  In an $n$-manifold $X$ each element of $H_{n-1}(X; \mathbb{Z}_2)$ can be
represented by an embedded codimension-1 submanifold. Hence for any two such
submanifolds there is a third one that represents the sum of their homology
classes. We construct such a representative explicitly. We describe the
analogous construction for codimension-2 co-oriented submanifolds, and examine
the special case of oriented and/or co-oriented submanifolds. We also give a
lower bound for the number of connected components of the intersection of two
oriented codimension-1 submanifolds in terms of the homology classes they
represent.
