Fundamental groups of some adjunction spaces:

The method used in the last few examples may be adapted to prove a general theorem about the fundamental group of the adjunction space $X\sqcup_f E^k$ obtained by attaching $E^k$ to a given space $X$ via a map $f:S^{k-1}\longrightarrow X$. As in the case of the projective plane, Klein's bottle and torus the crucial point is to obtain some specific information about the induced map $f_*$. We shall merely state the result and suppress the proof.