The intent of this article is to distinguish and study some
n
n
-dimensional compacta (such as weak
n
n
-manifolds) with respect to embeddability into products of
n
n
curves. We show that if
X
X
is a locally connected weak
n
n
-manifold lying in a product of
n
n
curves, then
rank
H
1
(
X
)
≥
n
\operatorname {rank} H^{1}(X)\ge n
. If
rank
H
1
(
X
)
=
n
\operatorname {rank} H^{1}(X)=n
, then
X
X
is an
n
n
-torus. Moreover, if
rank
H
1
(
X
)
>
2
n
\operatorname {rank} H^{1}(X)>2n
, then
X
X
can be presented as a product of an
m
m
-torus and a weak
(
n
−
m
)
(n-m)
-manifold, where
m
≥
2
n
−
rank
H
1
(
X
)
m\ge 2n-\operatorname {rank} H^{1}(X)
. If
rank
H
1
(
X
)
>
∞
\operatorname {rank} H^{1}(X)>\infty
, then
X
X
is a polyhedron. It follows that certain 2-dimensional compact contractible polyhedra are not embeddable in products of two curves. On the other hand, we show that any collapsible 2-dimensional polyhedron embeds in a product of two trees. We answer a question of Cauty proving that closed surfaces embeddable in a product of two curves embed in a product of two graphs. We construct a 2-dimensional polyhedron that embeds in a product of two curves but does not embed in a product of two graphs. This solves in the negative another problem of Cauty. We also construct a weak
2
2
-manifold
X
X
lying in a product of two graphs such that
H
2
(
X
)
=
0
H^{2}(X)=0
.