Given a smooth map
f
V
:
V
→
K
f_V\colon V \to K
with
f
V
∗
(
ν
K
)
=
ν
V
f^*_V (\nu _K) = \nu _V
, a general question arises: under which conditions there exists a smooth extension
f
:
M
→
N
f\colon M \to N
of
f
V
f_V
such that
f
f
is transverse to
K
K
and
f
−
1
(
K
)
=
V
f^{-1}(K) = V
, where
M
M
,
N
N
are smooth closed manifolds of dimension
m
m
and
n
n
,
V
V
,
K
K
are closed submanifolds of
M
M
and
N
N
, respectively, of same codimension and
ν
K
\nu _K
,
ν
V
\nu _V
are the normal bundles of
K
K
in
N
N
and
V
V
in
M
M
, respectively. In this paper, we give conditions to the existence of extensions, by using bordism intersection product. Moreover, we present an interesting and non-trivial example illustrating the systematic construction of such extensions, skeletonwise.