Kervaire and Milnor's paper "Groups of homotopy spheres: 1" in Annals, is the condition "$k>1$" necessary in Lemma 6.3? If yes, why? (I couldn't find any part requiring the condition in their progress to obtain the lemma.)
Lemma 6.3 is the following: Let $M$ be a framed $(k-1)$-connected manifold of dimension $2k+1$ with $k$ odd, $k>1$, such taht $H_k M$ is finite. Let $\chi(\varphi, F)$ be a framed modification of $M$ which replaces the element $\lambda\in H_k M$ of order $l>1$ by an element $\lambda_k' \in H_k M'$ of order $\pm l'$. If $l' \not\equiv 0 \mod l$ then it is possible to choose $(\alpha)\in \pi_k(SO_{k+1})$ so that the modification $\chi(\varphi_\alpha)$ can still be framed, and so that the group $H_k M'_\alpha$ is definitely smaller than $H_k M$.
In addition, please recommend good documents to study surgery on a map from a (3-)manifold (with boundary) to kill the kernel of $H_k(-)$.