You can show that M is homeomorphic to S^2 so it has no boundary and its Euler character is 2.

To show that M is homeomorphic to S^2={(x,y,z)|x^2+y^2+z^2=1}, define a map: f: M -> S^2 by f(p) = p/|p| and try to show that f is a homeomorphism( actually it is a diffeomorphism).