Prove that if $X$ and $Z$ are cobordant in $Y$, then for every compact manifold $C$ in $Y$ with dimension complementary to $X$ and $Z$, $I_2(X, C) = I_2(Z, C)$. [HINT: Let $f$ be the restriction to $W$ of the projection map $Y \times I \to Y$ and use the Boundary Theorem.]
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If $X$ and $Z$ cobound the manifold $W$, consider the intersection of $C$ with $Y$. After perturbing it to be in general position, $C\cap W$ must be a compact $1$-manifold with boundary. The boundary of $C\cap W$ must therefore be an even number of points lying in $X\cup Z$. The points that lie in $X$ are counted mod 2 by $I_2(X,C)$ and the points in $Z$ are counted mod 2 by $I_2(Z,C)$. Thus $I_2(X,C)+I_2(Z,C)$ is even, which proves the result.