Let
V
V
be a real vector space. An arrangement of hyperplanes in
V
V
is a finite set
A
\mathcal {A}
of hyperplanes through the origin. A chamber of
A
\mathcal {A}
is a connected component of
V
−
(
∪
H
∈
A
H
)
V - ({ \cup _{H \in \mathcal {A}}}H)
. The arrangement
A
\mathcal {A}
is called simplicial if
∩
H
∈
A
H
=
{
0
}
{ \cap _{H \in \mathcal {A}}}H = \{ 0\}
and every chamber of
A
\mathcal {A}
is a simplicial cone. For an arrangement
A
\mathcal {A}
of hyperplanes in
V
V
, we set
\[
M
(
A
)
=
V
C
−
(
⋃
H
∈
A
H
C
)
,
M(\mathcal {A}) = {V_\mathbb {C}} - \left ({\bigcup \limits _{H \in \mathcal {A}} {{H_\mathbb {C}}} } \right ),
\]
where
V
C
=
C
⊗
V
{V_\mathbb {C}} = \mathbb {C} \otimes V
is the complexification of
V
V
, and, for
H
∈
A
H \in \mathcal {A}
,
H
C
{H_\mathbb {C}}
is the complex hyperplane of
V
C
{V_\mathbb {C}}
spanned by
H
H
. Let
A
\mathcal {A}
be an arrangement of hyperplanes of
V
V
. Salvetti constructed a simplicial complex
Sal
(
A
)
\operatorname {Sal}(\mathcal {A})
and proved that
Sal
(
A
)
\operatorname {Sal}(\mathcal {A})
has the same homotopy type as
M
(
A
)
M(\mathcal {A})
. In this paper we give a new short proof of this fact. Afterwards, we define a new simplicial complex
Sal
^
(
A
)
\hat {\operatorname {Sal}}(\mathcal {A})
and prove that there is a natural map
p
:
Sal
^
(
A
)
→
Sal
(
A
)
p:\hat {\operatorname {Sal}}(\mathcal {A}) \to \operatorname {Sal}(\mathcal {A})
which is the universal cover of
Sal
(
A
)
\operatorname {Sal}(\mathcal {A})
. At the end, we use
Sal
^
(
A
)
\hat {\operatorname {Sal}}(\mathcal {A})
to give a new proof of Deligne’s result: "if
A
\mathcal {A}
is a simplicial arrangement of hyperplanes, then
M
(
A
)
M(\mathcal {A})
is a
K
(
π
,
1
)
K(\pi ,1)
space." Namely, we prove that
Sal
^
(
A
)
\hat {\operatorname {Sal}}(\mathcal {A})
is contractible if
A
\mathcal {A}
is a simplicial arrangement.