  We study the Borsuk-Ulam theorem for triple (M;\tau; \R^n), where M is a
compact, connected, 3-manifold equipped with a fixed-point-free involution
\tau. The largest value of n for which the Borsuk-Ulam theorem holds is called
the Z_2-index and in our case it takes value 1, 2 or 3. We fully discuss this
index according to cohomological operations applied on the characteristic class
x \in H^1(N; Z_2), where N = M/\tau is the orbit space. In oriented case, we
obtain an expression of the index from the linking matrix of a surgery
presentation of the orbit space. We illustrate our results with examples,
including a non orientable one.
