Wall realisation (Ex)
From Manifold Atlas
Exercise 0.1. Given a compact -manifold show that we can attach two -handles to in such a way that the geometric intersection number of immersed spheres representing these handles (i.e. the upper hemisphere, say, is given by the core of the handle und the lower on by a map into ) is for a given , where is some fixed point.
Hint 0.2. Such a construction is sketched in [Lück2001, pp.115 - 116] all that remains to be checked is the signs. In particular, you may assume that the attaching spheres of the -handles are homotopic in .