Suppose
f
:
(
M
n
,
∂
M
n
)
→
(
Q
2
n
−
1
,
∂
Q
2
n
−
1
)
f:({M^n},\partial {M^n}) \to ({Q^{2n - 1}},\partial {Q^{2n - 1}})
is a proper PL map between PL manifolds
M
n
{M^n}
and
Q
2
n
−
1
{Q^{2n - 1}}
of dimension
n
n
and
2
n
−
1
2n - 1
respectively,
M
M
compact. J. F. P. Hudson has shown that associated with each such map
f
f
that is an embedding on
∂
M
\partial M
is an element
α
¯
(
f
)
\bar \alpha (f)
in
H
1
(
M
;
Z
2
)
{H_1}(M;{Z_2})
when
n
n
is odd and an element
β
¯
(
f
)
\bar \beta (f)
in
H
1
(
M
;
Z
)
{H_1}(M;Z)
when
n
n
is even. These elements are invariant under a homotopy relative to
∂
M
\partial M
. We show that, under slight additional assumptions on
M
,
Q
M,Q
and
f
,
f
f,f
is homotopic to an embedding if and only if
α
¯
(
f
)
=
0
\bar \alpha (f) = 0
for
n
n
odd and
β
¯
(
f
)
=
0
\bar \beta (f) = 0
for
n
n
even. This result is used to give a sufficient condition for extending an embedding
f
:
∂
M
n
→
∂
B
2
n
−
1
f:\partial {M^n} \to \partial {B^{2n - 1}}
(
B
2
n
−
1
{B^{2n - 1}}
denotes
(
2
n
−
1
)
(2n - 1)
-dimensional ball) to an embedding
F
:
(
M
n
,
∂
M
n
)
→
(
B
2
n
−
1
,
∂
B
2
n
−
1
)
F:({M^n},\partial {M^n}) \to ({B^{2n - 1}},\partial {B^{2n - 1}})
.