AbstractLet $$\phi :X \multimap Y$$
ϕ
:
X
⊸
Y
be an n-valued map of connected finite polyhedra and let $$a \in Y$$
a
∈
Y
. Then, $$x \in X$$
x
∈
X
is a root of $$\phi $$
ϕ
at a if $$a \in \phi (x)$$
a
∈
ϕ
(
x
)
. The Nielsen root number $$N(\phi : a)$$
N
(
ϕ
:
a
)
is a lower bound for the number of roots at a of any n-valued map homotopic to $$\phi $$
ϕ
. We prove that if X and Y are compact, connected triangulated manifolds without boundary, of the same dimension, then given $$\epsilon > 0$$
ϵ
>
0
, there is an n-valued map $$\psi $$
ψ
homotopic to $$\phi $$
ϕ
within Hausdorff distance $$\epsilon $$
ϵ
of $$\phi $$
ϕ
such that $$\psi $$
ψ
has finitely many roots at a. We conjecture that if X and Y are q-manifolds without boundary, $$q \ne 2$$
q
≠
2
, then there is an n-valued map homotopic to $$\phi $$
ϕ
that has $$N(\phi : a)$$
N
(
ϕ
:
a
)
roots at a. We verify the conjecture when $$X = Y$$
X
=
Y
is a Lie group by employing a fixed point result of Schirmer. As an application, we calculate the Nielsen root numbers of linear n-valued maps of tori.
On the Nielsen root theory of n-valued maps
