\dagger}}\left(k^{0},\overrightarrow{k}\right)P^{-1}$
$\displaystyle=$
$\displaystyle\eta_{P}b_{c}^{{\dagger}}\left(k^{0},-\overrightarrow{k}\right)$
$\displaystyle Ta_{c}\left(k^{0},\overrightarrow{k}\right)T^{-1}$
$\displaystyle=$
$\displaystyle\eta_{T}b_{c}^{{\dagger}}\left(k^{0},-\overrightarrow{k}\right)$
$\displaystyle Tb_{c}^{{\dagger}}\left(k^{0},\overrightarrow{k}\right)T^{-1}$
$\displaystyle=$
$\displaystyle\eta_{T}a_{c}\left(k^{0},-\overrightarrow{k}\right)$
$\displaystyle Ca_{c}\left(k^{0},\overrightarrow{k}\right)C^{-1}$
$\displaystyle=$
$\displaystyle\eta_{C}a_{c}^{{\dagger}}\left(k^{0},\overrightarrow{k}\right)$
$\displaystyle Cb_{c}^{{\dagger}}\left(k^{0},\overrightarrow{k}\right)T^{-1}$
$\displaystyle=$
$\displaystyle\eta_{C}b_{c}\left(k^{0},\overrightarrow{k}\right)$ (9)

for chronoparticle operators, with $P$ and $T$ unitary and $C$ antiunitary
(for $CPT$ invariance). For simplicity the helicity labels were dropped.

Now it is clear in which way chronoparticles violate $T-$invariance. Let the
interaction term of ordinary matter with chronoparticles be

$\mathcal{L}_{I}\sim\Phi_{o}^{{\dagger}}\Phi_{o}\Phi_{c}$

Then, the simple vertex

$\left\langle\phi_{o\left(\overrightarrow{k}+\overrightarrow{v}\right)}\left|a_{o}^{{\dagger}}\left(\overrightarrow{k}+\overrightarrow{v}\right)a_{c}\left(\overrightarrow{v}\right)a_{o}\left(\overrightarrow{v}\right)\right|\phi_{o\left(\overrightarrow{k}\right)}\psi_{c\left(\overrightarrow{v}\right)}\right\rangle$

where an ordinary matter state $\phi_{o\left(\overrightarrow{k}\right)}$
absorbs a chronoparticle $\psi_{c\left(\overrightarrow{v}\right)}$ leading to
$\phi_{o\left(\overrightarrow{k}+\overrightarrow{v}\right)}$, becomes, under
the $T-$operation

$\left\langle\phi_{o\left(-\overrightarrow{k}\right)}\psi_{c\left(-\overrightarrow{v}\right)}^{{\dagger}}\left|a_{o}^{{\dagger}}\left(-\overrightarrow{k}\right)b_{c}^{{\dagger}}\left(-\overrightarrow{v}\right)a_{o}\left(-\overrightarrow{k}-\overrightarrow{v}\right)\right|\phi_{o\left(-\overrightarrow{k}-\overrightarrow{v}\right)}\right\rangle$

rather than

$\left\langle\phi_{o\left(-\overrightarrow{k}\right)}\psi_{c\left(-\overrightarrow{v}\right)}\left|a_{o}^{{\dagger}}\left(-\overrightarrow{k}\right)a_{c}^{{\dagger}}\left(-\overrightarrow{v}\right)a_{o}\left(-\overrightarrow{k}-\overrightarrow{v}\right)\right|\phi_{o\left(-\overrightarrow{k}-\overrightarrow{v}\right)}\right\rangle$

That is, the $T-$operation leads to a process different from the usual time-
reversed one and therefore does not establish an identification of the
couplings for the direct and the reversed processes. Notice that in this
example, because $C$ is chosen to be antiunitary for chronoparticles, one has
$CT$ and $P$ invariance but $CP$ and $T$ violation. Chronoparticles may induce
$T-$violating interactions, either by establishing interactions between matter
particles in different real slices or in each real slice by virtual
chronoparticle interactions.

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