class $\gamma(f)_{*}(e_{\alpha,U,V})$ is represented by the $G$-map

$S^{V\oplus U}\ \longrightarrow\ Y(V)\ ,\quad(v,u)\ \longmapsto\
f(V)(v\wedge[1\wedge\operatorname{Id}\wedge u])\ .$

Since this class is trivial, there is a $G$-representation $W$ such that the
stabilization

(A.18) $S^{V\oplus W\oplus U}\ \longrightarrow\ Y(W\oplus V)\ ,\quad(v,w,u)\
\longmapsto\ f(V\oplus W)(v\wedge[1\wedge i_{V}\wedge u])$

is $G$-equivariantly based null-homotopic. We choose a null-homotopy that
witnesses this fact and adjoint it to a morphism of orthogonal $C$-spectra

$H\ :\ \Sigma^{\infty}C\ltimes_{\alpha}(\mathbf{L}(V\oplus W,-)_{+}\wedge
S^{U})\wedge[0,1]\ \longrightarrow\ Y\ ;$

here the unit interval $[0,1]$ is based at 0, and the restriction of $H$ to
the point 1 is adjoint to (A.18). We arrive at a commutative diagram in
$Sp_{C}$:

$\textstyle{\Sigma^{\infty}C\ltimes_{\alpha}(\mathbf{L}(V,-)_{+}\wedge
S^{U})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\Sigma^{\infty}C\ltimes_{\alpha}(\mathbf{L}(V\oplus
W,-)_{+}\wedge
S^{U})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\wedge-}$$\scriptstyle{\Sigma^{\infty}C\ltimes_{\alpha}((\rho_{V,W})_{+}\wedge
S^{U})}$$\scriptstyle{\sim}$$\textstyle{Y}$$\textstyle{\Sigma^{\infty}C\ltimes_{\alpha}(\mathbf{L}(V,-)_{+}\wedge
S^{U})\wedge[0,1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H}$

Since the upper left morphism becomes invertible in the $C$-global stable
homotopy category, and because the lower left spectrum becomes a zero object
in ${\mathcal{GH}}_{C}$, this proves that the image of $f$ in
${\mathcal{GH}}_{C}$ is the zero morphism.

(ii) By Theorem A.11 (ii), the wedge of any family of orthogonal $C$-spectra
is a coproduct in ${\mathcal{GH}}_{C}$. The vertical maps in the commutative
square

$\textstyle{{\bigoplus_{i\in
I}}{\mathcal{GH}}_{C}(\Sigma^{\infty}_{+}B_{\operatorname{gl}}\alpha,\,X_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{GH}}_{C}(\Sigma^{\infty}_{+}B_{\operatorname{gl}}\alpha,\,\bigoplus_{i\in
I}X_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\bigoplus_{i\in
I}}\pi^{G}_{0}(\alpha^{*}(X_{i}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{0}^{G}\left(\bigvee_{i\in
I}\alpha^{*}(X_{i})\right)}$

are evaluation at the tautological class, and hence isomorphisms by part (i).
The lower horizontal map is an isomorphism, see for example [20, Corollary
3.1.37]; so the upper horizontal map is an isomorphism, too. This shows that
$\Sigma^{\infty}_{+}B_{\operatorname{gl}}\alpha$ is compact as an object of
the triangulated category ${\mathcal{GH}}_{C}$. ∎

In the special case where the $G$-representation $U$ is trivial, Theorem A.17
(i) says that the unreduced suspension spectrum
$\Sigma^{\infty}_{+}B_{\operatorname{gl}}\alpha$ represents the functor
$\pi_{0}^{G}\circ\alpha^{*}:{\mathcal{GH}}_{C}\longrightarrow{\mathcal{A}}b$.
So if $Y$ is an orthogonal $C$-spectrum such that the group
${\mathcal{GH}}_{C}(\Sigma^{\infty}_{+}(B_{\operatorname{gl}}\alpha)[k],\,Y)$
is trivial for every continuous homomorphism $\alpha:G\longrightarrow C$ from
a compact Lie group $G$ and all integers $k$, then $Y$ is $C$-globally
equivalent to the trivial orthogonal $C$-spectrum. So $Y$ is a zero object in
${\mathcal{GH}}_{C}$. This proves the following result.

######  Corollary A.19.

Let $C$ be a topological group. As $\alpha:G\longrightarrow C$ varies over a
set of representatives of the isomorphism classes of continuous homomorphism
from compact Lie groups to $C$, the suspension spectra
$\Sigma^{\infty}_{+}B_{\operatorname{gl}}\alpha$ form a set of compact weak
generators for the triangulated $C$-global stable homotopy category
${\mathcal{GH}}_{C}$. In particular, the $C$-global stable homotopy category
is compactly generated.

######  Remark A.20 (The $\infty$-category of global $C$-spectra).

We let $C$ be a topological group. The triangulated $C$-global stable homotopy
category is only the shadow of a more refined structure, namely an underlying
compactly generated stable $\infty$-category. We define the $\infty$-category
of $C$-global spectra as the $\infty$-categorical localization of the
1-category of orthogonal $C$-spectra at the class of $C$-global equivalences.
For example, the quasicategory of frames in the sense of Szumiło [22, Section
2] associated to the cofibration structure of Theorem A.11 on the category of
orthogonal $C$-spectra is a particular construction. This quasicategory is
cocomplete by [22, Theorem 2.3]. Stability of an $\infty$-category is detected
by the homotopy category, see [14, Corollary 1.4.2.27]; so the
$\infty$-category of $C$-global spectra is stable. Similarly, the property of
a stable $\infty$-category to be compactly generated is detected by the
homotopy category, compare [14, Remark 1.4.4.3]. So the stable
$\infty$-category of $C$-global spectra is compactly generated; in particular,
this stable $\infty$-category is also presentable and complete.

Suppose that $C$ is a Lie group (not necessarily compact); for example, $C$
could be an infinite discrete group. Then the $C$-global equivalences also
take part in several model category structures on the category of orthogonal
$C$-spectra. The essential ingredients for the construction of the model
structure are a synthesis of the arguments needed in the special case of the
trivial group, i.e., the global model structure of orthogonal spectra [20,
Theorem 4.3.18], and the arguments used in [6, Section 1.2] to set up the
proper stable homotopy theory for non-compact Lie groups. Since we don’t need
any model structures for the purposes of this paper, I won’t dwell on this any
further.

###  A.3. Global classifying spaces by complex and quaternion isometric
embeddings

As we explain in detail in Construction A.4, the global classifying $C$-space
of a continuous homomorphism $\alpha:G\longrightarrow C$ from a compact Lie
group uses real Stiefel manifolds, i.e., spaces of ${\mathbb{R}}$-linear
isometric embeddings from a faithful orthogonal $G$-representation. In our
applications to stable splitting of ${\mathbf{U}}/m$ and ${\mathbf{Sp}}/m$,
the naturally occurring objects are complex and quaternionic Stiefel
manifolds. We will thus need to know that we can also use spaces of
${\mathbb{C}}$-linear or ${\mathbb{H}}$-linear isometric embeddings from a
faithful unitary or symplectic $G$-representation to define global classifying
spaces. Moreover, we want to keep track of the natural symmetries,
parameterized by the groups of ${\mathbb{R}}$-algebra automorphisms of
${\mathbb{C}}$ and ${\mathbb{H}}$. In this subsection we explain the
connection in detail. A precursor of the results in this subsection already
occurs in [20, Proposition 1.3.11], which treats the complex case without any
mentioning of the Galois group $G({\mathbb{C}})$. The key ingredients are
already present in the proof of the precursor; our main work here is to
carefully adapt the arguments from ${\mathbb{C}}$ to ${\mathbb{H}}$, while
incorporating an augmentation to the Galois group.

In the rest of this section, we let ${\mathbb{K}}$ be one of the skew-fields
${\mathbb{C}}$ or ${\mathbb{H}}$. The arguments also work for
${\mathbb{K}}={\mathbb{R}}$, but then they are either tautological, or already
well-known and explicitly stated in [20]. As before we write
$G({\mathbb{K}})=\operatorname{Aut}_{\mathbb{R}}({\mathbb{K}})$ for the
‘Galois group’ of ${\mathbb{K}}$, i.e., the compact Lie group of
${\mathbb{R}}$-algebra automorphisms. Then $G({\mathbb{C}})$ is discrete of
order 2, and $G({\mathbb{H}})$ is abstractly isomorphic to $SO(3)$.

######  Definition A.21.

We let $G$ be a compact Lie group and $\epsilon:G\longrightarrow
G({\mathbb{K}})$ a continuous homomorphism. A $(G,\epsilon)$-representation is
a ${\mathbb{K}}$-inner product space $W$ endowed with a continuous
${\mathbb{R}}$-linear $G$-action such that

$g\cdot(x\lambda)\ =\ (gx)\cdot\epsilon(g)(\lambda)\text{\qquad
and\qquad}[gx,gy]\ =\ \epsilon(g)([x,y])$

for all $g\in G$, $\lambda\in{\mathbb{K}}$ and $x,y\in W$.

######  Remark A.22 (Extended isometry group).

The extended isometry group $\tilde{I}(W)$ of a ${\mathbb{K}}$-inner product
space $W$ is the group of pairs $(A,\tau)$ consisting of an
${\mathbb{R}}$-linear automorphism $A:W\longrightarrow W$ and a Galois
automorphism $\tau\in G({\mathbb{K}})$ such that

  * •

the morphism $A$ is $\tau$-semilinear, i.e.,
$A(x\lambda)=A(x)\cdot\tau(\lambda)$ for all $x\in W$ and
$\lambda\in{\mathbb{K}}$, and

  * •

the relation

(A.23) $[Ax,Ay]\ =\ \tau[x,y]$

holds for all $x,y\in W$.

If $W$ is non-zero, then the Galois automorphism $\tau$ is actually determined
by $A$; nevertheless, it is convenient to explicitly keep track of $\tau$.
Composition in $\tilde{I}(W)$ is componentwise, i.e,
$(A,\tau)\cdot(A^{\prime},\tau^{\prime})=(AA^{\prime},\tau\tau^{\prime})$. The
relation (A.23) in particular implies that for $(A,\tau)\in\tilde{I}(W)$, the
map $A$ is an isometry of the underlying euclidean vector space $uW$, i.e.,
the underlying ${\mathbb{R}}$-vector space of $W$ equipped with the euclidean
inner product $\langle x,y\rangle=\text{Re}[x,y]$. So $\tilde{I}(W)$ is a
closed subgroup of $O(uW)\times G({\mathbb{K}})$, and hence a compact Lie
group. The projection to the second factor $p:\tilde{I}(W)\longrightarrow
G({\mathbb{K}})$ is a continuous epimorphism whose kernel is isomorphic to
$I(W)$. Moreover, $\tilde{I}(W)$ is isomorphic to a semidirect product
$I(W)\rtimes G({\mathbb{K}})$, but not in a natural way.

The reader might want to convince himself or herself that a
$(G,\epsilon)$-representation with underlying inner product space $W$ can
equivalently be specified by a continuous homomorphism
$\rho:G\longrightarrow\tilde{I}(W)$ to the extended isometry group that covers
the augmentations, i.e., such that $p\circ\rho=\epsilon$.

For ${\mathbb{K}}={\mathbb{C}}$, every $(G,\epsilon)$-representation in
particular has an underlying orthogonal representation of the compact Lie
group $G$; and it has an underlying unitary representation of the kernel of
$\epsilon:G\longrightarrow G({\mathbb{C}})$. Compact Lie groups augmented to
the Galois group of ${\mathbb{C}}$ over ${\mathbb{R}}$ and their twisted
representations are studied as ‘augmented compact Lie groups’ by Karoubi [11],
and prior to that (in special cases) by Atiyah and Segal [1, Section 6]. For
${\mathbb{K}}={\mathbb{H}}$, every $(G,\epsilon)$-representation in particular
has an underlying orthogonal representation of the compact Lie group $G$; and
it has an underlying symplectic representation of the kernel of
$\epsilon:G\longrightarrow G({\mathbb{H}})$. I am not aware of a reference
where compact Lie groups augmented to $G({\mathbb{H}})$ and their ‘twisted’
representations are systematically investigated.

We write $V_{\mathbb{K}}=V\otimes_{\mathbb{R}}{\mathbb{K}}$ for the scalar
extension from ${\mathbb{R}}$ to ${\mathbb{K}}$ of a euclidean inner product
space $V$, with ${\mathbb{K}}$-inner product $[-,-]$ obtained from the
euclidean inner product $\langle-,-\rangle$ on $V$ by

(A.24) $[x\otimes\lambda,y\otimes\mu]\ =\ \bar{\lambda}\cdot\langle
x,y\rangle\cdot\mu$

for $x,y\in V$ and $\lambda,\mu\in{\mathbb{K}}$. The underlying euclidean
inner product space $uW$ of a ${\mathbb{K}}$-inner product space $W$ is the
underlying ${\mathbb{R}}$-vector space endowed with the euclidean inner
product

$\langle x,y\rangle\ =\ \text{Re}[x,y]\ ,$

the real part of the ${\mathbb{K}}$-valued inner product.

######  Construction A.25.

We let ${\mathbb{K}}$ be ${\mathbb{C}}$ or ${\mathbb{H}}$, and we let $W$ be a
${\mathbb{K}}$-inner product space. For several arguments we shall need a
specific natural ${\mathbb{K}}$-linear isometric embedding

(A.26) $\displaystyle\zeta\ $ $\displaystyle:\ W\ \longrightarrow\
(uW)_{\mathbb{K}}\ =\ (uW)\otimes_{\mathbb{R}}{\mathbb{K}}\ .$

To define it, we separate the two cases. For ${\mathbb{K}}={\mathbb{C}}$, the
map $\zeta$ is defined as

$\zeta\ :\ W\ \longrightarrow\ (uW)_{\mathbb{C}}\ =\
(uW)\otimes_{\mathbb{R}}{\mathbb{C}}\text{\qquad by\qquad}\zeta(x)\ =\
1/\sqrt{2}\cdot(x\otimes 1-xi\otimes i)\ .$

For ${\mathbb{K}}={\mathbb{H}}$, it is given by

$\displaystyle\zeta\ :\ W\ $ $\displaystyle\longrightarrow\ (uW)_{\mathbb{H}}\
=\ (uW)\otimes_{\mathbb{R}}{\mathbb{H}}\ ,$ $\displaystyle\zeta(x)\ $
$\displaystyle=\ 1/2\cdot(x\otimes 1-xi\otimes i-xj\otimes j-xk\otimes k)\ .$

######  Proposition A.27.

Let $\epsilon:G\longrightarrow G({\mathbb{K}})$ be a continuous homomorphism,
where ${\mathbb{K}}$ is ${\mathbb{C}}$ or ${\mathbb{H}}$. Let $W$ be a
$(G,\epsilon)$-representation. Then the map $\zeta$ is a ${\mathbb{K}}$-linear
isometric embedding that satisfies

(A.28) $\zeta\circ l_{g}\ =\ (l_{g}\otimes\epsilon(g))\circ\zeta$

for all $g\in G$, where $l_{g}:W\longrightarrow W$ is left multiplication by
$g$.

###### Proof.

We give the argument for ${\mathbb{K}}={\mathbb{H}}$. The proof in the complex
case is similar, but easier. The map $\zeta$ is clearly ${\mathbb{R}}$-linear,
and it commutes with right multiplication by the quaternion scalars $i$ and
$j$; so $\zeta$ is in fact ${\mathbb{H}}$-linear. To show that the map $\zeta$
is isometric we observe that

$\left[x\lambda\otimes\bar{\lambda},x\nu\otimes\bar{\nu}\right]=\
\lambda\cdot\langle x\lambda,x\nu\rangle\cdot\bar{\nu}\ =\
\lambda\cdot\text{Re}\left(\bar{\lambda}[x,x]\nu\right)\cdot\bar{\nu}\ =\
[x,x]\cdot\text{Re}\left(\bar{\lambda}\nu\right)\cdot\lambda\bar{\nu}$

for all $x\in W$ and $\lambda,\nu\in{\mathbb{H}}$. For
$\lambda,\nu\in\\{1,i,j,k\\}$ with $\lambda\neq\nu$ we have
$\text{Re}\left(\bar{\lambda}\nu\right)=0$. So

$\displaystyle[\zeta(x),\zeta(x)]\ =\
1/4\cdot[x,x]\cdot\sum_{\lambda\in\\{1,i,j,k\\}}\text{Re}(\bar{\lambda}\lambda)\cdot\lambda\bar{\lambda}\
=\ [x,x]\ .$

It remains to prove the relation (A.28). The tautological action of
$G({\mathbb{H}})$ on ${\mathbb{H}}$ is isometric for the euclidean inner
product $\langle x,y\rangle=\text{Re}(\bar{x}y)$. Since $(1,i,j,k)$ is an
orthonormal ${\mathbb{R}}$-basis for this inner product, the element $1\otimes
1+i\otimes i+j\otimes j+k\otimes k$ is fixed under the diagonal
$G({\mathbb{H}})$-action (and could have been specified by any other
orthonormal basis). Since $1\otimes 1$ is evidently $G({\mathbb{H}})$-fixed,
too, the element

$1\otimes 1-i\otimes i-j\otimes j-k\otimes k$

is fixed under the diagonal $G({\mathbb{H}})$-action. So we conclude that

$\displaystyle\zeta(gx)\ $ $\displaystyle=\ 1/2\cdot(gx)\cdot(1\otimes
1-i\otimes i-j\otimes j-k\otimes k)$ $\displaystyle=\
1/2\cdot(gx)\cdot(1\otimes
1-\epsilon(g)(i)\otimes\epsilon(g)(i)-\epsilon(g)(j)\otimes\epsilon(g)(j)-\epsilon(g)(k)\otimes\epsilon(g)(k))$
$\displaystyle=\ 1/2\cdot((gx)\otimes
1-g(xi)\otimes\epsilon(g)(i)-g(xj)\otimes\epsilon(g)(j)-g(xk)\otimes\epsilon(g)(k))\
=\ (l_{g}\otimes\epsilon(g))(\zeta(x))$

for all $g\in G$ and $x\in W$. ∎

A basic fact in the global homotopy theory of orthogonal spaces is that for
two compact Lie groups $K$ and $G$, and for any faithful $G$-representation
$V$, the infinite Stiefel manifold $\mathbf{L}(V,{\mathcal{U}}_{K})$ is a
universal $(K\times G)$-space for the family of graph subgroups, compare [20,
Proposition 1.1.26]. This property relies on the fact that for every finite-
dimensional $K$-representation $W$, the space
$\mathbf{L}^{K}(W,{\mathcal{U}}_{K})$ of $K$-equivariant ${\mathbb{R}}$-linear
isometric embeddings is contractible, compare [20, Propositions 1.1.21]. Both
statement have suitable analogs in the complex and quaternionic situations,
with easily adapted proofs. Since I do not know of references, I provide
proper statements and proofs now.

######  Proposition A.29.

Let ${\mathcal{U}}_{K}$ be a complete universe of a compact Lie group $K$. Let
$\beta:K\longrightarrow G({\mathbb{K}})$ be a continuous homomorphism, where
${\mathbb{K}}$ is either ${\mathbb{C}}$ or ${\mathbb{H}}$. Let $K$ act on
${\mathcal{U}}_{K}^{\mathbb{K}}={\mathcal{U}}_{K}\otimes_{\mathbb{R}}{\mathbb{K}}$
by

$k\cdot(x\otimes\lambda)\ =\ (kx)\otimes\beta(k)(\lambda)\ .$

Then for every $(K,\beta)$-representation $W$, the space
$\mathbf{L}^{{\mathbb{K}},K}(W,{\mathcal{U}}_{K}^{\mathbb{K}})$ of
${\mathbb{K}}$-linear $K$-equivariant isometric embeddings is weakly
contractible.

###### Proof.

We let $U$ be any $(K,\beta)$-representation, possibly of countably infinite
dimension. Then the homotopy

$H\ :\ [0,1]\times\mathbf{L}^{{\mathbb{K}},K}(W,U)\ \longrightarrow\
\mathbf{L}^{{\mathbb{K}},K}(W,U\oplus W)\ ,\quad H(t,\varphi)(w)\ =\
(\sqrt{1-t^{2}}\cdot\varphi(w),\,t\cdot w)$

witnesses the fact that the map

$i_{1}\circ-\ :\ \mathbf{L}^{{\mathbb{K}},K}(W,U)\ \longrightarrow\
\mathbf{L}^{{\mathbb{K}},K}(W,U\oplus W)$

(post-composition with $i_{1}:U\longrightarrow U\oplus W)$ is homotopic to a
constant map. The space

$\mathbf{L}^{{\mathbb{K}},K}(W,U\oplus W^{\infty})\ =\
\operatorname{colim}_{n\geq 0}\,\mathbf{L}^{{\mathbb{K}},K}(W,U\oplus W^{n})$

is the colimit along the post-composition maps with the direct sum embeddings
$U\oplus W^{n}\longrightarrow U\oplus W^{n+1}$. Every map in the colimit
system is homotopic to a constant map, by the previous paragraph. Since the
maps are also closed embeddings, the colimit is weakly contractible.

Now we can prove the proposition. Because ${\mathcal{U}}_{K}$ is a complete
$K$-universe, there is a $K$-equivariant ${\mathbb{R}}$-linear isometry
${\mathcal{U}}_{K}\cong{\mathcal{V}}\oplus(uW)^{\infty}$ for some orthogonal
$K$-representation ${\mathcal{V}}$ (typically infinite dimensional). The
${\mathbb{K}}$-linear $K$-equivariant isometric embedding
$\zeta:W\longrightarrow(uW)\otimes_{\mathbb{R}}{\mathbb{K}}$ provides an
isomorphism of $(K,\beta)$-representations

$(uW)\otimes_{\mathbb{R}}{\mathbb{K}}\ \cong\ C\oplus W\ ,$

where $C$ is the orthogonal complement of the image of $\zeta$. So
${\mathcal{U}}_{K}^{\mathbb{K}}$ is isomorphic to

$\left({\mathcal{V}}\oplus(uW)^{\infty}\right)\otimes_{\mathbb{R}}{\mathbb{K}}\
\cong\ ({\mathcal{V}}\otimes_{\mathbb{R}}{\mathbb{K}})\oplus C^{\infty}\oplus
W^{\infty}\ .$

The space $\mathbf{L}^{{\mathbb{K}},K}(W,{\mathcal{U}}_{K}^{\mathbb{K}})$ is
thus weakly contractible by the first paragraph. ∎

######  Construction A.30.

We let $K$ and $G$ be compact Lie groups, and we let $\beta:K\longrightarrow
G({\mathbb{K}})$ and $\epsilon:G\longrightarrow G({\mathbb{K}})$ be two
continuous homomorphisms. We write

$K\times_{G({\mathbb{K}})}G\ =\ \\{(k,g)\in K\times G\ :\
\beta(k)=\epsilon(g)\\}$

for the fiber product over $G({\mathbb{K}})$. We let $W$ be a
$(G,\epsilon)$-representation, and we let ${\mathcal{U}}_{K}$ be a complete
$K$-universe. As before we write
${\mathcal{U}}_{K}^{\mathbb{K}}={\mathcal{U}}_{K}\otimes_{\mathbb{R}}{\mathbb{K}}$
for the scalar extension. The group $K\times_{G({\mathbb{K}})}G$ then acts on
the space $\mathbf{L}^{\mathbb{K}}(W,{\mathcal{U}}_{K}^{\mathbb{K}})$ of
${\mathbb{K}}$-linear isometric embeddings by

(A.31) $(k,g)\cdot\varphi\ =\ (l_{k}\otimes\beta(k))\circ\varphi\circ
l_{g}^{-1}\ ,$

where $l_{k}:{\mathcal{U}}_{K}\longrightarrow{\mathcal{U}}_{K}$ and
$l_{g}:W\longrightarrow W$ are the translation maps. We exploit here that for
$(k,g)\in K\times_{G({\mathbb{K}})}G$, both maps $l_{k}\otimes\beta(k)$ and
$l_{g}$ are semilinear for the same ${\mathbb{R}}$-algebra automorphism of
${\mathbb{K}}$, so the composite (A.31) is again ${\mathbb{K}}$-linear. So
altogether the assignment (A.31) defines a continuous action of the compact
Lie group $K\times_{G({\mathbb{K}})}G$ on the space
$\mathbf{L}^{\mathbb{K}}(W,{\mathcal{U}}_{K}^{\mathbb{K}})$.

######  Proposition A.32.

Let ${\mathbb{K}}$ be ${\mathbb{C}}$ or ${\mathbb{H}}$. Let
$\epsilon:G\longrightarrow G({\mathbb{K}})$ be a continuous epimorphism from a
compact Lie group with kernel $G_{0}$. Let $W$ be a faithful
$(G,\epsilon)$-representation. Then for every continuous homomorphism
$\beta:K\longrightarrow G({\mathbb{K}})$ from another compact Lie group and
for every complete $K$-universe ${\mathcal{U}}_{K}$, the
$(K\times_{G({\mathbb{K}})}G)$-space
$\mathbf{L}^{\mathbb{K}}(W,{\mathcal{U}}_{K}^{\mathbb{K}})$ is a universal
$(K\times_{G({\mathbb{K}})}G)$-space for the family of those closed subgroups
that intersect $1\times G_{0}$ trivially.

###### Proof.

Essentially the same argument as in the real situation in [20, Proposition
1.1.19], which relies on Illman’s equivariant triangulation theorem for smooth
actions of compact Lie groups [9], shows that
$\mathbf{L}^{\mathbb{K}}(W,{\mathcal{U}}_{K}^{\mathbb{K}})$ is cofibrant as a
$(K\times_{G({\mathbb{K}})}G)$-space. Since $G_{0}$ acts faithfully on $W$,
the action on $\mathbf{L}^{\mathbb{K}}(W,{\mathcal{U}}_{K}^{\mathbb{K}})$ by
precomposition is free. So every subgroup of $K\times_{G({\mathbb{K}})}G$ with
fixed points on $\mathbf{L}^{\mathbb{K}}(W,{\mathcal{U}}_{K}^{\mathbb{K}})$
must intersect the group $1\times G_{0}$ trivially. Conversely, we now
consider a closed subgroup $\Delta$ of $K\times_{G({\mathbb{K}})}G$ such that
$\Delta\cap(1\times G_{0})=e$. Then $\Delta$ is the graph of some continuous
homomorphism

$\alpha\ :\ L\ \longrightarrow\ G$

defined on some closed subgroup $L$ of $K$, such that
$\epsilon\circ\alpha=\beta|_{L}$. Hence

$\left(\mathbf{L}^{\mathbb{K}}(W,{\mathcal{U}}_{K}^{\mathbb{K}})\right)^{\Delta}\
=\ \mathbf{L}^{{\mathbb{K}},L}(\alpha^{*}(W),{\mathcal{U}}_{K}^{\mathbb{K}})\
,$

the space of ${\mathbb{K}}$-linear and $L$-equivariant isometric embeddings
from $\alpha^{*}(W)$ into ${\mathcal{U}}_{K}^{\mathbb{K}}$, with $L$-action on
the target by

$l\cdot(x\otimes\lambda)\ =\ (lx)\otimes\beta(l)(\lambda)\ .$

Because ${\mathcal{U}}_{K}$ is a complete $K$-universe, the underlying
$L$-representation is a complete $L$-universe. So the space
$\mathbf{L}^{{\mathbb{K}},L}(\alpha^{*}(W),{\mathcal{U}}_{K}^{\mathbb{K}})$ is
weakly contractible by Proposition A.29. This completes the proof. ∎

######  Construction A.33.

Let $\epsilon:G\longrightarrow G({\mathbb{K}})$ be a continuous epimorphism
from a compact Lie group. Let $W$ be a $(G,\epsilon)$-representation. For a
euclidean inner product space $V$, we define a $G$-action on the Stiefel
manifold $\mathbf{L}^{\mathbb{K}}(W,V_{\mathbb{K}})$ by

${{}^{g}\varphi}\ =\ (V\otimes\epsilon(g))\circ\varphi\circ l_{g}^{-1}\ ,$

where $l_{g}:W\longrightarrow W$ is translation by $g\in G$. The orbit space

$\mathbf{L}^{\mathbb{K}}(W,V_{\mathbb{K}})/G_{0}$

by the action of the closed normal subgroup $G_{0}=\ker(\epsilon)$ inherits a
residual $G({\mathbb{K}})$-action; for varying $V$, this defines an orthogonal
$G({\mathbb{K}})$-space $\mathbf{L}^{\mathbb{K}}(W,(-)_{\mathbb{K}})/G_{0}$.
For example, for the augmentation $\epsilon_{k}:\tilde{I}(k)=I(k)\rtimes
G({\mathbb{K}})\longrightarrow G({\mathbb{K}})$ of the extended isometry group
of ${\mathbb{K}}^{k}$ and the tautological
$(\tilde{I}(k),\epsilon_{k})$-representation on ${\mathbb{K}}^{k}$, the
construction specializes to the Grassmannian $\mathbf{Gr}_{k}^{\mathbb{K}}$
from Example 2.12.

We let $\epsilon:G\longrightarrow G({\mathbb{K}})$ be a continuous
epimorphism, and we continue to write $G_{0}=\ker(\epsilon)$. We will now
argue that for every faithful $(G,\epsilon)$-representation $W$, the
orthogonal $G({\mathbb{K}})$-space
$\mathbf{L}^{\mathbb{K}}(W,(-)_{\mathbb{K}})/G_{0}$ is a
$G({\mathbb{K}})$-global classifying space, in the sense of Construction A.4,
of the epimorphism $\epsilon:G\longrightarrow G({\mathbb{K}})$. Because $G$
acts faithfully on $W$, we can use the underlying orthogonal
$G$-representation $uW$ to construct a $G({\mathbb{K}})$-global classifying
space

$B_{\operatorname{gl}}\epsilon\ =\
G({\mathbb{K}})\times_{\epsilon}\mathbf{L}(uW,-)\ =\ \mathbf{L}(uW,-)/G_{0}\
.$

The issue now is to compare this orthogonal $G({\mathbb{K}})$-space to
$\mathbf{L}^{\mathbb{K}}(W,(-)_{\mathbb{K}})/G_{0}$. To this end we recall
from (A.26) the ${\mathbb{K}}$-linear isometric embedding

$\zeta\ :\ W\ \longrightarrow\ (uW)\otimes_{\mathbb{R}}{\mathbb{K}}\ =\
(uW)_{\mathbb{K}}\ .$

The enriched Yoneda lemma provides a unique morphism of orthogonal spaces

$\zeta^{\flat}\ :\ \mathbf{L}(uW,-)\ \longrightarrow\
\mathbf{L}^{\mathbb{K}}(W,(-)_{\mathbb{K}})$

whose value at $uW$ takes the identity to $\zeta$. Since the map $\zeta$ is
$G_{0}$-equivariant, so is the morphism $\zeta^{\flat}$. For any $G$-space
$Z$, we can thus form the morphism of orthogonal spaces

$\zeta^{\flat}\times_{G_{0}}Z\ :\ \mathbf{L}(uW,-)\times_{G_{0}}Z\
\longrightarrow\ \mathbf{L}^{\mathbb{K}}(W,(-)_{\mathbb{K}})\times_{G_{0}}Z\
.$

The relation (A.28) implies that this morphism is
$G({\mathbb{K}})$-equivariant, and hence a morphism of orthogonal
$G({\mathbb{K}})$-spaces.

######  Theorem A.34.

Let $\epsilon:G\longrightarrow G({\mathbb{K}})$ be a continuous epimorphism
from a compact Lie group, where ${\mathbb{K}}$ is either ${\mathbb{C}}$ or
${\mathbb{H}}$. Let $W$ be a faithful $(G,\epsilon)$-representation, and let
$Z$ be a $G$-space.

  1. (i)

The morphism of orthogonal $G({\mathbb{K}})$-spaces
$\zeta^{\flat}\times_{G_{0}}Z$ is a $G({\mathbb{K}})$-global equivalence.

  2. (ii)

The morphism of orthogonal $G({\mathbb{K}})$-spectra

$\Sigma^{\infty}\zeta^{\flat}_{+}\wedge_{G_{0}}Z^{\diamond}\ :\
\Sigma^{\infty}\mathbf{L}(uW,-)_{+}\wedge_{G_{0}}Z^{\diamond}\
\longrightarrow\
\Sigma^{\infty}\mathbf{L}^{\mathbb{K}}(W,(-)_{\mathbb{K}})_{+}\wedge_{G_{0}}Z^{\diamond}$

is a $G({\mathbb{K}})$-global equivalence, where $Z^{\diamond}$ denotes the
unreduced suspension.

###### Proof.

(i) We consider a continuous homomorphism $\beta:K\longrightarrow
G({\mathbb{K}})$ from another compact Lie group. Then
$\mathbf{L}^{\mathbb{K}}(W,{\mathcal{U}}_{K}^{\mathbb{K}})$ is a universal
$(K\times_{G({\mathbb{K}})}G)$-space for the family of those closed subgroups
that intersect $1\times G_{0}$ trivially, by Proposition A.32. Also, the
$(K\times G)$-space $\mathbf{L}(uW,{\mathcal{U}}_{K})$ is a universal space
for the family of graph subgroups, i.e., those closed subgroups $\Delta$ of
$K\times G$ such that $\Delta\cap(1\times G)=e$, compare [20, Proposition
1.1.26]. For a subgroup of $K\times_{G({\mathbb{K}})}G$, the intersections
with $1\times G$ and $1\times G_{0}$ coincide; so the underlying
$(K\times_{G({\mathbb{K}})}G)$-space of $\mathbf{L}(uW,{\mathcal{U}}_{K})$ is
a universal space for the family of those closed subgroups $\Delta$ such that
$\Delta\cap(1\times G_{0})=e$. The continuous map

$\zeta^{\flat}({\mathcal{U}}_{K})\ :\ \mathbf{L}(uW,{\mathcal{U}}_{K})\
\longrightarrow\ \mathbf{L}^{\mathbb{K}}(W,{\mathcal{U}}_{K}^{\mathbb{K}})\
,\quad\varphi\ \longmapsto\ (\varphi\otimes{\mathbb{K}})\circ\zeta$

is $(K\times_{G({\mathbb{K}})}G)$-equivariant; since source and target are
universal spaces for the same family of subgroups, the map is a
$(K\times_{G({\mathbb{K}})}G)$-equivariant homotopy equivalence. The induced
map on $G_{0}$-orbit spaces

$\zeta^{\flat}({\mathcal{U}}_{K})\times_{G_{0}}Z\ :\
\mathbf{L}(uW,{\mathcal{U}}_{K})\times_{G_{0}}Z\ \longrightarrow\
\mathbf{L}^{\mathbb{K}}(W,{\mathcal{U}}_{K}^{\mathbb{K}})\times_{G_{0}}Z$

is thus an equivariant homotopy equivalence for the action of the group
$(K\times_{G({\mathbb{K}})}G)/G_{0}$. The projection to the first factor
identifies the group $(K\times_{G({\mathbb{K}})}G)/G_{0}$ with $K$; and the
$(K\times_{G({\mathbb{K}})}G)/G_{0}$-fixed points coincide with the fixed
points of the graph of $\beta$ on
$(\zeta^{\flat}\times_{G_{0}}Z)({\mathcal{U}}_{K})$. So we have verified that
the map $((\zeta^{\flat}\times_{G_{0}}Z)({\mathcal{U}}_{K}))^{\Gamma(\beta)}$
is a weak equivalence for every continuous homomorphism
$\beta:K\longrightarrow G({\mathbb{K}})$. Since the orthogonal spaces
underlying source and target of $\zeta^{\flat}\times_{G_{0}}Z$ are closed, the
morphism $\zeta^{\flat}\times_{G_{0}}Z$ is a $G({\mathbb{K}})$-global
equivalence by Proposition A.3.

(ii) The same argument to compare the long exact homotopy group sequences of
mapping cones as in Theorem A.15 applies here, but now the role of Proposition
A.5 is played by the first part of this theorem. ∎

We record a special case of Theorem A.34 that is particularly relevant for the
application to global stable splittings of Stiefel manifolds. We write

$\epsilon_{k}\ :\ \tilde{I}(k)=I(k)\rtimes
G({\mathbb{K}})=\mathbf{L}^{\mathbb{K}}({\mathbb{K}}^{k},{\mathbb{K}}^{k})\rtimes
G({\mathbb{K}})\ \longrightarrow\ G({\mathbb{K}})$

for the augmentation of the extended isometry group of ${\mathbb{K}}^{k}$,
i.e., the projection to the second factor. The tautological action makes
${\mathbb{K}}^{k}$ a faithful $(\tilde{I}(k),\epsilon_{k})$-representation in
the sense of Definition A.21. So for every orthogonal
$\tilde{I}(k)$-representation $U$, Theorem A.34 (ii) provides a
$G({\mathbb{K}})$-global equivalence of orthogonal $G({\mathbb{K}})$-spectra

$\Sigma^{\infty}\zeta^{\flat}_{+}\wedge_{I(k)}S^{U}\ :\
\Sigma^{\infty}(B_{\operatorname{gl}}\epsilon_{k})^{U}\ \xrightarrow{\ \simeq\
}\ \Sigma^{\infty}(\mathbf{Gr}_{k}^{\mathbb{K}})^{U}\ .$

We write

(A.35) $e_{k,U}\ =\
(\Sigma^{\infty}\zeta^{\flat}_{+}\wedge_{I(k)}S^{U})_{*}(e_{\epsilon_{k},U,u{\mathbb{K}}^{k}})\
\in\
\pi_{U}^{\tilde{I}(k)}(\epsilon_{k}^{*}(\Sigma^{\infty}(\mathbf{Gr}_{k}^{\mathbb{K}})^{U}))$

for the image under this $G({\mathbb{K}})$-global equivalence of the
tautological class defined in (A.16). Because
$\Sigma^{\infty}\zeta^{\flat}_{+}\wedge_{I(k)}S^{U}$ is a
$G({\mathbb{K}})$-global equivalence, Theorem A.17 (i) implies the following
result.

######  Corollary A.36.

Let ${\mathbb{K}}$ be ${\mathbb{C}}$ or ${\mathbb{H}}$. Let $U$ be an
orthogonal representation of the extended isometry group $\tilde{I}(k)$. Then
for every orthogonal $G({\mathbb{K}})$-spectrum $Y$, the evaluation map

${\mathcal{GH}}_{G({\mathbb{K}})}(\Sigma^{\infty}(\mathbf{Gr}_{k}^{\mathbb{K}})^{U},Y)\
\longrightarrow\ \pi_{U}^{\tilde{I}(k)}(\epsilon_{k}^{*}(Y))\ ,\quad f\
\longmapsto\ f_{*}(e_{k,U})$

is an isomorphism.

##  Appendix B Some linear algebra

The purpose of this appendix is to provide detailed proofs of the linear
algebra facts used in the main part of this paper. We set things up so that
everything works simultaneously over the fields ${\mathbb{R}}$ and
${\mathbb{C}}$ and over the skew-field ${\mathbb{H}}$ of quaternions. Linear
algebra over the quaternions comes with some additional caveats, many due to
the non-commutativity of the multiplication. So I felt the need to justify
that the relevant arguments can indeed be adapted to the quaternions
situation. I make absolutely no claim to originality for anything in this
appendix. If during the peer reviewing process a referee insists that the
material in this appendix is too trivial, I am willing to honor the tradition
in the subject, remove the appendix from the published version of this paper,
and declare the proofs as exercises to the reader.

Throughout this appendix, we will let ${\mathbb{K}}$ denote one of the three
skew-fields ${\mathbb{R}}$, ${\mathbb{C}}$ or ${\mathbb{H}}$. A
${\mathbb{K}}$-vector space is a right ${\mathbb{K}}$-module. Given two
${\mathbb{K}}$-vector spaces $V$ and $W$, we write
$\operatorname{Hom}_{\mathbb{K}}(V,W)$ for the ${\mathbb{R}}$-vector space of
right ${\mathbb{K}}$-linear maps. For ${\mathbb{K}}={\mathbb{C}}$ one can make
$\operatorname{Hom}_{\mathbb{C}}(V,W)$ a ${\mathbb{C}}$-vector space by
pointwise scalar multiplication, but we will not use this structure. Due to
the non-commutativity of the quaternions, for ${\mathbb{K}}={\mathbb{H}}$
there is no natural way to endow $\operatorname{Hom}_{\mathbb{H}}(V,W)$ with
an ${\mathbb{H}}$-action.

A ${\mathbb{K}}$-inner product space is a finite-dimensional
${\mathbb{K}}$-vector space equipped with a sesquilinear, hermitian and
positive-definite ${\mathbb{K}}$-valued inner product $[-,-]$, see Definition
2.2. An example of an inner product space is ${\mathbb{K}}^{k}$ with the
standard inner product

$[x,y]\ =\ \bar{x}_{1}\cdot y_{1}+\dots+\bar{x}_{k}\cdot y_{k}\ .$

Every ${\mathbb{K}}$-inner product space $W$ admits an orthonormal basis
$(w_{1},\dots,w_{k})$, i.e., such that

$[w_{i},w_{j}]\ =\ \begin{cases}1&\text{ for $i=j$, and}\\\ 0&\text{ for
$i\neq j$.}\end{cases}$

The familiar argument from real and complex linear algebra also works over the
quaternions: we choose a non-zero vector $w\in W$ and normalize it to
$w_{1}=w/|w|$; the orthogonal complement $W^{\perp}=\\{v\in W\ :\ [v,w]=0\\}$
is then a ${\mathbb{K}}$-subspace that has an orthonormal basis by induction
over the dimension. A choice of orthonormal basis of $W$ provides a
${\mathbb{K}}$-linear isometry ${\mathbb{K}}^{k}\cong W$, where
$k=\dim_{\mathbb{K}}(W)$. So up to ${\mathbb{K}}$-linear isometry, the
standard examples ${\mathbb{K}}^{k}$ are the only examples of
${\mathbb{K}}$-inner product spaces.

######  Remark B.1 (Adjoints).

Because inner products are positive-definite, they are in particular non-
degenerate, i.e., they provide an identification with the dual vector space.
Given a ${\mathbb{K}}$-vector space, we write

$W^{\vee}\ =\ \operatorname{Hom}_{{\mathbb{K}}}(W,{\mathbb{K}})$

for the ${\mathbb{R}}$-vector space of right ${\mathbb{K}}$-linear maps. We
give $W^{\vee}$ a right ${\mathbb{K}}$-action by

$(f\cdot\lambda)(w)\ =\ \bar{\lambda}\cdot f(w)$

for $f\in W^{\vee}$, $\lambda\in{\mathbb{K}}$ and $w\in W$. If $W$ is finite-
dimensional and $[-,-]$ is a ${\mathbb{K}}$-inner product on $W$, then the map

$W\ \longrightarrow\ W^{\vee}\ ,\quad w\ \longmapsto\ [w,-]$

is a ${\mathbb{K}}$-linear isomorphism.

Now we let $X:W\longrightarrow V$ be a ${\mathbb{K}}$-linear map between
${\mathbb{K}}$-inner product spaces. The adjoint of $X$ is the
${\mathbb{K}}$-linear map $X^{*}:V\longrightarrow W$ that makes the following
square commute:

$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{X^{*}}$$\scriptstyle{v\mapsto[v,-]}$$\scriptstyle{\cong}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w\mapsto[w,-]}$$\scriptstyle{\cong}$$\textstyle{V^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{X^{\vee}}$$\textstyle{W^{\vee}}$

So the adjoint is characterized by the relation

$[X^{*}v,w]\ =\ [v,Xw]$

for all $v\in V$ and $w\in W$. Passage to the adjoint is
${\mathbb{R}}$-linear, contravariantly functorial, and involutive.

Given two ${\mathbb{K}}$-inner product spaces $V$ and $W$, we write
$\mathbf{L}^{\mathbb{K}}(V,W)$ for the Stiefel manifold of
${\mathbb{K}}$-linear isometric embeddings, i.e., right ${\mathbb{K}}$-linear
maps $A:V\longrightarrow W$ that satisfy $[Av,Av^{\prime}]=[v,v^{\prime}]$ for
all $v,v^{\prime}\in V$. An equivalent condition is to demand that $A^{*}\cdot
A=\operatorname{Id}_{V}$. In the special case $V=W$, we also write

$I(W)\ =\ \mathbf{L}^{\mathbb{K}}(W,W)$

and refer to this as the isometry group of $W$. The traditional names for the
isometry group are of course the orthogonal group $O(W)$ in the case
${\mathbb{K}}={\mathbb{R}}$, the unitary group $U(W)$ in the case
${\mathbb{K}}={\mathbb{C}}$, and the symplectic group $Sp(W)$ in the case
${\mathbb{K}}={\mathbb{H}}$.

######  Remark B.2 (Adjoint representation).

We let $W$ be a finite-dimensional ${\mathbb{K}}$-vector space. The
exponential map

(B.3) $\exp\ :\ \operatorname{End}_{\mathbb{K}}(W)\ \longrightarrow\
GL_{\mathbb{K}}(W)\text{\qquad is given by\qquad}\exp(X)\ =\ \sum_{k\geq
0}X^{k}/k!\ .$

If the endomorphisms $X$ and $Y$ commute, then
$\exp(X+Y)=\exp(X)\cdot\exp(Y)$. For a ${\mathbb{K}}$-inner product space $W$,
we write

$\mathfrak{ad}(W)\ =\ \\{X\in\operatorname{End}_{\mathbb{K}}(W)\ :\
X^{*}=-X\\}$

for the ${\mathbb{R}}$-vector space of its skew-adjoint endomorphisms. Since
$(X^{*})^{k}=(X^{k})^{*}$ for all $k\geq 0$, also $\exp(X^{*})=\exp(X)^{*}$.
So if $X\in\mathfrak{ad}(W)$ is skew-adjoint, then

$\exp(X)^{*}\cdot\exp(X)\ =\ \exp(X^{*})\cdot\exp(X)\ =\ \exp(-X)\cdot\exp(X)\
=\ \exp(0)\ =\ \operatorname{Id}_{W}\ .$

In other words, the exponential map restricts to a smooth map

(B.4) $\exp\ :\ \mathfrak{ad}(W)\ \longrightarrow\ I(W)$

from the skew-adjoint endomorphisms to the isometry group. This map is in fact
a local diffeomorphism around the origin, and it exhibits $\mathfrak{ad}(W)$
as the adjoint representation of the compact Lie group $I(W)$, whence the
notation.

Due to the non-commutativity of the quaternions, the notions of ‘eigenvalues’
and ‘eigenvectors’ of ${\mathbb{H}}$-linear endomorphisms are somewhat
problematic. Indeed, if $X:W\longrightarrow W$ is an ${\mathbb{H}}$-linear
endomorphism and $\lambda\in{\mathbb{H}}$ a scalar, then the ‘eigenspace’

$\\{w\in W\ :\ Xw=w\lambda\\}$

is an ${\mathbb{R}}$-subspace of $W$, but typically not closed under
multiplication by scalars from ${\mathbb{H}}$. If the scalar $\lambda$ is
real, and hence central in ${\mathbb{H}}$, then the issue disappears and the
above is an ${\mathbb{H}}$-subspace of $W$. So in order to deal with
eigenspaces in a uniform way for ${\mathbb{R}}$, ${\mathbb{C}}$ and
${\mathbb{H}}$, we should – and will – restrict to real eigenvalues.

######  Proposition B.5.

Let $X$ be a self-adjoint endomorphism of a ${\mathbb{K}}$-inner product space
$W$, i.e., $X^{*}=X$. Then $X$ is diagonalizable with real eigenvalues and
pairwise orthogonal eigenspaces.

###### Proof.

The argument over the real and complex numbers can be found in many text books
on linear algebra. So we restrict to the less common case
${\mathbb{K}}={\mathbb{H}}$ of the quaternions. We argue over the dimension of
$W$; the induction starts with $W=0$, where there is nothing to show. Now we
suppose that $W\neq 0$. We treat $X$ as a ${\mathbb{C}}$-linear endomorphism
of the underlying ${\mathbb{C}}$-vector space of $W$. There is then an
eigenvector $w\in W\setminus\\{0\\}$ and a complex scalar
$\lambda\in{\mathbb{C}}$ such that $Xw=w\lambda$. Because $X$ is self-adjoint
we deduce that

$[w,w]\lambda\ =\ [w,w\lambda]\ =\ [w,Xw]\ =\ [Xw,w]\ =\ [w\lambda,w]\ =\
\bar{\lambda}[w,w]\ .$

Since $w\neq 0$, the inner product $[w,w]$ is a non-zero real number, so we
must have $\lambda=\bar{\lambda}$, i.e., the complex scalar $\lambda$ is in
fact real. Now we observe that the orthogonal complement $W^{\perp}$ of $w$ is
invariant under $X$. Indeed, for $v\in W^{\perp}$ we have

$[Xv,w]\ =\ [v,Xw]\ =\ [v,w\lambda]\ =\ [v,w]\lambda\ =\ 0\ .$

We can thus apply the inductive hypothesis to the restricted endomorphism
$X|_{W^{\perp}}:W^{\perp}\longrightarrow W^{\perp}$. Since this restriction is
diagonalizable with real eigenvalues and pairwise orthogonal eigenspaces, the
same is true for the original endomorphism $W$. ∎

We write

$\mathfrak{sa}(W)\ =\ \\{X\in\operatorname{End}_{\mathbb{K}}(W)\ :\
X^{*}=X\\}$

for the ${\mathbb{R}}$-vector space of self-adjoint endomorphisms of a
${\mathbb{K}}$-inner product space $W$. As mentioned above, the exponential
map (B.3) commutes with the passage to adjoints; so if $X$ is self-adjoint,
then so is $\exp(X)$. Hence for $X\in\mathfrak{sa}(X)$, both $X$ and $\exp(X)$
are diagonalizable with real eigenvalues and pairwise orthogonal eigenspaces,
by Proposition B.5. Moreover, if $Xw=w\lambda$, then

$\exp(X)w\ =\ \sum_{k\geq 0}X^{k}w\ =\ \sum_{k\geq 0}w\cdot\lambda^{k}\ =\
w\cdot\exp(\lambda)\ .$

So $\exp(X)$ has the same eigenspaces as $X$, but the corresponding
eigenvalues are exponentiated. This process can be reversed as long as the
eigenvalues are positive, so the restricted exponential map

(B.6) $\exp\ :\ \mathfrak{sa}(W)\ \xrightarrow{\ \cong\ }\
\mathfrak{sa}^{+}(W)\ =\ \\{X\in\mathfrak{sa}(W)\ :\ \text{$X$ is positive-
definite}\\}$

is a homeomorphism onto the subspace of positive-definite self-adjoint
endomorphisms, i.e, the ones with positive real eigenvalues.

######  Proposition B.7.

Let $V$ and $W$ be ${\mathbb{K}}$-inner product spaces. The smooth map

$\mathbf{L}^{\mathbb{K}}(W,V)\times\mathfrak{sa}(W)\ \longrightarrow\
\operatorname{Hom}_{\mathbb{K}}(W,V)\ ,\quad(A,Z)\ \longmapsto\
A\cdot\exp(-Z)$

is an open embedding with image the subspace of ${\mathbb{K}}$-linear
monomorphisms.

###### Proof.

Since the restricted exponential map (B.6) is a homeomorphism, it suffices to
show that the composition map

$\circ\ :\ \mathbf{L}^{\mathbb{K}}(W,V)\times\mathfrak{sa}^{+}(W)\
\longrightarrow\ \operatorname{Hom}^{\text{inj}}_{\mathbb{K}}(W,V)$

is a homeomorphism. If $B:W\longrightarrow V$ is any ${\mathbb{K}}$-linear
map, then $B^{*}\cdot B$ is self-adjoint and positive semi-definite. If $B$ is
a moreover injective, then $B^{*}\cdot B$ is even positive-definite. We write

$\sqrt{\quad}\ :\ \mathfrak{sa}^{+}(W)\ \longrightarrow\ \mathfrak{sa}^{+}(W)$

for the homeomorphism that sends a positive-definite self-adjoint endomorphism
$X$ to the unique positive-definite self-adjoint endomorphism such that
$\sqrt{X}\cdot\sqrt{X}=X$. Then the continuous map

$\ \operatorname{Hom}_{\mathbb{K}}^{\text{inj}}(W,V)\ \longrightarrow\
\mathbf{L}^{{\mathbb{K}}}(W,V)\times\mathfrak{sa}^{+}(W)\ ,\quad B\longmapsto\
\left(B\cdot\sqrt{B^{*}\cdot B}^{-1},\sqrt{B^{*}\cdot B}\right)$

is inverse to the composition map. ∎

The complex version of the next proposition is [5, Lemma 1.13]; Crabb leaves
the proof as an exercise in linear algebra, and we will do the exercise. Crabb
indicates how one can arrive at the formula for the open embedding
$\mathfrak{c}$ in the paragraph after [5, Lemma 1.15]. Propositions B.8 and
B.11 together recover Miller’s homeomorphism [18, Theorem A] that exhibits the
filtration stratum
$\mathbf{F}_{k}^{\mathbb{K}}(W;m)\setminus\mathbf{F}_{k-1}^{\mathbb{K}}(W;m)$
as the total space of the vector bundle over $\mathbf{Gr}_{k}^{\mathbb{K}}(W)$
associated to the $I(k)$-representation $\nu(k,m)\oplus\mathfrak{ad}(k)$.

######  Proposition B.8.

Let $V$ and $W$ be ${\mathbb{K}}$-inner product spaces. Then the map

$\mathfrak{c}\ :\
\operatorname{Hom}_{{\mathbb{K}}}(W,V)\oplus\mathfrak{ad}(W)\ \longrightarrow\
\mathbf{L}^{\mathbb{K}}(W,W\oplus V)\ ,\quad\mathfrak{c}(Y,X)\ =\ (g,h)$

with

$\displaystyle g\ $ $\displaystyle=\ (X/2+Y^{*}Y/4-1)(X/2+Y^{*}Y/4+1)^{-1}$
$\displaystyle:\ W\ \longrightarrow\ W$ $\displaystyle h\ $ $\displaystyle=\
Y(1-g)/2$ $\displaystyle:\ W\ \longrightarrow\ V$

is an open embedding onto the subspace of those
$f\in\mathbf{L}^{\mathbb{K}}(W,W\oplus V)$ such that $\ker(f-i_{1})=0$.

###### Proof.

It will be convenient to abbreviate

$C\ =\ X/2+Y^{*}Y/4\ ;$

we check that the endomorphism $C+1$ has a trivial kernel, and is thus
invertible. Indeed, suppose that $Cw=-w$ for some $w\in W$. Then

$-Xw/2\ =\ -(C-Y^{*}Y/4)w\ =\ w+Y^{*}Yw/4\ ,$

and hence

$\displaystyle-[w,w]\ $ $\displaystyle=\ [w,Cw]\ =\ [w,(X/2+Y^{*}Y/4)w]\ =\
1/2[X^{*}w,w]+1/4[Yw,Yw]$ $\displaystyle=\ -1/2[Xw,w]+1/4[Yw,Yw]$
$\displaystyle=\ [w+Y^{*}Yw/4,w]+1/4[Yw,Yw]\ =\ [w,w]+1/2[Yw,Yw]\geq 0\ .$

So we must have $w=0$. Since $C+1$ is invertible, the definitions of

(B.9) $g\ =\ (C-1)(C+1)^{-1}$

and $h$ make sense.

Now we show that $g^{*}g+h^{*}h=1$, so that $(g,h):W\longrightarrow W\oplus V$
is indeed a linear isometric embedding. Because $X$ is skew-adjoint and
$Y^{*}Y$ is self-adjoint, we have $C+C^{*}=Y^{*}Y/2$. Relation (B.9) implies

(B.10) $1-g\ =\ 2(C+1)^{-1}\ ,$

so

$h^{*}h\ =\ (1-g^{*})Y^{*}Y(1-g)/4\quad_{\eqref{eq:1-g}}=\
2(C^{*}+1)^{-1}(C^{*}+C)(C+1)^{-1}\ .$

Hence

$\displaystyle g^{*}g+h^{*}h\ $ $\displaystyle=\
(C^{*}+1)^{-1}\left[(C^{*}-1)(C-1)+2(C^{*}+C)\right](C+1)^{-1}$
$\displaystyle=\ (C^{*}+1)^{-1}(C^{*}+1)(C+1)(C+1)^{-1}\ =\ 1\ .$

The morphism $1-g$ is invertible by (B.10), and thus has trivial kernel. So
also the kernel of $(g,h)-i_{1}=(g-1,h):W\longrightarrow W\oplus V$ is
trivial. This concludes the verification that $\mathfrak{c}$ is a well-defined
map with image in the subspace of those linear isometric embeddings
$f:W\longrightarrow W\oplus V$ such that $f-i_{1}$ has trivial kernel.

Now we exhibit a continuous inverse to $\mathfrak{c}$. We define

$\lambda\ :\ \\{(g,h)\in\mathbf{L}^{\mathbb{K}}(W,W\oplus V)\ :\
\ker(1-g)=0\\}\ \longrightarrow\
\operatorname{Hom}_{\mathbb{K}}(W,V)\oplus\mathfrak{ad}(W)\text{\quad by
\quad}\lambda(g,h)\ =\ (Y,X)\ ,$

where

$\displaystyle Y\ $ $\displaystyle=\ 2h(1-g)^{-1}$ $\displaystyle X\ $
$\displaystyle=\ 2(1-g^{*})^{-1}(g-g^{*})(1-g)^{-1}\ .$

We show that $\lambda$ is indeed inverse to $\mathfrak{c}$. To verify the
relation $\mathfrak{c}(\lambda(g,h))=(g,h)$ in the first component we set

$\displaystyle C\ $ $\displaystyle=\ X/2+Y^{*}Y/4$ $\displaystyle=\
(1-g^{*})^{-1}(g-g^{*}+h^{*}h)(1-g)^{-1}$ $\displaystyle=\
(1-g^{*})^{-1}\left(g-g^{*}+(1-g^{*}g)\right)(1-g)^{-1}$ $\displaystyle=\
(1-g^{*})^{-1}(1-g^{*})(1+g)(1-g)^{-1}\ =\ (1+g)(1-g)^{-1}\ =\ 1+2g(1-g)^{-1}\
.$

Then

$\displaystyle(C-1)\cdot(C+1)^{-1}\ $ $\displaystyle=\
2g(1-g)^{-1}\cdot\left(2(1+g(1-g)^{-1})\right)^{-1}$ $\displaystyle=\
g(1-g)^{-1}\cdot\left(1+g(1-g)^{-1}\right)^{-1}\ =\ g(1-g)^{-1}\cdot(1-g)\ =\
g\ .$

In the second component, the desired equality is simply

$Y(1-g)/2\ =\ 2h(1-g)^{-1}(1-g)/2\ =\ h\ .$

The final check is the relation $\lambda(\mathfrak{c}(Y,X))=(Y,X)$. In the
second component we use that

$\displaystyle 1-g\ $ $\displaystyle=\ 1-(C-1)(C+1)^{-1}\ =\ 2(C+1)^{-1}$
$\displaystyle g-g^{*}$ $\displaystyle=\ 2(C^{*}+1)^{-1}-2(C+1)^{-1}\ =\
2(C^{*}+1)^{-1}(C-C^{*})(C+1)^{-1}\ ,$

and hence

$\displaystyle 2(1-g^{*})^{-1}(g-g^{*})(1-g)^{-1}\ $ $\displaystyle=\ C-C^{*}\
\ =\ 1/2(X-X^{*})\ =\ X\ .$

In the first component, the desired equality is simply

$2h(1-g)^{-1}\ =\ 2\left(Y(1-g)/2\right)(1-g)^{-1}\ =\ Y\ .\qed$

We let $W$ be a ${\mathbb{K}}$-inner product space. As in the body of this
paper, we denote the $k$-th term of the eigenspace filtration of the Stiefel
manifold $\mathbf{L}^{\mathbb{K}}(W,W\oplus{\mathbb{K}}^{m})$ by

$\mathbf{F}_{k}(W;m)\ =\ \\{f\in\
\mathbf{L}^{\mathbb{K}}(W,W\oplus{\mathbb{K}}^{m})\ :\
\dim_{\mathbb{K}}(\ker(f-i_{1})^{\perp})\leq k\\}\ ,$

where $i_{1}:W\longrightarrow W\oplus{\mathbb{K}}^{m}$ is the embedding of the
first summand. We write
$I(k)=\mathbf{L}^{\mathbb{K}}({\mathbb{K}}^{k},{\mathbb{K}}^{k})$ for the
isometry group of ${\mathbb{K}}^{k}$. Conjugation ${{}^{\psi}(-)}$ by a linear
isometric embedding $\psi$ was defined in (2.5).

######  Proposition B.11.

For every ${\mathbb{K}}$-inner product space $W$ and all $k,m\geq 0$, the map

$\mathbf{L}^{\mathbb{K}}({\mathbb{K}}^{k},W)\times_{I(k)}\mathbf{L}^{\mathbb{K}}({\mathbb{K}}^{k},{\mathbb{K}}^{k+m})\
\longrightarrow\ \mathbf{F}_{k}(W;m)\ ,\quad[\psi,f]\ \longmapsto\
{{}^{\psi}f}$

is a relative homeomorphism from the pair

$\left(\mathbf{L}^{\mathbb{K}}({\mathbb{K}}^{k},W)\times_{I(k)}\mathbf{L}^{\mathbb{K}}({\mathbb{K}}^{k},{\mathbb{K}}^{k+m}),\
\mathbf{L}^{\mathbb{K}}({\mathbb{K}}^{k},W)\times_{I(k)}\mathbf{F}_{k-1}({\mathbb{K}}^{k};m)\right)$

to the pair $(\mathbf{F}_{k}(W;m),\mathbf{F}_{k-1}(W;m))$.

###### Proof.

Since all spaces involved are compact, we only need to show that the map in
question restricts to a homeomorphism from

$\mathbf{L}^{\mathbb{K}}({\mathbb{K}}^{k},W)\times_{I(k)}(\mathbf{L}^{\mathbb{K}}({\mathbb{K}}^{k},{\mathbb{K}}^{k+m})\setminus\mathbf{F}_{k-1}({\mathbb{K}}^{k};m))$

to the complement of $\mathbf{F}_{k-1}(W;m)$ in $\mathbf{F}_{k}(W;m)$. We
consider a linear isometric embedding $g:W\longrightarrow
W\oplus{\mathbb{K}}^{m}$ in
$\mathbf{F}_{k}(W;m)\setminus\mathbf{F}_{k-1}(W;m)$, i.e., so that

$\dim_{\mathbb{K}}(\ker(g-i_{1})^{\perp})\ =\ k\ .$

We choose a ${\mathbb{K}}$-linear isometry

$\psi\ :\ {\mathbb{K}}^{k}\ \xrightarrow{\ \cong\ }\ \ker(g-i_{1})^{\perp}\ .$

Then $g$ restricts to a linear isometric embedding

$\bar{g}\ :\ \ker(g-i_{1})^{\perp}\ \longrightarrow\
\ker(g-i_{1})^{\perp}\oplus{\mathbb{K}}^{m}\ ,$

and we can define the linear isometric embedding
$f:{\mathbb{K}}^{k}\longrightarrow{\mathbb{K}}^{k+m}$ by

$f\ =\ (\psi^{-1}\oplus{\mathbb{K}}^{m})\circ\bar{g}\circ\psi\ .$

Then $g={{}^{\psi}f}$, and $f$ satisfies $\ker(f-i_{{\mathbb{K}}^{k}})=0$, so
that $f$ belongs to
$\mathbf{L}^{\mathbb{K}}({\mathbb{K}}^{k},{\mathbb{K}}^{k+m})\setminus\mathbf{F}_{k-1}({\mathbb{K}}^{k};m)$.
Since the only choice in this construction was the ${\mathbb{K}}$-linear
isometry $\psi$, and any two choices differ by precomposition with an element
of $I(k)$, this proves the claim. ∎

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11institutetext: L. T. George 22institutetext: National Centre for Radio
Astrophysics - Tata Institute of Fundamental Research
Ganeshkhind, Pune - 411007
22email<EMAIL_ADDRESS>33institutetext: R. Kale 44institutetext: NCRA-
TIFR
Ganeshkhind, Pune - 411007 55institutetext: Y. Wadadekar 66institutetext:
NCRA-TIFR
Ganeshkhind, Pune - 411007

# An upper limit calculator (UL-CALC) for undetected extended sources with
radio interferometers: radio halo upper limits

Lijo T. George  Ruta Kale  Yogesh Wadadekar

(Received: / Accepted: )

###### Abstract

Radio halos are diffuse, extended sources of radio emission detected primarily
in massive, merging galaxy clusters. In smaller and/or relaxed clusters, where
no halos are detected, one can instead place upper limits to a possible radio
emission. Detections and upper limits are both crucial to constrain
theoretical models for the generation of radio halos. The upper limits are
model dependent for radio interferometers and thus the process of obtaining
these is tedious to perform manually. In this paper, we present a Python based
tool to automate this process of estimating the upper limits. The tool allows
users to create radio halos with defined parameters like physical size,
redshift and brightness model. A family of radio halo models with a range of
flux densities, decided based on the rms noise of the image, is then injected
into the parent visibility file and imaged. The halo injected image and the
original image are then compared to check for the radio halo detection using a
threshold on the detected excess flux density. Injections separated by finer
differences in the flux densities are carried out once the coarse range where
the upper limit is likely to be located has been identified. The code
recommends an upper limit and provides a range of images for manual
inspection. The user may then decide on the upper limit. We discuss the
advantages and limitations of this tool. A wider usage of this tool in the
context of the ongoing and upcoming all sky surveys with the LOFAR and SKA is
proposed with the aim of constraining the physics of radio halo formation. The
tool is publicly available at https://github.com/lijotgeorge/UL-CALC.

###### Keywords:

galaxies:clusters:general galaxies: clusters: intracluster medium radiation
mechanisms: non-thermal radio continuum:general methods:data analysis
software: data analysis

††journal: Experimental Astronomy

##  1 Introduction

The radio interferometric technique used in producing images of the sky in
radio bands uses the Fourier transform relation between the visibilities and
the intensity distribution in the sky (Thompson et al.,, 2017, Chapter 15).

The image produced by inverse Fourier transforming the visibilities is limited
to the angular scales ($\theta$) which are sampled by the baselines of the
interferometer. For a range of baseline lengths, from $B_{min}$ to $B_{max}$,
the smallest angular scale is $\theta_{min}=\lambda/B_{max}$ and the largest
scale is $\theta_{min}=\lambda/B_{min}$, where $\lambda$ is the wavelength. It
has been shown that radio interferometers recover a fraction of the total flux
density at a given angular scale even when that scale is sampled. Good overall
uv-coverage with full synthesis imaging at that angular scale is needed for a
near complete flux recovery (Deo and Kale, (2017)). The effect of insufficient
sampling in the uv-plane is especially evident in the study of extended radio
sources - for example, radio halos in galaxy clusters (Kale et al., (2015))
and supernova remnants (Nayana et al., (2017)). In the case of non-detection
of such sources, the upper limit is dependent on the uv-coverage of that
particular observation and needs to be obtained using models of the brightness
distribution.

The method of injecting models of the expected extended source and quantifying
the recovery has been extensively used to obtain the upper limits for the
detections of radio halos. Radio halos are diffuse radio sources of
synchrotron origin that are associated, not with individual galaxies, but with
the intra-cluster medium (see van Weeren et al., (2019) for a review) in
galaxy clusters. They have linear extents of 1 - 2 Mpc that correspond to
angular extents of a few to tens of arcminutes in clusters at redshifts
$<0.4$. They are also rare and thus the rates of non-detection in samples of
clusters can exceed $70\%$. The non-detection of a radio halo is highly
dependent on the short spacings in the data in addition to the rms
sensitivity. The method of obtaining model dependent upper limits was first
used in the GMRT Radio Halo Survey (Venturi et al., (2007, 2008)) and then in
the Extended GMRT Radio Halo Survey (Kale et al., (2013, 2015)). The steps of
building a model radio halo, injecting it in the visibilities and imaging
those to check the recovery were carried out either manually or in a semi-
automated fashion making it a slow and tedious process.

In this work, we present a general tool that we call the “UL-CALC” to obtain
upper limits on extended sources of a model of the user’s choice. Application
of this tool is illustrated in the context of finding upper limits for radio
halos in galaxy clusters, and discussed in the context of other areas such as
mini-halos, radio relics, supernova remnants and radio galaxies. The paper is
organised as follows: the models implemented in literature for radio halos are
described in Section 2. In Section 3 the halo upper limit calculator is
presented and an illustration of its usage is provided in Section 4. We
discuss the advantages and caveats of the calculator and its applicability to
large surveys in Section. 5. Conclusions are presented in Section. 6.

Figure 1: The various halo models available in the upper limit calculator.
From left to right, they are the Gaussian, Exponential and Polynomial models.
A colorbar common to all three images is shown to the right as well. All the
halos have a physical size of 1000 kpc, total intensity of 1 Jy and are placed
at a redshift of 0.1. Figure 2: Radial profile comparison of the various halo
models as depicted in Figure 1. The dashed line corresponds to the radius of
the halo calculated in pixels ($139.33$) assuming a physical size of 1 Mpc and
a redshift of 0.1 and each pixel corresponds to $1.5"$

##  2 Models for radio halos

Radio halos are of synchrotron origin and are thus manifestations of the
cosmic ray electrons ($\sim$GeV) and magnetic fields ($\sim\mu$G) pervading
the ICM (Intra Cluster Medium). These sources are known to occur in about
$22\%$ of massive ($>5\times 10^{14}$ M⊙) and merging galaxy clusters (Kale et
al., (2015)). Although there is a large variety in the details of the
morphology of radio halos (Feretti et al., (2012)), radial profiles of well
known radio halos have been modelled.

In this work, we use three models of radio halo surface brightness taken from
the literature. These models are described below.

  * •

Gaussian:
The physical size of the radio halo is assumed to be the FWHM (Full Width at
Half Maximum) of a single Gaussian. The standard deviation of the model is
then estimated from the FWHM.

  * •

Polynomial:
A polynomial model of halo emission was first introduced by Johnston-Hollitt
and Pratley, (2017) where the authors fit the normalized brightness profile of
5 radio halos to a polynomial expression. They carried out a linear least
square fit to the radial brightness distribution with a quadratic function and
obtained a best fit of the form:

$I(r)=-0.719r^{2}+1.867r-0.141$ (1)

  * •

Exponential:
Murgia et al., (2009) found the brightness profile of radio halos to follow an
exponential law in the form of:

$I(r)=I_{0}e^{-r/r_{e}},$ (2)

where $r_{e}$ and $I_{0}$ are free parameters. Bonafede et al., (2017) fit the
brightness profiles of 8 known halos and estimated $r_{e}=2.6R_{H}$ where
$R_{H}$ is the measured radius of the halo as estimated from images.

Figure 1 shows all three models for a fixed value of $I_{0}$ (1 Jy), $R_{H}$
(1 Mpc) and $z$ (0.1). For our purposes, we have chosen the exponential model
to be the default model used while creating model radio halos. We would like
to mention that while we have developed the tool with these three models, it
is fairly straightforward to introduce a new model, should the need arise.

In Figure 2 we also show the radial profiles of all three models discussed
above. The profiles were calculated for the three images in Figure 1. Each
pixel corresponds to $1.5"$ and the radius was calculated to be 139.33 pixels
assuming $R_{H}$ = 1 Mpc and $z$ = 0.1.

##  3 The Halo Upper Limit Calculator

Radio halos being faint (surface brightnesses at 1.4 GHz typically $<1\mu$Jy
arcsec-2) and extended (a few to several arcminutes depending on the redshift)
sources, need high quality radio observations with radio interferometers to
sample the angular scales and also to resolve the discrete sources such as
point sources and radio galaxies. The non-detection of such a source is also
important to derive the implications to the physics of their origin (e.g.
Brunetti et al., (2007)). The non-detection is a function of the sensitivity
reached in the map and the uv-coverage on the angular scales where the
emission is expected. Here, an empirical approach of injecting a model
extended source in the visibilities and then checking for its detection is
needed. This process involves the steps of model preparation, injection of the
model in the visibilities and then making an image of the visibilities with
the injected model. Once the image is made, the detection of the source needs
to be checked for and this determines the direction of further model injection
- whether a fainter or a brighter model is to be used. We describe the upper
limit calculator that automates this process.

The implementation is described using the case of determining upper limits for
a radio halo in a galaxy cluster. The steps are schematically shown in Fig. 3
and described below.

  1. 1.

The input files required are the final visibility file after self-calibration
and the final image. The final image either already exists or is created as a
first step.

  2. 2.

An image of the model radio halo of given flux, redshift, linear size and
brightness profile is generated. The geometry of this image is identical to
the image in the previous step.

  3. 3.

The model image is Fourier transformed and added to the visibility file using
standard CASA tasks.

  4. 4.

The new visibility file is imaged using the same parameters as used earlier to
produce a new image.

  5. 5.

The difference in flux density around the central region between the image
with and without the radio halo is estimated to calculate the “recovered flux
density”.

  6. 6.

The process is repeated for different values of halo flux densities keeping
other parameters such as redshift, halo size fixed, and the new images are
manually inspected to find the value at which a radio halo is visible.

  7. 7.

This value can now be scaled to 1.4 GHz using an appropriate spectral index
value to estimate the upper limit to the radio halo power. This step is
performed because most scaling relations for radio halos are determined using
the radio power at 1.4 GHz (see for example van Weeren et al., (2019); Feretti
et al., (2012) for reviews).

In the following subsections, the steps are described in further detail. All
the tasks described below are performed using the Common Astronomy Software
Applications (CASA) software (McMullin et al., (2007)) and makes use of the
visibility and image architecture used by CASA.

Figure 3: A schematic representation of the steps followed in the upper limit
calculator.

###  3.1 Creating the model radio halo

The strength of the radio halo that needs to be injected in an image is
entirely dependent on the quality of the image itself as well as the frequency
of observation. The noise in the image can be used as a guide to decide on the
flux densities of the halo that will be injected. If the observation is also
deep with good short spacing coverage, then the upper limit of halo emission
will be correspondingly lower when compared to an image that is noisy. As a
result, the first step of the process is to estimate the RMS in the image
around the region where the source is expected to be. For our purposes, we are
using the Background And Noise Estimation (BANE) program which comes packaged
with the Aegean111https://github.com/PaulHancock/Aegean/wiki software.

The flux density of the injected radio halo is taken to be a factor times the
calculated RMS in the image. Alternatively, a manual list of flux densities
can also be provided to be injected.

The only other physical property of the halo that is required is the angular
size, $\theta$. This value depends on the redshift ($z$) of the target as well
as the assumed physical size of the radio halo. Based on literature, the
physical size of radio halos can be assumed to be $\sim 1$ Mpc and the angular
size ($\theta$) subsequently estimated. This task uses the cosmology from
Planck Collaboration et al., (2016) with $H_{0}=67.7$, $\Omega_{m}=0.307$ and
$\Omega_{total}=1$.

The model of surface brightness distribution for the halo has been previously
discussed in Section 2.

###  3.2 Injecting the radio halo

The radio halo created in the manner described above is only at a single
frequency: the central bandwidth frequency of the input visibility file.
Therefore we need to create additional images of the halo at all the other
frequencies.

This requires an assumption to be made regarding the frequency dependence of
the radio halo flux density. The average spectral index ($\alpha$) of radio
halos has been found to be $~{}-1.3$ (Feretti et al., (2012)).

We use this value to estimate the radio halo flux densities at other
frequencies before taking Fourier transforms of the images and adding them to
the input visibility file.

###  3.3 Estimating radio halo upper limit

Once all the halo visibilities have been added the next step is to image the
new visibility file, preferably to the same depth as the input image.
Currently, we use the tclean task provided in CASA for imaging.

The RMS estimated in the first step is used as a threshold during this
deconvolution procedure. In order to better retrieve the extended emission in
the image, we convolve the image with a wider beam to improve the signal to
noise ratio. This can be done in multiple ways: a larger beam can be specified
manually, or the beam can simply be a factor times the input beam, or the beam
can be such that the halo surface area contains at least nbeams number of
beams. The tool uses the last option by default with nbeams as 100.

Figure 4: Progress of upper limit estimator for the cluster RXC J1701.3+6414.
Contours start at $3\sigma(\sigma=0.284$ mJy/beam) and increase by $\sqrt{2}$
afterwards. A single negative contour at $-3\sigma$ is also shown in dotted
lines. The top left image is the original image created of the cluster. The
next two images in the top row have halos with injected flux densities 14.21
and 28.41 mJy, respectively. The images in the bottom row (from left to right)
have halos of injected flux densities 46.83, 85.24 and 142.07 mJy. Beam size
in all images is the same and is shown in the bottom left corner
($\theta=18^{\prime\prime}\times 18^{\prime\prime}$).

Based on previous experience by the authors (Kale et al., (2013, 2015)) most
halo upper limits were calculated when the excess flux in the halo region was
$\sim 10\%$ higher than in the input image which has been our experience as
well.

The above steps are performed iteratively for different values of the input
radio halo flux density until the excess flux exceeds $10\%$. At this stage
one still needs to manually inspect the images to deduce the flux density of
the halo when it becomes “visible”.

The program allows the user to generate and view contours at each iteration of
the process. By default contour levels start at $3\sigma$ (where $\sigma$ is
the RMS noise in the original image) and increase by $\sqrt{2}$ afterwards. A
single negative contour at $-3\sigma$ is also plotted.

Parameter name | Description
---|---
bane_pth | Path to BANE executable
srcdir | Source Directory
visname | Reference visibility file
imgname | Reference image file made from ’visname’
cluster | Cluster name (optional)
z | Redshift of source †
l | Size of halo to be injected (kpc)
alpha | Spectral index for frequency scaling ($S\propto\nu^{-\alpha}$)
ftype | Radial profile of halo. Options: (G)aussian, (P)olynomial, (E)xponential
theta | Angular size (in arcsec) for halo (size=l) at redshift z †
x0, y0 | Halo injection position
cell | Pixel separation (in arcsec)
hsize | Size of halo (in pixels) †
do_fac | Whether calculate injection halo flux density or use manually (True or False)
flx_fac | Flux level factors
flx_lst | Flux list (in Jy)
cln_task | Clean task to use (’tclean’, ’wsclean’)
N | No. of iterations
isize | Image size †
csize | Cell size †
weight | Weighting to be used
dcv | Deconvolver to use
scle | Multi-scale options
thresh_f | Cleaning threshold factor
radius | Radius of halo (in arcsec) †
rms_reg | Region from which to estimate rms †
bopts | Method of image convolution (’num_of_beams’, ’factor’, ’beam’)
nbeams | No. of synthesized beams in halo
bparams | Beam size (bmaj("), bmin("), bpa(deg))
smooth_f | Factor to smooth input beam
recv_th | Threshold of Excess flux recovery at which to fine tune (in percent)
n_split | Number of levels to split during fine tuning
do_contours | Whether to make contours at the end of each iteration (True or False)

Table 1: Brief description of all options available in params.py. Parameters
with the $\dagger$ superscript are automatically calculated based on other
user provided paramters.

###  3.4 Options

Table 1 gives the list of various parameters that can be modified in
params.py. Below we will briefly go through some of those options and their
default values.

  * •

l: Giant radio halos are typically $\sim$1 Mpc in size. Default value: 1000
kpc

  * •

alpha: Spectral index of the radio halo spectrum. Default value: $-1.3$

  * •

flx_fac: Factor times the RMS used to calculate injection halo flux density.
Default values: [50, 100, 200, 300, 500, 1000, 2000]

  * •

cln_task: specifies which task should be used for deconvolution. Currently
works best with CASA’s tclean task.

  * •

thresh_f: Threshold to which to clean to. Default value: $3\sigma$

  * •

radius: Calculated halo radius based on l and z

  * •

rms_reg: Radius of region in which to estimate RMS. Default value:
$3\times$radius

  * •

recv_th: Excess flux density recovery threshold. Default value: 10%.

  * •

n_split: How many extra images to produce during fine-tuning (Default: 6)

Figure 5: Sub images created between images 4 and 5 of Figure 3. Injected flux
densities (left to right) are 34.1 and 39.78 mJy (top row) and 45.46 and 51.15
mJy (bottom row).

##  4 Illustration of using the UL calculator tool

Figure 4 shows this process for the cluster RXC J1701.3+6414. The cluster was
observed using the GMRT (Giant Metrewave Radio Telescope) on 21-08-2002 at 325
MHz for 189 minutes as part of the GMRT Key Project on galaxy clusters (George
et al., in prep). The top left panel shows the original image (produced using
the SPAM (Source Peeling and Atmospheric Modeling) software (Intema et al.,
(2009); Intema, (2014))). The rest of the images have radio halos of various
flux densities added to them at the cluster centre. In the rest of the figures
in the top panel the halo is still not visible. The figures in the bottom
panel clearly detect the injected radio halo. The injected flux densities of
the radio halos in the three images (from left to right) are 56.83, 85.24 and
142.07 mJy.

Quantitatively, this can also be seen through the contours seen in the figure.
The RMS in the central region of the original image was estimated to be
$0.284$ mJy/beam. We plot contours on the images starting at $3\sigma$ to
trace the emergence of the radio halo in the subsequent images. Plotting these
contours provides another visual indication of the detection of a radio halo
of flux density 56.83 mJy.

At this point fine-tuning can be performed between the flux range 28 and 56
mJy. Figure 5 shows the images of injected halos with flux densities between
28 and 56 mJy. Based on these images the injected halo with flux density 51.15
mJy is the best image to claim detection of the radio halo.

This can now be extrapolated to 1.4 GHz using an average spectral index of
$-1.3$ to get the radio halo upper limit at 1.4 GHz. Table 2 also shows the
injected flux densities and the recovered flux densities in the halo region
for the cluster.

##  5 Discussion

Figure 6: Image of the injected halo with flux density 14.21 mJy with contours of recovered halo of flux densities 34.1 mJy (cyan), 45.46 mJy (magenta), 56.83 mJy (green) and 142.07 mJy (white) plotted over. Figure 7: Plot of the fractional halo flux recovered (in percentage) vs injected flux for the cluster RXC J1701.3+6414. Injected | Recovered | Percentage
---|---|---
(mJy) | (mJy) | (%)
14.21 | -4.68 | –
28.41 | 0.31 | 1.09
34.1 | 2.05 | 6.01
39.78 | 4.13 | 10.38
45.46 | 6.38 | 14.03
51.15 | 7.71 | 15.07
56.83 | 9.15 | 16.1
85.24 | 19.31 | 22.65
142.07 | 36.02 | 25.35
Table 2: Injected and recovered flux densities for the cluster RXC
J1701.3+6414. All values are in mJy. Column 3 is the fractional flux density
recovered. (Note that the negative flux recovered for the first injection is
merely due to an improper deconvolution process and can be ignored.)

The main advantage of this tool is the control provided to the user to inject
halos of various configurations and subsequently estimate upper limits in a
quick and automated manner. The software can also be used to estimate upper
limits to emission of other extended radio sources such as mini-halos. This is
because the models of their emission can also be approximated to one of the
three models we have used to model radio halos (Murgia et al., (2009); Basu et
al., (2016)). In practice, the tool has been used to estimate upper limits for
20 clusters which were part of the sample in the GMRT Key Project on Galaxy
clusters (George et al. in preparation).

###  5.1 Uncertainty on the obtained upper limit

It should be noted that at the current stage the process for upper limit
estimation is still dependent on human intervention at the final step. While
most of the steps of this analysis are automated, confirmation of detection
still requires manual input. This is because a robust detection of a radio
halo in a cluster is dependent on a lot of factors. These include the depth of
the observations, the quality of the image and even the local environment
around the central regions of the cluster. As a result manual inspection on
this final step is still recommended to confirm the “appearance” of an
injected radio halo. The recovery threshold value of 10% is still accurate in
terms of estimating the approximate range of flux density values at which the
radio halo appears.

Figure 6 shows an image of a model halo with exponential profile and flux
density 14.21 mJy. Plotted on top of the image are contours of recovered halos
with flux densities 34.1 mJy (cyan), 45.46 mJy (magenta), 56.83 mJy (green)
and 142.07 mJy (white). These are the same contours as shown in Figures 4 and
5. This shows that the recovered halo image is smaller in size compared to the
injected halo until it crosses a certain flux density threshold.

In Figure 7 we have plotted the fractional flux “recovered” vs the injected
flux density of the halo for the cluster RXC J1701.3+6414. The knee of the
plot occurs when the injected flux density is $40-50$ mJy. This is in
agreement with what we have seen from Figures 4 and 5. We believe that the 10%
recovery threshold chosen is a reasonable estimator to confirm the detection
limit of the halo and fine-tuning around that range can provide a more
accurate estimate of the upper limit. Please note that first data point from
Table 2 is not plotted in the figure for the sake of clarity.

Bonafede et al., (2017) also performed a similar analysis on a sample of
clusters but used a different measure to confirm the detection of an injected
halo. A halo was considered to be detected if the $2\sigma$ size of the source
($D_{2\sigma}$) was greater than half the size of the injected halo, or
($R_{H}$). However, they did not inject their halos in the central regions of
the cluster, instead choosing a location near the cluster, free from any
confusion due to unresolved sources. We do not believe this is an accurate
estimation for a cluster’s upper limit. Cluster radio halos are almost always
found in the central regions of the cluster and have to contend with the
presence of existing sources. This could be unresolved radio sources such as
the BCGs (Brightest Cluster Galaxies) of the cluster or even tailed radio
galaxies (e.g. A2163, Feretti et al., (2001)). In order to get the best
estimate of the halo upper limit, we believe that the artificial halo should
be injected at the cluster centre and the total flux density in the new image
compared with the original image.

###  5.2 Application to all sky surveys

Large area sky surveys such as the NVSS (NRAO Very large Sky Survey, Condon et
al., (1998)) were crucial in the discovery of the earliest samples of cluster
radio halos and relics (Giovannini et al., (1999)). This software can also be
used to estimate average detection limits for the entire survey by selecting
representative fields in the survey and estimating the upper limits in those
surveys. Upper limits calculated in such a manner can represent the detection
limit for extended sources in those surveys.

This will have implications to next generation radio surveys like the LOFAR
Two-metre Sky Survey (LoTSS, Shimwell et al., (2017, 2019)) and surveys in the
the future using the Square Kilometre Array (SKA). The sensitivity that these
surveys are expected to reach is below those of current generation radio
telescopes (e.g. $<100\mu$Jy at 150 MHz for LoTSS). These surveys are expected
to result in the detection of many more new sources of diffuse extended
emission (Bonafede et al., (2015); Johnston-Hollitt et al., (2015)) including
previously undetected radio halos. Radio halo detections as well as non-
detections from these next generation of telescopes will have implications on
the physics of their origin. (Brunetti et al., (2007); Brunetti and Lazarian,
(2016)).

##  6 Conclusions

The vast majority of galaxy clusters do not contain detectable radio halos;
diffuse and extended sources of synchrotron radio emission produced in the
intra-cluster gas. It is unclear if the lack of detection is due to a genuine
absence of emission or if the emission is too weak to be detected by the
current generation of telescopes. However, it is still possible to estimate
upper limits to a possible radio emission. These upper limits can help put
constraints on the various competing models of halo emission.

In this paper, we have presented a Python based tool that can be used to
automate the process of estimating upper limits to radio halo emission. A user
defined model of a radio halo is generated, Fourier transformed and then
injected into the input visibility file and imaged. The halo injected image is
then compared to the original image to detect the presence of a radio halo. On
the basis of this comparison, the software decides either to increase or
decrease the injection flux density so as to best reach the true “upper limit”
of detection. At every step, an image can also be produced for the user to
check the progress of the analysis.

This tool has been successfully tested on a large number of galaxy cluster
observations with reasonable results (George et al. in preparation).
Currently, the tool is optimised for the injection and detection of radio
halos although other extended sources such as mini-halos, radio relics and
supernovae remnants could also be modelled and their upper limits estimated.
The python code for estimating the upper limits can be found
here222https://github.com/lijotgeorge/UL-CALC.

###### Acknowledgements.

We acknowledge the support of the Department of Atomic Energy, Government of
India, under project no. 12-R&D-TFR-5.02-0700. We thank the staff of the GMRT
that made these observations possible. GMRT is run by the National Centre for
Radio Astrophysics of the Tata Institute of Fundamental Research. RK
acknowledges support from the DST-INSPIRE Faculty Award of the Government of
India. This research made use of Astropy,333http://www.astropy.org a
community-developed core Python package for Astronomy (Astropy Collaboration
et al., (2013); Price-Whelan et al., (2018)). This research made use of
Astroquery, an astropy affiliated package that contains a collection of tools
to access online Astronomical data (Ginsburg et al., (2019)). We would also
like to thank the anonymous referee for their comments and suggestions that
have helped improve this paper.

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