constructs a Morse complex, whose homology we call the $\SO(3)$ instanton homology. It is defined with coefficients $\F=\Z/2$. We use the notation $J(\check Y;\mu)$. If $(Y,K)$ is a pair consisting of a $3$-manifold and an embedded web, we similarly write $J(Y,K;\mu)$.

Let $\check X$ be an oriented bifold cobordism from $\check{Y_{1}}$ to $\check Y_{2}$, let $\nu$ be marking data for $\check X$, and let $\mu_{i}$ be the restriction of $\nu$ to $\check Y_{i}$. If $\mu_{i}$ is strong, for $i=1, 2$, then $(\check X, \nu)$ gives rise to a linear map $$J(\check X;  \nu) : J(\check Y_{1};\mu_{1}) \to J(\check Y_{2};\mu_{2}).$$ In general, the map which $J$ assigns to composite cobordism may not be the composite map. However, the composition law does hold if the marking data $\nu$ on the two cobordisms satisfies an extra condition. In this paper, our cobordisms will always contain product cobordisms in the neighborhoods of the marking data, and $\nu$ will always be a product $[0,1]\times \mu_{1}$. This restriction is sufficient to ensure that the composition law holds.

#### The construction of $\Jsharp$.

Let $K$ be a compact web in $\R^{3}$. From $K$, we form a new web $K^{\sharp}\subset S^{3}$ as the disjoint union of $K$ and 