In this paper, we view the equivariant orientation theory of equivariant vector bundles from the lenses of equivariant Picard spectra. This viewpoint allows us to identify, for a finite group $\mathrm{G}$, a precise condition under which an $\mathrm{R}$-orientation of a $\mathrm{G}$-equivariant vector bundle is encoded by a Thom class. Consequently, we are able to construct a generalization of the first Stiefel–Whitney class of a “homogeneous" $\mathrm{G}$-equivariant bundle with respect to an $\mathbb{E}_\infty^{\mathrm{G}}$-ring spectrum $\mathrm{R}$. As an application, we show that the $2$-fold direct sum of any homogeneous bundle is $\mathrm{H}\underline{\mathcal{A}}_{\mathrm{G}}$-orientable, where $\underline{\mathcal{A}}_{\mathrm{G}}$ is the Burnside Mackey functor. We notice that  $\mathrm{H}\underline{\mathcal{A}}_{\mathrm{G}}$-orientability is equivalent to $\mathrm{H}\underline{\mathbb{Z}}$-orientability when the order of $\mathrm{G}$ is odd. When the order of $\mathrm{G}$ is even, we show that a $\mathrm{G}$-equivariant analog of the tautological line bundle over $\mathbb{RP}^\infty$ is $\mathrm{H}\underline{\mathbb{Z}}$-orientable but not $\mathrm{H}\underline{\mathcal{A}}_{\mathrm{G}}$-orientable.

§ INTRODUCTION

In differential topology, an orientation of a vector bundle is an
orientation of each fiber which is compatible with the topology of the base space.
In homotopy theory, an orientation is expressed as a Thom class in the ordinary cohomology with $\ZZ$-coefficients of its Thom space.  The concept of Thom class
extends to extraordinary cohomology theories:
For any ring spectrum $\mr{R}$,  an $\mr{R}$-Thom class is a cohomology class
in the  $\mr{R}$-cohomology of the Thom space $\upxi$

\[ {\sf u}_{\upxi} \in  [\mr{Th}(\upxi),  \Sigma^{\dim \upxi} \mr{R} ] = \mr{R}^{\dim \upxi}(\mr{Th}(\upxi)) \]

such that its restriction to the Thom space  over any point is a unit in  the homotopy group of $\mr{R}$.  This idea led to many interesting ideas and results, which are now foundational in homotopy theory and its application to geometry.

Equivariant orientation theory was systematically developed in the  eighties and nineties, geometrically in [9, 8],
and cohomologically in [17, 9, 20, 21]. Unlike the nonequivariant orientation theory, the geometric theory fails to coincide with the cohomological theory (see <cit.>), although a comparison  appears in [21].
One of the challenges of these developments was that many important results,
such as the equivariant Poincare duality (see <cit.> and [10]),  were best understood when restricted to $\mr{G}$-connected spaces. This is because the equivariant dimension of a vector bundle is not well defined unless the base space is $\mr{G}$-connected. Two different ideas came in as a remedy: Costenoble and Waner [9] used representations of the fundamental groupoid of the base space, instead of representation ring of $\mr{G}$, to index  cohomology theories, whereas May [21] dropped the idea of a single Thom class and advocated  a “family of Thom classes" instead.
The difficulty in glueing such a “family of Thom classes”
lies in the fact that, unlike  nonequivariant spaces, a $\mr{G}$-equivariant space is not equivalent to the disjoint union of its  $\mr{G}$-connected components (see <Ref>).
[$\mr{G}$-connected components]
The seminal work of Elmendorf [11]  was extended in [28]  to show  that $\Top^{\mr{G}}$,  the category   of $\mr{G}$-equivariant spaces with $\mr{G}$-equivariant continuous maps,  is Quillen equivalent to the category of
$\mathcal{O}_{\mr{G}}$-spaces. Given a $\mr{G}$-equivariant space $\mr{X}$ we get an $\mathcal{O}_{\mr{G}}$-space

\[
\begin{tikzcd}
\Phi(\mr{X}): \cal{O}_{\mr{G}}^{\mr{op}} \rar & \Top,
\end{tikzcd}
\]

that maps $\mr{G}/\mr{H}$ to the $\mr{H}$-fixed points of $\mr{X}$ for any subgroup $\mr{H}$.
By compatibly choosing  a path component of $\mr{X}^{\mr{H}}$ for each $\mr{H} \subset \mr{G}$, one can often form  an $\cal{O}_{\mr{G}}$-space. The corresponding $\mr{G}$-equivariant space  obtained using <cit.> is then a $\mr{G}$-connected component of $\mr{X}$.

Let $\mr{S}^{\upsigma}$ denote the one-point compactification of $\upsigma$, the sign representation of $\mr{C}_2$. Note that the $\mr{C}_2$-fixed point of $\mr{S}^{\upsigma}$ is $\mr{S}^0$ which has two components. Therefore,
$\mr{S}^{\upsigma}$ has two different $\mr{C}_2$-connected components both equivalent to $\Sigma (\mr{EC}_2)_+$, where $\mr{EC}_2$ is the total space of the universal  $\mr{C}_2$-principal bundle. The space $\mr{S}^{\upsigma}$ is not equivalent to the disjoint union of its $\mr{G}$-connected components as the underlying space of $\mr{S}^{\upsigma}$ is connected.

In this paper, we propose a different approach to equivariant orientation theory,  where  the existence of an  orientation of an equivariant bundle is determined through a two-step obstruction theory.  As a consequence, we are able to make explicit calculations to check whether a given equivariant bundle admits an orientation or not.

Taking advantage of the classification theory of equivariant vector bundles [32] (also see [19, 36]), equivariant $\mr{J}$-homomorphism (see <Ref>), and  the recent construction of  equivariant Picard spectra [15],  we introduce the notion of homogeneity of  $\mr{G}$-equivariant vector bundles relative to any $\mathbb{E}_\infty^{\mr{G}}$-ring spectrum $\R$  (see <Ref>). In our work,  $\R$-homogeneity is a requirement for an equivariant vector bundle  to be $\R$-orientable, hence should be thought of as the first obstruction. Our first result is an equivariant generalization of the classical Thom isomorphism theorem where  the base space of  $\mr{G}$-equivariant vector bundles need not be $\mr{G}$-connected.

Let $\mr{G}$ be a finite group and let $\R$ be an $\mathbb{E}_\infty^{\mr{G}}$-ring spectrum. Suppose that $\upxi$ is an $\mr{R}$-homogeneous $\mr{G}$-equivariant real vector bundle over a $\mr{G}$-space $\mr{B}$,   then the following statements are equivalent:

* The bundle $\upxi$ is $\mr{R}$-orientable.

* There exists an $\mr{R}$-Thom class
\[ {\sf u}_{\upxi} \in [\mr{Th}(\upxi), \cal{I}  ]^{\mr{G}} \]
for some invertible $\R$-module $\cal{I}$.

* There exists a $\mr{G}$-equivariant weak equivalence

\begin{equation} \label{eqn:thomiso}
\begin{tikzcd}
\mr{Th}(\upxi) \sma \mr{R} \simeq \mr{B}_+ \sma \cal{I}
\end{tikzcd}
\end{equation}

for some invertible $\R$-module $\cal{I}$.

[Finiteness assumption on $\mr{G}$]
We let $\mr{G}$ to be a finite group in <Ref> and in the rest of the paper to avoid complications of the equivariant infinite loop space theory otherwise (see <cit.>).  Further,  our main application <Ref>  makes sense only when $|\mr{G}|$ is finite.

The real regular representation of any finite group $\mr{G}$ will be denoted by  $\uprho_{\mr{G}}$. When the underlying group $\mr{G}$ is  clear from the context, we will drop the subscript.

The notion of $\R$-homogeneity of an equivariant bundle is an extension of the concept of homogeneity (see <Ref>) which already existed in the literature. A $\mr{G}$-equivariant homogeneous bundle is referred to as  a “$\mr{V}$-bundle" in the work of  Costenoble and Waner [10], where $\mr{V}$ is the $\mr{G}$-representation that encodes the equivariant dimension of the bundle. There are  many  important  and interesting examples of  homogeneous $\mr{G}$-equivariant vector bundles.

Let $\Pi$ be a compact Lie group, and let $\mr{E}_{\mr{G}}\Pi$ denote the total space of the universal principal  $\mr{G}$-$\Pi$ bundle (see (<ref>) for details). For a finite dimensional real $\Pi \times \mr{G}$  representation $\mr{V}$, the  $\mr{G}$-equivariant bundle $\uplambda_{\mr{V}}$  whose projection map is

\begin{equation} \label{eqn:generalbundle}
\begin{tikzcd}
\uppi_{\uplambda_{\mr{V}}}: \mr{E}_{\mr{G}} \Pi  \times_{ \Pi} \mr{V} \rar & \mr{E}_{\mr{G}} \Pi  \times_{ \Pi} {\bf 0} := \mr{B}_{\mr{G}} \Pi
\end{tikzcd}
\end{equation}

has a base space which is not $\mr{G}$-connected (see [16] and <Ref>).
We show that:
Suppose $\mr{V}$ is a $\Pi \times \mr{G}$-representation isomorphic to
\[ \mr{V} \cong  n \uprho_{\mr{G}} \otimes \mr{W},\]
where $\mr{W}$ is an arbitrary finite real representation of $\Pi$, and $n \in \NN$.
Then $\uplambda_{\mr{V}}$ as defined in <Ref>  is  a homogeneous $\mr{G}$-equivariant vector bundle.

When $\Pi =\mr{O}(n)$, the space $\mr{B}_{\mr{G}}\mr{O}(n)$ is a classifying space for $\mr{G}$-equivariant vector bundles [32].

When $\mr{G} = \mr{C}_2$, $\Pi = \Sigma_2$ and $\mr{V} = \uprho \otimes \uptau$, the tensor product of the regular representation of $\mr{C}_2$ and the sign representation of $\Sigma_2$, then $\uplambda_{\mr{V}}$ is a  $\mr{C}_2$-equivariant bundle which is used in the  geometric construction of the $\mr{C}_2$-equivariant Steenrod operations (see <cit.>).

When $\mr{G} = \mr{C}_2$, $\Pi = \Sigma_2$ and $\mr{V} = \mr{U} \otimes \uptau$, where the $\mr{C}_2$-representation $\mr{U}$ is not in the form of $n\uprho$, it follows from
<Ref> that $\uplambda_{\mr{V}}$ is not homogeneous.

The second and final obstruction to $\R$-orientability is an equivariant analog of the first Stiefel–Whitney class. Classically,  the first Stiefel–Whitney class is  the obstruction to $\mr{H}\mathbb{Z}$-orientability of vector bundles. In <Ref>, we  define for
an $\R$-homogeneous bundle  $\upxi$,   a cohomology class
${\sf w}_1^{\R}(\upxi)$, which can be thought of as
the first $\mr{G}$-equivariant Stiefel–Whitney class  of $\upxi$ with respect to

An $\mr{R}$-homogeneous $\mr{G}$-equivariant bundle $\upxi$ is $\R$-orientable if and only if ${\sf w}_1^{\R}(\upxi) =0$.

Similar to the classical  first Stiefel–Whitney class, our generalization also satisfies the additivity formula

\begin{equation} \label{eqn:additivity}
{\sf w}_1^\R(\upxi_1 \oplus \upxi_2) = {\sf w}_1^\R(\upxi_1) + {\sf w}_1^\R(\upxi_2)
\end{equation}

for a pair of $\R$-homogeneous vector bundles (as usual).
Using the additivity property, one may  define the $\R$-orientation order of an $\mr{R}$-homogeneous vector bundle as the smallest number $n$ for which the $n$-fold direct sum of $\upxi$ is $\R$-orientable.
The additivity is also fundamental in the proof of <Ref> and <Ref>.
Throughout this paper

* $\ull{\mr{A}}$ denotes the constant $\mr{G}$-Tambara functor at the abelian group $\mr{A}$,

* $\ull{\mathcal{A}}_{\mr{G}}$ denotes the Burnside Tambara functor of $\mr{G}$, and

* $\mr{H}\cal{M}$ denotes  the  Eilenberg–MacLane spectrum corresponding to a Mackey functor $\cal{M}$.

A Tambara functor $\mathcal{T}$ is a  $\mr{G}$-commutative monoid object in the category of Mackey functors, and therefore,
the corresponding Eilenberg–MacLane spectrum  $\mr{H} \cal{T}$ is an $\Einfty^{\mr{G}}$-ring spectrum. The first Stiefel–Whitney class of an $\mr{H} \cal{T}$-homogeneous  bundle $\upxi$ over $\mr{B}$ is a class in

\[ {\sf w}_1^{\cal{T}}(\upxi):= {\sf w}_1^{\mr{H}\cal{T}}(\upxi)  \in \mr{H}^1(\mr{B}; \cal{T}^{\times}),\]
where $\cal{T}^{\times}$ is the unit Mackey functor for $\cal{T}$.

In $\mr{G}$-equivariant homotopy theory, $\mr{H}\underline{\mathbb{Z}}$ is often regarded as a generalization of the integral Eilenberg–MacLane spectrum. In this paper we show that:
For any $\mr{H}\ull{\mathbb{Z}}$-homogeneous $\mr{G}$-equivariant vector bundle, its $2$-fold direct sum is $\mr{H}\ull{\mathbb{Z}}$-orientable.

Although $\mr{H}\underline{\mathbb{Z}}$ is a perfectly valid generalization, many
consider the Burnside Eilenberg–MacLane  spectrum
$\mr{H}\ull{\cal{A}}_{\mr{G}}$ as a more appropriate one. This is because, just like $\mr{H}\mathbb{Z}$ in the classical case,   $\mr{H}\ull{\cal{A}}_{\mr{G}}$ is the zeroth Postnikov approximation of the sphere spectrum $\mathbb{S}_{\mr{G}}$. Further, $\ull{\cal{A}}_{\mr{G}}$ is the unit in the symmetric monoidal category of Mackey functors (see [31, 22]), which is an equivariant generalization of the fact that $\mathbb{Z}$  is the unit for the tensor product of abelian groups.

Studying $\mr{H}\ull{\cal{A}}_{\mr{G}}$-orientation of equivariant bundles is an important problem as it is a step towards understanding the genuine stable equivalence class (see <Ref>) of  $\mr{H}\underline{\mathbb{Z}}$-orientable vector bundles.
This is because the unit map for $\mr{H}\underline{\mathbb{Z}}$ factors through that of $\mr{H}\ull{\cal{A}}_{\mr{G}}$

\[
\begin{tikzcd}
\mathbb{S}_{\mr{G}} \rar & \mr{H} \ull{\cal{A}}_{\mr{G}} \rar & \mr{H} \ull{\mathbb{Z}}.
\end{tikzcd}
\]

Moreover, $\mr{H}\ull{\cal{A}}_{\mr{G}}$-orientation of an equivariant vector bundle has a precise geometric interpretation as explained in  [21], because of which many important results in the subject, such as equivariant Poincare duality <cit.>, are discussed in terms of $\mr{H}\ull{\cal{A}}_{\mr{G}}$-cohomology.
We show:
For any $\mr{H} \ull{\cal{A}}_{\mr{G}}$-homogeneous $\mr{G}$-equivariant vector bundle, its $2$-fold direct sum is  $\mr{H} \ull{\cal{A}}_{\mr{G}}$-orientable.

We notice that an $\mr{H}\underline{\mathbb{Z}}$-orientation is sufficient to obtain an $\mr{H}\ull{\cal{A}}_{\mr{G}}$-orientation when $|\mr{G}|$ is odd.
Let  $\mr{G}$ be a finite group of odd order. Then
\[ {\sf w}_1^{\ull{\cal{A}}_{\mr{G}}} (\upxi) = {\sf w}_1^{\ull{\mathbb{Z}}} (\upxi)\]
for any $\mr{H}\ull{\cal{A}}_{\mr{G}}$-homogeneous $\mr{G}$-equivariant bundle $\upxi$.

The above theorem does not hold when $\mr{G}$ has an even order because the kernel of the  map

\[
\begin{tikzcd}
\ull{\cal{A}}_{\mr{G}}^{\times} \rar & \ull{\mathbb{Z}}^{\times}
\end{tikzcd}
\]

of $\mr{G}$-Mackey functors is nonzero.  We use this fact to define a set
$\mathfrak{G}_{1}(-)$ (see <Ref>),  and refer to its elements as
“ghosts[The term ghost is used  to indicate that the set $\mathfrak{G}_1(\upxi)$
is not defined unless ${\sf w}_1^{\ull{\mathbb{Z}}}(\upxi)$ is dead  (i.e. ${\sf
w}_1^{\ull{\mathbb{Z}}}(\upxi) =0$).] of the first Stiefel–Whitney class". These elements can be regarded as  obstructions to $\mr{H}\ull{\cal{A}}_{\mr{G}}$-orientability for
$\mr{H} \ull{\mathbb{Z}}$-orientable vector bundles:
An $\mr{H}\ull{\cal{A}}_{\mr{G}}$-homogeneous $\mr{H}\ull{\mathbb{Z}}$-orientable vector bundle $\upxi$ is $\mr{H}\ull{\cal{A}}_{\mr{G}}$-orientable if and only if $0 \in \mathfrak{G}_1(\upxi) $.

We show that  the trivial element always appears as a ghost when $\mr{G}$ acts freely on the base space of a bundle.  Consequently, we  have the following result.

Let $\upxi$ is a $\mr{H}\ull{\cal{A}}_{\mr{G}}$-homogeneous bundle such that $\mr{G}$ acts freely on
its base space. Then $\upxi$  is $\mr{H}\ull{\cal{A}}_{\mr{G}}$-orientable if and only if it is $\mr{H}\ull{\mathbb{Z}}$-orientable.

In 1966, M.F.  Atiyah  [2] introduced a subclass of $\mr{C}_2$-equivariant vector bundles whose underlying nonequivariant bundle is a complex vector bundle and the action of $\mr{C}_2$ on the fibers is compatible with the complex conjugation action on $\mathbb{C}$. These bundles are referred to as Atiyah Real bundles. His work implies that the classifying space of Atiyah Real bundles is a “genuine" $\mr{C}_2$-equivariant infinite loop space, and hence deloops to a  genuine $\mr{C}_2$-spectrum $\kr$  called the connective Atiyah Real $\mr{K}$-theory.

The tautological Atiyah Real line bundle  $\hat{\upgamma}$ is the
tautological complex line bundle over the  infinite complex projective space
equipped with the complex conjugation action of $\mr{C}_2$.
One of the results in [2], which is crucial to many applications,  is that the bundle $\hat{\upgamma}$ is $\kr$-orientable. This automatically implies that $\hat{\upgamma}$ is orientable with respect to $\mr{H}\ull{\mathbb{Z}}$,  the zeroth Postnikov approximation of $\kr$. We  show  that:
The  tautological Atiyah Real bundle $\hat{\upgamma}$  is not $\mr{H} \ull{\cal{A}}_{\mr{C}_2}$-orientable.

Let $\upgamma_{\uprho}$ denote the $\uprho$-dimensional bundle over $\mr{B}_{\mr{G}}\Sigma_2$ obtained by setting  $\Pi = \Sigma_2$ and $\mr{V} = \uprho \otimes \uptau$ in (<ref>). In some sense, $\upgamma_{\uprho}$
is  the $\mr{G}$-equivariant analog of the real tautological line bundle.

When $\mr{G} = \mr{C}_2$ then there exist a map of $\mr{C}_2$-equivariant vector bundle (see <cit.>)

\begin{equation} \label{C2bundleMap}
\begin{tikzcd}
\upgamma_{\uprho} \rar & \hat{\upgamma},
\end{tikzcd}
\end{equation}

thus an $\mr{H}\ull{\mathbb{Z}}$-orientation of $\hat{\upgamma}$ leads to an  $\mr{H}\ull{\mathbb{Z}}$-orientation of $\upgamma_{\uprho}$. However, <Ref> implies that $\hat{\upgamma}$ is not  $\mr{H} \ull{\cal{A}}_{\mr{C}_2}$-orientable. We end the paper by generalizing this result to all finite groups of even order:
If $\mr{G}$ is a finite group of even order, then $\upgamma_{\uprho}$ is an $\mr{H}\ull{\mathbb{Z}}$-orientable bundle which does not admit an $\mr{H}\ull{\cal{A}}_{\mr{G}}$-orientation.

Morally, the results in this paper are independent of the model of $\Sp^{\mr{G}}$ (a modern poinset category of genuine $\mr{G}$-spectra) and  $\Top^{\mr{G}}$ (a category of $\mr{G}$-spaces); they
rely on the loop-suspension adjunction

\begin{equation}  \label{loop-suspension}
\begin{tikzcd}
\Sigma^{\infty}_{\mr{G}}:{\bf Ho}(\Top^{\mr{G}}_*) \rar[shift right] &  {\bf Ho}(\Sp^{\mr{G}}) : \Omega^{\infty}_{\mr{G}} \lar[shift right],
\end{tikzcd}
\end{equation}

which always exists between homotopy categories. However, to expose a construction of $\mr{G}$-equivariant Picard spectra (in <Ref>), the authors  find the modern infinity apparatus convenient. Therefore, we set

* $\Top^{\mr{G}}$ – the $\infty$-category associated with simplicial model category of topological $\mr{G}$-spaces,

* $\Top^{\mr{G}}_*$  – the $\infty$-category of pointed topological $\mr{G}$-spaces, and

*   $\Sp^{\mr{G}}$ – the stable $\infty$-category of genuine $\mr{G}$-spectra.

We will use $ [- , -]^{\mr{G}} $ to denote the the homotopy class of $\mr{G}$-equivariant maps (as usual) in
$\Top^{\mr{G}}_*$  as well as $\Sp^{\mr{G}}$.

Throughout the paper, $\Top^{\mr{G}}$ denotes the $\infty$-category associated with simplicial model category of topological $\mr{G}$-spaces,  $\Top^{\mr{G}}_*$  denotes the $\infty$-category of pointed topological $\mr{G}$-spaces, and  $\Sp^{\mr{G}}$ denotes the stable $\infty$-category of genuine $\mr{G}$-spectra. Let

\[
\begin{tikzcd}
\Sigma^{\infty}_{\mr{G}}:{\bf Ho}(\Top^{\mr{G}}_*) \rar[shift right] &  {\bf Ho}(\Sp^{\mr{G}}) : \Omega^{\infty}_{\mr{G}} \lar[shift right]
\end{tikzcd}
\]

denote the usual loop-suspension adjoint pair of functors between their homotopy categories. For a pair of objects $\mr{X}, \mr{Y}$ in
either $\Top^{\mr{G}}_*$ or $\Sp^{\mr{G}}$, $[\mr{X}, \mr{Y}]^{\mr{G}}$ will denote the homotopy class of $\mr{G}$-equivariant map (as usual).

§.§ Acknowledgement.

This paper has  greatly benefitted  from many insightful exchanges with J.P. May on the topic of equivariant orientation theory.  We have also benefitted from our conversations with Vigleik Angelveit, Agnes Beaudry, Hood Chatham, Christy Hazel, Mike Hill, Clover May,  Mike Mandell, and Doug Ravenel.

The first author would also like to thank Tata Institute of Fundamental Research at Mumbai for their hospitality during the period when a part of this work was done.

§.§ Organization of the paper
In <Ref>, we first expose the notion of homogeneity, discuss examples, and prove <Ref>. Then we introduce the concept of relative homogeneity, coordinate of homogeneity,  and define $\mr{R}$-orientation of equivariant bundles, with respect to an $\mathbb{E}_\infty$-ring spectrum $\mr{R}$. We end this section with a proof of <Ref>.

In <Ref>, we define the equivariant generalization of Stiefel–Whitney class, and prove <Ref>, <Ref>, <Ref>, <Ref>, <Ref>, <Ref>  and <Ref>.

The purpose of  <Ref> is to record the Mackey functor structure of the units of the Burnside Tambara functor $\ull{\cal{A}}_{\mr{G}}^{\times}$ for a few familiar abelian $2$-groups.

§ EQUIVARIANT ORIENTATION THEORY

A $\mr{G}$-equivariant vector bundle $\upxi$ over a base space $\mr{B}$ consists of a map $\uppi_{\upxi}: \mr{E} \longrightarrow \mr{B}$ in $\Top_{\mr{G}}$ such that the fiber $\mr{E}_b$ over $b \in \mr{B}$ is a real vector space, and the fiberwise action of  ${\sf g} \in \mr{G}$

\[
\begin{tikzcd}
{\sf g}: \mr{E}_{b} \rar & \mr{E}_{{\sf g} \cdot  b}
\end{tikzcd}
\]

is a linear isomorphism.

Two $\mr{G}$-equivariant vector bundles, $\upxi_1$ and $\upxi_2$,  are
genuinely stably equivalent  if
there exists some $\mr{G}$-representation $\mr{V}$ such that

\[ \upxi_1 \oplus \upepsilon_{\mr{V}} \cong  \upxi_2 \oplus \upepsilon_{\mr{V}},\]

where $\upepsilon_{\mr{V}}$ is the trivial bundle $\mr{B} \times \mr{V}$ over $\mr{B}$.

Genuine stable equivalence classes of vector bundles are classified by the infinite loop space of the genuine $\mr{G}$-equivariant real $\mr{K}$-theory

\begin{equation} \label{repnkoG}
\{ \text{$\mr{G}$-equivariant vector bundle over $\mr{B}$} \} /(\sim_{\mr{G}}) \cong [\mr{B}_+,\Omega^{\infty}_{\mr{G}} \ko_{\mr{G}}]^{\mr{G}},
\end{equation}

where $\sim_{\mr{G}}$ stands for genuine stable equivalence <cit.>.

Let  ${\sf f}_{\upxi}: \mr{B} \longrightarrow \Omega_{\mr{G}}^{\infty}\ko_{\mr{G}}$ denote the
classifying map  (in the homotopy category of $\Top_*$) for a
$\mr{G}$-equivariant bundle $\upxi$ over $\mr{B}$. We will use the same notation
to  denote the corresponding map of spectra under the  adjunction (<ref>).

When $\mr{G}$ is the trivial group, the map (in $\Set$)

\begin{equation} \label{eqn:dimconst}
\begin{tikzcd}
\pi_0({\sf f}_{\upxi}) : \{ \text{path components of $\mr{B}$}\} \rar & \ZZ
\end{tikzcd}
\end{equation}

assigns each path component the dimension of $\upxi$ over it.
In order to identify an orientation  using a single Thom class, we require $\pi_0({\sf f}_{\upxi})$ to be  a constant map.

We will now introduce the notion of homogeneity for $\mr{G}$-equivariant vector bundles,  which  is essentially a equivariant generalization of   “${\sf f}_{\upxi}$ inducing a constant map on the zeroth homotopy”.

§.§ Homogeneity of vector bundles

When $\mr{G}$ is a  nontrivial group,  there are a variety of morphisms in
$\cal{O}_{\mr{G}}$-$\Set$ which can be regarded as constant maps. In order to
describe them, we first recall that a set $\cal{F}$ consisting of subgroups of
$\mr{G}$ is called a family if it is closed under inclusions up to
conjugations. For a family $\cal{F}$, define the $\cal{O}_{\mr{G}}$-$\Set$

\[
{\sf c}_{\cal{F}}(\mr{G}/\mr{H}) = \left\lbrace
\begin{array}{ccccc}
\ast  & \text{if $\mr{H} \in \cal{F}$} \\
\emptyset & \text{otherwise.}
\end{array}
\right.
\]

An $\cal{F}$-point of an  $ \cal{O}_{\mr{G}}$-$\Set$  $\mr{X}$ is an $\cal{O}_{\mr{G}}$-map

\[
\begin{tikzcd}
\mr{p}: {\sf c}_{\cal{F}} \rar & \mr{X}.
\end{tikzcd}
\]

Thus, an $\cal{F}$-point $\mr{p}$ consists of  a collection of points $\mr{p}_{\mr{H}}\in \mr{X}(\mr{G}/\mr{H}) $ for  each $\mr{H} \in \cal{F}$, such that $ \mr{X}(\alpha)(\mr{p}_{\mr{H}_1}) =\mr{p}_{\mr{H}_2}$ whenever $\mr{H}_1, \mr{H}_2 \in \cal{F}$ and $\alpha:\mr{G}/\mr{H}_2 \to \mr{G}/\mr{H}_1$ is a morphism in $\cal{O}_{\mr{G}}$.

Let $\All_{\mr{G}}$ denote the family consisting of all subgroups of $\mr{G}$.

For a $\mr{G}$-equivariant space $\mr{B}$, without a designated basepoint, we let

\[
\begin{tikzcd}
\ull{\pi}_0(\mr{B}): \mathcal{O}_{\mr{G}}^{\mr{op}} \rar & \Set
\end{tikzcd}
\]

be the $\mathcal{O}_{\mr{G}}$-set which assigns $\mr{G}/\mr{H}$ the set of path components of $\mr{B}^{\mr{H}}$ for any subgroup $\mr{H}$ of $\mr{G}$.

For a $\mr{G}$-spectrum $\mr{R} \in \Sp^{\mr{G}}$, the $\mathcal{O}_{\mr{G}}$-set
$\ull{\pi}_0(\Omega^{\infty}_{\mr{G}} \R)$ (defined after forgetting the basepoint of $\Omega^{\infty}_{\mr{G}} \R$) has a Mackey functor structure as it is isomorphic to   the zeroth homotopy group of $\R$ given by

\[ \ull{\pi}_0(\R)(\mr{G}/\mr{H}):= [\mathbb{S}_{\mr{G}} \sma (\mr{G}/\mr{H})_+, \R]^{\mr{G}},\]

for any subgroup $\mr{H}$ of $\mr{G}$.

We say a $\mr{G}$-vector bundle  $\upxi$ is  homogeneous  if $\underline{\pi}_0({\sf f}_{\upxi})$ admits a factorization

\begin{equation} \label{eqn:factor}
\begin{tikzcd}
&& {\sf c}_{\All_{\mr{G}}} \dar["\mr{p}"] \\
\underline{\pi}_0(\mr{B}) \ar[rru, dashed] \ar[rr,"\ull{\pi}_0 ({\sf f}_{\upxi})"'] &&\underline{\pi}_0( \Omega^{\infty}_{\mr{G}} \ko_{\mr{G}}),
\end{tikzcd}
\end{equation}

through   an $\All_{\mr{G}}$-point $\mr{p}$ of $\underline{\pi}_0\ko_{\mr{G}}$. We then call $\mr{p}$ a coordinate of homogeneity of $\upxi$.

It follows from the  definition of $\ko_{\mr{G}}$   (see  (<ref>))
that  its zeroth homotopy is the real representation Mackey functor
$\underline{\RO}_{\mr{G}}$ given by
\[
\underline{\RO}_{\mr{G}} (\mr{G}/\mr{H}) \cong \RO(\mr{H})
\]

for any subgroup $\mr{H} \subset \mr{G}$. Note that  an $\All_{\mr{G}}$-point $\mr{p}$ of $\underline{\pi}_0(\Omega^{\infty}_{\mr{G}}\ko_{\mr{G}})$ is determined by $\mr{p}_{\mr{G}} \in \RO(\mr{G})$ as $\mr{p}_{\mr{H}}$ is simply the restriction of $\mr{p}_\mr{G}$ to the subgroup $\mr{H}$ of $ \mr{G}$.  Therefore, we declare  $\mr{p}_{\mr{G}}$ as the dimension

\[
\dim (\upxi, \mr{p}) := \mr{p}_{\mr{G}}
\]

of the pair $(\upxi, \mr{p})$.

[Geometric interpretation]
For a homogeneous $\mr{G}$-equivariant vector bundle $\upxi$ whose coordinate of homogeneity is $\mr{p}$, the fiber over any point in $\mr{B}^{\mr{H}}$ is an $\mr{H}$-representation representing the class  $\mr{p}_{\mr{H}} \in \RO(\mr{H})$ (also see [10]).

The coordinate of homogeneity of a $\mr{G}$-equivariant bundle may not be unique.  Such situations may arise, when the $\mr{H}$-fixed points of the base space is empty for some subgroup $\mr{H} \subset \mr{G}$ as demonstrated by the following example.
Let $\upxi$ denote the $\mr{C}_2$-equivariant bundle

\[
\begin{tikzcd}
\uppi_{\upxi}:
\mr{S}(2 \upsigma) \times \upsigma \rar &
\mr{S}(2 \upsigma),
\end{tikzcd}
\]

where $\upsigma$ is the sign representation of $\mr{C}_2$. Then  $\underline{\pi}_0({\sf f}_{\upxi})$ induces a map of $\cal{O}_{\mr{C}_2}$-$\Set$ given by the diagram

\[
\begin{tikzcd}
\emptyset \rar \dar & \RO(\mr{C}_2) \dar["\res"]  \\
\{ \ast \} \rar & \ZZ
\end{tikzcd}
\]

that maps $\ast$ to  $1$. Thus, there are infinitely many coordinates of homogeneity.

§.§ Examples of homogeneous bundles

Products and pullbacks of homogeneous bundles are homogeneous, and therefore, the interest lies in knowing which among the universal examples of  $\mr{G}$-equivariant bundles defined in (<ref>) are homogeneous.
We begin by recalling a result of  Lashof and May  [16] which identifies all $\mr{G}$-connected components of the base space of these bundles. We also sketch a proof following [16] as the details are necessary in the  proof of <Ref>.

For a subgroup $\mr{H}$ of $\mr{G}$, the $\mr{H}$-fixed
points of $\mr{B}_{\mr{G}} \Pi$  is equivalent to

\begin{equation} \label{eqn:fixequiv}
(\mr{B}_{\mr{G}} \Pi)^{\mr{H}} \simeq \bigsqcup_{[\uptheta] \in \Theta} \mr{B}\cal{Z} (\uptheta)
\end{equation}

* $\Theta$ is the set of $\Pi$-conjugacy classes of group homomorphism $\uptheta: \mr{H} \to \Pi$,

* $\cal{Z} (\uptheta)$ is the subgroup  of $\Pi$ consisting of elements that commute with the image
of $\uptheta$, which, up to conjugation, is independent of the choice of
the representative $\uptheta$.

Let $\Lambda_{\uptheta} := (\uptheta(h),h) $ denote the graph subgroup of the group homomorphism $\uptheta: \mr{H} \to \Pi$. The universal property of $\mr{E}_{\mr{G}}\Pi$ implies

\begin{equation}
\label{eq:universal}
(\mr{E}_{\mr{G}}\Pi)^{\Lambda} \simeq
\begin{cases}
* & \Lambda = \Lambda_{\uptheta} \text{ for some } \mr{H},\uptheta\\
\emptyset & \text{ o.w. }
\end{cases}
\end{equation}

\[ \uppi: \mr{E}_{\mr{G}}\Pi \longrightarrow  \mr{B}_{\mr{G}}\Pi\]

is the universal $\mr{G}$-equivariant principal $\Pi$ bundle.
The fiber $\uppi^{-1}(b)$ is a $\Pi \times \mr{H}$-space for each point $b \in  (\mr{B}_{\mr{G}}\Pi)^{\mr{H}}$.

Notice that for each point $z \in \uppi^{-1}(b)$, the isotropy subgroup is precisely $\Lambda_{\uptheta}$ for some

\begin{equation} \label{eqn:assohomo}
\begin{tikzcd}
\uptheta: \mr{H} \rar &  \Pi.
\end{tikzcd}
\end{equation}

It is easy to check that the $\Pi$-conjugacy  class  $[\uptheta]$ is independent of the choice of $z$, and therefore,
is constant on each of the connected components of $(\mr{B}_{\mr{G}}\Pi)^{\mr{H}}$. Consequently, we get a map

\begin{equation} \label{eqn:alpha}
\begin{tikzcd}
\upalpha:\pi_0^{\mr{H}}(\mr{B}_{\mr{G}}\Pi) \rar & \Theta,
\end{tikzcd}
\end{equation}

which is a bijection because of (<ref>).

Let $\mr{B}$ be a connected component of $(\mr{B}_{\mr{G}}\Pi)^{\mr{H}}$, and let $\uptheta \in \upalpha([\mr{B}])$. Using (<ref>), one concludes  that  $(\uppi^{-1}(\mr{B}))^{\Lambda_{\uptheta}}$ is a contractible space with a free action of $\cal{Z}(\uptheta)$. Therefore, $(\uppi^{-1}(\mr{B}))^{\Lambda_{\uptheta}} \longrightarrow \mr{B}$ is a universal $\cal{Z}(\uptheta)$-bundle and $\mr{B} \simeq \mr{B}\cal{Z}(\uptheta)$.

Given a $(\mr{G}\times \Pi)$-space  $\mr{F}$, one can construct a $\mr{G}$-equivariant fiber bundle $\uplambda_{\mr{F}}$ whose projection map

\[
\begin{tikzcd}
\uppi_{\uplambda_{\mr{F}}}: \mr{E}_{\mr{G}}\Pi \times_{\Pi} \mr{F} \rar & \mr{B}_{\mr{G}} \Pi
\end{tikzcd}
\]

is induced by the collapsing map from $\mr{F}$ to a point.

\begin{tikzcd}
{\sf a}_{\mr{F}} : \Pi \times \mr{G} \rar & \mr{Aut}(\mr{F})
\end{tikzcd}
is the action map on $\mr{F}$, $\mr{H}$ is a subgroup of $\mr{G}$ and $b \in (\mr{B}_{\mr{G}} \Pi)^{\mr{H}}$, then the action of $\mr{H}$ on $\mr{F}' := \uppi_{\uplambda_{\mr{F}}}^{-1}(b) \cong \mr{F}$ is given by the composite

\begin{equation} \label{eq:modifyaction}
\begin{tikzcd}
{\sf a}_{\mr{F}'}:\mr{H} \ar[rr,"\text{$(\uptheta, \iota)$}"] && \Pi \times \mr{G} \rar["{\sf a}_{\mr{F}}"] & \mr{Aut}(\mr{F}) \rar[cong] & \mr{Aut}(\mr{F}'),
\end{tikzcd}
\end{equation}

where $\iota: \mr{H} \hookrightarrow \mr{G}$ is the inclusion of $\mr{H}$ into $\mr{G}$ and $\uptheta \in \upalpha(\text{\emph{component of $b$}})$, for $\upalpha$ defined in (<ref>).

Fix $z \in \uppi^{-1}(b)$, and let $\uptheta$ be the corresponding group homomorphism  as in  (<ref>). It follows from the construction of $\uptheta$ that

\[ h \cdot z = \uptheta(h)^{-1} \cdot z \]

for all $h \in \mr{H}$.
Let ${\sf r}$ denote the composite

\[
\begin{tikzcd}
{\sf r} :\mr{F}' := \uppi_{\uplambda_{\mr{F}}}^{-1}(b) \rar["\cong", "{\sf r}_1"'] &\uppi^{-1}(b) \times_{\Pi} \mr{F}  \rar["\cong", "{\sf r}_2"'] & \mr{F}
\end{tikzcd}
\]

and  set the last isomorphism in (<ref>) as conjugation by ${\sf r} $.
Then the result follows from the observation that

\begin{eqnarray*}
h \cdot x &=& {\sf r}^{-1} (h \cdot {\sf r}( x))\\
&=& {\sf r}^{-1}({\sf r}_2(h \cdot {\sf r}_1(x)  ))  \\
&=& {\sf r}^{-1}({\sf r}_2(  h \cdot [z,{\sf r}(x)]  ))  \\
&=& {\sf r}^{-1}({\sf r}_2( [\uptheta(h)^{-1} \cdot z, h \cdot {\sf r}(x)] ) ) \\
&=& {\sf r}^{-1}({\sf r}_2([z, \uptheta(h)h \cdot {\sf r}(x)  ])) \\
&=& {\sf r}^{-1} (\uptheta(h)h \cdot {\sf r}(x) )
\end{eqnarray*}

for all $h \in \mr{H}$.

Let  $\mr{F}= \mr{U} \otimes \mr{W}$, where
$\mr{U}$ and $\mr{W}$   are real representations of
$\mr{G}$ and  $\Pi$  respectively.  Let $b \in (\mr{B}_{\mr{G}} \Pi)^{\mr{H}}$, and let

\[
\begin{tikzcd}
\uptheta: \mr{H} \rar & \Pi
\end{tikzcd}
\]

be a group homomorphism such that $[\uptheta] = \upalpha(\text{component of $b$})$. Then it follows from
<Ref> that

\[
\uppi_{\uplambda_{\mr{F}}}^{-1}(b) \cong  \uptheta^*(\mr{W}) \otimes  \Res_{\mr{H}}^{\mr{G}} \mr{V}
\]

as an $\mr{H}$-representation.

The regular representation  $\uprho$ of any compact Lie group group  has the property that

\[ \uprho \otimes \mr{W} = \bigoplus_{\dim \mr{W}} \uprho  \]

for a finite dimensional representation $\mr{W}$. Therefore, when $\mr{V}$ is a direct sum of regular representations of $\mr{G}$ then $\Res_{\mr{H}}^{\mr{G}} \mr{V}$ is a direct sum of regular representations of $\mr{H}$ and
$\uppi_{\uplambda_{\mr{F}}}^{-1}(b)$ is independent of the choice of $b$ for
all $\mr{H} \subset \mr{G}$. Hence, the result follows (also see <Ref>).

§.§ Relative homogeneity of vector bundles

The notion of homogeneity can be defined relative to any $\EE_\infty^{\mr{G}}$-ring spectrum $\mr{R}$ using the genuine stable $\mr{J}$-homomorphism $\mJJ_{\mr{G}}$ defined in <Ref>. For any $\mr{G}$-equivariant vector bundle $\upxi$ over $\mr{B}$, let ${\sf f}_{\upxi, \mr{R}}$ denote  the composite

\begin{equation} \label{vectrelR}
\begin{tikzcd}
{\sf f}_{\upxi, \mr{R}}: \mr{B} \rar["{\sf f}_{\upxi}"] & \Omega^{\infty}_{\mr{G}} \ko_{\mr{G}} \ar[rr, "\Omega^{\infty}_{\mr{G}}\mJJ_{\mr{G}}"] && \Omega^{\infty}_{\mr{G}} {\pic(\SS_{\mr{G}})} \ar[rr,"\Omega^{\infty}_{\mr{G}}\pic(\iota_{\mr{R}})"] &&\Omega^{\infty}_{\mr{G}} {\pic(\mr{R})},
\end{tikzcd}
\end{equation}

where $\pic(\SS_{\mr{G}})$ and $\pic(\R)$ are the picard spectrum of
$\SS_{\mr{G}}$ and $\R$ respectively (see <Ref>), and $\iota_{\R}$ is the unit map of $\R$.
A $\mr{G}$-equivariant vector bundle  $\upxi$ is  homogeneous relative to  $\R$  (abbrev.  $\R$-homogeneous) if $\underline{\pi}_0( {\sf f}_{\upxi, \mr{R}})$ factors

\begin{equation} \label{eqn:factor}
\begin{tikzcd}
&&& {\sf c}_{\All_{\mr{G}}} \dar["\mr{p}"] \\
\underline{\pi}_0(\mr{B}) \ar[rrru, dashed] \ar[rrr,"\underline{\pi}_0({\sf f}_{\upxi, \mr{R}})"'] &&&\underline{\pi}_0(  \pic(\R))
\end{tikzcd}
\end{equation}

through an $\All_{\mr{G}}$-point $\mr{p}$ of $\underline{\pi}_0 (\pic(\R))$.
We call $\mr{p}$ a coordinate of $\R$-homogeneity, and set
$\mr{p}_{\mr{G}}$ as the relative dimension

\[\dim_\R (\upxi, \mr{p}) :=  \mr{p}_{\mr{G}}
\]

of the pair $(\upxi, \mr{p})$ with respect to $\R$.

Any homogeneous bundle is automically $\R$-homogeneous for any $\EE_\infty^{\mr{G}}$-ring spectrum $\R$. Conversely,  an $\R$-homogeneous bundle $\upxi$ is homogeneous if and only if  the relative coordinate admits a lift along $\ull{\pi}_0(  \pic(\iota_\R) \circ \mJJ_{\mr{G}} )$

\[
\begin{tikzcd}
& \ull{\pi}_0 (\ko_{\mr{G}}) \ar[d, "\ull{\pi}_0( \pic(\iota_\R) \circ \mJJ_{\mr{G}} )"] \\
{\sf c}_{\All_{\mr{G}}} \ar[ur, dashed]  \ar[r, "\mr{p}"'] &  \ull{\pi}_0 \pic(\mr{R}).
\end{tikzcd}
\]

However, the map $\ull{\pi}_0\mJJ_{\mr{G}}$ of Mackey functors is not well understood for a general group  $\mr{G}$ (but see [1, 12] for partial answers).

[Equivariant Picard spectrum]   The recent work
of  [15]  refines the definition of the equivariant Picard group (see [12])
to construct the Picard spectrum $\pic(\SS_{\mr{G}})$. Their construction
involves foundations of parametrized $\infty$-category introduced in [4, 30].
Roughly speaking, a $\mr{G}$-$\infty$-category, or a coCartesian fibration

\[
\begin{tikzcd}
{\sf p}: \underline{\mathcal{C}} \rar & \mathcal{O}_{\mr{G}}^{\mr{op}},
\end{tikzcd}
\]
models genuine $\mr{G}$-objects. A $\mr{G}$-symmetric monoidal $\mr{G}$-$\infty$-category is a coCartesian fibration over $\FG$

\[
\begin{tikzcd}
{\sf p}: \underline{\mathcal{M}}^{\otimes} \rar & \FG
\end{tikzcd}
\]

satisfying the equivariant Segal condition <cit.>, where
the category $\FG$ itself is a coCartesian fibration  over
$\mathcal{O}_{\mr{G}}^{\mr{op}}$. The  fiber $\ull{\cal{M}}_{[\mr{G}/\mr{H}]}$ over
$\mr{G}/\mr{H}$ spanned by invertible objects assemble to form a
$\mr{G}$-$\infty$-groupoid $ \ull{\mr{Pic}}(\ull{\cal{M}}),$ called the Picard
groupoid of $\ull{\cal{M}}$. It inherits a $\mr{G}$-symmetric monoidal
structure from $\ull{\cal{M}}^{\otimes}$, and therefore
$\ull{\mr{Pic}}(\ull{\cal{M}})$  models an $\EE_\infty^{\mr{G}}$-space which
deloops to a genuine $\mr{G}$-spectrum using the functor $\mr{B}$ in <cit.> (which is a version  of a equivariant delooping machine originally introduced in [13]).
For an $\EE_{\infty}^{\mr{G}}$-ring spectrum $\R$, we let $\pic(\mr{R})$ denote the
$\mr{G}$-spectrum obtained by setting  $\ull{\cal{M}}$ to be  the category of $\R$-modules.

[Genuine stable $\mr{J}$-homomorphism]
Let $\cal{R}\mr{ep}$ denote the  symmetric monoidal topological category  whose objects are $\mathbb{R}^n$ and morphisms are $\bigsqcup_{n \in \mathbb{N}} \mr{Gl}_n(\mathbb{R})$. One can consider “topological $\mr{G}$-objects” in $\cal{R}\mr{ep}$ <cit.> and obtain a $\mr{G}$-symmetric monoidal $\mr{G}$-$\infty$-category

\[
\begin{tikzcd}
\ull{\cal{R}\mr{ep}}^{\otimes} \rar &  \ull{\mathrm{Fin}}_{*}^{\mr{G}}
\end{tikzcd}
\]

which deloops to  $\ko_{\mr{G}}$. Explicitly, the fiber $\infty$-category
$\ull{\cal{R}\mr{ep}}_{[\mr{G/H}]}$ is equivalent to the nerve of
the category of finite dimensional $\mr{H}$-representations and $\mr{H}$-equivariant
linear isomorphisms.
By sending an $\mr{H}$-representation $\mr{V}$ to $\Sigma^{\infty}_{\mr{H}}\mr{S}^{\mr{V}}$ via one-point compactification, we obtain the $\mr{G}$-equivariant $\mr{J}$-homomorphism $\mJJ_{\mr{G}}$ (also see <cit.>).

§.§ Orientation  and Thom isomorphism

The classifying map of a  $\mr{G}$-equivariant vector bundle $\upxi$ can be regarded as a map of $\mr{G}$-$\infty$-groupoids

\[
\begin{tikzcd}
\mr{F}_{\upxi}:\mr{B} \rar & \ull{\cal{R}\mr{ep}}_{\mr{G}},
\end{tikzcd}
\]

where $\mr{B}$ is the  fundamental $\mr{G}$-$\infty$-groupoid of the base space of $\upxi$. In fact,  ${\sf f}_{\upxi}$  (as in  <Ref>) is  the homotopy class of $\mr{F}_{\upxi}$. Likewise, the map ${\sf f}_{\upxi, \R}$ defined in (<ref>) can also be constructed as a map  of $\mr{G}$-$\infty$-groupoids

\[
\begin{tikzcd}
\mr{F}_{\upxi, \mr{R}}: \mr{B} \rar & \ull{\cal{R}\mr{ep}}_{\mr{G}}  \rar & \ull{\mr{Pic}}(\Sp^{\mr{G}}) \rar & \ull{\mr{Pic}}(\mr{R}\text{-}\mr{Mod}) .
\end{tikzcd}
\]

An $\R$-orientation of  a $\mr{G}$-equivariant vector bundle $\upxi$  is a homotopy between the composite $\mr{F}_{\upxi, \mr{R}}$ and
a constant map.

The existence of an $\R$-orientation of a $\mr{G}$-equivariant bundle $\upxi$
automatically implies $\R$-homogeneity. If ${\sf c}$ denote a constant map equivalent to $\mr{F}_{\upxi, \R}$,

\[
\begin{tikzcd}
\ull{\pi}_0({\sf c}): \ull{\pi}_0(\mr{B})  \rar & \underline{\pi}_0( \Omega^{\infty}_{\mr{G}} \pic(\R))
\end{tikzcd}
\]

serves as a coordinate of $\R$-homogeneity of $\upxi$.

The  $\mr{R}$-Thom spectrum  of $\upxi$ is defined as the colimit

\begin{equation} \label{eqn:RThom}
\mr{Th}(\upxi, \mr{R}) := \Gcolim_{ \ \ \mr{B}} \ \mr{F}_{\upxi, \mr{R}}
\end{equation}

as in <cit.>.
The commuting diagram of $\mr{G}$-$\infty$-categories

\[
\begin{tikzcd}
\ull{\mr{Pic}}(\Sp^{\mr{G}}) \rar \dar & \ull{\mr{Pic}}( \mr{R}\text{-}\mr{Mod}) \dar \\
\Sp^{\mr{G}} \rar["-\sma \R"'] &  \mr{R}\text{-}\mr{Mod}
\end{tikzcd}
\]

means $\mr{Th}(\xi, \mr{R})$ is weakly equivalent to $\mr{Th}(\upxi, \SS_{\mr{G}}) \sma \mr{R}$.

We  abbreviate  $\mr{Th}(\upxi) := \mr{Th} (\upxi, \SS_{\mr{G}})$.

Let $\cal{I}$ denote the dimension of $\upxi$ relative to $\R$.  The proof of  $\textit{(1)} \Leftrightarrow \textit{(2)}$ is essentially a tautology.

The proof of $\textit{(2)} \Leftrightarrow \textit{(3)}$ is classical:
An $\R$-Thom class ${\sf u}_{\upxi} $ leads to an $\R$-Thom isomorphism

\[
\begin{tikzcd}
\omega: \mr{Th}(\upxi) \sma \R \ar[rr, "\Delta \sma 1_{\R}"] && \mr{Th}(\upxi) \sma \mr{B}_+ \sma \R \rar["(2)"] & \cal{I}  \sma \mr{B}_+ \sma \R \rar["(3)"] & \mr{B}_+ \sma \cal{I} ,
\end{tikzcd}
\]

where $\Delta$ is the Thom diagonal map, the map $(2)$ is induced by  ${\sf u}_{\upxi}$, and the map $(3)$ is induced by the $\mr{R}$-module structure of $\cal{I}$. Conversely, an $\R$-Thom isomorphism  $\omega$ leads to an $\R$-Thom-class

\[
\begin{tikzcd}
{\sf u}_{\upxi}:\mr{Th}(\upxi)  \ar[r, "(4)"] & \mr{Th}(\upxi) \sma   \R \rar["\omega"] &  \mr{B}_+ \sma \cal{I}  \rar["(6)"] & \cal{I},
\end{tikzcd}
\]

where the map $(4)$ is induced by the unit of $\R$, and the map $(6)$ collapses $\mr{B}$ to a point.

§ THE FIRST EQUIVARIANT STIEFEL–WHITNEY CLASS

For an $\R$-homogeneous bundle $\upxi$ of relative dimension $\cal{I}$, the corresponding relative dimension zero virtual bundle  is classified by the map

\[
\begin{tikzcd}
{\sf f}_{\upxi, \R} - {\sf c}_{\mathcal{I}}: \mr{B}_+ \rar & {\pic(\R)},
\end{tikzcd}
\]

where  ${\sf c}_{\cal{I}}$ is the constant map at $\cal{I}$.  By construction, this map admits a lift

\begin{equation} \label{dimzero}
\begin{tikzcd}
&& \bgl(\R) \dar \\
\mr{B}_+ \ar[rr, "{\sf f}_{\upxi, \R} - {\sf c}_{\mathcal{I}}"']   \ar[urr, dashed, "{\sf f}^{(0)}_{\upxi, \R}"] && \text{$\pic(\R)$},
\end{tikzcd}
\end{equation}

where $\bgl(\R)$ is the $0$-connected cover of $\pic(\R)$.
For an $\R$-homogeneous $\mr{G}$-equivariant bundle $\upxi$  define its  first  Steifel Whitney class with respect to $\R$  as the homotopy class

\[ {\sf w}_1^{\R}(\upxi):= [{\sf f}^{(0)}_{\upxi, \R}] \in [\mr{B}_+, \bgl(\R)]^{\mr{G}}. \]

Let $\ull{\mr{BGL}}( \mr{R}\text{-}\mr{Mod})$ denote the sub $\mr{G}$-$\infty$-groupoid of  $\ull{\mr{Pic}}( \mr{R}\text{-}\mr{Mod})$ generated by $\R$. In fact, $\ull{\mr{BGL}}( \mr{R}\text{-}\mr{Mod})$ is symmetric monoidal and deloops to a spectrum, which we call $\bgl(\R)$. It is easy to see that $\bgl(\R)$ is the fiber of the zeroth Postnikov approximation map

\[
\begin{tikzcd}
{\pic(\R)} \rar & \mr{H} \ull{\pi}_0 \pic(\R),
\end{tikzcd}
\]

and  hence, it is the $0$-connected cover of $\pic(\R)$.

[Uniqueness of ${\sf w}_1^{\R}(-)$]
The homotopy class of the map ${\sf f}^{(0)}_{\upxi, \R}$ is well-defined as the indeterminacies lie in $\mr{H}^{-1}(\mr{B}; \ull{\pi}_0\pic(\R))$  (see <Ref>), which is a trivial group for any $\mr{G}$-equivariant space $\mr{B}$.

By definition,  ${\sf w}_1^{\R}(\upxi)$ is the obstruction to $\R$-orientability of an $\R$-homogeneous bundle $\upxi$.

§.§   $\mr{H}\ull{\ZZ}$-orientability of homogenous bundles

When $\R = \mr{H}\ull{\ZZ}$ then $\bgl(\R) \simeq \Sigma\mr{H} \ull{\FF}_2$. Therefore, the Stiefel–Whitney class ${\sf w}_1^{\ull{\ZZ}}(\upxi)$ for an $\mr{H}\ull{\ZZ}$-homogeneous bundle $\upxi$ is an element in

\[
{\sf w}_1^{\ull{\ZZ}}(\upxi) \in \mr{H}^1(\mr{B}; \ull{\FF}_2),
\]

where $\mr{B}$ is the base space of $\upxi$.  Thus, when $\mr{H}^1(\mr{B}; \ull{\FF}_2) = 0$, $\upxi$ is automatically $\mr{H}\ull{\ZZ}$-oriented.

Using the additivity formula  (<ref>), we see that

\[ {\sf w}_1^{\ull{\ZZ}}( \upxi \oplus \upxi) = {\sf w}_1^{\ull{\ZZ}}( \upxi) + {\sf w}_1^{\ull{\ZZ}}( \upxi) = 2 {\sf w}_1^{\ull{\ZZ}}( \upxi) = 0    \]

as $\mr{H}^1(\mr{B}; \ull{\FF}_2)$ is an $\FF_2$-vector space.

§.§   $\mr{H}\ull{\cal{A}}_{\mr{G}}$-orientability of homogenous bundles

For any finite group $\mr{G}$, the units of the Burnside Mackey functor
$\ull{\cal{A}}_{\mr{G}}^{\times}$ consist of only simple $2$-torsions [18] (also see <Ref>). Therefore, $\mr{H}^\star_{\mr{G}}(\mr{B}; \ull{\cal{A}}_{\mr{G}}^{\times})$ is a graded vector space.

Using  the additivity condition (<ref>), we conclude

\[ {\sf w}_1^{\ull{\cal{A}}_{\mr{G}}}( \upxi \oplus \upxi) = {\sf w}_1^{\ull{\cal{A}}_{\mr{G}}}( \upxi) + {\sf w}_1^{\ull{\cal{A}}_{\mr{G}}}( \upxi) = 2 {\sf w}_1^{\ull{\cal{A}}_{\mr{G}}}( \upxi) = 0    \]

for any $\mr{H}\ull{\mathcal{A}}_{\mr{G}}$-homogeneous vector bundle. The  result then follows from <Ref>.

A Tambara functor $\mathcal{T}$ is an
$\ull{\cal{A}}_{\mr{G}}$-algebra. Therefore, one may wonder  if <Ref> implies $2$-fold direct sum of an $\mr{H}\mathcal{T}$-homogeneous
bundle is always $\mr{H}\mathcal{T}$-orientable. The statement is true when $\mr{G}$ is trivial. However, the statement does not immediately generalize to nontrivial groups  because an $\mr{H}\cal{T}$-homogeneous bundle need not be $\mr{H}\ull{\cal{A}}_{\mr{G}}$-homogeneous. However, the authors are not aware of any counterexample.

The natural map $
\upiota: \ull{\cal{A}}_{\mr{G}} \longrightarrow \ull{\ZZ}
$ of Tambara functors
induces a map of Mackey functors   on the units

\begin{equation} \label{AGtoZunit}
\begin{tikzcd}
\upiota^{\times}: \ull{\cal{A}}_{\mr{G}}^{\times} \rar & \ull{\ZZ}^{\times} \cong \ull{\FF}_2
\end{tikzcd}
\end{equation}

which is an isomorphism when $|\mr{G}|$ is odd (see <Ref>). Consequently,  an $\ull{\cal{A}}_{\mr{G}}$-homogeneous bundle is $\ull{\cal{A}}_{\mr{G}}$-orientable if and only if  it is $\mr{H}\ull{\ZZ}$-orientable.

When  $|\mr{G}|$ is even,  (<ref>) is not an isomorphism, and  therefore,

\[ \ull{\cal{A}}_{\mr{G}}^{\circ} := \ker(\upiota^{\times})\]

is a nontrivial Mackey functor (see <Ref>). Note that $\Sigma\mr{H}\ull{\cal{A}}_{\mr{G}}^{\circ}$ is the fiber of  the map

\[
\begin{tikzcd}
\bgl(\upiota): \Sigma \mr{H } \ull{\cal{A}}_{\mr{G}}^{\times} \simeq \bgl(\mr{H}\ull{\cal{A}}_{\mr{G}})  \ar[rr] && \bgl( \mr{H}\ull{\ZZ}) \simeq \Sigma\mr{H} \ull{\FF}_2.
\end{tikzcd}
\]

Thus, when an $\mr{H}\ull{\cal{A}}_{\mr{G}}$-homogeneous  $\mr{G}$-equivariant
vector bundle $\upxi$  is $\mr{H}\ull{\mathbb{Z}}$-orientable,  we have a lift

\begin{equation} \label{fiberg}
\begin{tikzcd}
& \mr{B}_+ \ar[dl, dashed, bend right ,"{\sf f}^{{\sf g}}_{\upxi, \mr{H} \ull{\cal{A}}_{\mr{G}} } "' ]
\ar[d, "{\sf f}^{(0)}_{\upxi, \mr{H} \ull{\cal{A}}_{\mr{G}} } "']
\ar[drr, "{\sf f}^{(0)}_{\upxi, \mr{H} \ull{\ZZ} }"]  \\
\Sigma\mr{H} \ull{\cal{A}}_{\mr{G}}^{\circ} \rar & \bgl(\mr{H} \ull{\cal{A}}_{\mr{G}}) \ar[rr]  && \bgl(\mr{H} \ull{\ZZ})
\end{tikzcd}
\end{equation}

because ${\sf f}^{(0)}_{\upxi, \mr{H} \ull{\ZZ} } \simeq 0$. The map ${\sf f}^{{\sf
g}}_{\upxi, \mr{H} \ull{\cal{A}}_{\mr{G}} }$ is not unique. In fact the homotopy class of ${\sf f}^{{\sf g}}_{\upxi, \mr{H} \ull{\cal{A}}_{\mr{G}} }$ can be varied by elements in the image of
\[
\begin{tikzcd}
\updelta_*: \mr{H}^0(\mr{B}; \ull{\mathbb{F}}_2) \rar & \mr{H}^1(\mr{B}; \ull{\cal{A}}_{\mr{G}}^{\circ}),
\end{tikzcd}
\]

where $\updelta:\mr{H}\ull{\mathbb{F}}_2 \longrightarrow \Sigma\mr{H}\ull{\cal{A}}_{\mr{G}}^{\circ}$ is the connecting map of the  fiber sequence in (<ref>) (the bottom horizontal row). In other words, the homotopy classes of all possible ${\sf f}^{{\sf g}}_{\upxi, \mr{H} \ull{\cal{A}}_{\mr{G}} }$ form a coset of the subgroup $\mr{D} = {\sf im}(\updelta_*)$ within $\mr{H}^1(\mr{B}; \ull{\cal{A}}_{\mr{G}}^{\circ})$.

Let $\upxi$ be a $\mr{H}\ull{\cal{A}}_{\mr{G}}$-homogeneous  $\mr{G}$-equivariant vector bundle. Then a ghost of the first Stiefel–Whitney class ${\sf w}_1^{\ull{\mathbb{Z}}}(\upxi)$ is the homotopy class of any map ${\sf f}^{{\sf g}}_{\upxi, \mr{H} \ull{\cal{A}}_{\mr{G}} }$ that fits in the diagram (<ref>). Let

\[ \mathfrak{G}_1(\upxi) := \{ [{\sf f}^{{\sf g}}_{\upxi, \mr{H} \ull{\cal{A}}_{\mr{G}} }] \in \mr{H}^1(\mr{B}; \ull{\cal{A}}_{\mr{G}}^{\circ}) : \text{diagram  \eqref{fiberg} is homotopy commutative}\} \]

denote the $\mr{D}$-coset consisting of all possible ghosts of  the first Stiefel–Whitney class.

It follows  from
(<ref>) and <Ref> that when $0 \in \mathfrak{G}_1(\upxi)$
there is ${\sf w}_1^{\ull{\cal{A}}_{\mr{G}} }(\upxi) := [ {\sf f}^{(0)}_{\upxi, \mr{H} \ull{\cal{A}}_{\mr{G}} }]= 0$.

Next we will show that $0 \in \mathfrak{G}_1(\upxi)$ if $\mr{G}$ acts freely on the base space $\mr{B} $ of $\upxi$.

Let $\upxi$ denote an $\mr{H}\ull{\mathcal{A}}_{\mr{G}}$-homogeneous bundle and let $\mr{B}$ denote the base space of $\upxi$. Since there is an $\EE_\infty^{\mr{G}}$-ring map

\[
\begin{tikzcd}
\mr{H} \ull{\mathcal{A}}_{\mr{G}} \rar & \mr{H} \ull{\mathbb{Z}},
\end{tikzcd}
\]

an $\mr{H}\ull{\mathcal{A}}_{\mr{G}}$-orientation leads to an $\mr{H} \ull{\mathbb{Z}}$-orientation. We will now show that the converse is true when $\mr{G}$ acts freely on $\mr{B}$.

As $\mr{B}$ has a free action,  we get a principal  $\mr{G}$-bundle
\mr{p} :\mr{B} \longrightarrow \mr{B}/\mr{G}
whose classifying map is

\[
\begin{tikzcd}
f:\mr{B} \rar &  \mathrm{EG}
\end{tikzcd}
\]

on the total space. Notice that the composite

\[
\begin{tikzcd}
\mr{B} \rar["\Delta"] & \mr{B} \times \mr{B} \rar["1_{\mr{B}} \times f"] & \mr{B} \times \mr{EG}
\end{tikzcd}
\]

is a $\mr{G}$-equivalence. Therefore,

\begin{eqnarray*}
\mr{H}^n_{\mr{G}}(\mr{B}; \ull{\cal{A}}_{\mr{G}}^{\circ})
[\mr{B}_+, \Sigma^n\mr{H}\ull{\cal{A}}_{\mr{G}}^{\circ}]^{\mr{G}} \\
[(\mr{B} \times \mathrm{EG})_+, \Sigma^n\mr{H}\ull{\cal{A}}_{\mr{G}}^{\circ}]^{\mr{G}} \\
& \cong&  [\mr{B}_+, \Sigma^n\mathrm{F}(\mathrm{EG}_+, \mr{H}\ull{\cal{A}}_{\mr{G}}^{\circ})]^{\mr{G}}
\end{eqnarray*}

for any natural number $n \in \mathbb{N}$. By <cit.>,

\[
\begin{tikzcd}
\mathrm{F}(\mathrm{EG}_+, \mr{H}\ull{\cal{A}}_{\mr{G}}) \rar &
\mathrm{F}(\mathrm{EG}_+, \mr{H}\ull{\mathbb{Z}})
\end{tikzcd}
\]

is a $\mr{G}$-equivalence, which means $\mathrm{F}(\mathrm{EG}_+, \mr{H}\ull{\cal{A}}_{\mr{G}}^{\circ})\simeq_{\mr{G}} *$. This means $\mr{H}^n(\mr{B}; \ull{\cal{A}}_{\mr{G}}^{\circ}) \cong 0$ for all $n \in \mathbb{N}$,  and in particular,
$\mathfrak{G}_1(\upxi) = \{ 0\}$.  Thus, the result follows from <Ref>.

Any Atiyah Real bundle of a given dimension is  a homogeneous $\mr{C}_2$-equivariant bundle, and therefore it is also a $\mr{H}\ull{\mathcal{A}}_{\mr{C}_2}$-homogeneous bundle (see <Ref>).

Recall that the $\mr{C}_2$-fixed points of the tautological Atiyah Real bundle $\hat{\upgamma}$ is the tautological line bundle $\upgamma_1$ over  $\mathbb{RP}^\infty$

\[ \hat{\gamma}^{\mr{C}_2} \cong \upgamma_1 .\]

Consequently, we have a commutative diagram

\[
\begin{tikzcd}
\mr{S}^1 \dar[hook, "\mr{V}_1"'] \rar[hook, "\mr{U}_1"] & \mathbb{RP}^{\infty} \dar[hook,"\mr{V}_2"'] \rar["\mr{U}_2"] &  \Omega^{\infty} \bo \dar[hook, "\mr{V}_3"]  \\
\hat{\mathbb{C}}\mathbb{P}^1 \rar[hook, "\mr{L}_1"] &  \hat{\mathbb{C}}\mathbb{P}^\infty \rar["\mr{L}_2"] & \Omega^{\infty}_{\mr{C}_2} {\sf br} ,
\end{tikzcd}
\]

where $\mr{V}_i$ are the inclusion of the fixed points, $\mr{U}_1$ and $\mr{L}_1$ are the skeletal inclusions, and,  $\mr{U}_2$ and $\mr{L}_2$ are the classifying maps for $\upgamma_1$ and $\hat{\upgamma}$ respectively. Note that

* $\mr{U}_2 \circ \mr{U}_1$ is the classifying map for the möbius bundle and therefore represents the generator in $\pi_1\bo \cong \mathbb{Z}/2$, and,

* $\mr{V}_{3\ast}: \pi_1\bo \longrightarrow \pi_1^{\mr{C}_2} {\sf br}$ is an isomorphism.

Consequently, the composition $\mr{L}_2 \circ \mr{L}_1 \circ \mr{V}_1$ represents the nonzero element
\[ \eta \in \pi_1^{\mr{C}_2} {\sf br} \cong \mathbb{Z}/2.\]

Greenlees <cit.> showed that
\[ \pi_1^{\mr{C}_2}( \bo_{\mr{C}_2}) \cong \mr{RO}(\mr{C}_2)/\mr{RU}(\mr{C}_2) \cong \mathbb{Z}/2\lbrace [1], [\upsigma] \rbrace,\]
and it is a standard fact that the natural map
$\Omega^{\infty}_{\mr{C}_2} {\sf br} \longrightarrow \Omega^{\infty}_{\mr{C}_2} \bo_{\mr{C}_2}$
sends $\eta$ to $[1 + \upsigma]$ at the level of fundamental groups. Since the $\mr{J}$-homomorphism induces  an isomorphism $\ull{\pi}_1(\bo_{\mr{C}_2}) \cong \ull{\pi}_1( \bgl(\mathbb{S}_{\mr{C}_2}))$
of Mackey functors, it follows that the composition

\[
\begin{tikzcd}
\mr{S}^1  \rar["\mr{L}_1 \circ \mr{V}_1"']   &  \hat{\mathbb{C}}\mathbb{P}^\infty \rar["\mr{L}_2"'] \ar[rrrr, bend left, dashed, "{\sf f}^{(0)}_{\hat{\upgamma}, \mr{H}\ull{\mathcal{A}}_{\mr{C}_2} }"'] & \Omega^{\infty}_{\mr{C}_2} {\sf br} \rar[] & \Omega^{\infty}_{\mr{C}_2} \bo_{\mr{C}_2} \rar[" \mJJ^{\circ}_{\mr{C}_2}"'] & \bgl(\mathbb{S}_{\mr{C}_2}) \rar["\simeq"']  & \bgl(\mr{H}\ull{\mathcal{A}}_{\mr{C}_2}),
\end{tikzcd}
\]

is not null. Thus, ${\sf w}_1^{\ull{\mathcal{A}}_{\mr{C}_2}}(\hat{\upgamma}) := [{\sf f}^{(0)}_{\hat{\upgamma}, \mr{H}\ull{\mathcal{A}}_{\mr{C}_2} }]$ cannot equal zero.

§.§ Nonexistence of $\mr{H}\ull{\mathcal{A}}_{\mr{G}}$-orientation of $\upgamma_{\uprho}$

The underlying nonequivariant bundle of $\upgamma_{\uprho}$ is isomorphic to  $
\upgamma_1^{\oplus |\mr{G}|}$, the
$|\mr{G}|$-fold direct sum of the tautological line bundle $\upgamma_1$ over
$\mathbb{RP}^{\infty}$. Therefore, the restriction map

\[
\begin{tikzcd}
{\sf res}^{\mr{G}}_{\sf e}: \mr{H}^{1}_{\mr{G}}(\mr{B}_{\mr{G}} \Sigma_2; \ull{\mathbb{F}}_2) \ar[rr] && {\mr{H}^{1}(\mathbb{RP}^\infty; \mathbb{F}_2) }
\end{tikzcd}
\]

sends the first  $\mr{G}$-equivariant Stiefel–Whitney class ${\sf w}_1^{\ull{\mathbb{Z}}}(\upgamma_{\uprho})$ to

\[
{\sf  w}_1^{\mathbb{Z}}(\upgamma_1^{\oplus |\mr{G}|})   = |\mr{G}|\cdot {\sf  w}_1^{\mathbb{Z}}(\upgamma_1) ,\]

which is nonzero when $|\mr{G}|$ is odd. Thus,  ${\sf w}_1^{\ull{\mathbb{Z}}}(\upgamma_{\uprho})$  must also be nonzero in this case, and by <Ref>, we conclude that $\upgamma_{\uprho}$ is not $\mr{H} \ull{\mathbb{Z}}$-orientable.

For a group $\mr{G}$ and its subgroup $\mr{H}$, let

\[
\begin{tikzcd}
\upiota_{\mr{H}}: \Top^{\mr{G}} \rar & \Top^{\mr{H}}
\end{tikzcd}
\]

denote the functor that restricts the action of $\mr{G}$ to $\mr{H}$. The right adjoint of $\upiota_{\mr{H}}$ is the functor $\mr{Map}_{\mr{H}}(\mr{G}, -)$ which converts an $\mr{H}$-equivariant space $\mr{X}$  to a $\mr{G}$-equivariant space  which consists of $\mr{H}$-equivariant maps from $\mr{G}$ to $\mr{X}$. Let

\[
\begin{tikzcd}
\upeta: \mathbbm{1}_{\Top^{\mr{G}}} \rar & \mr{Map}_{\mr{H}}(\mr{G}, \upiota_{\mr{H}}(-))
\end{tikzcd}
\]

denote the unit of this adjunction. We also use $\upiota_{\mr{H}}$ to denote the corresponding restriction functor $\Sp^{\mr{G}}$ to $\Sp^{\mr{H}}$.

Let $\mr{B}$ be  a $\mr{G}$-equivariant space   such that its restriction
$\upiota_{\mr{H}} (\mr{B})$ is the base space of the  $\mr{H}$-equivariant
bundle $\upxi$. Then $\mr{Map}_{\mr{H}}(\mr{G}, \upxi)$  is a
$\mr{G}$-equivariant bundle over  $\mr{Map}_{\mr{H}}(\mr{G},
\upiota_{\mr{H}}(\mr{B}))$.
We let

\[ {\sf Ind}_{\mr{H}}^{\mr{G}}(\upxi) := \upeta_{\mr{B}}^{\ast} \mr{Map}_{\mr{H}}(\mr{G}, \upxi) \]

denote the $\mr{G}$-equivariant bundle over $\mr{B}$ obtaining by
pulling back $\mr{Map}_{\mr{H}}(\mr{G}, \upxi)$ along the natural map $\upeta_{\mr{B}}: \mr{B} \longrightarrow \mr{Map}_{\mr{H}}(\mr{G}, \upiota_{\mr{H}}(\mr{B}))$.

Let $\R$ be an $\mathbb{E}^{\mr{G}}_{\infty}$-ring spectrum. Suppose $\upxi$ is an
$\mr{H}$-equivariant $\upiota_{\mr{H}}(\mr{R})$-homogeneous bundle over  $\upiota_{\mr{H}} (\mr{B})$ where $\mr{B}$ is a $\mr{G}$-equivariant space. Then

\[ {\sf w}_1^{\mr{R}}({\sf Ind}_{\mr{H}}^{\mr{G}}(\upxi)) = {\sf tr} ({\sf w}_1^{\upiota_\mr{H}(\mr{R})}(\upxi)), \]

where ${\sf tr}: [\upiota_{\mr{H}}(\mr{B})_+, \bgl(\upiota_{\mr{H}}(\mr{R})) ]^{\mr{H}} \longrightarrow [\mr{B}_+, \bgl(\mr{R})]^{\mr{G}} $ is the transfer map.

The classifying map of $\mr{Map}_{\mr{H}}(\mr{G},\upxi)$

corresponds to that of $\upxi$ under the isomorphism

\begin{eqnarray*}
[ \mr{Map}_{\mr{H}}(\mr{G}, \upiota_{\mr{H}}(\mr{B}))_+, \Omega_{\mr{G}}^{\infty} \ko_{\mr{G}}]^{\mr{G}} &\cong& [\Sigma^{\infty}_{\mr{G}} \mr{Map}_{\mr{H}}(\mr{G}, \upiota_{\mr{H}}(\mr{B}))_+,  \ko_{\mr{G}}]^{\mr{G}} \\
&\cong& [ \mr{Map}_{\mr{H}}(\mr{G}_+, \Sigma^{\infty}_{\mr{H}}\upiota_{\mr{H}}(\mr{B}_+)) , \ko_{\mr{G}}]^{\mr{G}} \\
&\cong & [  \Sigma^{\infty}_{\mr{H}} \upiota_{\mr{H}}(\mr{B})_+ , \ko_{\mr{H}}]^{\mr{H}}  \\
&\cong & [\upiota_{\mr{H}}(\mr{B})_+, \Omega_{\mr{H}}^{\infty} \ko_{\mr{H}}]^{\mr{H}},
\end{eqnarray*}

and its precomposition  with
$\upeta_{\mr{B}}: \mr{B}_+ \to \mr{Map}_{\mr{H}}(\mr{G},
\upiota_{\mr{H}}(\mr{B}))_+$ is the classifying map of ${\sf Ind}_{\mr{H}}^{\mr{G}}(\upxi)$.
However, the isomorphism above composed with  $\upeta_{\mr{B}}^{\ast}$ is   simply the transfer map

\[
\begin{tikzcd}
{\sf tr} : [ \upiota_{\mr{H}}(\mr{B})_+ , \ko_{\mr{H}}]^{\mr{H}} \rar
&  {[ \mr{B}_+, \ko_{\mr{G}} ]^{\mr{G}}, }
\end{tikzcd}
\]

of the Mackey functor $\ull{\ko}_{\mr{G}}^{0} (\mr{B}_+ )$, and
the commutative diagram

\[
\begin{tikzcd}
{[  \upiota_{\mr{H}}(\mr{B})_+ , \ko_{\mr{H}}]^{\mr{H}}} \rar["{\sf tr}"] \dar[" \pic(\iota_{\upiota_{\mr{H}}(\mr{R})} ) \circ \mJJ_{\mr{G} \ast}"'] &  {[ \mr{B}_+, \ko_{\mr{G}} ]^{\mr{G}}, } \dar[" \pic(\iota_{\mr{R}}) \circ \mJJ_{\mr{G} \ast} "] \\
{[  \upiota_{\mr{H}}(\mr{B})_+ , \pic(\upiota_{\mr{H}}(\mr{R}) )]^{\mr{H}}} \rar["{\sf tr}"] &  {[ \mr{B}_+, \pic(\mr{R}) ]^{\mr{G}}, }
\end{tikzcd}
\]

implies $[{\sf f}_{{\sf Ind}_{\mr{H}}^{\mr{G}}(\upxi), \mr{R}}] = {\sf tr}([{\sf f}_{\upxi, \upiota_{\mr{H}}(\mr{R}) } ])$.  Consequently, ${\sf Ind}_{\mr{H}}^{\mr{G}}(\upxi)$ is $\mr{R}$-homogeneous when $\upxi$ is $\upiota_{\mr{H}}(\mr{R})$-homogeneous and ${\sf w}_1^{\mr{R}}({\sf Ind}_{\mr{H}}^{\mr{G}}(\upxi)) = {\sf tr} ({\sf w}_1^{\upiota_\mr{H}(\mr{R})}(\upxi))$.

First note that the result is true for a group of order $2$ because of <Ref> and (<ref>).

Now let $\mr{G}$ be an  arbitrary group of even order. Note $\mr{G}$ is even it admits a subgroup of order $2$. Indeed, a simple counting argument shows that there exists an element (other than $1$) which is its own inverse. Fix a subgroup  $\mr{T}$ of $\mr{G}$ whose order is $2$. Observe that the bundle $\upgamma_{\uprho_{\mr{G}}}$ is isomorphic to  ${\sf Ind}_{\mr{T}}^{\mr{G}}( \upgamma_{\uprho_{\mr{T}}})$ (see <Ref>).
Since $\upgamma_{\mr{T}}$ is $\mr{H}\ull{\mathbb{Z}}$-orientable, it follows from
<Ref> that

\[ {\sf w}_1^{\ull{\mathbb{Z}}}(\upgamma_{\uprho_{\mr{G}}}) = {\tr} ({\sf w}_1^{\ull{\mathbb{Z}}}(\upgamma_{\uprho_{\mr{T}}}) ) ={\sf tr}( 0) = 0,   \]

thus $\upgamma_{\uprho_{\mr{G}}}$ is $\mr{H}\ull{\mathbb{Z}}$-orientable.

Now we will show that $\uprho_{\upgamma_{\mr{G}}}$ is not $\mr{H} \ull{\cal{A}}_{\mr{G}}$-orientable. By construction  $\upgamma_{\uprho_{\mr{G}}}$ is a homogeneous bundle, therefore it $\mr{H}\ull{\cal{A}}_{\mr{G}}$-homogeneous (see <Ref>). By <Ref>, each of the components of $\mr{G}$-fixed points of $\mr{B}_{\mr{G}}\Sigma_2$  is isomorphic to $\mr{B}\Sigma_2$. The restriction of $\upgamma_{\uprho_{\mr{G}}} \cong {\sf Ind}_{\sf e}^{\mr{G}}(\upgamma_1)$ to the $1$-skeleton of any of its component
is isomorphic to ${\sf Ind}_{\sf e}^{\mr{G}}(\mathfrak{m})$,

\[
\begin{tikzcd}
{\sf Ind}_{\sf e}^{\mr{G}}(\mathfrak{m}) \ar[rrr] \dar[] &&& {\sf Ind}_{\sf e}^{\mr{G}}(\upgamma_{1}) \dar[] \\
\mr{S}^1  \rar[hook] & \mr{B}\Sigma_2   \rar[hook] & (\mr{B}_{\mr{G}}\Sigma_2)^{\mr{G}} \rar[hook] &  \mr{B}_{\mr{G}}\Sigma_2
\end{tikzcd}
\]

where $\mathfrak{m}$ is the mobius bundle over $\mr{S}^1$. To show that $\upgamma_{\uprho_{\mr{G}}}$ is not $\mr{H}\ull{\cal{A}}_{\mr{G}}$-orientable, it is enough to show ${\sf Ind}_{\sf e}^{\mr{G}}(\mathfrak{m})$ is not  $\mr{H}\ull{\cal{A}}_{\mr{G}}$-orientable.

It is well-known that  ${\sf w}_1^{\mathbb{Z}}(\mathfrak{m})$ is the element

\[-1 \in \mathbb{Z}^{\times} \cong \mr{H}^{1} ( \mr{S}^1; \mathbb{F}_2) \cong \ull{\cal{A}}_{\mr{G}}^{\times}(\mr{G}/{\sf e}).\]

A  standard formula for the norm of the sum of two elements (see <cit.>) implies that

\[ \mathrm{norm}(-1) \equiv -1 \mod \langle [\mr{G}/\mr{H}]: \text{$\mr{H} \subsetneq \mr{G}$}\rangle \]

in $\ull{\cal{A}}_{\mr{G}}(\mr{G}/\mr{G})$.  In particular, $\mathrm{norm}(-1) \neq 1$, thus

\[ \tr(-1) \in \mr{H}_{\mr{G}}^1(\mr{S}^1 ; \ull{\cal{A}}_{\mr{G}}^{\times}) \cong \ull{\cal{A}}_{\mr{G}}^{\times}(\mr{G}/\mr{G})  \]

is a nontrivial element.
Therefore, by <Ref> the first Stiefel–Whitney class for ${\sf Ind}_{\sf e}^{\mr{G}}(\mathfrak{m})$

\[ {\sf w}^{\ull{\cal{A}}_{\mr{G}}}_1({\sf Ind}_{\sf e}^{\mr{G}}(\mathfrak{m})) = {\sf tr}(-1) \in \mr{H}^1(\mr{B}; \ull{\cal{A}}^{\times}_{\mr{G}}) \]

is nonzero. Thus ${\sf Ind}_{\sf e}^{\mr{G}}(\mathfrak{m})$, and consequently  $\upgamma_{\uprho_{\mr{G}}}$, does not admit an $\mr{H}\ull{\cal{A}}_{\mr{G}}$-orientation.

§ UNITS OF BURNSIDE TAMBARA FUNCTORS

In anticipation of future applications of <Ref>, we record the Mackey functor structure of  $\ull{\mathcal{A}}_{\mr{G}}^{\times}$ for a few familiar  abelian $2$-groups.
The starting point of  these calculations is a fundamental result due to Matsuda [18] which identifies $ \ull{\mathcal{A}}_{\mr{G}}^{\times}(\mr{G}/\mr{H}) := \mr{A}_{\mr{H}}^{\times}$, where $\mr{A}_{\mr{H}}$ is the Burnside ring of the group $\mr{H}$.
For a finite group $\mr{G}$, the units $\mr{A}_{\mr{G}}^{\times} \cong (\mathbb{Z}/2)^{m +1}$,  where $m$ is the cardinality of the set

\[ \mathcal{I} =  \{ \mr{H}: \text{$\mr{H} \subset \mr{G}$ has index $2$} \}/(\text{conjugation}).\]

The elements  $[\mr{G}/\mr{H}] - 1$ for $\mr{H} \in \mathcal{I}$, along with $-1$, form a basis for $\mr{A}_{\mr{G}}^{\times}$.

While the restriction maps of $\ull{\mathcal{A}}_{\mr{G}}^{\times}$  is straightforward to determine, the  transfer maps are deduced from  the formula of  norm map $\mr{N}_{\mr{H}}^{\mr{G}}: \ull{\mathcal{A}}_{\mr{G}}(\mr{G}/\mr{H}) \longrightarrow  \ull{\mathcal{A}}_{\mr{G}}(\mr{G}/\mr{G})
$ given by the formula (see <cit.> as well as <cit.>)

\begin{equation} \label{norm}
\begin{tikzcd}
\mr{N}_{\mr{H}}^{\mr{G}}(\mr{X}) =  \mr{Map}_{\mr{H}}(\mr{G}, \mr{X}),
\end{tikzcd}
\end{equation}

where $\mr{X}$ is   a finite $\mr{H}$-set.  The Mackey functor structure of $\ull{\cal{A}}_{\mr{G}}^{\times}$  can be complicated depending on the structure of $\mr{G}$ and
has been the subject of research for several decades [18, 24, 35, 34, 6, 3, 7, 33].

When $\mr{G} = \mr{C}_{2^n}$, then the subgroups of $\mr{G}$ are cyclic of order $2^k$, in fact, there is exactly one for each $0 \leq k \leq n$. We will denote this subgroup by $\mr{C}_{2^k}$. From <Ref>, we get

\[ \ull{\mathcal{A}}_{\mr{C}_{2^n}}^{\times}(\mr{C}_{2^n}/\mr{C}_{2^k}) \cong \mathbb{Z}/2\{ {\sf a}_k, {\sf b}_k \}, \]

where ${\sf a}_k = -1$ and ${\sf b}_k = [\mr{C}_{2^k}/\mr{C}_{2^{k-1}}] -1$.
The restriction and the transfer map in the diagram

\[
\begin{tikzcd}
\ull{\mathcal{A}}_{\mr{C}_{2^n}}^{\times}(\mr{C}_{2^n}/\mr{C}_{2^k}) \ar[dd, bend right, "{\sf res}_k"'] \\
\\
\ull{\mathcal{A}}_{\mr{C}_{2^n}}^{\times}(\mr{C}_{2^n}/\mr{C}_{2^{k-1}}) \ar[uu, bend right, "{\sf tr}_k"']
\end{tikzcd}
\]

$ {\sf tr}_k({\sf a}_{k-1}) = {\sf tr}_k({\sf b}_{k-1}) = {\sf b}_{k}, \  {\sf res}_k({\sf a}_k) = {\sf a}_{k-1}, \  {\sf res}_k({\sf b}_k) = 0. $

The map ${\sf res}_k$ is immediate from restricting the actions. The transfer is  obtained by calculating the norm in the Burnside Tambara functor $\ull{\mathcal{A}}_{G}$ using the formula (<ref>).
When $\mr{H} = \mr{C}_{2^{k-i}}$ and $\mr{G}=\mr{C}_{2^k}$ in (<ref>), we get

\begin{equation*}
\begin{array}{clll}
\mathrm{N}_{\mr{C}_{2^{k-i}}}^{\mr{C}_{2^k}} (l) &=& l + \frac{l^2-l}{2}[\mr{C}_{2^k}/\mr{C}_{2^{k-1}}] +
\frac{l^4 - l^2}{4} [\mr{C}_{2^k}/\mr{C}_{2^{k-2}}] + \dots   \\
&& \hspace{150pt}  \cdots  + \frac{l^{2^{k-i}} - l^{2^{k-i-1}}}{2^{i}} [\mr{C}_{2^k}/ \mr{C}_{2^{k-i}} ].
\end{array}
\end{equation*}

By setting $i= 1$ and $l =-1$ in the above equation, we get $ {\sf tr}_k({\sf a}_{k-1}) =  {\sf b}_{k}$, and by setting $i=k$ and $l =-1$,   we get $ {\sf tr}_e^{\mr{C}_{2^k}}({\sf a}_0) = {\sf b}_k$ for all $0 \leq k \leq n$.

\begin{eqnarray*}
{\sf tr}_k({\sf b}_{k-1}) & =& {\sf tr}_k ( {\sf tr}_e^{\mr{C}_{2^{k-1}}}({\sf a}_0)) \\
&=&  {\sf tr}_e^{\mr{C}_{2^k}}({\sf a}_0) \\
&=& {\sf b}_k
\end{eqnarray*}

as desired.

When $ \mr{G} = \mr{C}_2 \times \mr{C}_2,$ there  are three different order $2$ subgroups $\mr{G}_{01} := \mr{C}_2 \times \{ 0\} $, $\mr{G}_{10} = \{ 0\} \times \mr{C}_2$ and $\mr{G}_{11} = \{(x,x): x \in \mr{C}_2\}$. Using <Ref>, we note:

\begin{align*}
\ull{\mathcal{A}}_{\mr{G}}^{\times}(\mr{G}/\mr{G})
&= \mathbb{Z}/2\{{\sf a}_{2}, {\sf b}_{2,10}, {\sf b}_{2,01}, {\sf b}_{2,11}\}
\\
\ull{\mathcal{A}}_{\mr{G}}^{\times}(\mr{G}/\mr{G}_{10})
&= \mathbb{Z}/2\{{\sf a}_{10}, {\sf b}_{10}\}\\
\ull{\mathcal{A}}_{\mr{G}}^{\times}(\mr{G}/\mr{G}_{01})
&= \mathbb{Z}/2\{{\sf a}_{01}, {\sf b}_{01}\}\\
\ull{\mathcal{A}}_{\mr{G}}^{\times}(\mr{G}/\mr{G}_{11})
&= \mathbb{Z}/2\{{\sf a}_{11}, {\sf b}_{11}\}\\
\ull{\mathcal{A}}_{\mr{G}}^{\times}(\mr{G}/{\sf e})
&= \mathbb{Z}/2\{{\sf a}_{0}\},
\end{align*}

where ${\sf a}_0, {\sf a}_{10}, {\sf a}_{01}, {\sf a}_{11}$ and  ${\sf a}_{2}$  denote $-1$ in their respective rings, ${\sf b}_{ij}$ denotes $[\mr{G}_{ij}] -1$, and, ${\sf b}_{2, ij}$ denotes $[\mr{G}/ \mr{G}_{ij}] - 1$.
The restriction and transfer maps as indicated in the diagrams

\[
\begin{tikzcd}
\ull{\mathcal{A}}_{\mr{G}}^{\times}(\mr{G}/\mr{G}) \ar[dd, bend right, "{\sf res}_{2, ij}"']  &&  \ull{\mathcal{A}}_{\mr{G}}^{\times}(\mr{G}/\mr{G}_{ij}) \ar[dd, bend right, "{\sf res}_{ij}"']  \\
\\
\ull{\mathcal{A}}_{\mr{G}}^{\times}(\mr{G}/\mr{G}_{ij}) \ar[uu, bend right, "{\sf tr}_{2, ij}"'] && \ull{\mathcal{A}}_{\mr{G}}^{\times}(\mr{G}/{\sf e})  \ar[uu, bend right, "{\sf tr}_{ij}"']
\end{tikzcd}
\]

is determined by the formulas ${\sf res}_{2, ij}({\sf a}_2) = {\sf a}_{ij}, \ {\sf res}_{ij}({\sf a}_{ij}) = {\sf a}_0$,

\[
{\sf res}_{2, ij}({\sf b}_{2,kl}) = \left\lbrace \begin{array}{ccc}
0 & \text{if $ij = kl$,} \\
{\sf b}_{ij}  &\text{otherwise},
\end{array} \right.
\]

$ {\sf tr}_{ij}({\sf a}_{0}) = {\sf b}_{ij} $, ${\sf tr}_{ij}({\sf a}_{ij}) ={\sf b}_{ij} $ and  $ {\sf tr}_{2, ij}({\sf b}_{ij}) = {\sf b}_{2,10} + {\sf b}_{2,01} + {\sf b}_{2,11} $.

All claims, except for the formula for ${\sf tr}_{2, ij}({\sf b}_{ij})$,   can be verified directly using arguments very similar to that in the proof of <Ref>.

To see the formula for ${\sf tr}_{2, ij}({\sf b}_{ij})$, we first observefrom (<ref>) that

\begin{equation*}
\mr{N}_{\sf e}^{\mr{G}} (-1) = -1 + [\mr{G}/ \mr{G}_{01}] + [\mr{G}/\mr{G}_{10}] + [\mr{G}/\mr{G}_{11}] - [\mr{G}/{\sf e}].
\end{equation*}

If  ${\sf tr}_{2, ij}({\sf b}_{ij}) =  \epsilon_1{\sf a}_2 + \epsilon_2 {\sf  b}_{2,10} +\epsilon_3 {\sf b}_{2,01}+ \epsilon_4{\sf b}_{2,11}$, it means

\begin{equation*}
\mr{N}_{\sf e}^{\mr{G}} (-1) = (-1)^{\epsilon_1}([\mr{G}/ \mr{G}_{01}]-1)^{\epsilon_2}([\mr{G}/ \mr{G}_{10}]-1)^{\epsilon_3}([\mr{G}/ \mr{G}_{11}]-1)^{\epsilon_4}.
\end{equation*}

Comparing the two equations for $\mr{N}_{\sf e}^{\mr{G}} (-1)$, we conclude  $\epsilon_2 = \epsilon_3 = \epsilon_4 = 1$ and $\epsilon_1 =0$.

<Ref> can be readily extended to $\mathbb{T}_n :=  \mr{C}_2^{\times n}$. Any subgroup of $\mathbb{T}_n$ is isomorphic to $\mathbb{T}_k$ for some $k \in \{ 0,1, \dots, n \}$, and there are exactly
\[ \frac{(2^n -1)\dots(2^n - 2^{k-1})}{(2^k-1) \dots (2^{k} - 2^{k-1})}\]
of them.
Fix a subgroup $\mr{H}$ isomorphic to $\mathbb{T}_k$. The above formula implies there are exactly $2^k -1$ index $2$ subgroups of $\mr{H}$. Therefore, by <Ref>,

\[ \ull{\cal{A}}^{\times}_{\mathbb{T}_n}(\mathbb{T}_n/\mr{H}) \cong (\mathbb{Z}/2)^{\times 2^k}\]

generated by ${\sf a}_{\mr{H}} = -1$ and ${\sf b}_{\mr{H}/\mr{K}'} = [\mr{H}/\mr{K}'] -1 $ where $\mr{K}'$ is an index $2$ subgroup of $\mr{H}$.  Likewise, for a given index $2$ subgroup $\mr{K}$ of $\mr{H}$

\[ \ull{\cal{A}}^{\times}_{\mathbb{T}_n}(\mathbb{T}_n/\mr{K}) \cong (\mathbb{Z}/2)^{\times 2^{k-1}}\]
generated by ${\sf a}_{\mr{K}} = -1$ and ${\sf b}_{\mr{K}/\mr{L}'} -1 $ where $\mr{L}'$ is an index $2$ subgroup of $\mr{K}$.

It is easy to see that the restriction map

\[
\begin{tikzcd}
{\sf res}_{\mr{K}}^{\mr{H}}: \ull{\cal{A}}^{\times}_{\mathbb{T}_n}(\mathbb{T}_n/\mr{H}) \rar & \ull{\cal{A}}^{\times}_{\mathbb{T}_n}(\mathbb{T}_n/\mr{K})
\end{tikzcd}
\]

sends ${\sf a}_{\mr{H}}$ to ${\sf a}_{\mr{K}}$, ${\sf b}_{\mr{H}/\mr{K}'}$ to ${\sf b}_{\mr{K}/ (\mr{K} \cap \mr{K}')}$ if $\mr{K}' \neq \mr{K}$, and ${\sf b}_{\mr{H}/\mr{K}} $ to $0$. The transfer map

\[
\begin{tikzcd}
{\sf tr}_{\mr{K}}^{\mr{H}}: \ull{\cal{A}}^{\times}_{\mathbb{T}_n}(\mathbb{T}_n/\mr{K}) \rar & \ull{\cal{A}}^{\times}_{\mathbb{T}_n}(\mathbb{T}_n/\mr{H})
\end{tikzcd}
\]

sends ${\sf a}_{\mr{K}}$ to ${\sf b}_{\mr{H}/\mr{K}}$, and ${\sf b}_{\mr{K}/\mr{L}'}$ to ${\sf b}_{\mr{H}/\mr{K}} +{\sf b}_{\mr{H}/\mr{K}_1} + {\sf b}_{\mr{H}/\mr{K}_2}$, where $\mr{K}_1$ and $\mr{K}_2$ are the two index $2$ subgroups of $\mr{H}$ whose intersection with $\mr{K}$ equals $\mr{L}'$.

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