Homology groups of adjunction spaces:

We shall now consider the space $Y = X\sqcup_fE^k$ obtained by attaching a $k-$cell $E^n$ to $X$ via an attaching map

$$f:S^{k-1}\longrightarrow X.$$

We shall closely follow the method used in lecture 26 to compute the fundamental groups of the projective plane and Klein's bottle. We do not have to keep track of base points and use the Mayer Vietoris sequence instead of the Seifert Van Kampen theorem. We shall use the same notations and denote by $p$ the center of $E^k$, the interior of $E^k$ by $U$ and the space $Y - \{p\}$ by $V$. The space $U\cap V$ deformation retracts to a space homeomorphic to $S^{k-1}$. Since $V$ deformation retracts to $X$, the spaces $V$ and $X$ have the same homology groups and $H_n(U) = \{0\}$ when $n\geq 1$. in We are ready to prove the following result: