# Cobordism of $G$-manifolds

Jack Carlisle  Department of Mathematics, the University of Notre Dame, South
Bend, Indiana 46616<EMAIL_ADDRESS>
(Date: June 30, 2023)

###### Abstract.

We survey some results in the field of equivariant cobordism. In particular,
we use methods from equivariant stable homotopy theory to calculate the
unoriented $C_{2}$-equivariant bordism ring $\Omega^{C_{2}}_{*}$, which was
originally calculated by Alexander using other methods. Our proof method
generalizes well to other settings, such as equivariant complex cobordism, and
affords a formal group theoretic interpretation of Alexander’s calculation.

###### Key words and phrases:

Equivariant cobordism

###### 2020 Mathematics Subject Classification:

Primary 55-02, 55N22, 55P91

###### Contents

  1. 1 Representing cobordism of $G$-manifolds
  2. 2 The Tate square for $\Omega^{C_{2}}_{*}$
  3. 3 Generators and relations for $\Omega^{C_{2}}_{*}$
  4. 4 The extended cobordism ring $\Omega^{C_{2}}_{\diamond}$

## Introduction

Suppose $M_{1}$ and $M_{2}$ are smooth, closed manifolds. We say $M_{1}$ is
cobordant to $M_{2}$ if there exists a smooth manifold $W$ whose boundary is

$\partial W\cong M_{1}\sqcup M_{2}.$

Equivalence classes of manifolds under this equivalence relation form the
unoriented cobordism ring

$\Omega_{*}=\dfrac{\\{\text{Smooth closed manifolds}\\}}{\text{cobordism}}.$

While there are important variants of this construction such as oriented
cobordism and complex cobordism, we restrict our attention to the unoriented
case. The ring $\Omega_{*}$ is calculable because it arises as the coefficient
ring of a spectrum, namely the Thom spectrum $MO$. As an orthogonal spectrum,
the component spaces

$(MO)_{n}=\text{Thom}\left(\gamma\to BO(n)\right)$

of $MO$ are the Thom complexes of universal vector bundles. The Pontrjagin-
Thom construction determines a ring map

${\Omega_{*}}$${MO_{*},}$

which turns out to be an isomorphism. This identification allows one to
calculate $\Omega_{*}$ using techniques from stable homotopy theory. More
precisely, Thom [T54] proved that the unoriented cobordism ring $\Omega_{*}$
is given by

$\Omega_{*}=\mathbb{F}_{2}[x_{i}:i\neq 2^{n}-1],$

for certain classes $x_{i}$ of degree $i$.

Suppose now that $G$ is a finite group. We wish to understand cobordism in the
$G$-equivariant setting. The appropriate analogue of $\Omega_{*}$ is the
unoriented $G$-cobordism ring

$\Omega^{G}_{*}=\dfrac{\\{\text{Smooth closed }G\text{ -
manifolds}\\}}{G\text{ - cobordism}}.$

In order to calculate $\Omega^{G}_{*}$, we hope to represent this ring by a
$G$-spectrum. There is a natural $G$-equivariant analogue of $MO$, which
denote $MO_{G}$. As an orthogonal $G$-spectrum, the component spaces

$(MO_{G})_{n}=\text{Thom}\left(\gamma_{G,n}\to BO_{G}(n)\right)$

of $MO_{G}$ are the Thom complexes of universal $G$-equivariant vector
bundles. The $G$-equivariant Pontrjagin-Thom construction (which we explicate
in section 2) determines a ring map

${\Omega^{G}_{*}}$${MO^{G}_{*},}$

but if $G$ is a non-trivial group, then this map is not an isomorphism. This
is related to the fact that transversality is not a generic property in the
$G$-equivariant setting [W69].

While the unoriented $G$-cobordism ring is not represented by the expected
$G$-spectrum $MO_{G}$, it is represented by another $G$-spectrum, which we
suggestively denote $\Omega_{G}$.111Some other sources use the notation
$mO_{G}$ for the spectrum $\Omega_{G}$ As an orthogonal $G$-spectrum, the
component spaces

$(\Omega_{G})_{n}=\text{Thom}\left(\gamma_{n}\to BO(n)\right)$

of $\Omega_{G}$ are the Thom complexes of universal (non-equivariant) vector
bundles. While $G$ acts trivially on $(\Omega_{G})_{n}$ for each $n\geq 0$,
the component spaces $(\Omega_{G})_{V}$ for $G$ a non-trivial
$G$-representation carry a non-trivial $G$-action, so $\Omega_{G}$ is not a
trivial $G$-spectrum. The component space $(\Omega_{G})_{V}$ is the Thom
complex of a certain Tautological vector bundle $\gamma_{V}$ which is, in a
precise sense, intermediate between the universal vector bundle over $BO(n)$,
and the universal $G$-equivariant vector bundle over $BO_{G}(n)$.

The identification of $\Omega^{G}_{*}$ with the coefficients of a $G$-spectrum
allows one to calculate $\Omega^{G}_{*}$ using techniques from equivariant
stable homotopy theory. We will use such techniques to calculate the
$C_{2}$-cobordism ring $\Omega^{C_{2}}_{*}$, where $C_{2}$ is the group of
order $2$ (see Theorem 3.5). The ring $\Omega^{C_{2}}_{*}$ was originally
calculated by Alexander [A72] using other methods. Our proof method is of
independent interest, and generalizes well to other settings, such as
equivariant complex cobordism [C21]. In addition to our calculation of
$\Omega^{C_{2}}_{*}$, we calculate a presentation of the extended cobordism
ring $\Omega^{C_{2}}_{\diamond}$ (see Theorem 4.3), of which
$\Omega^{C_{2}}_{*}$ is a subring, and of which $MO^{C_{2}}_{*}$ is a
localization. This provides algebraic insight into a theorem of Brocker and
Hook [BH72].

##  1\. Representing cobordism of $G$-manifolds

Suppose $G$ is a finite group. Let $\Omega^{G}_{*}$ denote the unoriented
$G$-cobordism ring. Elements of $\Omega^{G}_{*}$ are represented by smooth
compact $G$-manifolds, and two such manifolds $M_{1}$ and $M_{2}$ represent
the same class in $\Omega^{G}_{*}$ if there is a $G$-manifold $W$ whose
boundary is

$\partial W\cong M_{1}\sqcup M_{2}.$

The addition in $\Omega^{G}_{*}$ is determined by

$[M_{1}]+[M_{2}]=[M_{1}\sqcup M_{2}],$

and the product in $\Omega^{G}_{*}$ is determined by

$[M_{1}][M_{2}]=[M_{1}\times M_{2}],$

where $G$ acts diagonally on $M_{1}\times M_{2}$.

The bordism rings $\Omega^{H}_{*}$ for $H\leq G$ assemble to form a $G$-Mackey
functor. In particular, this means that for any subgroup inclusion $H\leq K$,
we have restriction and transfer maps

${\Omega^{K}_{*}}$${\Omega^{H}_{*},}$$\scriptstyle{\text{res}^{K}_{H}}$
${\Omega^{H}_{*}}$${\Omega^{K}_{*}.}$$\scriptstyle{\text{tr}^{K}_{H}}$

The restriction $\text{res}^{K}_{H}$ send the class of a $K$-manifold $M$ to
the class of its underlying $H$-manifold. The transfer $\text{tr}_{H}^{K}$
sends the class of a $H$-manifold $M$ to the class of the induced $K$-manifold
$K\times_{H}M$. We write $\underline{\Omega}^{G}_{*}$ when we wish to
emphasize the Mackey functor structure of geometric cobordism.

Next, we define the $G$-spectrum $\Omega_{G}$ which represents geometric
cobordism. We use orthogonal $G$-spectra as our model of spectra, and refer
the interested reader to [HHR16], section 2.2.2, for background on orthogonal
$G$-spectra. If $V$ is an orthogonal $G$-representation, let
$BO(V)=\text{Gr}_{|V|}(V\oplus\mathbb{R}^{\infty})$ denote the Grassmannian of
$|V|$-dimensional subspaces of $V\oplus\mathbb{R}^{\infty}$, where $C_{2}$
acts trivially on $\mathbb{R}^{\infty}$. This $G$-space carries a tautological
$G$-equivariant vector bundle $\gamma_{V}$, and as an orthogonal $G$-spectrum,
we define $\Omega_{G}$ by

$(\Omega_{G})_{V}=\text{Thom}\left(\gamma_{V}\to BO(V)\right).$

In order to define our ring homomorphism
$\Omega^{G}_{*}\to\pi^{G}_{*}(\Omega_{G})$, we review the $G$-equivariant
Pontrjagin-Thom construction.

Suppose we are given a class $[M]\in\Omega^{G}_{n}$ which is represented by an
$n$-dimensional $G$-manifold $M$. Choose a $G$-equivariant embedding of $M$
into an orthogonal $G$-representation of the form $\mathbb{R}^{n}\oplus V$.
Let $S^{n+V}$ denote the one-point compactification of $\mathbb{R}^{n}\oplus
V$. By [W69], we may choose a $G$-equivariant tubular neighborhood $N$ of $M$
in $\mathbb{R}^{n}\oplus V$, so that the quotient

$S^{n+V}/(S^{n+V}\setminus N)\cong\text{Thom}(\nu\to M)$

is $G$-equivariantly homeomorphic to the Thom complex of the normal bundle
$\nu$ of $M$ in $\mathbb{R}^{n}\oplus V$. The normal bundle $\nu$ is equipped
with a bundle map to the tautological vector bundle over
$\text{Gr}_{|V|}(V\oplus\mathbb{R}^{n})$, which itself maps to the
tautological vector bundle over $BO(V)$. The corresponding composite

${S^{V+n}}$${\dfrac{S^{V+n}}{S^{V+n}\setminus N}}$${\text{Thom}(\nu\to
M)}$${\text{Thom}\left(\gamma_{V}\to
BO(V)\right)}$${(\Omega_{G})_{V}}$$\scriptstyle{\cong}$$\scriptstyle{=}$

represents a class in $\pi_{n}^{G}(\Omega_{G})$.

Having explicated the equivariant Pontryagin-Thom construction, we may now
state the following theorem, which identifies the coefficient ring of the
$G$-spectrum $\Omega_{G}$ with the (geometrically defined) $G$-cobordism ring
$\Omega^{G}_{*}$.

######  Theorem 1.1.

([S18], Theorem 6.2.33) For any finite group $G$, the assignment

$[M]\mapsto[S^{n+V}\to(\Omega_{G})_{V}],$

as defined above, determines a ring isomorphism

${\Omega^{G}_{*}}$${\pi^{G}_{*}(\Omega_{G}).}$$\scriptstyle{\cong}$

In fact, this holds true at the level of $H$-fixed points for any $H\leq G$,
so the construction above determines an isomorphism of $G$-Mackey functors

${\underline{\Omega}^{G}_{*}}$${\underline{\pi}_{*}(\Omega_{G}).}$$\scriptstyle{\cong}$

##  2\. The Tate square for $\Omega^{C_{2}}_{*}$

In this section we begin our calculation of $\Omega^{C_{2}}_{*}$. Since, we’ve
identified the $C_{2}$-cobordism ring with the coefficients of the
$C_{2}$-spectrum $\Omega_{C_{2}}$, we may use techniques from equivariant
stable homotopy to calculate $\Omega^{C_{2}}_{*}$. For background on
equivariant stable homotopy theory, see [LMS86]. Our primary tool is the Tate
square associated to a $C_{2}$-spectrum $E_{C_{2}}$, which, at the level of
coefficients, has the form

${E^{C_{2}}_{*}}$${E^{\Phi C_{2}}_{*}}$${E^{hC_{2}}_{*}}$${E^{tC_{2}}_{*}.}$

Here, $E^{C_{2}}_{*}$, $E^{hC_{2}}_{*}$, $E^{\Phi C_{2}}_{*}$, and
$E^{tC_{2}}_{*}$ are called the categorical, homotopy, geometric, and Tate
fixed points of $E_{C_{2}}$, respectively. Note that for
$E_{C_{2}}=\Omega_{C_{2}}$, we have identified the categorical fixed points
$\Omega^{C_{2}}_{*}$ of $\Omega_{C_{2}}$ with the geometrically defined
cobordism ring of $C_{2}$-manifolds, so our notation is consistent.

We refer the interested reader to [GM95] for a more expert understanding of
the Tate square. For our purposes, it suffices to understand that the other
flavors of fixed points in this diagram are often more easily computable than
$E^{C_{2}}_{*}$, and that if the map

$E^{hC_{2}}_{*}\oplus E^{\Phi C_{2}}_{*}\to E^{tC_{2}}_{*}$

is surjective, then this square is a pullback (see [K98] Lemma 2.1). When
$E_{C_{2}}=\Omega_{C_{2}}$, we will see by direct computation that this
condition is satisfied, so that the Tate square

${\Omega^{C_{2}}_{*}}$${\Omega^{\Phi
C_{2}}_{*}}$${\Omega^{hC_{2}}_{*}}$${\Omega^{tC_{2}}_{*},}$

is a pullback of rings. We begin our calculation by identifying the homotopy,
geometric, and Tate fixed points of $\Omega_{C_{2}}$.

  * •

Homotopy fixed points: If $E_{C_{2}}$ is a $C_{2}$-spectrum, then the homotopy
fixed points of $E_{C_{2}}$ are defined by

$E^{hC_{2}}_{*}=\pi_{*}^{C_{2}}(F(EC_{2+},E_{C_{2}}))$

where $F(EC_{2+},E_{C_{2}})$ denotes the spectrum of maps from the free, non-
equivariantly contractible $C_{2}$-space $EC_{2}$ to the spectrum $E_{C_{2}}$.
The skeletal filtration of the $C_{2}$-space $EC_{2}$ determines a spectral
sequence which converges to $E^{hC_{2}}_{*}$, called the homotopy fixed point
spectral sequence. If $E_{C_{2}}$ is a split222 We say a $C_{2}$-spectrum
$E_{C_{2}}$ is split if there is a map of non-equivariant spectra
$(E_{C_{2}})^{e}\to(E_{C_{2}})^{C_{2}}$ such that the composite
$(E_{C_{2}})^{e}\to(E_{C_{2}})^{C_{2}}\to(E_{C_{2}})^{e}$ is homotopic to the
identity, Here, $(E_{C_{2}})^{e}$ is the non-equivariant spectrum underlying
$E_{C_{2}}$, and $(E_{C_{2}})^{C_{2}}$ is the fixed point spectrum of
$E_{C_{2}}$. $C_{2}$-spectrum, then the homotopy fixed point spectral sequence
collapses, which results in an isomorphism

$E^{hC_{2}}_{*}\cong H^{-*}(C_{2},E_{*})$

between the homotopy fixed points of $E_{C_{2}}$, and the group cohomology of
$C_{2}$ with coefficients in the (trivial) $C_{2}$-module $E_{*}$. The
$C_{2}$-spectrum $\Omega_{C_{2}}$ is known to be split [GM95], which leads to
the identification

$\displaystyle\Omega^{hC_{2}}_{*}$ $\displaystyle\cong
H^{-*}(C_{2};\Omega_{*})$ $\displaystyle=\Omega_{*}[[e]],$

where $|e|=-1$. The element $e\in\Omega^{hC_{2}}_{*}$ is called the Euler
class associated to the sign representation $\sigma$ of $C_{2}$, and plays an
important role in the theory of equivariant formal group laws as defined in
[CGK00], and in equivariant stable homotopy theory more generally.

  * •

Tate fixed points: The Tate fixed points may be obtained from the homotopy
fixed points by inverting the Euler class $e$, which gives

$\Omega^{tC_{2}}_{*}=\Omega_{*}((e)),$

the ring of Laurent series in $e$ with finitely many negative powers of $e$.

  * •

Geometric fixed points: The geometric fixed points of $\Omega_{C_{2}}$, which
were originally calculated by tom Dieck [tD70], are given by

$\Omega^{\Phi C_{2}}_{*}=\bigoplus_{k=0}^{\infty}\Omega_{*-k}(BO(k)).$

Elements of this ring are cobordism classes of pairs $(F,\xi)$ of a manifold
$F$ equipped with a vector bundle $\xi\to F$. The ring $\Omega^{\Phi
C_{2}}_{*}$ has a simple algebraic description. The $\Omega$ homology of
$BO(k)$ is known to be

$\Omega_{*}(BO(k))=\Omega_{*}\\{\beta_{i_{1}}\dots\beta_{i_{k}}:0\leq
i_{1}\leq\dots\leq i_{k}\\},$

where

$\beta_{i_{1}}\dots\beta_{i_{k}}=[\mathbb{R}P^{i_{1}}\times\dots\times\mathbb{R}P^{i_{k}},\gamma\times\cdots\times\gamma],$

and where $\gamma\to\mathbb{R}P^{n}$ denotes the tautological vector bundle
(see, for instance [M74]). If we set

$d_{i}=\beta_{i}=[\mathbb{R}P^{i},\gamma]\in\Omega_{i}(BO(1)),$

then

$\displaystyle\Omega^{\Phi C_{2}}_{*}$ $\displaystyle\cong\bigoplus_{k\geq
0}\Omega_{*-k}(BO(k))$ $\displaystyle=\Omega_{*}[d_{0},d_{1},d_{2},\dots].$

We’ve identified the geometric, homotopy and Tate fixed points of
$\Omega^{C_{2}}_{*}$, and our next step is to identify the maps in this
diagram. We begin with the maps whose target is $\Omega_{*}((e))$. As we
mentioned before, the map $\Omega_{*}[[e]]\to\Omega_{*}((e))$ is localization
at $e$, so we turn our attention to the map
$\Omega_{*}[d_{0},d_{1},\dots]\to\Omega_{*}((e))$. In order to describe this
map, we review the necessary background on formal group laws.

Recall that a formal group law over a commutative ring $A$ is a power series
$F(y,z)\in A[[y,z]]$ satisfying an associativity, unitality, and commutativity
axiom. If $A$ is an $\mathbb{F}_{2}$-algebra and the formal group law $F$
satisfies $F(x,x)=0$, we say $F$ is a $[2]$-torsion formal group law over $A$.
These arise in homotopy theory in the following way. Consider the map

$\otimes:\mathbb{R}P^{\infty}\times\mathbb{R}P^{\infty}\to\mathbb{R}P^{\infty}$

which classifies the tensor product of line bundles. If $E$ is an $MO$-algebra
spectrum, then $E^{*}(\mathbb{R}P^{\infty})$ is equal to $E_{*}[[x]]$,
$E^{*}(\mathbb{R}P^{\infty}\times\mathbb{R}P^{\infty})$ is equal to
$E_{*}[[y,z]]$, and the image of $x$ under $E^{*}(\otimes)$ is a $[2]$-torsion
formal group law $F(y,z)\in E_{*}[[y,z]]$ over $E_{*}$. In particular, this
construction yields a $[2]$-torsion formal group law

$F_{MO}(y,z)=\sum_{i,j\geq 0}a_{i,j}y^{i}z^{j}$

over $MO_{*}$, and Quillen [Q69] proved that this is in fact the universal
$[2]$-torsion formal group law. Having made the necessary definitions, we may
now identify the map $\Omega^{\Phi C_{2}}_{*}\to\Omega^{tC_{2}}_{*}$.

######  Lemma 2.1.

The composite

${\phi\colon\Omega_{*}[d_{0},d_{1},d_{2},\dots]}$${\Omega^{\Phi
C_{2}}_{*}}$${\Omega^{tC_{2}}_{*}}$${\Omega((e))}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

is given by

$\phi(d_{i})=\sum_{j\in\mathbb{Z}}c_{i,j}e^{j}=\text{coeff}\left(x^{j},\frac{1}{F_{MO}(e,x)}\right),$

where

$\dfrac{1}{F_{MO}(e,y)}=\sum_{i\geq
0}\sum_{j\in\mathbb{Z}}c_{i,j}y^{i}e^{j}\in\Omega_{*}((e))[[y]]$

is the multiplicative inverse of the universal $[2]$-torsion formal group law
$F_{MO}(e,y)=\sum a_{i,j}e^{i}y^{j}$.

###### Proof.

Write

$\Omega_{*}(BO)=\Omega_{*}[\beta_{1},\beta_{2},\dots]$

where $\beta_{i}=[\mathbb{R}P^{i},\gamma]$. The map $\phi$ factors as the
composite

$\bigoplus_{k\geq
0}\Omega_{*-k}(BO(k))\to\bigoplus_{k\in\mathbb{Z}}\Omega_{*-k}(BO)\to\bigoplus_{k\in\mathbb{Z}}\Omega_{*-k}(BO)$

where the first map is induced by the inclusions $BO(k)\to BO$, and the second
map is induced by the endomorphism $\iota:BO\to BO$ which classifies inverses
of stable vector bundles. We know that the map

$\bigoplus_{k\in\mathbb{Z}}\Omega_{*-k}(BO)=\Omega_{*}[e^{\pm
1},\beta_{1},\beta_{2},\dots]\to\Omega_{*}((e))$

is defined so that the induced map

$\Omega_{*}[e^{\pm 1},\beta_{1},\beta_{2},\dots][[y]]\to\Omega_{*}((e))[[y]]$

satisfies

$e(1+\beta_{1}y+\beta_{2}y^{2}+\dots)\mapsto F_{MO}(e,y),$

where $F_{MO}$ is the universal $2$-torsion formal group law. This is the
unoriented analogue of a result proved by Kriz in the complex case (see
[K98]). Set

$\bar{\beta}_{i}=[\mathbb{R}P^{i},-\gamma],$

so that the antipode on the Hopf algebra $\Omega_{*}(BO)$ is given by
$\beta_{i}\mapsto\bar{\beta}_{i}$. It follows that the map

$\Omega_{*}[e^{\pm 1},\beta_{1},\beta_{2},\dots][[y]]\to\Omega_{*}((e))[[y]]$

satisfies

$e^{-1}(1+\bar{\beta}_{1}y+\bar{\beta}_{2}y^{2}+\dots)\cdots\mapsto\dfrac{1}{F_{MO}(e,y)}.$

We conclude by observing that
$\nu\mid_{\mathbb{R}P^{i}}^{\mathbb{R}P^{i+1}}=\gamma$, so that
$\iota_{*}(d_{i})=e^{-1}\bar{\beta}_{i}$, and so our map satisfies

$d_{0}+d_{1}y+d_{2}y^{2}+\dots\mapsto\dfrac{1}{F_{MO}(e,y)}.$

∎

Next, we need to identify the maps whose domain is $\Omega^{C_{2}}_{*}$. We
will give a geometric description of each of these. We start with the
geometric fixed points $\Omega^{\Phi C_{2}}_{*}$, whose elements are cobordism
classes of pairs $(F,\xi)$ of a (non-equivariant) manifold $F$ equipped with a
vector bundle $\xi\to F$. In these terms, the map

${\Omega^{C_{2}}_{*}}$${\Omega^{\Phi C_{2}}_{*}}$

is given by the formula

${\left[M\right]}$${\left[M^{C_{2}},\nu\mid_{M^{C_{2}}}^{M}\right]}$

where $M^{C_{2}}$ is the fixed point submanifold of the $C_{2}$-manifold $M$,
and where $\nu\mid_{M^{C_{2}}}^{M}$ is the normal bundle of $M^{C_{2}}$ in
$M$. The map $\Omega^{C_{2}}_{*}\to\Omega_{*}[[e]]$ may be described in terms
of the Conner-Floyd operation $\Gamma$, which we review now (see [CF64] for
more details). Let $S(1+\sigma)$ denote the unit circle in the regular
representation $\mathbb{R}^{1+\sigma}$ of $C_{2}$. If $M$ is a
$C_{2}$-manifold, we define $\Gamma M$ to be the balanced product

$\Gamma M=M\times_{C_{2}}S(1+\sigma).$

If we define $\Gamma^{n}=\Gamma\circ\cdots\circ\Gamma$, then the map from
$\Omega^{C_{2}}_{*}$ to $\Omega^{hC_{2}}_{*}$ is given by

${\Omega^{C_{2}}_{*}}$${\Omega_{*}[[e]]}$${\left[M\right]}$${\sum_{n\geq
0}\left[(\Gamma^{n}M)^{e}\right]e^{n}.}$

where we write $M^{e}$ for the underlying non-equivariant manifold of
$M$.333Note that there are two different instances of the symbol $e$ in the
expression above. On one hand, we write $e$ for the trivial group, so that the
underlying class of a $C_{2}$-manifold $M$ is $[M^{e}]$. On the other hand, we
write $e$ for the euler class $e\in\Omega^{hC_{2}}_{-1}$. The meaning of each
instance of the symbol $e$ should be clear from context. This is the
unoriented analogue of [H05], Theorem 6.3, and may be verified analogously.

##  3\. Generators and relations for $\Omega^{C_{2}}_{*}$

In this section, we will complete our calculation of $\Omega^{C_{2}}_{*}$
using the Tate square

${\Omega^{C_{2}}_{*}}$${\Omega_{*}[d_{0},d_{1},d_{2},\dots]}$${\Omega_{*}[[e]]}$${\Omega_{*}((e)).}$$\scriptstyle{\phi}$

Our description of $\Omega^{C_{2}}_{*}$ is stated in Theorem 3.5. Our proof
strategy is as follows. Since $\phi$ maps $d_{0}$ to the unit
$e\in\Omega_{*}((e))$, we may factor the map $\phi$ as

${\Omega_{*}[d_{0},d_{1},d_{2},\dots]}$${\Omega_{*}[d_{0}^{\pm
1},d_{1},d_{2},\dots]}$${\Omega_{*}((e))}$$\scriptstyle{\iota}$$\scriptstyle{\psi}$

where $\iota$ is localization at $d_{0}$. The Tate square for
$\Omega^{C_{2}}_{*}$ then also factors as

(3.1)
${\Omega^{C_{2}}_{*}}$${\Omega_{*}[d_{0},d_{1},d_{2},\dots]}$${R}$${\Omega_{*}[d_{0}^{\pm
1},d_{1},d_{2},\dots]}$${\Omega_{*}[[e]]}$${\Omega_{*}((e)),}$$\scriptstyle{\iota}$$\scriptstyle{\psi}$

where both squares are pullbacks. We will first calculate the pullback $R$,
then use this information to calculate $\Omega^{C_{2}}_{*}$.

######  Remark 3.1.

It is worth mentioning that the bottom square in (3.1) is isomorphic to the
Tate square for the stable cobordism spectrum $MO_{C_{2}}$. In particular, the
ring $R$ is isomorphic to the stable cobordism ring $MO^{C_{2}}_{*}$. The
relationship between geometric and homotopy theoretic cobordism has been
investigated by Sinha [Si01] in the unoriented, $C_{2}$-equivariant case, and
by Hanke [H05] in the complex, torus-equivariant case.

######  Lemma 3.2.

The square

${d_{i,j}}$${d_{i}e^{-j}-\sum_{\ell<j}c_{i,\ell}e^{\ell-j}}$${d_{i,j}}$${\dfrac{\Omega_{*}[e,d_{i,j}:i-1,j\geq
0]}{d_{i,j}-c_{i,j}-ed_{i,j+1}}}$${\Omega_{*}[e^{\pm
1},d_{1},d_{2},\dots]}$${\sum_{\ell\geq
0}c_{i,j+\ell}e^{\ell}}$${\Omega_{*}[[e]]}$${\Omega_{*}((e))}$$\scriptstyle{\psi}$

is a pullback of rings.

###### Proof.

Following Strickland [St01], it suffices to show that the map

${\Omega_{*}[e,d_{i,j}]/(d_{i,j}-c_{i,j}-ed_{i,j+1})}$${\Omega_{*}[e^{\pm
1},d_{i}]}$

is an isomorphism after inverting $e$, that

${\Omega_{*}[e,d_{i,j}]/(d_{i,j}-c_{i,j}-ed_{i,j+1})}$${\Omega_{*}[[e]]}$

is an isomorphism after completing at $e$, and that

$e\in\Omega_{*}[e,d_{i,j}]/(d_{i,j}-c_{i,j}-ed_{i,j+1})$

is a regular element. The first two assertions are easily verified, and the
third follows from the fact that we may write

$\Omega_{*}[d_{i,j},e]/(d_{i,j}-c_{i,j}-ed_{i,j+1})=\varinjlim Q_{k}$

where $Q_{k}=\Omega_{*}[e,d_{i,k}:i\geq 1]$, and clearly $e$ is a regular
element of each of the rings $Q_{k}$. ∎

Having calculated $R$ using the bottom pullback square in 3.1, we may now use
the pullback square

${\Omega^{C_{2}}_{*}}$${\Omega_{*}[d_{0},d_{1},d_{2},\dots]}$${\Omega_{*}[e,d_{i,j}]/(d_{i,j}-c_{i,j}-ed_{i,j+1})}$${\Omega_{*}[d_{0}^{\pm
1},d_{1},d_{2},\dots].}$

to calculate $\Omega^{C_{2}}_{*}$.

######  Theorem 3.3.

The $C_{2}$-equivariant geometric bordism ring is given by

${\Omega^{C_{2}}_{*}\cong\dfrac{\Omega_{*}[d_{i,j}:i-1,j\geq
0]}{(d_{i,j}-c_{i,j})d_{k,\ell+1}=d_{i,j+1}(d_{k,\ell}-c_{k,\ell})}.}$

where $c_{i,j}\in\Omega_{i+j+1}$ is the coefficient of $e^{j}y^{i}$ in
$\dfrac{1}{F_{MO}(e,y)}$.

###### Proof.

First, we claim that

$\Omega^{C_{2}}_{*}\to\Omega_{*}[e,d_{i,j}]/(d_{i,j}-c_{i,j}-ed_{i,j+1})=R$

identifies $\Omega^{C_{2}}_{*}$ with the $\Omega_{*}$ sub-algebra of $R$
generated by $\\{d_{i,j}:i-1,j\geq 0\\}$. If $f\in R$, then, using the
relations in $R$, we may write

$f=f_{0}(d_{i,j})+ef_{1}(d_{i,0},e)$

for some $f_{0}\in\Omega_{*}[d_{i,j}]$ and $f_{1}\in\Omega_{*}[d_{i,0},e]$. By
considering the highest power of $e=d_{0}^{-1}$ that may occur in $\phi(f)$,
we can deduce that $\phi(f)$ lifts to $\Omega_{*}[d_{0},d_{1},d_{2},\dots]$ if
and only if $f_{1}(d_{i,0},e)=0$, i.e., if and only if $f$ may be written as
$f=f_{0}(d_{i,j})$.

We are left to compute the kernel of the map

$\Omega_{*}[d_{i,j}]\to\dfrac{\Omega_{*}[e,d_{i,j}]}{d_{i,j}-c_{i,j}-ed_{i,j+1}}.$

In $R$, we may write the element $ed_{i,j+1}d_{k,\ell+1}$ as

$\displaystyle(d_{i,j}-c_{i,j})d_{k,\ell+1}$

or as

$\displaystyle d_{i,j+1}(d_{k,\ell}-c_{k,\ell}).$

This gives us the relation

$(d_{i,j}-c_{i,j})d_{k,\ell+1}=d_{i,j+1}(d_{k,\ell}-c_{k,\ell})$

for any $i,k\geq 1$ and $j,\ell\geq 0$. We claim that these form a complete
set of relations for $\Omega^{C_{2}}_{*}$. To see this, we may choose an
ordering of the generators $\\{d_{i,j},e\\}$ such that $e$ is the largest
element in the ordering. This induces a partial ordering on the set of
monomials in the variables $\\{d_{i,j},e\\}$. We may then apply Buchberger’s
algorithm, which allows us to eliminate all instances of the variable $e$.
This is a purely algebraic result, and is an immediate consequence of [C21],
Lemma 7.1. ∎

We have calculated $\Omega^{C_{2}}_{*}$ in terms of certain generators
$d_{i,j}\in\Omega^{C_{2}}_{*}$. Since every class in $\Omega^{C_{2}}_{*}$ is
represented by some $C_{2}$-manifold, it is natural to ask for
$C_{2}$-manifold representatives of the classes
$d_{i,j}\in\Omega^{C_{2}}_{*}$. We conclude our calculation of
$\Omega^{C_{2}}_{*}$ by identifying manifold representatives of the classes
$d_{i,j}$. For any $n\geq 0$, we define the twisted projective space
$\mathbb{R}P^{n}_{\sigma}$ to be the $C_{2}$-manifold whose underlying space
is $\mathbb{R}P^{n}$, and whose $C_{2}$-action is given by

$(x_{0}:x_{1}:\cdots:x_{n})\mapsto(-x_{0},x_{1}:\cdots:x_{n}).$

For example, the twisted projective space $\mathbb{R}P^{1}_{\sigma}$ is
$C_{2}$-equivariantly homeomorphic to the $C_{2}$-space $S(1+\sigma)$, which
is the unit sphere in the regular representation $\mathbb{R}^{1+\sigma}$ of
$C_{2}$. This evidently bounds the disc $D(1+\sigma)$, so
$[\mathbb{R}P^{1}_{\sigma}]=0$ in $\Omega^{C_{2}}_{1}$. It turns that if
$n\geq 2$, then the class $[\mathbb{R}P^{n}_{\sigma}]\in\Omega^{C_{2}}_{n}$ is
non-zero, and these classes will form part of our generating set. The rest of
the generating set will be obtained from the twisted projective spaces
$\mathbb{R}P^{n}_{\sigma}$ by (iteratively) applying the Conner-Floyd
operation $\Gamma$.

######  Lemma 3.4.

For any $i\geq 1$ and $j\geq 0$, we have

$d_{i,j}=[\Gamma^{j}\mathbb{R}P^{i+1}_{\sigma}]\in\Omega^{C_{2}}_{*}$

###### Proof.

Since the map $\Omega^{C_{2}}_{*}\to\Omega^{\Phi C_{2}}_{*}$ is injective, it
suffices to verify the equality in $\Omega^{\Phi C_{2}}_{*}$. Recall that the
image of $[M]\in\Omega^{C_{2}}_{*}$ in $\Omega^{\Phi C_{2}}_{*}$ is given by
$[M^{C_{2}},\nu\mid_{M^{C_{2}}}^{M}]$. If $M=\mathbb{R}P^{i+1}_{\sigma}$, then

$[M^{C_{2}},\nu\mid_{M^{C_{2}}}^{M}]=[\mathbb{R}P^{i},\nu\mid_{\mathbb{R}P^{i}}^{\mathbb{R}P^{i+1}}]+[*,\mathbb{R}^{i+1}],$

and by definition of the classes $d_{i}\in\Omega^{\Phi C_{2}}_{*}$, this is
exactly the element $d_{i}-d_{0}^{i+1}$. On the other hand, the image of
$d_{i,0}$ in $\Omega^{\Phi C_{2}}_{*}$ is

$d_{i}-\sum_{\ell<0}c_{i,\ell}e^{\ell}.$

But since $d_{0}=e^{-1}$, and

$c_{i,\ell}=\begin{cases}-1&\ell=-i-1\\\ 0&0>\ell\neq-i-1,\end{cases}$

we can deduce that $d_{i,0}=[\mathbb{R}P^{i+1}_{\sigma}]$. The case $j>0$
follows from the fact that $d_{i,j+1}$ is obtained from $d_{i,j}$ by applying
the Conner-Floyd operation $\Gamma$. ∎

Combining our previous results, we may state our calculation of
$\Omega^{C_{2}}_{*}$.

######  Theorem 3.5.

The $C_{2}$-equivariant unoriented cobordism ring $\Omega^{C_{2}}_{*}$ is
generated over $\Omega_{*}$ by the classes

${d_{i,j}=[\Gamma^{j}\mathbb{R}P^{i+1}_{\sigma}]}$${i\geq 1\text{ and }j\geq
0.}$

A complete set of relations is given by

${(d_{i,j}-c_{i,j})d_{k,\ell+1}-d_{i,j+1}(d_{k,\ell}-c_{k,\ell})}$${i,k\geq
1\text{ and }j,\ell\geq 0.}$

where $c_{i,j}\in\Omega_{i+j+1}$ is the coefficient of $e^{j}y^{i}$ in
$\dfrac{1}{F_{MO}(e,y)}$. The restriction

$\Omega^{C_{2}}_{*}\to\Omega_{*}$

is determined by

$d_{i,j}\mapsto c_{i,j},$

and the transfer $\Omega_{*}\to\Omega^{C_{2}}_{*}$ is zero.

##  4\. The extended cobordism ring $\Omega^{C_{2}}_{\diamond}$

We have seen two $C_{2}$-equivariant analogues of cobordism, namely the
geometric cobordism ring $\Omega^{C_{2}}_{*}$, and the stable cobordism ring
$MO^{C_{2}}_{*}$. It turns out that the structure of each of these rings is
subsumed by the structure of the extended cobordism ring
$\Omega^{C_{2}}_{\diamond}$, which is the subject of the present section. This
ring can be viewed from two different perspectives, and we pause to discuss
each of these.

On one hand, the ring $\Omega^{C_{2}}_{\diamond}$ has a geometric
interpretation. For any $k\geq 0$, let $D(k\sigma)$ denote the unit disc in
the orthogonal $C_{2}$-representation $\mathbb{R}^{k\sigma}$, whose boundary
$\partial D(k\sigma)=S(k\sigma)$ is the unit sphere in $\mathbb{R}^{k\sigma}$.
Then $\Omega^{C_{2}}_{*}(D(k\sigma),S(k\sigma))$ is the cobordism group of
$C_{2}$-manifolds $(M,\partial M)$ equipped with a map to
$(D(k\sigma),S(k\sigma))$. This is the standard geometric intepretation of the
cobordism homology of a pair of spaces. From this perspective, the extended
cobordism ring $\Omega^{C_{2}}_{\diamond}$ is the direct sum

$\displaystyle\Omega^{C_{2}}_{\diamond}=\bigoplus_{k\geq
0}\Omega^{C_{2}}_{*}(D(k\sigma),S(k\sigma)).$

with product structure specified by

$[M\to D(k\sigma)][M^{\prime}\to D(k^{\prime}\sigma)]=[M\times M^{\prime}\to
D((k+k^{\prime})\sigma)].$

Here, we have identified the $C_{2}$-spaces $D(k\sigma)\times
D(k^{\prime}\sigma)\cong D((k+k^{\prime})\sigma)$.

On the other hand, the ring $\Omega^{C_{2}}_{\diamond}$ has a homotopical
interpretation: it is a portion of the $RO(C_{2})$-graded coefficient ring
$\Omega^{C_{2}}_{\star}$ of $\Omega_{C_{2}}$. To see this, we note that if
$A\subset X$ is an inclusion of (unbased) $C_{2}$-spaces, then the Pontrjagin-
Thom construction extends to an identification

$\Omega^{C_{2}}_{*}(X,A)\cong\pi_{*}^{C_{2}}(\Omega_{C_{2}}\wedge X/A).$

If $(X,A)$ is the pair $(D(k\sigma),S(k\sigma))$, then
$D(k\sigma)/S(k\sigma)\cong S^{k\sigma}$, the representation sphere associated
to the $C_{2}$-representation $k\sigma$, and the preceding isomorphism takes
the form

$\displaystyle\Omega^{C_{2}}_{*}(D(k\sigma),S(k\sigma))$
$\displaystyle\cong\pi_{*}^{C_{2}}(\Omega_{C_{2}}\wedge S^{k\sigma})$
$\displaystyle\cong\Omega^{C_{2}}_{*-k\sigma}.$

This prespective allows us to view

$\Omega^{C_{2}}_{\diamond}=\bigoplus_{k=0}^{\infty}\Omega^{C_{2}}_{*-k\sigma},$

as a subring of the $RO(C_{2})$-graded coefficients of $\Omega_{C_{2}}$. This
perspective is useful because, for instance, it affords us access to long
exact sequences in $RO(C_{2})$-graded homotopy groups induced by cofiber
sequences of $C_{2}$-spectra. We refer the interested reader to [LMS86] for
further information about $RO(C_{2})$-graded homotopy groups, and to chapter
XV of [Co96] for further details regarding the equivariant Pontrjagin-Thom
construction in the relative case.

There are two especially important examples of relative $C_{2}$-manifolds. The
first is the shifted Euler class

$a=[*\to D(\sigma)]\in\Omega^{C_{2}}_{-\sigma},$

which is represented by the inclusion of the origin in the unit disc
$D(\sigma)$. The second is the orientation class

$u=[D(\sigma)\to D(\sigma)]\in\Omega^{C_{2}}_{1-\sigma},$

which represented by the identity map on $D(\sigma)$. It turns out that the
extended cobordism ring $\Omega^{C_{2}}_{\diamond}$ is generated over
$\Omega^{C_{2}}_{*}$ by the classes $a$ and $u$. In the following lemma, we
identify the relationship between geometric classes, the shifted Euler class,
and the orientation class.

######  Lemma 4.1.

If $M$ is a $C_{2}$-manifold, then the equation

$u[M]=u[M^{e}]+a[\Gamma M]$

holds in $\Omega^{C_{2}}_{*-\sigma}$.

###### Proof.

Recall that elements of $\Omega^{C_{2}}_{*-\sigma}$ are represented by
$C_{2}$-manifolds $(X,\partial X)$ equipped with a map to
$(D(\sigma),S(\sigma))$. The classes $u[M]$, $u[M^{e}]$, and $a[\Gamma M]$ are
represented geometrically as follows:

$\displaystyle u[M]$ $\displaystyle=[M\times D(\sigma)\to D(\sigma)]$
$\displaystyle u[M^{e}]$ $\displaystyle=[M^{e}\times D(\sigma)\to D(\sigma)]$
$\displaystyle a[\Gamma M]$ $\displaystyle=[\Gamma M\to*\to D(\sigma)].$

Since the map

${\Omega^{C_{2}}_{*-\sigma}}$${\Omega^{\Phi C_{2}}_{*-\sigma}}$${\Omega^{\Phi
C_{2}}_{*}}$$\scriptstyle{\cong}$$\scriptstyle{a^{-1}}$

is injective, it suffices to prove the relation in the fixed point bordism
ring $\Omega^{\Phi C_{2}}_{*}$. The image of the classes $u[M]$ and $u[M^{e}]$
in $\Omega^{\Phi C_{2}}_{*}$ are given by
$[M^{C_{2}},\nu\mid_{M^{C_{2}}}^{M}\oplus\mathbb{R}]$ and
$[M^{e},\mathbb{R}]$. The image of the class $a[\Gamma M]$ in $\Omega^{\Phi
C_{2}}_{*}$ is

$[M^{C_{2}}\sqcup
M^{e},\nu\mid_{M^{C_{2}}}^{M}\oplus\mathbb{R}\sqcup\mathbb{R}]=[M^{C_{2}},\nu\mid_{M^{C_{2}}}^{M}\oplus\mathbb{R}]+[M^{e},\mathbb{R}].$

∎

It turns out that this relation completely determines the structure of the
extended bordism ring $\Omega^{C_{2}}_{\diamond}$ as an algebra over
$\Omega^{C_{2}}_{*}$.

######  Theorem 4.2.

As an algebra over the $C_{2}$-cobordism ring $\Omega^{C_{2}}_{*}$, the
extended cobordism ring $\Omega^{C_{2}}_{\diamond}$ is generated by the
classes $a\in\Omega^{C_{2}}_{-\sigma}$ and $u\in\Omega^{C_{2}}_{1-\sigma}$. A
complete set of relations is given by

$u[M]=u[M^{e}]+a[\Gamma M]$

for each $[M]\in\Omega^{C_{2}}_{*}$.

###### Proof.

It suffices to prove the $\Omega_{*}$-module isomorphism

$\displaystyle\Omega^{C_{2}}_{*-n\sigma}$
$\displaystyle\cong\dfrac{\Omega^{C_{2}}_{*}\\{a^{n},a^{n-1}u,\dots,au^{n-1},u^{n}\\}}{a^{i}u^{n-i}[M]=a^{i}u^{n-i}[M^{e}]+a^{i+1}u^{n-i-1}[\Gamma
M]}\hskip 36.135pt1\leq i\leq n$

for every $n\geq 0$. We prove the claim by induction on $n\geq 0$, with the
case $n=0$ holding trivially. Suppose that we have proved the claim for some
$n\geq 0$. We apply $\Omega^{C_{2}}_{*-n\sigma}(-)$ to the cofiber sequence

$C_{2+}\to S^{0}\to S^{\sigma}$

to obtain the exact sequence

${\cdots}$${\Omega_{*+1-(n+1)\sigma}}$${\Omega^{C_{2}}_{*-n\sigma}}$${\Omega^{C_{2}}_{*-(n+1)\sigma}}$${\Omega_{*-(n+1)\sigma}}$${\cdots.}$$\scriptstyle{\text{tr}_{e}^{C_{2}}}$$\scriptstyle{a}$$\scriptstyle{\text{res}^{C_{2}}_{e}}$

By induction we may assume that the map $\text{tr}_{e}^{C_{2}}$ above is zero,
and the map $\text{res}^{C_{2}}_{e}$ above is surjective, which implies that
our exact sequence is short:

${0}$${\Omega^{C_{2}}_{*-n\sigma}}$${\Omega^{C_{2}}_{*-(n+1)\sigma}}$${\Omega_{*-(n+1)\sigma}}$${0.}$$\scriptstyle{a}$

The righthand term $\Omega_{*-(n+1)\sigma}$ is a free $\Omega_{*}$-module, and
the short exact sequence above is split by $u^{n+1}$, which leads to the
isomorphism

$\displaystyle\Omega^{C_{2}}_{*-(n+1)\sigma}$
$\displaystyle\cong\Omega_{*}\\{u^{n+1}\\}\oplus\Omega^{C_{2}}_{*-n\sigma}\\{a\\}$
$\displaystyle\cong\dfrac{\Omega^{C_{2}}_{*}\\{a^{n+1},a^{n}u,\dots,au^{n},u^{n+1}\\}}{a^{i}u^{n+1-i}[M]=a^{i}u^{n+1-i}[M^{e}]+a^{i+1}u^{n-i}[\Gamma
M]}\hskip 36.135pt1\leq i\leq n+1$

Taking the direct sum over $n\geq 0$ yields the desired result. ∎

Combining the previous result with Theorem 3.5, we obtain the following
description of $\Omega^{C_{2}}_{\diamond}$.

######  Theorem 4.3.

The extended $C_{2}$-bordism ring $\Omega^{C_{2}}_{\diamond}$ is generated
over $\Omega_{*}$ by the classes

${d_{i,j}=[\Gamma^{j}\mathbb{R}P^{i+1}_{\sigma}]}$${i\geq 1\text{ and }j\geq
0,}$

together with the shifted Euler class

$a=[*\to D(\sigma)]$

and the orientation class

$u=[D(\sigma)\to D(\sigma)].$

A complete set of relations is given by

${(d_{i,j}-c_{i,j})d_{k,\ell+1}-d_{i,j+1}(d_{k,\ell}-c_{k,\ell})}$${i,k\geq
1\text{ and }j,\ell\geq 0,}$${u(d_{i,j}-c_{i,j})-ad_{i,j+1}}$${i\geq 1\text{
and }j\geq 0,}$

where $c_{i,j}\in\Omega_{i+j+1}$ is the coefficient of $e^{j}y^{i}$ in
$\dfrac{1}{F_{MO}(e,y)}$. The restriction

$\Omega^{C_{2}}_{\diamond}\to\Omega_{\diamond}=\Omega_{*}[u]$

is determined by

$\displaystyle d_{i,j}$ $\displaystyle\mapsto c_{i,j}$ $\displaystyle a$
$\displaystyle\mapsto 0,$ $\displaystyle u$ $\displaystyle\mapsto u.$

and the transfer $\Omega_{\diamond}\to\Omega^{C_{2}}_{\diamond}$ is zero.

## References

  * [A72] J. C. Alexander, The Bordism Ring of Manifolds with Involution, Proc. Amer. Math. Soc. 31  (2) (1972), 536–542.
  * [BH72] T. Brocker, E.C. Hook, Stable equivariant bordism, Math. Z. 129 (1972), 269–277.
  * [C21] J. H. Carlisle, Complex cobordism with involutions and geometric orientations, preprint (2021).
  * [CGK00] M. Cole, J. P. C. Greenlees, and I. Kriz, Equivariant formal group laws, Proc. London Math. Soc. (3), 81 (2) (2000), 355–386.
  * [CF64] P. E. Conner, E. E. Floyd, Differentiable periodic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 33, Springer, Berlin (1964).
  * [Co96] S. R. Costenoble, An introduction to equivariant cobordism, in: J. P. May (ed.) Equivariant homotopy and cohomology theory, CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI (1996).
  * [GM95] J. P. C. Greenlees, J. P. May, Generalized Tate cohomology, Mem. Amer. Math. Soc. 113 (1995), no. 543, viii + 178pp.
  * [H05] B. Hanke, Geometric versus homotopy theoretic equivariant bordism, Math. Ann., 332 (3) (2005), 677–696.
  * [HHR16] M. A. Hill, M. J. Hopkins, D. C. Ravenel, On the nonexistence of elements of Kervaire invariant one, Annals of Math. 184 (1) (2016) 1–262.
  * [K98] I. Kriz, The $\mathbb{Z}/p$-equivariant complex cobordism ring, Homotopy invariant algebraic structures (Baltimore, MD, 1998), Contemp. Math., 239 (1999), 217–223. Amer. Math. Soc., Providence, RI.
  * [LMS86] L.G. Lewis, J. P. May, M. Steinberger, Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, New York (1986).
  * [M74] J. W. Milnor, J. H. Stasheff, Characteristic Classes, Princeton Unviersity Press and University of Tokyo Press, Princeton, NJ (1974)
  * [Q69] D. Quillen, On the formal group laws of unoriented and complex cobordism theory, Bulletin of the American Mathematical Society 75 (1969), 1293–1298.
  * [S18] S. Schwede, Global homotopy theory, New Mathematical Monographs 34 (2018), Cambridge University Press, Cambridge.
  * [Si01] D. Sinha, Real equivariant bordism and stable transversality obstructions for $\mathbb{Z}/2$, Proc. Amer. Math. Soc., 130 (1) (2001), 271–281.
  * [St01] N. P. Strickland, Complex cobordism of involutions, Geom. Topol., 5 (1) (2001), 335–345.
  * [T54] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86.
  * [tD70] T. tom Dieck, Bordism of $G$-manifolds and integrality theorems, Topology 9 (1970), 345–358.
  * [W69] D. Wasserman, Equivariant differential topology, Topology 8  (1969), 127–150.