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Assume \(V\) is a real finite dimensional orthogonal
representation of a compact Lie group \(G\).
Let \(\Omega\) be an open invariant subset of \(V\).
We say that \(f\) is an equivariant local map if
the domain of \(f\) is an open invariant subset of \(\Omega\),
\(f\) is continuous equivariant and \(f^{-1}(0)\) is compact.
The space of equivariant local maps will be denoted by
\(\mathcal{F}_G(\Omega)\).
Let \(I=[0,1]\). We assume that the action
of \(G\) on \(I\) is trivial.
Let \(\Lambda\subset I\times\Omega\) be an open invariant subset.
Any equivariant map \(h\colon \Lambda\to V\)
such that \(h^{-1}(0)\) is compact is called an otopy.
Of course, otopy gives an equivalence relation
on \(\mathcal{F}_G(\Omega)\).
The set of otopy classes will be denoted by
\(\mathcal{F}_G[\Omega]\).
Assume that \(H\) is a closed subgroup of \(G\).
Recall that \((H)\) stands for a conjugacy class of \(H\)
and \(WH=NH/H\), where \(NH\) is a normalizer of \(H\) in \(G\).
Let
\begin{align*}
\Phi(G) &=\{(H)\mid \text{\(H\) is a closed subgroup of \(G\)}\},\\
\Phi_0(G)&=\{(H)\in\Phi(G)\mid\dim WH=0\},\\
Iso(\Omega) &=\{(H)\in\Phi(G)\mid\Omega_{(H)}\neq\emptyset\}.
\end{align*}
It is well-known that the set
\(Iso(\Omega)\) is finite and so is
\(Iso(\Omega)\cap\Phi_0(G)\).
We can now formulate our main result,
which may be viewed as a local equivariant version
of the well-known Hopf theorem.
Theorem:
There is a natural bijection
\[
\mathcal{F}_G[\Omega]\approx
\prod_{(H)}\left(\sum_{i=1}^{n(H)}\mathbb{Z}\right),
\]
where the product is taken over
the set \(\text{Iso}(\Omega)\cap\Phi_0(G)\)
and each direct sum is indexed by
the set of connected components
of the quotient \(\Omega_H/H\).
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