If f: X->Y is transversal to Z and dim X + dim Z = dim Y, then we can at least define I_2(f, Z) as #f^(-1)(Z) mod 2, as long as f^(-1)(Z) is finite. Let us explore how useful this definition is without the two assumptions made in the text: X compact and Z closed. Find Examples to show:
(a) I_2(f, Z) may not be a homotopy invariant if Z is not closed.
(b) I_2(f, Z) may not be a homotopy invariant if X is not compact.
(c) The Boundary Theorem is false without the requirement that Z be closed.
(d) The Boundary Theorem is false without the requirement that X be compact.
(e) The Boundary Theorem is false without the requirement that W be compact, even if X = dW is compact and Z is closed. (Hint: Look at the cylinder S^1 * R.)
Reference: Ex 2.4.13 of Guillemin